https://icme.hpc.msstate.edu/mediawiki/api.php?action=feedcontributions&user=Madabhushi&feedformat=atomEVOCD - User contributions [en]2020-02-22T07:36:39ZUser contributionsMediaWiki 1.19.1https://icme.hpc.msstate.edu/mediawiki/index.php/ICME_Student_Contributions_2019_-_MsStateICME Student Contributions 2019 - MsState2019-04-28T15:33:51Z<p>Madabhushi: /* Student 6 */</p>
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<div>[[ICME 8373 Student Contributions (Spring 2019)|< ICME 2019 Student Contributions]]<br />
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=Student Contributions=<br />
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===Student 1===<br />
Student Contribution 1<br />
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* Added the following page to the ICME website [[Proposal: Quenched and Partitioned Steel Strength/Ductility versus Volume Fraction of Retained Austenite]]<br />
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Student Contribution 2<br />
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* Added the following tutorial to the ICME website https://www.youtube.com/watch?v=VsqUBnpqJu0&feature=youtu.be<br />
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Student Contribution 3<br />
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===Student 2===<br />
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Student Contribution 1<br />
* Added instructions on modifying and running the [[Gsfe curve]] python script.<br />
* Added [[Gsfe curve]] to Repository of Codes.<br />
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Student Contribution 2<br />
* Created [[Code: Ternary Plot]] page.<br />
* Linked [[Code: Ternary Plot]] in Repository of Codes.<br />
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Student Contribution 3<br />
* Created [[Pure Chromium]] page.<br />
* Linked [[Pure Chromium]] in Metals Category page.<br />
* Added GSFE curves from class assignments to [[Pure Chromium]].<br />
* Intend to add CPFEM and other information from class assignments to [[Pure Chromium]].<br />
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===Student 3===<br />
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Created page to begin putting information about intermediate strain rate testing capabilities at CAVS: [[Intermediate Strain Rate Bar]]<br />
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Created page detailing the general capabilities of high rate testing at CAVS: [[Split-Hopkinson Pressure Bars| Split-Hopkinson Pressure Bars]] & [[Tension Hopkinson Bars|Tension Bars]]<br />
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Organized pages from [[:Special:UncategorizedPages| uncategorized pages]]<br />
*[[SSC Steel: HY100 steel alloy]]<br />
*[[SSC Steel: 1020 steel alloy]]<br />
*[[SSC Steel: 10b22 steel alloy]]<br />
*[[SSC Steel: 300 Maraging Steel Alloy]]<br />
*[[SSC Steel: 304L SS alloy]]<br />
*[[SSC Steel: 321 SS alloy]]<br />
*[[SSC Steel: 4340 steel alloy]]<br />
*[[SSC Steel: A286 steel alloy]]<br />
*[[SSC Steel: AF steel alloy]]<br />
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===Student 4===<br />
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Student Contribution 1<br />
*[[Python Scripting in Abaqus]]<br />
Student Contribution 2<br />
*[[Towards an Open-Source, Preprocessing Framework for Simulating Material Deposition for a Directed Energy Deposition Process]]<br />
Student Contribution 3<br />
*[[Multi-Scale Modeling of Pure Vanadium]]<br />
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===Student 5===<br />
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Student Contribution 1<br />
*[[Media:MDDP_PostProcessing_Tecplot.zip|MDDP Post-Processing Tecplot Tutorial]] <br />
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Student Contribution 2<br />
*Page creation - [[Piezoelectrically Controlled Actuator]] & [[Serpentine Transmitted Bar]]<br />
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Student Contribution 3<br />
*Organized pages from [[:Special:UncategorizedPages| uncategorized pages]]<br />
**[[Stainless Steel: 17-7 PH TH1050]]<br />
**[[Stress Strain Curves: Brass]]<br />
**[[SSC Steel: 1006 steel alloy]]<br />
**[[SSC Steel: C1008 steel alloy]]<br />
**[[SSC Steel: FC0205 steel alloy]]<br />
**[[SSC Steel: HY130 steel alloy]]<br />
**[[SSC Steel: HY80 steel alloy]]<br />
**[[SSC Steel: Mild steel alloy]]<br />
**[[SSC Steel: S7tool steel alloy]]<br />
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===Student 6===<br />
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*Student Contribution 1: Uploaded ICME research proposal, [[Residual Stress & Distortion Modelling for Additively Manufactured Ti6Al4V Parts]]. The page is properly linked to relevant categories<br />
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*Student Contribution 2: <br />
**Uploaded a page describing the Additive Manufacturing method Powder Bed Fusion describing its basic outline [[Powder Bed Fusion]]. The page is properly linked to relevant categories <br />
**Uploaded a page describing Metal Matrix Composites (MMCs) and metal matrix Nanocomposites (MMNCs) [[Metal Matrix Composites]]. The page is properly linked to relevant categories<br />
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*Student Contribution 3: The ICME HWs are unified into a single journal article style report, which can be found [[Multi-Scale Modeling of Pure Vanadium|here]]. The page is properly linked to relevant categories<br />
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===Student 7===<br />
*CLAIMED*<br />
Student Contribution 1: [[A Goal-Oriented, Inverse Decision-Based Design Method for Multi-Component Product Design]] Personal research paper upload.<br />
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Student Contribution 2: [[PyDEM]] Design software upload.<br />
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Student Contribution 3: class assignment [[Pure Chromium]]<br />
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===Student 8===<br />
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Student Contribution 1<br />
*Added the following page [[Structure Optimization]] under Quantum espresso at Electronic Scale.<br />
*Added the following page [[relax]] under Quantum espresso at Electronic Scale.<br />
*Added the following page [[vc-relax]] under Quantum espresso at Electronic Scale.<br />
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Student Contribution 2<br />
*Added the following page [[How to make Supercell for Quantum ESPRESSO]] under Quantum espresso at Electronic Scale.<br />
Student Contribution 3<br />
*Added the following page [[ICME overview of shape memory effect on Bismuth Ferrite ceramic]] on Electronic Scale.<br />
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===Student 9===<br />
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Student Contribution 1<br />
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i have made a section in the microscale category about a tutorial for porous Microsctucture Analysis (PuMA), here is the link of the contributions, https://icme.hpc.msstate.edu/mediawiki/index.php/Category:Microscale#Microscale_oxidation_simulation_PuMA.<br />
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and here is the video added in the section https://www.youtube.com/watch?v=l9NrCsXmtBU.<br />
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Student Contribution 2<br />
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this is an MSF model for the additive manufacturing 17-4 PH stainless steel.<br />
https://icme.hpc.msstate.edu/mediawiki/index.php/17-4_PH_SS#MSF_Calibration<br />
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Student Contribution 3<br />
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Added a page for Research proposal about 17-4 PH SS https://icme.hpc.msstate.edu/mediawiki/index.php/Proposal_for_Multiscale_Modeling_of_17-4_PH_and_life_prediction_using_MSF_model<br />
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===Student 10===<br />
Student Contribution 1<br />
- [[Fatigue Life Prediction of Aluminum Alloy 6063 for Vertical Axis Wind Turbine Blade Application]] (Research proposal)<br />
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Student Contribution 2<br />
- [[Characterization and Modeling of the Fatigue Behavior of 304L Stainless Steel Using the MultiStage Fatigue (MSF) Model]] (Co-authored journal article.)<br />
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Student Contribution 3<br />
- [[Pure Chromium]] (Co-authored journal article. Main sections include: theoretical models, MEAM potential calibration, and single crystal plasticity.)<br />
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===Student 11===<br />
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Student Contribution 1<br><br />
Added [[Intermediate Strain-Rate Testing of ASTM A992 and A572 Grade 50 Steel]]<br />
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Student Contribution 2<br><br />
Added "Tungsten" to [[Metals]]<br><br />
Added [[W]] to "Tungsten" in [[Metals]]<br><br />
Added [[The effect of Fe atoms on the absorption of a W atom on W(100) surface]] to "Tungsten" in [[Metals]]<br><br />
Added [[Nanoscale]] and "Category: Tutorial" to [[Code: WARP - Description]]<br><br />
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Student Contribution 3<br><br />
Added [[Proposal for Multiscale Modeling of Tungsten Heavy Alloy (WHA) for Kinetic Energy Perpetrators]]<br />
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===Student 12===<br />
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Student Contribution 1<br />
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Categorized [[DFT Assignment]] and [[K-Point Variation]]<br />
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Student Contribution 2<br />
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Added video to [[K-Point Variation]] and linked to the VASP wiki site for more K-Point information.<br />
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Student Contribution 3<br />
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===Student 13===<br />
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Student Contribution 1<br />
*Added the following page to the ICME website: [[Porosity in Cast Bronze Pump Impeller]]<br />
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Student Contribution 2:<br />
*Added the following tutorial to the ICME website: Installing Linux on Window 10 - Compiling LAMMPS package from the source (https://icme.hpc.msstate.edu/mediawiki/index.php/Category:Nanoscale)<br />
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Student Contribution 3:<br />
*Added the following tutorial to the ICME website: Learn Python - Full Course for Beginners (https://icme.hpc.msstate.edu/mediawiki/index.php/Python)<br />
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===Student 14===<br />
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Student Contribution 1: [[Calculating Dislocation Mobility]]<br />
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Student Contribution 2: [[Multiscale Modeling of Hydrogen Porosity Formation During Solidification of Al-H]]<br />
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Student Contribution 3: [[Multi-Scale Modeling of Pure Vanadium]]<br />
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===Student 15===<br />
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Student Contribution 1: Upload Journal Article: Interatomic Potential for Hydrocarbons on the Basis of the Modified Embedded-Atom Method with Bond-Order (MEAM-BO): https://icme.hpc.msstate.edu/mediawiki/index.php/Interatomic_Potential_for_Hydrocarbons_on_the_Basis_of_the_Modified_Embedded-Atom_Method_with_Bond_Order_(MEAM-BO) <br />
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Student Contribution 2: ICME Research Proposal to Evaluate Multi-Scale Property Relations for Moisture Absorption of Carbon Fiber Reinforced Plastic Bicycle Wheel: https://icme.hpc.msstate.edu/mediawiki/index.php/ICME_Research_Proposal_to_Evaluate_Multi-Scale_Property_Relations_for_Moisture_Absorption_of_Carbon_Fiber_Reinforced_Plastic_Bicycle_Wheel<br />
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Student Contribution 3: Dynamic Dislocation Plasticity Background Information: https://icme.hpc.msstate.edu/mediawiki/index.php/Dynamic_Dislocation_Plasticity. This page describes key research that was conducted in the background in Dislocation Dynamics and cautions with applying these methods to BCC metals.<br />
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===Student 16===<br />
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Student Contribution 1<br />
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Student Contribution 2<br />
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Student Contribution 3<br />
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===Student 17===<br />
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Student Contribution 1<br />
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Student Contribution 2<br />
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Student Contribution 3<br />
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===Student 18===<br />
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Student Contribution 1<br />
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Student Contribution 2<br />
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Student Contribution 3<br />
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===Student 19===<br />
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Student Contribution 1<br />
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Student Contribution 2<br />
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Student Contribution 3<br />
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===Student 20===<br />
<br />
Claimed* in progress<br />
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Student Contribution 1<br />
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Created multiple post processing codes for plotting data from DFT calculations that can be found at:<br />
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* [[EvA_EvV_plot.py | Python code for post-processing EvsA and EvsV files from running Quantum Espresso simulations using the ev_curve.bash script to generate plots for the EvV and EvA curves ]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
* [[convergence_plots.py | Python code for post-processing <code> SUMMARY</code> files from running Quantum Espresso simulations using the ev_curve.bash script to generate a plot for a convergence study ]] for [[Code: Quantum Espresso | Quantum Espresso ]] <br />
* [[ecut_conv.py | Python code for post-processing .out files files from running Quantum Espresso simulations to generate a plot for the ecut convergencerate ]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
* [[EOS_comp_plot.py | Python code for post-processing <code> SUMMARY</code>, <code> EsvA </code>, <code> EsvV</code>, and <code> evfit.#</code> files from running Quantum Espresso simulations using the ev_curve.bash script to generate a plot comparing the effect of using the different equations of state in the evfit code ]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
* [[EOS_plot.py | Python code for post-processing <code> evfit.#</code> files from running Quantum Espresso simulations and using the evfit.f routine to fit to multiple equations of state]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
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Student Contribution 2<br />
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Uploaded research proposal for method of creating nanocrystalline/amorphous metals using femtosecond laser induced ablation. Found at: [[Laser induced microstructure]]<br />
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Student Contribution 3<br />
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Added link to software for generating high order finite elements to be used in codes that solve PDE's using discretization methods. Works for both continuous and discontinuous methods. Found at:<br />
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* [[DIY-FEA]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:MacroscaleCategory:Macroscale2019-04-26T20:46:21Z<p>Madabhushi: /* Aluminum */</p>
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<br />
= Overview =<br />
<br />
The Macroscale is a continuum point, where one develops the constitutive model for the [[Structural Scale|structural scale]] finite element simulations and is able to downscale by defining the requirements and admitting the subscale information with the use of internal state variables. We are concerned here with model calibration, model validation, and experimental stress-strain curves. Model calibration is related to correlating constitutive model constants with experimental data from homogeneous stress states like uniaxial compression. Model validation is related to comparing predictive results with experimental results that arise from heterogeneous stress states like a notch tensile test. Experimental stress-strain curves can include different strain rates, temperatures, and stress states (compression, tension, and torsion). <br />
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The macroscale can also be thought to apply to cyclic behavior like fatigue. Here there is also model calibration, model validation, and experimental data. Model calibration is related to strain-life curves (or stress-life curves). Model validation is related to mean stress effects and multi-axial stress states.<br />
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Finally, to garner more information about the information bridges between length scales go to the [[Mississippi State University| MSU Education page]].<br />
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=Material Models=<br />
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===Human Head/Brain Models===<br />
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Several options for the modeling of the head and brain exist. Samaka and Tarlochan [2013]<ref>H. Samaka and F. Tarlochan, “Finite Element (FE) Human Head Models/Literature Review”, Internat. J. Sci. Tech. Res., 2(7), 17-31., 2013.[http://http://www.ijstr.org/final-print/july2013/Finite-Element-Fe-Human-Head-Models--Literature-Review.pdf]</ref> and Caroline and Remy [2009],<ref name="Caroline"> D. Caroline and W. Remy, "Head Injury Prediction Tool for Protective Systems Optimisation”, 7th European LS-Dyna Users Conference, 2009.[https://www.dynamore.de/en/downloads/papers/09-conference/papers/E-II-01.pdf]</ref> provide an overview of these models and the softwares used to implement the models:<br />
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* EEVC WG17 Adult Headform Finite Element Model <ref> S. Huanga and J. Yanga, "Optimization of a Reversible Hood for Protecting a Pedestrian's Head during car collisions", Accident Analysis and Prevention, 42(4), 1136-1143., 2010.</ref><br />
* Wayne State University (WSU) Head (Brain) Injury Model (WSUHIM) <ref> L. Zhang, K. H. Yang, and A. I. King, "Comparison of Brain Responses between Frontal and Lateral Impacts by Finite Element Modeling", J. of Neurotrauma, 18(1), 21-30., 2004.</ref><br />
* TNO Head FE Model <ref>[https://www.tno.nl/en/]</ref> [http://www.lstc.com/products/ls-dyna]<br />
* Head Brain Model <ref> M. Iwamoto, Y. Nakahira, A. Tamura, H. Kimpara, I. Watanabe, and K. Miki, "Development of Advanced Human Models in THUMS", 6th European LS-Dyna Users Conference, 2007, 47-56. [http://www.dynalook.com/european-conf-2007/development-of-advanced-human-models-in-thums.pdf]</ref> [http://www.lstc.com/products/ls-dyna]<br />
* University of Louis Pasteur Finite Element Model (ULP FEM)of the Human Head <ref> H. S. Kang, R. Willinger, F. Turquier, A. Domont, X. Trosseille, C. Tarriere, and F. Lavaste, "Evaluation Study of a 3D Human Head Model Against Experimental Data", Proceed. 40th Stapp Car Crash Conference, 339-366., 1996 </ref><br />
* Politecnico di Torino University Finite Element Model of the Human Head <ref> G. Belingardi,G. Chiandussi, and I. Gaviglio, "Development and Validation of a New Finite Element Model of Human Head", 1-9., [http://www-nrd.nhtsa.dot.gov/pdf/esv/esv19/05-0441-o.pdf]</ref> [http://www.altairhyperworks.com/]<br />
* Harvard Medical School Finite Element Model <ref> L. M. Vigneron, J. G. Verly, and S. K. Warfield, "Modelling Surgical Cuts, Retractions, and Resections via Extended Finite Element Method", in C. Barillot, D. R. Haynor, and P. Hellier (eds.): MICCAI 2004, LNCS 3217, 311-318., 2004.[https://www.spl.harvard.edu/archive/spl-pre2007/pages/papers/vigneron/vigneron-miccai2004.pdf]</ref><br />
* University College Dublin Brain Trauma Model (UCDBTM) <ref> T. J. Horgan and M. D. Gilchrist, "Influence of FE Model Variability in Predicting Brain Motion and Intracranial Pressure Changes in Head Impact Simulations", I. J. Crash, 9(4), 401-418., 2004, [http://www.truegrid.com/Gilchrist_2004e.pdf]</ref> [http://www.vtk.org/]; [http://www.mscsoftware.com/product/patran]<br />
* Strasbourg University Finite Element Head Model (SUFEHM) <ref name="Caroline" /> [http://icube-mmb.unistra.fr/index.php/Human_model]; [http://www.lstc.com/products/ls-dyna]<br />
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===[[Code: DMG | Internal State Variable Plasticity-Damage (MSU ISV-DMG) Model]]===<br />
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The Mississippi State University Internal State Variable (ISV) plasticity-damage model ([[Code: DMG|DMG]]) production version 1.0 is being released along with its model calibration tool (DMGfit). The <br />
model equations and material model fits are explained in [https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/DMG/trunk/doc/MSU.CAVS.CMD.2009-R0010.pdf?view=co CAVS Technical Report: MSU.CAVS.CMD.2009-R0010.pdf]. This model is based upon Bammann, DJ, Chiesa, ML, Horstemeyer, MF, Weingarten, LI, "Failure in Ductile Materials Using Finite Element Methods," Structural Crashworthiness and Failure, eds. Wierzbicki and Jones, Elsevier Applied Science, The Universities Press (Belfast) Ltd, 1993 and Horstemeyer, MF, Lathrop, J, Gokhale, AM, and Dighe, M, "Modeling Stress State Dependent Damage Evolution in a Cast Al-Si-Mg Aluminum Alloy," Theoretical and Applied Mech., Vol. 33, pp. 31-47, 2000. This model will predict the plasticity and failure in a metal alloy. It can be initialized to have different heterogeneous microstructures within the finite element mesh.<br />
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===Johnson-Cook Flow Stress (JC) Model===<br />
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The Johnson-Cook (JC) constitutive model is an empirically based flow model originally intended for the prediction of inelastic deformation in solid materials <ref>Johnson, G.R., Cook, W.H., “A Constitutive Model and Data for Metals Subjected To Large Strains, High Strain Rates and High Temperatures.” Proceedings of the 7th International Symposium on Ballistics. Vol. 21. 1983.</ref>. The Johnson-Cook plasticity model has terms that account for the strain hardening, strain rate, and temperature sensitivity of a material. The Johnson-Cook model has been extended to account for damage progression based upon strain rate, temperature, and pressure conditions<ref>2. Johnson, G.R., Cook, W.H., “Fracture characteristics of three metals subjected to various strains, strain rates, temperatures, and pressures.” Engineering Fracture Mechanics 21.1 (1985): 31-48.</ref>.<br />
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===Mechanical Threshold Stress (MTS) Model===<br />
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The Mechanical Threshold Stress (MTS) Model is a flow stress model that considers the effects of dislocation motion and interaction on macroscale deformation<ref>Follansbee, P. S., and U. F. Kocks. "A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable." Acta Metallurgica 36.1 (1988): 81-93.</ref>. The MTS model proposes the use of the mechanical threshold stress (described as the material flow stress at 0K) as an internal state variable. The MTS is formulated as a combination of dislocation mechanisms generation and recovery, strain rate, and temperature terms. The MTS variable is related to the flow stress of the material in conjunction with strain-rate dependent scaling factors thus capturing and relating the internal microscale evolution of the material to the macroscale stress-strain material behavior.<br />
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===[[MSF_v2 | MultiStage Fatigue (MSF) Model]]===<br />
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The multi-stage fatigue (MSF) model predicts the amount of fatigue cycling required to cause the appearance of a measurable crack, the crack size as a function of and loading cycles. The model incorporates microstructural features to the fatigue life predictions for incubation, microstructurally small crack growth, and long crack growth stages in both high cycle and low cycle regimes.<br />
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===ThermoPlastic Internal State Variable (TP-ISV) Model===<br />
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The Mississippi State University Internal State Variable (ISV) model for thermoplastics (TPISV) version 1.0 is being released along with its model calibration tool (TPfit). The model equations and material model fits are decribed in Bouvard et al. [2010]<ref>J.L. Bouvard, D.K. Ward, D. Hossain, E.B. Marin, D.J. Bammann, and M.F. Horstemeyer, “A General Inelastic Internal State Variables Model for amorphous glassy polymers”, Acta Mechanica, 213(1), 71-96., 2010.[https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/TPgui/trunk/doc/ISV_MODEL_POLYMER_PAPER_ACTAMECHANICA_2010.pdf ISV_MODEL_POLYMER_PAPER_ACTAMECHANICA]</ref>. This polymer based ISV model is able to capture the history effects of a thermoplastic polymer tested under different stress states and strain rates. The modeling approach follows current methodologies used for metals <ref>Bamman, D.J., Chiesa, M.L., Johnson, G.C. : Modeling large deformation and failure in manufacturing processes. In: Tatsumi, T.,Wanatabe, E.,Kambe, T. (eds.) Theoretical and Applied Mechanics, pp. 359–376. Elsevier Science,USA(1996)</ref> based on a thermodynamic approach with internal state variables. Thus, the material departs from spring-dashpot based models generally used to predict the mechanical behavior of polymers. To select the internal state variables, we have used a hierarchical multiscale approach for bridging mechanisms from the molecular scale (see * [[MD_PE_deformation | Atomistic Deformation of Amorphous Polyethylene]]) to the continuum scale. The continuum constitutive model applied a formalism using a three-dimensional large deformation kinematics and thermodynamics framework. The 3D constitutive equations of the model were implemented in [[Thermoplastic Modeling|ABAQUS Explicit]] using a VUMAT subroutine. These equations were then simplified to the one-dimensional case in order to fit the model parameters using MATLAB software.<br />
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===WARP3D - Open Source Code for 3D Nonlinear Analysis of Solids===<br />
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[[Code:WARP3D]] is a research code for the solution of large-scale, 3-D solid models subjected to static and dynamic loads. The capabilities of the code focus on fatigue & fracture analyses primarily in metals.<br />
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===Zerilli-Armstrong Flow Stress (ZA) Model===<br />
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The Zerilli-Armstrong (ZA) model is a flow stress model based upon dislocation mechanics <ref>Zerilli, F.J., Armstrong, R.W., “Dislocation-mechanics-based constitutive relations for material dynamcis calculations.” Journal of Applied Physics 61.5 (1987): 1816-1825.</ref>. The ZA plasticity model accounts for the effects of temperature and strain rate while also considering contribution of dislocation density, microstructural stress intensity, and material grain size. Material parameters within the ZA model are dependent upon the crystalline structure of the material.<br />
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= Tutorials =<br />
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==DMG v1.0==<br />
[[Code: DMG|MSU DMG v1.0]] is an example of a plasticity-damage internal state variable model<ref>Bammann, D. J., Chiesa, M. L., Horstemeyer, M. F., Weingarten, L. I., "Failure in Ductile Materials Using Finite Element Methods," Structural Crashworthiness and Failure, eds. T. Wierzbicki and N. Jones, Elsevier Applied Science, The Universities Press (Belfast) Ltd, 1993</ref> <ref>Horstemeyer, M.F., Lathrop, J., Gokhale, A.M., and Dighe, M., “Modeling Stress State Dependent Damage Evolution in a Cast Al-Si-Mg Aluminum Alloy,” Theoretical and Applied Fracture Mechanics, Vol. 33, pp. 31-47, 2000</ref>, which admits different grain sizes, particles sizes, particle volume fractions, pore sizes, and pore volume fractions within each continuum element. Experiments for calibration (homogeneously applied stress and strain states) and validation (heterogeneous stress and strain states) are used to determine the material constants and particular microstructures. The microstructures can be garnered from the [[Particle Characterization with ImageJ|Image Analysis]] tools, which require only a picture with a measured scale bar for the picture. Once the material model is calibrated and validated at the macroscale, it can then be used for the structural scale simulations.<br />
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A video tutorial for calibrating DMG is found [https://www.youtube.com/watch?v=6VMMCSZVyXo| here].<br />
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An in depth written tutorial of DMGfit can be found [[DMGfit 55p v1p1|here]].<br />
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A guide to setting up the DMG UMAT in [[Code: CALCULIX|Calculix]] can be found [[DMG in Calculix|here]].<br />
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The following single element finite element input decks should be used to verify the material point simulator (DMGfit) determinations:<br />
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* one element compression for aluminum A356 (ABAQUS-Implicit)[https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/inputDecks/abaqus_fem_decks/single_element_Linux/A356_Implicit_Compression/ here]<br />
* one element compression for aluminum A356 (ABAQUS-Explicit)[https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/inputDecks/abaqus_fem_decks/single_element_Linux/A356_Explicit_Compression/ here] <br />
* one element tension for aluminum A356 (ABAQUS-Implicit) [https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/inputDecks/abaqus_fem_decks/single_element_Linux/A356_Implicit_Tension/ here]<br />
* one element tension for aluminum A356 (ABAQUS-Explicit)[https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/inputDecks/abaqus_fem_decks/single_element_Linux/A356_Explicit_Tension/ here]<br />
* one element simple shear for aluminum A356 (ABAQUS-Implicit)[https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/inputDecks/abaqus_fem_decks/single_element_Linux/A356_Implicit_Shear/ here]<br />
* one element simple shear for aluminum A356 (ABAQUS-Explicit)[https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/inputDecks/abaqus_fem_decks/single_element_Linux/A356_Explicit_Shear/ here]<br />
<br />
[[Media:OneElement-compression-A356-Abaqus-explicit.tar|One element explicit compression A356 input decks]]<br />
Model Validation simulations include the following:<br />
notch tensile specimen (1/8 space mesh) for aluminum A356 (ABAQUS-Implicit)<br />
<br />
The ABAQUS input decks and instruction on how to one element simulations can be downloaded ('Download GNU tarball') [https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/inputDecks/abaqus_fem_decks/ here], or can be viewed online by clicking 'view' for each of the files.<br />
<br />
==MSF==<br />
The macroscale tools also include fatigue analysis tools. One example is the [[MSF v2|MultiStage Fatigue (MSF) model]], which also admits microstructural information to help quantify the number of cycles for crack incubation, the number of cycles in the Microstructurally Small Crack (MSC) regime, and the number of cycles in the long crack (LC) regime. The amount of cycles that is experienced in each regime depends on the manufacturing process and the type of material.<br />
<br />
== Thermoplastic Internal State Variable (TPISV) Model ==<br />
MSU TPISV a damage model that is implemented into a glassy, amorphous thermoplastic thermomechanical inelastic internal state variable framework. Internal state variable evolution equations are defined through thermodynamics, kinematics, and kinetics for isotropic damage arising from two different inclusion types: pores and particles. The damage arising from the particles and crazing is accounted for by three processes of damage: nucleation, growth, and coalescence. Nucleation is defined as the number density of voids/crazes with an associated internal state variable rate equation and is a function of stress state, molecular weight, fracture toughness, particle size, particle volume fraction, temperature, and strain rate. The damage growth is based upon a single void growing as an internal state variable rate equation that is a function of stress state, rate sensitivity, and strain rate. The coalescence internal state variable rate equation is an interactive term between voids and crazes and is a function of the nearest neighbor distance of voids/crazes and size of voids/crazes, temperature, and strain rate. The damage arising from the pre-existing voids employs the Cocks–Ashby void growth rule. The total damage progression is a summation of the damage volume fraction arising from particles and pores and subsequent crazing. The modeling results compare well to experimental findings garnered from the literature. Finally, this formulation can be readily implemented into a finite element analysis.<br />
<br />
* One element tension for Polycarbonate (ABAQUS-Explicit) -[[Media:OE_Tension.zip | here]]<br />
<br />
To learn how to use the parameters fitting routine, you can refer to the documentation ([https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/TPgui/trunk/doc/MSU.CAVS.CMD.2010-R0008-TPgui-v1.1.doc?view=co TPGui User Manual]; [https://icme.hpc.msstate.edu/viewvc/CMD%20Codes%20Repository/TPgui/trunk/doc/TPgui-Tutorial.pptx?view=co TPGui Tutorial]).<br />
<br />
A stand-alone TP tool is available from the [[CodeRepository:TP|online code repository]]. Please refer to the documentation ([https://icme.hpc.msstate.edu/cmddocs/TP/TPgui.html online help and tutorial]) to learn how to use this tool.<br />
<br />
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<td valign="top" width="50%" style="border: 1px solid black; background-color:#FFFFFF;"><br />
<br />
=Macroscale Research=<br />
== [[Biomaterials|Biomaterials]]==<br />
<br />
== Ceramics ==<br />
[[Constitutive Models for ferroelectric materials under combined mechanical and electrical loading]]<br />
== Geomaterials==<br />
==Corrosion==<br />
* [[Corrosion]]<br />
** [[Quantification of corrosion mechanisms on an extruded AZ31 magnesium alloy]]<br />
** [[Corrosion Behaviour of Extruded AM30 Magnesium Alloy]]<br />
** [[Corrosion Fatigue Behavior of Extruded AM30 Magnesium Alloy]]<br />
** [[Structure–property quantification of corrosion pitting under immersion and salt-spray environments on an extruded AZ61 magnesium alloy]]<br />
<br />
==Metals==<br />
===Aluminum===<br />
* 1100 - [[SSC Aluminum: Al 1100 alloy |DMGdata]]<br />
* 1060 - [[SSC Aluminum: Al 1060-T0 alloy |DMGdata]]<br />
* 2024-T0 - [[SSC Aluminum: Al 2024-T0 alloy |DMGdata]]<br />
* 2024-T3 - [[Aluminum 2024-T3 Stress-Strain and Fatigue Life Data| Stress-Strain Data]] | [[Aluminum 2024-T3 Stress-Strain and Fatigue Life Data| Fatigue-Life Data]]<br />
* 2024-T351 - [[SSC Aluminum: Al 2024-T351 alloy |DMGdata]]<br />
* 2024-T6 - [[Al 2024-T6 | Al 2024-T6 Stress-Strain Data]]<br />
* 2024-T4 - [[SSC Aluminum: Al 2024-T4 alloy |DMGdata]]<br />
* 6061-T0 - [[SSC_Aluminum:_Al_6061-T0_alloy|DMGdata]]<br />
* 6061-T6 - [http://www.matweb.com/search/datasheettext.aspx?matguid=1b8c06d0ca7c456694c7777d9e10be5b Matweb]<br />
* 6061-T651 - [[SSC Aluminum: Al 6061-T651 alloy|DMGdata]]<br />
* 6063-T6 - [[Al 6063-T6 | Stress-Strain Data]] | [[Al 6063-T6 Fatigue | Fatigue Stress-Life Data]]<br />
* 7039 - [[SSC Aluminum: Al 7039 alloy |DMGdata]]<br />
* 7075-T0 - [[SSC Aluminum: Al 7075-T0 alloy |DMGdata]]<br />
* 7075-T6 - [[SSC Aluminum: Al 7075-T6 alloy |DMGdata]]<br />
* 7075-T651 - [[SSC Aluminum: Al 7075-T651 alloy |DMGdata]]<br />
* 7475-T7351 - [[Al 7475-T7351 Stress-Strain data | Stress-Strain Data]] | [[Al 7475-T7351 Fatigue Life cycle data | Fatigue Stress-Life Data]]<br />
* 99% Pure Alloy - [[SSC Aluminum: Al 99% pure alloy |DMGdata]]<br />
<br />
===Copper===<br />
* OFHC Alloy - [[SSC Copper: OFHC copper alloy |DMGdata]]<br />
* ETP Alloy - [[SSC Copper: ETP copper alloy |DMGdata]]<br />
===Inconel===<br />
* 600 - [[Inconel 600 | Stress-Strain & Fatigue Life Data]]<br />
===Iron===<br />
* Armco alloy - [[SSC Iron: Armco Iron alloy |DMGdata]]<br />
[[Category:Iron]]<br />
* Cast Iron - [[Mechanical Properties of Nodular Cast Iron|Stress-Strain Data]]<br />
* Cast Iron - [[Mechanical Properties of Nodular Cast Iron|Fatigue Stress-Life Data]]<br />
* Compacted Graphite Iron - [[Cast Iron: Compacted Graphite Iron | DMGdata]]<br />
<br />
===Magnesium===<br />
* AE42 - [[SSC Magnesium: AE42 Mg alloy |DMGdata]]<br />
* AE44 - [[SSC Magnesium: AE44 Mg alloy |DMGdata]]<br />
* AM30 - [[SSC Magnesium: AM30 Mg alloy |DMGdata]]<br />
* AM50 - [[SSC Magnesium: AM50 Mg alloy |DMGdata]]<br />
* AM60 - [[SSC Magnesium: AM60 Mg alloy |DMGdata]]<br />
* AZ31 - [[SSC Magnesium: AZ31 Mg alloy |DMGdata]]<br />
* AZ61 - [[AZ61: Fatigue Life data|MSFdata]]<br />
* AZ91 - [[SSC Magnesium: AZ91 Mg alloy |DMGdata]]<br />
* ZE41 - [[Material Properties of ZE41 Magnesium Alloy#Stress-Strain_Data | Stress-Strain Data]] | [[Material Properties of ZE41 Magnesium Alloy#Fatigue_Life_Data | Fatigue Stress-Life Data]]<br />
* AZ61A - [[AZ61A]]<br />
<br />
===Nickel===<br />
* 200 - [[SSC Nickel: 200 nickel |DMGdata]]<br />
* B-1900 Superalloy [[B-1900 Ni Superalloy | Stress-Strain & Fatigue Life Data]]<br />
* In718 - [[SSC Nickel: In718 nickel alloy |DMGdata]]<br />
===Steel===<br />
* 1006 - [[SSC Steel: 1006 steel alloy |DMGdata]]<br />
* 1010 - [[1010 Steel | Stress-Strain Data]]<br />
* 1010 - [[1010 Steel | Fatigue Stress-Life Data]]<br />
* 1020 - [[SSC Steel: 1020 steel alloy |DMGdata]]<br />
* 10b22 - [[SSC Steel: 10b22 steel alloy |DMGdata]]<br />
* 300 Maraging - [[SSC Steel: 300 Maraging Steel Alloy |DMGdata]]<br />
* 301 SS - [[Stress-strain data for 301 SS | Stress-Strain Data]] | [[Stress-strain data for 301 SS|Fatigue Stress-Life Data]]<br />
* 304L SS - [[SSC Steel: 304L SS alloy |DMGdata]]<br />
* 316 SS - [[316 Stainless Steel | Stress-Strain Data]] | [[316 Stainless Steel#Fatigue-life curves| Fatigue Stress-Life Data]]<br />
* 321 SS - [[SSC Steel: 321 SS alloy |DMGdata]]<br />
* 3140 - [[3140 Steel| Stress-Strain & Fatigue Life Data]]<br />
* 4130 - [[Stress Strain data-4130 Steel | Stress-Strain Data]] | [[AISI 4130 Steel | Fatigue Stress-Life Data]] | [[Fatigue-life curve-4130 Steel | Additional Stress-Life Data]]<br />
* 4140 - [[Mechanical properties of 4140 steel|Stress-Strain Data]] | [[Mechanical properties of 4140 steel|Fatigue Stress-Life Data]]<br />
* 4340 - [[SSC Steel: 4340 steel alloy |DMGdata]]<br />
* A286 - [[SSC Steel: A286 steel alloy |DMGdata]]<br />
* AF - [[SSC Steel: AF steel alloy |DMGdata]]<br />
* C1008 - [[SSC Steel: C1008 steel alloy |DMGdata]]<br />
* Dual Phase - [[Dual Phase (DP) Steel: Mechanical Testing Data#Stress-Strain Data| Stress-Strain data]] | [[Dual Phase (DP) Steel: Mechanical Testing Data#Fatigue Life Data| Fatigue Life data]]<br />
* FC0205 - [[SSC Steel: FC0205 steel alloy |DMGdata]]<br />
* HY80 - [[SSC Steel: HY80 steel alloy |DMGdata]]<br />
* HY100 - [[SSC Steel: HY100 steel alloy |DMGdata]]<br />
* HY130 - [[SSC Steel: HY130 steel alloy |DMGdata]]<br />
* Mild Steel - [[SSC Steel: Mild steel alloy |DMGdata]]<br />
* Q&P Steel- [[Quench and Partitioned Steels]]<br />
* RHA - [[Rolled Homogeneous Armor Alloy |Experimental Data]]<br />
* S7tool - [[SSC Steel: S7tool steel alloy |DMGdata]]<br />
* SS 17-7 PH TH1050 - [[Stainless Steel: 17-7 PH TH1050 | DMGdata]]<br />
* TWIP - [[Mechanical Behavior of TWIP steels | Stress-Strain Data]]<br />
* TWIP - [[Mechanical Behavior of TWIP steels | Fatigue-life Data]]<br />
* HSLA Steel - [[HSLA Steel]]<br />
<br />
===Titanium===<br />
* CP-Ti - [[SSC Titanium: CP-Ti | Stress-Strain & Fatigue Life Data]] | [[SSC Titanium: CP-Ti | DMGdata]]<br />
* Ti0Al6V4 - [[SSC Titanium: Ti0Al6V4 titanium alloy | DMGdata]]<br />
* Ti6Al6V2Sn - [[SSC Titanium: Ti6Al6V2Sn titanium alloy | DMGdata]]<br />
* Ti7Al4Mo - [[SSC Titanium: Ti7Al4Mo titanium alloy |DMGdata]]<br />
* Ti8Al1Mo1V - [[SSC Titanium: Ti8Al1Mo1V titanium alloy | DMGdata]]<br />
===Zinc===<br />
* Ze20 [[ZE20: Stress-Strain data in Tension and Compression]]<br />
== Ceramics ==<br />
[[Constitutive Models for ferroelectric materials under combined mechanical and electrical loading]]<br />
== Polymers==<br />
* [[:File:CAVS_MSU_polymer_model_v1.pdf|On Developing a Viscoelastic–Viscoplastic Model for Polymeric Materials]] <br />
* [[A_general_inelastic_internal_state_variable_model_for_amorphous_glassy_polymers | A general inelastic internal state variable model for amorphous glassy polymers]]<br />
*[[An_internal_state_variable_material_model_for_predicting_the_time_thermomechanical_and_stress_state_dependence_of_amorphous_glassy_polymers_under_large_deformation | An internal state variable material model for predicting the time, thermomechanical, and stress state dependence of amorphous glassy polymers under large deformation]]<br />
* [[Formulation_of_a_damage_internal_state_variable_model_for_amorphous_glassy_polymers | Formulation of a damage internal state variable model for amorphous glassy polymers]]<br />
* [http://doi.org/10.1002/app.40882 Microstructure-based fatigue modeling of an acrylonitrile butadiene styrene (ABS) copolymer]<br />
<br />
== [[Equipment]] ==<br />
<br />
</td><br />
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</table><br />
<br />
== References==<br />
<references/><br />
<br />
<br />
[[Category:Overview]]<br />
[[Category:Multiscale Simulations]]<br />
[[Category:Polymers]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Category:MetalsCategory:Metals2019-04-26T20:41:04Z<p>Madabhushi: /* Metal Systems */</p>
<hr />
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<td colspan="2" style="border: 1px solid black; background-color:#FFFFFF;"><br />
<br />
=== Overview ===<br />
<br />
As shown on the periodic table of elements, the majority of the chemical elements in pure form are classified as metals. Physical properties show that metals are good electrical conductors and heat conductors, and exhibit good ductility and strength. Shown in chemical properties, metals usually have 1-3 electrons in their outer shell, and loose their valence electrons easily. <br />
<br />
Metals are composed of atoms held together by strong, delocalized bonds called metallic bonding: arrangement of positive ions surrounded by a cloud of delocalized electrons. Above their melting point, metals are liquids, and their atoms are randomly arranged and relatively free to move. However, when cooled below their melting point (solidification), metals rearrange to form ordered, crystalline structures. The smallest repeating array of atoms in a crystal is called a unit cell. In a unit cell, atoms are packed together as closely as possible to form the strongest metallic bonds. Typical packing or stacking arrangements are: face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close packing (HCP). <br />
As atoms of a melted metal begin to pack together to form a crystal lattice at the freezing point, groups of these atoms form tiny crystals. These tiny crystals increase in size by the progressive addition of atoms. The resulting solid is not one crystal but actually many smaller crystals, called grains. These grains grow until they impinge upon adjacent growing crystals. The interface formed between them is called a grain boundary. Metallic crystals are not perfect. Sometimes there are empty spaces called vacancies, where an atom is missing. Another common defect in metals are dislocations, which are lines of defective bonding. These and other imperfections, as well as the existence of grains and grain boundaries, determine many of the mechanical properties of metals. When a stress is applied to a metal, dislocations are generated and move, allowing the metal to deform.<br />
<br />
When loads (stresses) are applied to metals they deform. If the load is small, metals experience elastic deformation, which involves temporary stretching or bending of bonds between atoms. When higher stresses are applied, permanent (plastic) deformation occurs. This plastic deformation involves the breaking of bonds, often by the motion of dislocations. If placed under too large of a stress, metals will mechanically fail, or fracture. The most common reason for metal failure is fatigue, i.e., a fracture process resulting from the application and release of small stresses and re-application of the load (as many as millions of times).<br />
<br />
In industry, molten metal is cooled to form the solid ([[Casting|casting]]). The solid metal is then thermomechanically shaped to form a particular product. Processes such as extrusion and sheet forming are used for this purpose. During this shaping process, the application of heat and plastic deformation can strongly affect the mechanical properties of a metal. Heat treating induces microstructure changes, such as grain growth, that modify the properties of some metals. Annealing is a softening process in which metals are heated and then allowed to cool slowly. Most steels may be hardened by heating and quenching (cooling rapidly). Quenching produces a metal that is very hard but also brittle. Because plastic deformation results from the movement of dislocations, metals can be strengthened by preventing this motion. When a metal is shaped, dislocations are generated and move. As the number of dislocations in the crystal increases, they will get tangled or pinned and will not be able to move. This will strengthen the metal. This process is known as cold working. At higher temperatures the dislocations can rearrange, so little strengthening occurs. Heating removes the effects of cold-working. When cold worked metals are heated, recrystallization occurs, a process where new grains form and grow to consume the cold worked portion. The new grains have fewer dislocations and the original properties are restored.<br />
<br />
At CAVS at Mississippi State University, we perform research and application work for metals in two branches of materials - lightweight materials of magnesium and aluminum, and steel materials. The material research around these two branches is broad enough to attract various funding sources, from federal agencies to local manufaturers. We form interdisciplinary teams to support the material research. The team includes physicists, chemists, material scientists, mechancial/aerospace/civil engineers to develop multiscale material length scale models for use that are validated using a wide range of [[Equipment|experimental equipment]].<br />
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<br />
=== Metal Systems ===<br />
<br />
[[Powder Metallurgy| Powder Metallurgy]] <br><br />
[[Animations List|Animations List of Metals and other Materials]] <br><br />
[[Metal Matrix Composites]]<br />
<br />
<br />
==== Aluminum ====<br />
<br />
Aluminum alloys have been a focus in lightweight designs. Understanding the energy absorption, mechanical behavior and strength, creep resistance, and corrosion resistance are key research opportunities.<br />
<br />
* [[Structural Scale Research for Aluminum|Structural Scale]]<br />
* Macroscale<br />
** Plasticity-Damage Internal State Variable (DMG) Model<br />
** [[MSF Calibrations for Metals | MultiStage Fatigue (MSF) Model Calibrations ]]<br />
* [[MaterialModels:_Mesoscale#Metals|Mesoscale]]<br />
** [[Yield surface prediction of Aluminum on rolling]]<br />
** [[Visco-Plastic Self-Consistent (VPSC) Deformation Simulation of Polycrystalline FCC Aluminum]]<br />
** [[Code:_ABAQUS_CPFEM#Crystal_Plasticity_Finite_Element_Method|One element deformation of Aluminum]] <br />
* Microscale<br />
**[[Microstructural Inclusion Influence on Fatigue of a Cast A356 Aluminum Alloy|Fatigue of a Cast A356 Aluminum Alloy]]<br />
* Nanoscale<br />
** [[Al-Mg | Modified Embedded Atom Method (MEAM) potential for Al-Mg]]<br />
** [http://arxiv.org/abs/1107.0544 MEAM potential for Al, Si, Mg, Cu, and Fe alloys] (see also: [http://code.google.com/p/ase-atomistic-potential-tests/ routines to reproduce the results])<br />
** [[GB_Gen | Grain Boundary Generation of Aluminum]]<ref name="Tsc2007a">Tschopp, M. A., & McDowell, D.L. (2007). Structures and energies of Sigma3 asymmetric tilt grain boundaries in Cu and Al. Philosophical Magazine, 87, 3147-3173 ([http://dx.doi.org/10.1080/14786430701455321 http://dx.doi.org/10.1080/14786430701455321]).</ref><ref name="Tsc2007b">Tschopp, M. A., & McDowell, D.L. (2007). Asymmetric tilt grain boundary structure and energy in copper and aluminum. Philosophical Magazine, 87, 3871-3892 ([http://dx.doi.org/10.1016/j.commatsci.2010.02.003 http://dx.doi.org/10.1016/j.commatsci.2010.02.003]).</ref><br />
** [[Aluminum_Dislocation_Nucleation | Dislocation Nucleation in Single Crystal Aluminum]]<ref>Spearot, D.E., Tschopp, M.A., Jacob, K.I., McDowell, D.L., "Tensile strength of <100> and <110> tilt bicrystal copper interfaces," Acta Materialia 55 (2007) p. 705-714 ([http://dx.doi.org/10.1016/j.actamat.2006.08.060 http://dx.doi.org/10.1016/j.actamat.2006.08.060]).</ref><ref>Tschopp, M.A., Spearot, D.E., McDowell, D.L., "Atomistic simulations of homogeneous dislocation nucleation in single crystal copper," Modelling and Simulation in Materials Science and Engineering 15 (2007) 693-709 ([http://dx.doi.org/10.1088/0965-0393/15/7/001 http://dx.doi.org/10.1088/0965-0393/15/7/001]).</ref><ref name="Tsc2008a">Tschopp, M.A., McDowell, D.L., "Influence of single crystal orientation on homogeneous dislocation nucleation under uniaxial loading," Journal of Mechanics and Physics of Solids 56 (2008) 1806-1830. ([http://dx.doi.org/10.1016/j.jmps.2007.11.012 http://dx.doi.org/10.1016/j.jmps.2007.11.012]).</ref><br />
** [[Uniaxial_Tension | Uniaxial Tension in Single Crystal Aluminum]]<ref name="Tsc2008a" /><br />
**[[Uniaxial_Compression | Uniaxial Compression in Single Crystal Aluminum]]<ref name="Tsc2008a" /><br />
** Electronic Structure<br />
<br />
==== Cobalt ====<br />
<br />
* Structural Scale<br />
* Macroscale<br />
** Plasticity-Damage Internal State Variable (DMG) Model<br />
** [[MSF Calibrations for Metals | MultiStage Fatigue (MSF) Model Calibrations ]]<br />
* Mesoscale<br />
* Microscale<br />
* Nanoscale<br />
* Electronic Structure<br />
<br />
<br />
==== Copper ====<br />
<br />
* Structural Scale<br />
**[[Stress Strain Curves: Brass]]<br />
* Macroscale<br />
** Plasticity-Damage Internal State Variable (DMG) Model<br />
** [[MSF Calibrations for Metals | MultiStage Fatigue (MSF) Model Calibrations ]]<br />
* Mesoscale<br />
* Microscale<br />
* Nanoscale<br />
* Electronic Structure<br />
*[[Porosity in Cast Bronze Pump Impeller|Bronze Pump Impeller]]<br />
<br />
==== Chromium ====<br />
<br />
* Structural Scale<br />
* Macroscale<br />
** [[Pure Chromium]]<br />
* Mesoscale<br />
* Microscale<br />
* Nanoscale<br />
* Electronic Structure<br />
<br />
==== Manganese ====<br />
<br />
* Structural Scale<br />
* Macroscale<br />
* Mesoscale<br />
* Microscale<br />
* Nanoscale<br />
* Electronic Structure<br />
** [[First principles calculations of doped MnBi compounds|First principles calculations of doped MnBi compounds]]<br />
<br />
==== Magnesium ====<br />
<br />
Magnesium alloys have been a focus in lightweight designs. Understanding the energy absorption, mechanical behavior and anisotropy, creep resistance, and corrosion resistance are key research opportunities.<br />
<br />
* Structural Scale<br />
* Macroscale<br />
** Plasticity-Damage Internal State Variable (DMG) Model<br />
** [[MSF Calibrations for Metals | MultiStage Fatigue (MSF) Model Calibrations ]]<br />
** [[Three-point bending behavior of a ZEK100 Mg alloy at room temperature]]<br />
** [[Corrosion]]<br />
*** [[Quantification of corrosion mechanisms on an extruded AZ31 magnesium alloy]]<br />
*** [[Corrosion Behaviour of Extruded AM30 Magnesium Alloy]]<br />
*** [[Corrosion Fatigue Behavior of Extruded AM30 Magnesium Alloy]]<br />
*** [[Structure–property quantification of corrosion pitting under immersion and salt-spray environments on an extruded AZ61 magnesium alloy]]<br />
*** [[Comparison of corrosion pitting under immersion and salt-spray environments on an as-cast AE44 magnesium alloy]]<br />
** [[ZE20: Stress-Strain data in Tension and Compression]]<br />
** [[AZ31B-O: Stress-Strain data in Tension and Compression]]<br />
** [[AZ61: Fatigue Life data]]<br />
** [[Multistage Fatigue of a Cast Magnesium Subframe]]<br />
* Mesoscale<br />
** [[A channel die compression simulation on Mg AM30]]<br />
** [[Twinning and double twinning upon compression of prismatic textures in an AM30 magnesium alloy]]<br />
** [[Code:_ABAQUS_CPFEM#Crystal_Plasticity_Finite_Element_Method|One element deformation of Magnesium]] <br />
* Microscale<br />
**[[Three-point bending behavior of a ZEK100 Mg alloy at room temperature]]<br />
* Nanoscale<br />
** [[Al-Mg | Modified Embedded Atom Method (MEAM) potential for Mg-Al]]<br />
** [[Grain boundary generation in Mg | Grain boundary generation in Mg]]<ref name="Tsc2007a" /><ref name="Tsc2007b" /><br />
** [[MD_Fatigue_Crack_Growth | Fatigue Crack Growth Simulation]]<ref>Tang, T., Kim, S., & Horstemeyer, M. (2010). Fatigue Crack Growth in Magnesium Single Crystals under Cyclic Loading: Molecular Dynamics Simulation. Computational Materials Science, 48, 426., 48, 426-439 ([http://dx.doi.org/10.1080/14786430701255895 http://dx.doi.org/10.1080/14786430701255895]).</ref><br />
** [[Single Crystal Tensile Deformation | Uniaxial Tension MD]]<ref>Barrett, C.D., El Kadiri, H., Tschopp, M.A. (2011). Breakdown of the Schmid Law in Homogenous and Heterogenous Nucleation Events of Slip and Twinning in Magnesium. Journal of Mechanics and Physics of Solids, in review.</ref><br />
* Electronic Structure<br />
** [[Modified embedded-atom method interatomic potentials for the Mg-Al alloy system]]<ref> B. Jelinek, J. Houze, Sungho Kim, M. F. Horstemeyer, M. I. Baskes, and Seong-Gon Kim, "Modified embedded-atom method interatomic potentials for the Mg-Al alloy system" Phys. Rev. B 75, 054106 (2007)</ref><br />
** [[ICME Overview for Wrought Magnesium Alloys|ICME Overview for Wrought Magnesium Alloys]]<br />
** [[ICME Overview of the Chemo-mechanical Effects on Magnesium Alloys|ICME Overview of the Chemo-mechanical Effects on Magnesium Alloys]]<br />
<br />
==== Nickel ====<br />
<br />
Nickel has been in use since 3500BCE, is one of the few room temperature ferromagnetic elements, and today is utilized in alloys, superalloys and catalysis. <br />
<br />
* Structural Scale<br />
* Macroscale<br />
** Plasticity-Damage Internal State Variable (DMG) Model<br />
** [[MSF Calibrations for Metals | MultiStage Fatigue (MSF) Model Calibrations ]]<br />
** [[Inconel 600]]<br />
** [[Pure Nickel]]<br />
* Mesoscale<br />
* Microscale<br />
* Nanoscale<br />
** [[Atomistic simulations of Bauschinger effects of metals with high angle and low angle grain boundaries]]<br />
* Electronic Structure<br />
<br />
<br />
==== Tin ====<br />
<br />
* Structural Scale<br />
* Macroscale<br />
** Plasticity-Damage Internal State Variable (DMG) Model<br />
** [[MSF Calibrations for Metals | MultiStage Fatigue (MSF) Model Calibrations ]]<br />
* Mesoscale<br />
* Microscale<br />
* Nanoscale<br />
* Electronic Structure<br />
<br />
<br />
==== Titanium ====<br />
* [[Residual Stress & Distortion Modelling for Additively Manufactured Ti6Al4V Parts|ICME 2019 Research Proposal]]<br />
*Structural Scale<br />
* Macroscale<br />
** Plasticity-Damage Internal State Variable (DMG) Model<br />
** [[MSF Calibrations for Metals | MultiStage Fatigue (MSF) Model Calibrations ]]<br />
* Mesoscale<br />
* Microscale<br />
* Nanoscale<br />
* Electronic Structure<br />
<br />
==== Tungsten ====<br />
<br />
* Structural Scale<br />
**[[Proposal for Multiscale Modeling of Tungsten Heavy Alloy (WHA) for Kinetic Energy Perpetrators|ICME Research Proposal: Tungsten Heavy Alloy for Kinetic Energy Perpetrators]]<br />
* Macroscale<br />
* Mesoscale<br />
* Microscale<br />
* Nanoscale<br />
**[[W]]<br />
* Electronic Structure<br />
**[[The effect of Fe atoms on the absorption of a W atom on W(100) surface]]<br />
<br />
==== Solder ====<br />
<br />
* Structural Scale<br />
* Macroscale<br />
** Plasticity-Damage Internal State Variable (DMG) Model<br />
** [[MSF Calibrations for Metals | MultiStage Fatigue (MSF) Model Calibrations ]]<br />
* Mesoscale<br />
* Microscale<br />
* Nanoscale<br />
* Electronic Structure<br />
<br />
<br />
==== Steel ====<br />
<br />
Here we can discuss applications to iron with links to projects.<br />
<br />
* Structural Scale<br />
**[[Proposal for Multiscale Modeling of 17-4 PH and life prediction using MSF model]]<br />
**[[Civil Engineering Materials]]<br />
* Macroscale<br />
**[[Stainless Steel: 17-7 PH TH1050]]<br />
**[[SSC Steel: 1006 steel alloy]]<br />
**[[SSC Steel: HY100 steel alloy]]<br />
**[[SSC Steel: 1020 steel alloy]]<br />
**[[SSC Steel: 10b22 steel alloy]]<br />
**[[SSC Steel: 300 Maraging Steel Alloy]]<br />
**[[SSC Steel: 304L SS alloy]]<br />
**[[SSC Steel: 321 SS alloy]]<br />
**[[SSC Steel: 4340 steel alloy]]<br />
**[[SSC Steel: A286 steel alloy]]<br />
**[[SSC Steel: AF steel alloy]]<br />
**[[SSC Steel: C1008 steel alloy]]<br />
**[[SSC Steel: FC0205 steel alloy]]<br />
**[[SSC Steel: HY130 steel alloy]]<br />
**[[SSC Steel: HY80 steel alloy]]<br />
**[[SSC Steel: Mild steel alloy]]<br />
**[[SSC Steel: S7tool steel alloy]]<br />
** Plasticity-Damage Internal State Variable (DMG) Model<br />
** [[MSF Calibrations for Metals | MultiStage Fatigue (MSF) Model Calibrations ]]<br />
**[[Quench and Partitioned Steels]]<br />
**[[Rolled Homogeneous Armor]]<br />
**[[Intermediate Strain-Rate Testing of ASTM A992 and A572 Grade 50 Steel]]<br />
**[[Corrosion]]<br />
**[[LENS_316L_SS_heat_treat|Effect of process time and heat treatment on the mechanical and microstructural properties of LENS fabricated 316L Stainless Steel]]<br />
***Direct laser deposition/LENS (Laser Engineered Net Shaping)<br />
* Mesoscale<br />
* Microscale<br />
** [[Dry Sliding Wear Analysis Using Low Cycle Fatigue and Finite Element Analysis|low cycle fatigue]]<br />
** [[media:PlasticityFractureModelingStudyPorousMetal Allison Grewal Hammi.pdf|Plasticity and Fracture Modeling/Experimental Study of a Porous Metal]]<br />
* Nanoscale<br />
** [[FeHe | Fe-He MEAM Interatomic Potential Development]]<br />
** [[Grain_boundary_generation| Grain boundary structure generation]]<br />
* Electronic Structure<br />
====Pure Vanadium====<br />
*[[Multi-Scale Modeling of Pure Vanadium]]<br />
<br />
</td><br />
</table><br />
<br />
== References ==<br />
<references/></div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/ICME_Student_Contributions_2019_-_MsStateICME Student Contributions 2019 - MsState2019-04-26T20:38:09Z<p>Madabhushi: /* Student 6 */</p>
<hr />
<div>[[ICME 8373 Student Contributions (Spring 2019)|< ICME 2019 Student Contributions]]<br />
<br />
=Student Contributions=<br />
<br />
===Student 1===<br />
Student Contribution 1<br />
<br />
* Added the following page to the ICME website [[Proposal: Quenched and Partitioned Steel Strength/Ductility versus Volume Fraction of Retained Austenite]]<br />
<br />
Student Contribution 2<br />
<br />
* Added the following tutorial to the ICME website https://www.youtube.com/watch?v=VsqUBnpqJu0&feature=youtu.be<br />
<br />
Student Contribution 3<br />
<br />
===Student 2===<br />
<br />
Student Contribution 1<br />
* Added instructions on modifying and running the [[Gsfe curve]] python script.<br />
* Added [[Gsfe curve]] to Repository of Codes.<br />
<br />
Student Contribution 2<br />
* Created [[Code: Ternary Plot]] page.<br />
* Linked [[Code: Ternary Plot]] in Repository of Codes.<br />
<br />
Student Contribution 3<br />
* Created [[Pure Chromium]] page.<br />
* Linked [[Pure Chromium]] in Metals Category page.<br />
* Added GSFE curves from class assignments to [[Pure Chromium]].<br />
* Intend to add CPFEM and other information from class assignments to [[Pure Chromium]].<br />
<br />
===Student 3===<br />
<br />
Created page to begin putting information about intermediate strain rate testing capabilities at CAVS: [[Intermediate Strain Rate Bar]]<br />
<br />
Created page detailing the general capabilities of high rate testing at CAVS: [[Split-Hopkinson Pressure Bars| Split-Hopkinson Pressure Bars]] & [[Tension Hopkinson Bars|Tension Bars]]<br />
<br />
Organized pages from [[:Special:UncategorizedPages| uncategorized pages]]<br />
*[[SSC Steel: HY100 steel alloy]]<br />
*[[SSC Steel: 1020 steel alloy]]<br />
*[[SSC Steel: 10b22 steel alloy]]<br />
*[[SSC Steel: 300 Maraging Steel Alloy]]<br />
*[[SSC Steel: 304L SS alloy]]<br />
*[[SSC Steel: 321 SS alloy]]<br />
*[[SSC Steel: 4340 steel alloy]]<br />
*[[SSC Steel: A286 steel alloy]]<br />
*[[SSC Steel: AF steel alloy]]<br />
<br />
===Student 4===<br />
<br />
Student Contribution 1<br />
*[[Python Scripting in Abaqus]]<br />
Student Contribution 2<br />
*[[Towards an Open-Source, Preprocessing Framework for Simulating Material Deposition for a Directed Energy Deposition Process]]<br />
Student Contribution 3<br />
*Homework submission compilation STILL IN PROGRESS<br />
**[[Multi-Scale Modeling of Pure Vanadium]]<br />
<br />
===Student 5===<br />
<br />
Student Contribution 1<br />
*[[Media:MDDP_PostProcessing_Tecplot.zip|MDDP Post-Processing Tecplot Tutorial]] <br />
<br />
Student Contribution 2<br />
*Page creation - [[Piezoelectrically Controlled Actuator]] & [[Serpentine Transmitted Bar]]<br />
<br />
Student Contribution 3<br />
*Organized pages from [[:Special:UncategorizedPages| uncategorized pages]]<br />
**[[Stainless Steel: 17-7 PH TH1050]]<br />
**[[Stress Strain Curves: Brass]]<br />
**[[SSC Steel: 1006 steel alloy]]<br />
**[[SSC Steel: C1008 steel alloy]]<br />
**[[SSC Steel: FC0205 steel alloy]]<br />
**[[SSC Steel: HY130 steel alloy]]<br />
**[[SSC Steel: HY80 steel alloy]]<br />
**[[SSC Steel: Mild steel alloy]]<br />
**[[SSC Steel: S7tool steel alloy]]<br />
<br />
===Student 6===<br />
<br />
*Student Contribution 1: Uploaded ICME research proposal, [[Residual Stress & Distortion Modelling for Additively Manufactured Ti6Al4V Parts]]. The page is properly linked to relevant categories<br />
<br />
*Student Contribution 2: <br />
**Uploaded a page describing the Additive Manufacturing method Powder Bed Fusion describing its basic outline [[Powder Bed Fusion]]. The page is properly linked to relevant categories <br />
**Uploaded a page describing Metal Matrix Composites (MMCs) and metal matrix Nanocomposites (MMNCs) [[Metal Matrix Composites]]. The page is properly linked to relevant categories<br />
<br />
*Student Contribution 3: The ICME HWs are unified into a single journal article style report, which can be found [[Multi-Scale Modeling of Pure Vanadium|here]]. The Page is properly linked to relevant categories<br />
<br />
===Student 7===<br />
*CLAIMED*<br />
Student Contribution 1: [[A Goal-Oriented, Inverse Decision-Based Design Method for Multi-Component Product Design]] Personal research paper upload.<br />
<br />
Student Contribution 2: [[PyDEM]] Design software upload.<br />
<br />
Student Contribution 3: class assignment [[Pure Chromium]]<br />
<br />
===Student 8===<br />
<br />
Student Contribution 1<br />
*Added the following page [[Structure Optimization]] under Quantum espresso at Electronic Scale.<br />
*Added the following page [[relax]] under Quantum espresso at Electronic Scale.<br />
*Added the following page [[vc-relax]] under Quantum espresso at Electronic Scale.<br />
<br />
Student Contribution 2<br />
*Added the following page [[How to make Supercell for Quantum ESPRESSO]] under Quantum espresso at Electronic Scale.<br />
Student Contribution 3<br />
*Added the following page [[ICME overview of shape memory effect on Bismuth Ferrite ceramic]] on Electronic Scale.<br />
<br />
===Student 9===<br />
<br />
Student Contribution 1<br />
<br />
i have made a section in the microscale category about a tutorial for porous Microsctucture Analysis (PuMA), here is the link of the contributions, https://icme.hpc.msstate.edu/mediawiki/index.php/Category:Microscale#Microscale_oxidation_simulation_PuMA.<br />
<br />
and here is the video added in the section https://www.youtube.com/watch?v=l9NrCsXmtBU.<br />
<br />
Student Contribution 2<br />
<br />
this is an MSF model for the additive manufacturing 17-4 PH stainless steel.<br />
https://icme.hpc.msstate.edu/mediawiki/index.php/17-4_PH_SS#MSF_Calibration<br />
<br />
Student Contribution 3<br />
<br />
Added a page for Research proposal about 17-4 PH SS https://icme.hpc.msstate.edu/mediawiki/index.php/Proposal_for_Multiscale_Modeling_of_17-4_PH_and_life_prediction_using_MSF_model<br />
<br />
===Student 10===<br />
Student Contribution 1<br />
- [[Fatigue Life Prediction of Aluminum Alloy 6063 for Vertical Axis Wind Turbine Blade Application]] (Research proposal)<br />
<br />
Student Contribution 2<br />
- [[Characterization and Modeling of the Fatigue Behavior of 304L Stainless Steel Using the MultiStage Fatigue (MSF) Model]] (Co-authored journal article.)<br />
<br />
Student Contribution 3<br />
- [[Pure Chromium]] (Co-authored journal article. Main sections include: theoretical models, MEAM potential calibration, and single crystal plasticity.)<br />
<br />
===Student 11===<br />
<br />
Student Contribution 1<br><br />
Added [[Intermediate Strain-Rate Testing of ASTM A992 and A572 Grade 50 Steel]]<br />
<br />
Student Contribution 2<br><br />
Added "Tungsten" to [[Metals]]<br><br />
Added [[W]] to "Tungsten" in [[Metals]]<br><br />
Added [[The effect of Fe atoms on the absorption of a W atom on W(100) surface]] to "Tungsten" in [[Metals]]<br><br />
Added [[Nanoscale]] and "Category: Tutorial" to [[Code: WARP - Description]]<br><br />
<br />
Student Contribution 3<br><br />
Added [[Proposal for Multiscale Modeling of Tungsten Heavy Alloy (WHA) for Kinetic Energy Perpetrators]]<br />
<br />
===Student 12===<br />
<br />
Student Contribution 1<br />
<br />
Categorized [[DFT Assignment]] and [[K-Point Variation]]<br />
<br />
Student Contribution 2<br />
<br />
Added video to [[K-Point Variation]] and linked to the VASP wiki site for more K-Point information.<br />
<br />
Student Contribution 3<br />
<br />
===Student 13===<br />
<br />
Student Contribution 1<br />
*Added the following page to the ICME website: [[Porosity in Cast Bronze Pump Impeller]]<br />
<br />
Student Contribution 2:<br />
*Added the following tutorial to the ICME website: Installing Linux on Window 10 - Compiling LAMMPS package from the source (https://icme.hpc.msstate.edu/mediawiki/index.php/Category:Nanoscale)<br />
<br />
Student Contribution 3:<br />
*Added the following tutorial to the ICME website: Learn Python - Full Course for Beginners (https://icme.hpc.msstate.edu/mediawiki/index.php/Python)<br />
<br />
===Student 14===<br />
<br />
Student Contribution 1: [[Calculating Dislocation Mobility]]<br />
<br />
Student Contribution 2: [[Multiscale Modeling of Hydrogen Porosity Formation During Solidification of Al-H]]<br />
<br />
Student Contribution 3: [[Multi-Scale Modeling of Pure Vanadium]]<br />
<br />
===Student 15===<br />
<br />
Student Contribution 1: Upload Journal Article: Interatomic Potential for Hydrocarbons on the Basis of the Modified Embedded-Atom Method with Bond-Order (MEAM-BO): https://icme.hpc.msstate.edu/mediawiki/index.php/Interatomic_Potential_for_Hydrocarbons_on_the_Basis_of_the_Modified_Embedded-Atom_Method_with_Bond_Order_(MEAM-BO) <br />
<br />
Student Contribution 2: ICME Research Proposal to Evaluate Multi-Scale Property Relations for Moisture Absorption of Carbon Fiber Reinforced Plastic Bicycle Wheel: https://icme.hpc.msstate.edu/mediawiki/index.php/ICME_Research_Proposal_to_Evaluate_Multi-Scale_Property_Relations_for_Moisture_Absorption_of_Carbon_Fiber_Reinforced_Plastic_Bicycle_Wheel<br />
<br />
Student Contribution 3: Dynamic Dislocation Plasticity Background Information: https://icme.hpc.msstate.edu/mediawiki/index.php/Dynamic_Dislocation_Plasticity. This page describes key research that was conducted in the background in Dislocation Dynamics and cautions with applying these methods to BCC metals.<br />
<br />
===Student 16===<br />
<br />
Student Contribution 1<br />
<br />
Student Contribution 2<br />
<br />
Student Contribution 3<br />
<br />
===Student 17===<br />
<br />
Student Contribution 1<br />
<br />
Student Contribution 2<br />
<br />
Student Contribution 3<br />
<br />
===Student 18===<br />
<br />
Student Contribution 1<br />
<br />
Student Contribution 2<br />
<br />
Student Contribution 3<br />
<br />
===Student 19===<br />
<br />
Student Contribution 1<br />
<br />
Student Contribution 2<br />
<br />
Student Contribution 3<br />
<br />
===Student 20===<br />
<br />
Claimed* in progress<br />
<br />
<br />
Student Contribution 1<br />
<br />
Created multiple post processing codes for plotting data from DFT calculations that can be found at:<br />
<br />
* [[EvA_EvV_plot.py | Python code for post-processing EvsA and EvsV files from running Quantum Espresso simulations using the ev_curve.bash script to generate plots for the EvV and EvA curves ]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
* [[convergence_plots.py | Python code for post-processing <code> SUMMARY</code> files from running Quantum Espresso simulations using the ev_curve.bash script to generate a plot for a convergence study ]] for [[Code: Quantum Espresso | Quantum Espresso ]] <br />
* [[ecut_conv.py | Python code for post-processing .out files files from running Quantum Espresso simulations to generate a plot for the ecut convergencerate ]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
* [[EOS_comp_plot.py | Python code for post-processing <code> SUMMARY</code>, <code> EsvA </code>, <code> EsvV</code>, and <code> evfit.#</code> files from running Quantum Espresso simulations using the ev_curve.bash script to generate a plot comparing the effect of using the different equations of state in the evfit code ]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
* [[EOS_plot.py | Python code for post-processing <code> evfit.#</code> files from running Quantum Espresso simulations and using the evfit.f routine to fit to multiple equations of state]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
<br />
Student Contribution 2<br />
<br />
Uploaded research proposal for method of creating nanocrystalline/amorphous metals using femtosecond laser induced ablation. Found at: [[Laser induced microstructure]]<br />
<br />
<br />
<br />
Student Contribution 3<br />
<br />
Added link to software for generating high order finite elements to be used in codes that solve PDE's using discretization methods. Works for both continuous and discontinuous methods. Found at:<br />
<br />
* [[DIY-FEA]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/ICME_Student_Contributions_2019_-_MsStateICME Student Contributions 2019 - MsState2019-04-26T20:37:06Z<p>Madabhushi: /* Student 6 */</p>
<hr />
<div>[[ICME 8373 Student Contributions (Spring 2019)|< ICME 2019 Student Contributions]]<br />
<br />
=Student Contributions=<br />
<br />
===Student 1===<br />
Student Contribution 1<br />
<br />
* Added the following page to the ICME website [[Proposal: Quenched and Partitioned Steel Strength/Ductility versus Volume Fraction of Retained Austenite]]<br />
<br />
Student Contribution 2<br />
<br />
* Added the following tutorial to the ICME website https://www.youtube.com/watch?v=VsqUBnpqJu0&feature=youtu.be<br />
<br />
Student Contribution 3<br />
<br />
===Student 2===<br />
<br />
Student Contribution 1<br />
* Added instructions on modifying and running the [[Gsfe curve]] python script.<br />
* Added [[Gsfe curve]] to Repository of Codes.<br />
<br />
Student Contribution 2<br />
* Created [[Code: Ternary Plot]] page.<br />
* Linked [[Code: Ternary Plot]] in Repository of Codes.<br />
<br />
Student Contribution 3<br />
* Created [[Pure Chromium]] page.<br />
* Linked [[Pure Chromium]] in Metals Category page.<br />
* Added GSFE curves from class assignments to [[Pure Chromium]].<br />
* Intend to add CPFEM and other information from class assignments to [[Pure Chromium]].<br />
<br />
===Student 3===<br />
<br />
Created page to begin putting information about intermediate strain rate testing capabilities at CAVS: [[Intermediate Strain Rate Bar]]<br />
<br />
Created page detailing the general capabilities of high rate testing at CAVS: [[Split-Hopkinson Pressure Bars| Split-Hopkinson Pressure Bars]] & [[Tension Hopkinson Bars|Tension Bars]]<br />
<br />
Organized pages from [[:Special:UncategorizedPages| uncategorized pages]]<br />
*[[SSC Steel: HY100 steel alloy]]<br />
*[[SSC Steel: 1020 steel alloy]]<br />
*[[SSC Steel: 10b22 steel alloy]]<br />
*[[SSC Steel: 300 Maraging Steel Alloy]]<br />
*[[SSC Steel: 304L SS alloy]]<br />
*[[SSC Steel: 321 SS alloy]]<br />
*[[SSC Steel: 4340 steel alloy]]<br />
*[[SSC Steel: A286 steel alloy]]<br />
*[[SSC Steel: AF steel alloy]]<br />
<br />
===Student 4===<br />
<br />
Student Contribution 1<br />
*[[Python Scripting in Abaqus]]<br />
Student Contribution 2<br />
*[[Towards an Open-Source, Preprocessing Framework for Simulating Material Deposition for a Directed Energy Deposition Process]]<br />
Student Contribution 3<br />
*Homework submission compilation STILL IN PROGRESS<br />
**[[Multi-Scale Modeling of Pure Vanadium]]<br />
<br />
===Student 5===<br />
<br />
Student Contribution 1<br />
*[[Media:MDDP_PostProcessing_Tecplot.zip|MDDP Post-Processing Tecplot Tutorial]] <br />
<br />
Student Contribution 2<br />
*Page creation - [[Piezoelectrically Controlled Actuator]] & [[Serpentine Transmitted Bar]]<br />
<br />
Student Contribution 3<br />
*Organized pages from [[:Special:UncategorizedPages| uncategorized pages]]<br />
**[[Stainless Steel: 17-7 PH TH1050]]<br />
**[[Stress Strain Curves: Brass]]<br />
**[[SSC Steel: 1006 steel alloy]]<br />
**[[SSC Steel: C1008 steel alloy]]<br />
**[[SSC Steel: FC0205 steel alloy]]<br />
**[[SSC Steel: HY130 steel alloy]]<br />
**[[SSC Steel: HY80 steel alloy]]<br />
**[[SSC Steel: Mild steel alloy]]<br />
**[[SSC Steel: S7tool steel alloy]]<br />
<br />
===Student 6===<br />
<br />
*Student Contribution 1: Uploaded ICME research proposal, [[Residual Stress & Distortion Modelling for Additively Manufactured Ti6Al4V Parts]]<br />
<br />
*Student Contribution 2: <br />
**Uploaded a page describing the Additive Manufacturing method Powder Bed Fusion describing its basic outline [[Powder Bed Fusion]]<br />
**Uploaded a page describing Metal Matrix Composites (MMCs) and metal matrix Nanocomposites (MMNCs) [[Metal Matrix Composites]]<br />
<br />
*Student Contribution 3: The ICME HWs are unified into a single journal article style report, which can be found [[Multi-Scale Modeling of Pure Vanadium|here]].<br />
<br />
===Student 7===<br />
*CLAIMED*<br />
Student Contribution 1: [[A Goal-Oriented, Inverse Decision-Based Design Method for Multi-Component Product Design]] Personal research paper upload.<br />
<br />
Student Contribution 2: [[PyDEM]] Design software upload.<br />
<br />
Student Contribution 3: class assignment [[Pure Chromium]]<br />
<br />
===Student 8===<br />
<br />
Student Contribution 1<br />
*Added the following page [[Structure Optimization]] under Quantum espresso at Electronic Scale.<br />
*Added the following page [[relax]] under Quantum espresso at Electronic Scale.<br />
*Added the following page [[vc-relax]] under Quantum espresso at Electronic Scale.<br />
<br />
Student Contribution 2<br />
*Added the following page [[How to make Supercell for Quantum ESPRESSO]] under Quantum espresso at Electronic Scale.<br />
Student Contribution 3<br />
*Added the following page [[ICME overview of shape memory effect on Bismuth Ferrite ceramic]] on Electronic Scale.<br />
<br />
===Student 9===<br />
<br />
Student Contribution 1<br />
<br />
i have made a section in the microscale category about a tutorial for porous Microsctucture Analysis (PuMA), here is the link of the contributions, https://icme.hpc.msstate.edu/mediawiki/index.php/Category:Microscale#Microscale_oxidation_simulation_PuMA.<br />
<br />
and here is the video added in the section https://www.youtube.com/watch?v=l9NrCsXmtBU.<br />
<br />
Student Contribution 2<br />
<br />
this is an MSF model for the additive manufacturing 17-4 PH stainless steel.<br />
https://icme.hpc.msstate.edu/mediawiki/index.php/17-4_PH_SS#MSF_Calibration<br />
<br />
Student Contribution 3<br />
<br />
Added a page for Research proposal about 17-4 PH SS https://icme.hpc.msstate.edu/mediawiki/index.php/Proposal_for_Multiscale_Modeling_of_17-4_PH_and_life_prediction_using_MSF_model<br />
<br />
===Student 10===<br />
Student Contribution 1<br />
- [[Fatigue Life Prediction of Aluminum Alloy 6063 for Vertical Axis Wind Turbine Blade Application]] (Research proposal)<br />
<br />
Student Contribution 2<br />
- [[Characterization and Modeling of the Fatigue Behavior of 304L Stainless Steel Using the MultiStage Fatigue (MSF) Model]] (Co-authored journal article.)<br />
<br />
Student Contribution 3<br />
- [[Pure Chromium]] (Co-authored journal article. Main sections include: theoretical models, MEAM potential calibration, and single crystal plasticity.)<br />
<br />
===Student 11===<br />
<br />
Student Contribution 1<br><br />
Added [[Intermediate Strain-Rate Testing of ASTM A992 and A572 Grade 50 Steel]]<br />
<br />
Student Contribution 2<br><br />
Added "Tungsten" to [[Metals]]<br><br />
Added [[W]] to "Tungsten" in [[Metals]]<br><br />
Added [[The effect of Fe atoms on the absorption of a W atom on W(100) surface]] to "Tungsten" in [[Metals]]<br><br />
Added [[Nanoscale]] and "Category: Tutorial" to [[Code: WARP - Description]]<br><br />
<br />
Student Contribution 3<br><br />
Added [[Proposal for Multiscale Modeling of Tungsten Heavy Alloy (WHA) for Kinetic Energy Perpetrators]]<br />
<br />
===Student 12===<br />
<br />
Student Contribution 1<br />
<br />
Categorized [[DFT Assignment]] and [[K-Point Variation]]<br />
<br />
Student Contribution 2<br />
<br />
Added video to [[K-Point Variation]] and linked to the VASP wiki site for more K-Point information.<br />
<br />
Student Contribution 3<br />
<br />
===Student 13===<br />
<br />
Student Contribution 1<br />
*Added the following page to the ICME website: [[Porosity in Cast Bronze Pump Impeller]]<br />
<br />
Student Contribution 2:<br />
*Added the following tutorial to the ICME website: Installing Linux on Window 10 - Compiling LAMMPS package from the source (https://icme.hpc.msstate.edu/mediawiki/index.php/Category:Nanoscale)<br />
<br />
Student Contribution 3:<br />
*Added the following tutorial to the ICME website: Learn Python - Full Course for Beginners (https://icme.hpc.msstate.edu/mediawiki/index.php/Python)<br />
<br />
===Student 14===<br />
<br />
Student Contribution 1: [[Calculating Dislocation Mobility]]<br />
<br />
Student Contribution 2: [[Multiscale Modeling of Hydrogen Porosity Formation During Solidification of Al-H]]<br />
<br />
Student Contribution 3: [[Multi-Scale Modeling of Pure Vanadium]]<br />
<br />
===Student 15===<br />
<br />
Student Contribution 1: Upload Journal Article: Interatomic Potential for Hydrocarbons on the Basis of the Modified Embedded-Atom Method with Bond-Order (MEAM-BO): https://icme.hpc.msstate.edu/mediawiki/index.php/Interatomic_Potential_for_Hydrocarbons_on_the_Basis_of_the_Modified_Embedded-Atom_Method_with_Bond_Order_(MEAM-BO) <br />
<br />
Student Contribution 2: ICME Research Proposal to Evaluate Multi-Scale Property Relations for Moisture Absorption of Carbon Fiber Reinforced Plastic Bicycle Wheel: https://icme.hpc.msstate.edu/mediawiki/index.php/ICME_Research_Proposal_to_Evaluate_Multi-Scale_Property_Relations_for_Moisture_Absorption_of_Carbon_Fiber_Reinforced_Plastic_Bicycle_Wheel<br />
<br />
Student Contribution 3: Dynamic Dislocation Plasticity Background Information: https://icme.hpc.msstate.edu/mediawiki/index.php/Dynamic_Dislocation_Plasticity. This page describes key research that was conducted in the background in Dislocation Dynamics and cautions with applying these methods to BCC metals.<br />
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===Student 16===<br />
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Student Contribution 1<br />
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Student Contribution 2<br />
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Student Contribution 3<br />
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===Student 17===<br />
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Student Contribution 1<br />
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Student Contribution 2<br />
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Student Contribution 3<br />
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===Student 18===<br />
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Student Contribution 1<br />
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Student Contribution 2<br />
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Student Contribution 3<br />
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===Student 19===<br />
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Student Contribution 1<br />
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Student Contribution 2<br />
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Student Contribution 3<br />
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===Student 20===<br />
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Claimed* in progress<br />
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Student Contribution 1<br />
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Created multiple post processing codes for plotting data from DFT calculations that can be found at:<br />
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* [[EvA_EvV_plot.py | Python code for post-processing EvsA and EvsV files from running Quantum Espresso simulations using the ev_curve.bash script to generate plots for the EvV and EvA curves ]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
* [[convergence_plots.py | Python code for post-processing <code> SUMMARY</code> files from running Quantum Espresso simulations using the ev_curve.bash script to generate a plot for a convergence study ]] for [[Code: Quantum Espresso | Quantum Espresso ]] <br />
* [[ecut_conv.py | Python code for post-processing .out files files from running Quantum Espresso simulations to generate a plot for the ecut convergencerate ]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
* [[EOS_comp_plot.py | Python code for post-processing <code> SUMMARY</code>, <code> EsvA </code>, <code> EsvV</code>, and <code> evfit.#</code> files from running Quantum Espresso simulations using the ev_curve.bash script to generate a plot comparing the effect of using the different equations of state in the evfit code ]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
* [[EOS_plot.py | Python code for post-processing <code> evfit.#</code> files from running Quantum Espresso simulations and using the evfit.f routine to fit to multiple equations of state]] for [[Code: Quantum Espresso | Quantum Espresso ]]<br />
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Student Contribution 2<br />
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Uploaded research proposal for method of creating nanocrystalline/amorphous metals using femtosecond laser induced ablation. Found at: [[Laser induced microstructure]]<br />
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Student Contribution 3<br />
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Added link to software for generating high order finite elements to be used in codes that solve PDE's using discretization methods. Works for both continuous and discontinuous methods. Found at:<br />
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* [[DIY-FEA]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T20:33:51Z<p>Madabhushi: </p>
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<div>=Abstract=<br />
In this study, the stress/strain behavior of pure vanadium in the elastic and plastic regimes are simulated using a multiscale approach. Density Functional Theory (DFT) calculations were performed to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, bulk modulus, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. Three second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results to quantify uncertainty in GSFE result. The MEAM potentials were used in a molecular dynamics code to determine the dislocation mobility. The calculated dislocation mobility was upscaled to Dislocation Dynamics (DD) scale to study the stress-strain response of the material, specifically the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity scale to simulate the stress/strain response of a single and polycrystalline material. Finally, the stress/strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale.<br />
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Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
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a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
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b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
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=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity calculations to simulate the stress/strain response of a single and polycrystalline material. Finally, the stress/strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale. <br />
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This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
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While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
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=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
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[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
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===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
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==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
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[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
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==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
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All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
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== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
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== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
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[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
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=Results= <br />
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==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
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<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
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==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
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[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
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==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
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A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
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[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
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===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
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<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
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==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
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Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
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<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure A5 in the appendix. All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations. Polycrystalline simulations run with 500 grains were also simulated in tension, compression, and shear for all three cases. The stress-strain plots for these simulations are presented in Figure 7. It is apparent that the different dislocation hardening parameters for each case have affected these results since the yield stress obtained from this exercise is significantly different in comparison with the yield stress observed in BCC Vanadium.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure 7. CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best-fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
The ISV model for the three loading conditions was calibrated using the DMGfit 55 v1 p1 tool. The three datasets for each case are loaded on the calibration tool with the prescribed material constants and the stress-strain information from CPFEM. Since the data is temperature and strain rate independent, the fitting of the parameters is done in the following order: Compression, Tension, and Torsion. In the compression dataset, the first constant to be fitted is C3 which relates to Yield, followed by the Kinematic hardening and recovery constants C9 and C7, and finally the Isotropic hardening and recovery constants C15 and C13. After this, the Tension dataset is restored and the process is repeated for the Torsion data set. For the torsion data set, the torsion, compression and tension differentiation constants Ca and Cb are adjusted. This is done for the Best fit, Upper bound and the Lower bound cases. The fitted constants are then added to an input deck to be used to run single element calculations in ABAQUS. The following table lists the ISV fitted constants for the best-fit, upper bound and lower bound cases.<br />
<div align="center">Table 6. Verification results where best fit, lower bound and upper bound columns are organized left to right.</div><br />
[[Image:DMGfittable.JPG|center|800px]]<br />
<br><br />
Verification simulations ran with the calibrated model in single element simulations are presented in Figure 8 alongside the calibration plots from the DMGfit tool. These plots show reasonable agreement between the two codes, but differences are apparent. This could be due to several different reasons. Apart from the uncertainty in the GSFE from the previous scales, the current ISV model does not consider temperature and strain rate dependencies. Moreover, the accuracy of the Voce Hardening law parameters obtained in the previous section is the primary concern, since it would take a significantly higher amount of computational power and time to capture the entire scope of the MDDP simulations. These issues could be handled for a more robust study which is outside the scope of this report. <br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure 8: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
In this study, we have demonstrated a multiscale approach for the study of stress-strain response of a material, using pure vanadium for illustration. Starting from the electronic scale, DFT calculations provided information about energy verse volume curves, coherence energy, bulk modulus, GSFE, and lattice constant. This information was used in the parameterization of three Nearest-neighbor (2NN) MEAM interatomic potential to capture uncertainty in GSFE result. The MEAM potentials were used in MD code (LAMMPS) to determine the dislocation mobility. The calculated dislocation mobility was upscaled to the DD scale to study the stress/strain response of the material, specifically the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity scale to simulate the stress-strain response of a single and polycrystalline material. Finally, the stress-strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale.<br />
A more robust study can be performed later for more accuracy leading to the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The page will be updated in due course of time as we obtain newer and better results that accurately predict the continuum scale stress-strain behavior of pure BCC Vanadium.<br />
<br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T20:32:03Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
In this study, the stress/strain behavior of pure vanadium in the elastic and plastic regimes are simulated using a multiscale approach. Density Functional Theory (DFT) calculations were performed to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, bulk modulus, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. Three second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results to quantify uncertainty in GSFE result. The MEAM potentials were used in a molecular dynamics code to determine the dislocation mobility. The calculated dislocation mobility was upscaled to Dislocation Dynamics (DD) scale to study the stress-strain response of the material, specifically the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity scale to simulate the stress/strain response of a single and polycrystalline material. Finally, the stress/strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale.<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity calculations to simulate the stress/strain response of a single and polycrystalline material. Finally, the stress/strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale. <br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure A5 in the appendix. All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations. Polycrystalline simulations run with 500 grains were also simulated in tension, compression, and shear for all three cases. The stress-strain plots for these simulations are presented in Figure 7. It is apparent that the different dislocation hardening parameters for each case have affected these results since the yield stress obtained from this exercise is significantly different in comparison with the yield stress observed in BCC Vanadium.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure 7. CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best-fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
The ISV model for the three loading conditions was calibrated using the DMGfit 55 v1 p1 tool. The three datasets for each case are loaded on the calibration tool with the prescribed material constants and the stress-strain information from CPFEM. Since the data is temperature and strain rate independent, the fitting of the parameters is done in the following order: Compression, Tension, and Torsion. In the compression dataset, the first constant to be fitted is C3 which relates to Yield, followed by the Kinematic hardening and recovery constants C9 and C7, and finally the Isotropic hardening and recovery constants C15 and C13. After this, the Tension dataset is restored and the process is repeated for the Torsion data set. For the torsion data set, the torsion, compression and tension differentiation constants Ca and Cb are adjusted. This is done for the Best fit, Upper bound and the Lower bound cases. The fitted constants are then added to an input deck to be used to run single element calculations in ABAQUS. The following table lists the ISV fitted constants for the best-fit, upper bound and lower bound cases.<br />
<div align="center">Table 6. Verification results where best fit, lower bound and upper bound columns are organized left to right.</div><br />
[[Image:DMGfittable.JPG|center|800px]]<br />
<br><br />
Verification simulations ran with the calibrated model in single element simulations are presented in Figure 8 alongside the calibration plots from the DMGfit tool. These plots show reasonable agreement between the two codes, but differences are apparent. This could be due to several different reasons. Apart from the uncertainty in the GSFE from the previous scales, the current ISV model does not consider temperature and strain rate dependencies. Moreover, the accuracy of the Voce Hardening law parameters obtained in the previous section is the primary concern, since it would take a significantly higher amount of computational power and time to capture the entire scope of the MDDP simulations. These issues could be handled for a more robust study which is outside the scope of this report. <br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure 8: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
In this study, we have demonstrated a multiscale approach for the study of stress-strain response of a material, using pure vanadium for illustration. Starting from the electronic scale, DFT calculations provided information about energy verse volume curves, coherence energy, bulk modulus, GSFE, and lattice constant. This information was used in the parameterization of three Nearest-neighbor (2NN) MEAM interatomic potential to capture uncertainty in GSFE result. The MEAM potentials were used in MD code (LAMMPS) to determine the dislocation mobility. The calculated dislocation mobility was upscaled to the DD scale to study the stress/strain response of the material, specifically the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity scale to simulate the stress-strain response of a single and polycrystalline material. Finally, the stress-strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale.<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The page will be updated in due course of time as we obtain newer and better results that accurately predict the continuum scale stress-strain behavior of pure BCC Vanadium.<br />
<br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T20:27:14Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
In this study, the stress/strain behavior of pure vanadium in the elastic and plastic regimes are simulated using a multiscale approach. Density Functional Theory (DFT) calculations were performed to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, bulk modulus, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. Three second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results to quantify uncertainty in GSFE result. The MEAM potentials were used in a molecular dynamics code to determine the dislocation mobility. The calculated dislocation mobility was upscaled to Dislocation Dynamics (DD) scale to study the stress-strain response of the material, specifically the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity scale to simulate the stress/strain response of a single and polycrystalline material. Finally, the stress/strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale.<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity calculations to simulate the stress/strain response of a single and polycrystalline material. Finally, the stress/strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale. <br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure A5 in the appendix. All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations. Polycrystalline simulations run with 500 grains were also simulated in tension, compression, and shear for all three cases. The stress-strain plots for these simulations are presented in Figure 7. It is apparent that the different dislocation hardening parameters for each case have affected these results since the yield stress obtained from this exercise is significantly different in comparison with the yield stress observed in BCC Vanadium.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure 7. CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best-fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
The ISV model for the three loading conditions was calibrated using the DMGfit 55 v1 p1 tool. The three datasets for each case are loaded on the calibration tool with the prescribed material constants and the stress-strain information from CPFEM. Since the data is temperature and strain rate independent, the fitting of the parameters is done in the following order: Compression, Tension, and Torsion. In the compression dataset, the first constant to be fitted is C3 which relates to Yield, followed by the Kinematic hardening and recovery constants C9 and C7, and finally the Isotropic hardening and recovery constants C15 and C13. After this, the Tension dataset is restored and the process is repeated for the Torsion data set. For the torsion data set, the torsion, compression and tension differentiation constants Ca and Cb are adjusted. This is done for the Best fit, Upper bound and the Lower bound cases. The fitted constants are then added to an input deck to be used to run single element calculations in ABAQUS. The following table lists the ISV fitted constants for the best-fit, upper bound and lower bound cases.<br />
<div align="center">Table 6. Verification results where best fit, lower bound and upper bound columns are organized left to right.</div><br />
[[Image:DMGfittable.JPG|center|800px]]<br />
<br><br />
Verification simulations ran with the calibrated model in single element simulations are presented in Figure 8 alongside the calibration plots from the DMGfit tool. These plots show reasonable agreement between the two codes, but differences are apparent. This could be due to several different reasons. Apart from the uncertainty in the GSFE from the previous scales, the current ISV model does not consider temperature and strain rate dependencies. Moreover, the accuracy of the Voce Hardening law parameters obtained in the previous section is the primary concern, since it would take a significantly higher amount of computational power and time to capture the entire scope of the MDDP simulations. These issues could be handled for a more robust study which is outside the scope of this report. <br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure 8: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
In this study, we have demonstrated a multiscale approach for the study of stress/strain response of a material, using pure vanadium for illustration. Starting from the electronic scale, DFT calculations provided information about energy verse volume curves, coherence energy, bulk modulus, GSFE, and lattice constants. This information was used in the parameterization of three 2NN MEAM interatomic potential to capture uncertainty in GSFE result. The MEAM potentials were used in MD code (LAMMPS) to determine the dislocation mobility. The calculated dislocation mobility was upscaled to the DD scale to study the stress/strain response of the material, specifically the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity scale to simulate the stress/strain response of a single and polycrystalline material. Finally, the stress/strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale.<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium.<br />
<br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T20:26:19Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
In this study, the stress/strain behavior of pure vanadium in the elastic and plastic regimes are simulated using a multiscale approach. Density Functional Theory (DFT) calculations were performed to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, bulk modulus, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. Three second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results to quantify uncertainty in GSFE result. The MEAM potentials were used in a molecular dynamics code to determine the dislocation mobility. The calculated dislocation mobility was upscaled to Dislocation Dynamics (DD) scale to study the stress-strain response of the material, specifically the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity scale to simulate the stress/strain response of a single and polycrystalline material. Finally, the stress/strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale.<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity calculations to simulate the stress/strain response of a single and polycrystalline material. Finally, the stress/strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale. <br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure A5 in the appendix. All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations. Polycrystalline simulations run with 500 grains were also simulated in tension, compression, and shear for all three cases. The stress-strain plots for these simulations are presented in Figure 7. It is apparent that the different dislocation hardening parameters for each case have affected these results since the yield stress obtained from this exercise is significantly different in comparison with the yield stress observed in BCC Vanadium.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure 7. CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best-fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
The ISV model for the three loading conditions was calibrated using the DMGfit 55 v1 p1 tool. The three datasets for each case are loaded on the calibration tool with the prescribed material constants and the stress-strain information from CPFEM. Since the data is temperature and strain rate independent, the fitting of the parameters is done in the following order: Compression, Tension, and Torsion. In the compression dataset, the first constant to be fitted is C3 which relates to Yield, followed by the Kinematic hardening and recovery constants C9 and C7, and finally the Isotropic hardening and recovery constants C15 and C13. After this, the Tension dataset is restored and the process is repeated for the Torsion data set. For the torsion data set, the torsion, compression and tension differentiation constants Ca and Cb are adjusted. This is done for the Best fit, Upper bound and the Lower bound cases. The fitted constants are then added to an input deck to be used to run single element calculations in ABAQUS. The following table lists the ISV fitted constants for the best-fit, upper bound and lower bound cases.<br />
<div align="center">Table XXX Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
[[Image:DMGfittable.JPG|center|1000px]]<br />
<br><br />
Verification simulations ran with the calibrated model in single element simulations are presented in Figure 8 alongside the calibration plots from the DMGfit tool. These plots show reasonable agreement between the two codes, but differences are apparent. This could be due to several different reasons. Apart from the uncertainty in the GSFE from the previous scales, the current ISV model does not consider temperature and strain rate dependencies. Moreover, the accuracy of the Voce Hardening law parameters obtained in the previous section is the primary concern, since it would take a significantly higher amount of computational power and time to capture the entire scope of the MDDP simulations. These issues could be handled for a more robust study which is outside the scope of this report. <br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure 8: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
In this study, we have demonstrated a multiscale approach for the study of stress/strain response of a material, using pure vanadium for illustration. Starting from the electronic scale, DFT calculations provided information about energy verse volume curves, coherence energy, bulk modulus, GSFE, and lattice constants. This information was used in the parameterization of three 2NN MEAM interatomic potential to capture uncertainty in GSFE result. The MEAM potentials were used in MD code (LAMMPS) to determine the dislocation mobility. The calculated dislocation mobility was upscaled to the DD scale to study the stress/strain response of the material, specifically the hardening behavior. From the DD calculations, hardening law parameters were upscaled to crystal plasticity scale to simulate the stress/strain response of a single and polycrystalline material. Finally, the stress/strain curves calculated from crystal plasticity are used for the parameterization of the internal state variable (ISV) model to enable simulation on a continuum scale.<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium.<br />
<br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/File:DMGfittable.JPGFile:DMGfittable.JPG2019-04-26T20:25:33Z<p>Madabhushi: </p>
<hr />
<div></div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:41:39Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure A5 in the appendix. All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations. Polycrystalline simulations run with 500 grains were also simulated in tension, compression, and shear for all three cases. The stress-strain plots for these simulations are presented in Figure 7. It is apparent that the different dislocation hardening parameters for each case have affected these results since the yield stress obtained from this exercise is significantly different in comparison with the yield stress observed in BCC Vanadium.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure 7. CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best-fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
The ISV model for the three loading conditions was calibrated using the DMGfit 55 v1 p1 tool. The three datasets for each case are loaded on the calibration tool with the prescribed material constants and the stress-strain information from CPFEM. Since the data is temperature and strain rate independent, the fitting of the parameters is done in the following order: Compression, Tension, and Torsion. In the compression dataset, the first constant to be fitted is C3 which relates to Yield, followed by the Kinematic hardening and recovery constants C9 and C7, and finally the Isotropic hardening and recovery constants C15 and C13. After this, the Tension dataset is restored and the process is repeated for the Torsion data set. For the torsion data set, the torsion, compression and tension differentiation constants Ca and Cb are adjusted. This is done for the Best fit, Upper bound and the Lower bound cases. The fitted constants are then added to an input deck to be used to run single element calculations in ABAQUS. The following table lists the ISV fitted constants for the best-fit, upper bound and lower bound cases.<br />
<div align="center">Table XXX Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
[[Image:|center|1000px]]<br />
<br><br />
Verification simulations ran with the calibrated model in single element simulations are presented in Figure 8 alongside the calibration plots from the DMGfit tool. These plots show reasonable agreement between the two codes, but differences are apparent. This could be due to several different reasons. Apart from the uncertainty in the GSFE from the previous scales, the current ISV model does not consider temperature and strain rate dependencies. Moreover, the accuracy of the Voce Hardening law parameters obtained in the previous section is the primary concern, since it would take a significantly higher amount of computational power and time to capture the entire scope of the MDDP simulations. These issues could be handled for a more robust study which is outside the scope of this report. <br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure 8: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:40:44Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure A5 in the appendix. All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations. Polycrystalline simulations run with 500 grains were also simulated in tension, compression, and shear for all three cases. The stress-strain plots for these simulations are presented in Figure 7. It is apparent that the different dislocation hardening parameters for each case have affected these results since the yield stress obtained from this exercise is significantly different in comparison with the yield stress observed in BCC Vanadium.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure 7. CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best-fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
The ISV model for the three loading conditions was calibrated using the DMGfit 55 v1 p1 tool. The three datasets for each case are loaded on the calibration tool with the prescribed material constants and the stress-strain information from CPFEM. Since the data is temperature and strain rate independent, the fitting of the parameters is done in the following order: Compression, Tension, and Torsion. In the compression dataset, the first constant to be fitted is C3 which relates to Yield, followed by the Kinematic hardening and recovery constants C9 and C7, and finally the Isotropic hardening and recovery constants C15 and C13. After this, the Tension dataset is restored and the process is repeated for the Torsion data set. For the torsion data set, the torsion, compression and tension differentiation constants Ca and Cb are adjusted. This is done for the Best fit, Upper bound and the Lower bound cases. The fitted constants are then added to an input deck to be used to run single element calculations in ABAQUS. The following table lists the ISV fitted constants for the best-fit, upper bound and lower bound cases.<br />
<div align="center">Table XXX Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
[[Image:|center|1000px]]<br />
Verification simulations ran with the calibrated model in single element simulations are presented in Figure 8 alongside the calibration plots from the DMGfit tool. These plots show reasonable agreement between the two codes, but differences are apparent. This could be due to several different reasons. Apart from the uncertainty in the GSFE from the previous scales, the current ISV model does not consider temperature and strain rate dependencies. Moreover, the accuracy of the Voce Hardening law parameters obtained in the previous section is the primary concern, since it would take a significantly higher amount of computational power and time to capture the entire scope of the MDDP simulations. These issues could be handled for a more robust study which is outside the scope of this report. <br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure 8: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:36:53Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure A5 in the appendix. All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations. Polycrystalline simulations run with 500 grains were also simulated in tension, compression, and shear for all three cases. The stress-strain plots for these simulations are presented in Figure 7. It is apparent that the different dislocation hardening parameters for each case have affected these results since the yield stress obtained from this exercise is significantly different in comparison with the yield stress observed in BCC Vanadium.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure 7. CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best-fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:35:49Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure A5 in the appendix. All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure 7. CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best-fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
Polycrystalline simulations run with 500 grains were also simulated in tension, compression, and shear for all three cases. The stress-strain plots for these simulations are presented in Figure 7. It is apparent that the different dislocation hardening parameters for each case have affected these results since the yield stress obtained from this exercise is significantly different in comparison with the yield stress observed in BCC Vanadium.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:35:19Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure 7. All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure 7. CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best-fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
Polycrystalline simulations run with 500 grains were also simulated in tension, compression, and shear for all three cases. The stress-strain plots for these simulations are presented in Figure 7. It is apparent that the different dislocation hardening parameters for each case have affected these results since the yield stress obtained from this exercise is significantly different in comparison with the yield stress observed in BCC Vanadium.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:34:12Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure 7. All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure 7. CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:33:43Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
The single crystal CPFEM simulations ran for all three cases are presented below in Figure . All three cases look identical to the same loading condition. This is expected for single crystal simulations, as the effects of dislocation hardening will not be seen for single crystal simulations.<br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:32:55Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|400px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:32:31Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|350px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:32:05Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|500px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:31:22Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|800px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|700px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:31:07Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|700px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|1000px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|700px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:30:04Z<p>Madabhushi: /* Introduction */</p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the generalized stacking fault energy; which is propagated from the nanoscale to the MD scale calculations of dislocation mobility, which is then upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|800px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|1000px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|700px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:26:04Z<p>Madabhushi: /* Voce Hardening */</p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the uncertainty in the generalized stacking fault energy which is propagated to the MD scale calculations of dislocation mobility, which is upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|800px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|1000px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table 6 along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|700px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T17:25:05Z<p>Madabhushi: /* Voce Hardening */</p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the uncertainty in the generalized stacking fault energy which is propagated to the MD scale calculations of dislocation mobility, which is upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|800px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|1000px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table XXX along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<div align="center">Table 6. Fitted Voce Hardening parameters for Best fit, Upper Bound and Lower Bound for 10 FRS.</div><br />
[[Image:VOCE parameters.JPG|center|600px]]<br />
<br><br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|700px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/File:VOCE_parameters.JPGFile:VOCE parameters.JPG2019-04-26T17:21:13Z<p>Madabhushi: </p>
<hr />
<div></div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T16:50:02Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the uncertainty in the generalized stacking fault energy which is propagated to the MD scale calculations of dislocation mobility, which is upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|800px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|1000px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table XXX along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
=Appendix= <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|700px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T16:47:47Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the uncertainty in the generalized stacking fault energy which is propagated to the MD scale calculations of dislocation mobility, which is upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|800px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|1000px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table XXX along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
==Appendix== <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|700px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T16:46:44Z<p>Madabhushi: /* Voce Hardening */</p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the uncertainty in the generalized stacking fault energy which is propagated to the MD scale calculations of dislocation mobility, which is upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|800px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|1000px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <!-- Results Section --><br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
===Voce Hardening===<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table XXX along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
==Appendix== <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|700px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T16:46:14Z<p>Madabhushi: </p>
<hr />
<div>=Abstract=<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
=Introduction=<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the uncertainty in the generalized stacking fault energy which is propagated to the MD scale calculations of dislocation mobility, which is upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
=Methodology= <!-- Methodology Section --><br />
==Electronic Scale Simulations== <!-- Methodology from Homework #1 --><br />
===Ab Initio calculations===<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
===MEAM parameter calibration===<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
==Molecular Dynamics Simulations== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|800px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
==MDDP Simulations== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
== CPFEM Simulations== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
== Macroscale ISV Model Calibration== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|1000px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
=Results= <!-- Results Section --><br />
<br />
==Electronic Scale Simulations==<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
==MEAM Parameter Calibraion== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
==LAMMPS Simulations== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
===Calculation of Dislocation Mobility===<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
===Sensitivity Analysis===<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
==MDDP Simulations== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
====Voce Hardening====<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table XXX along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<br />
== CPFEM Simulations== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
== Macroscale ISV Model Calibration== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
=Conclusion=<!-- Conclusion Section --><br />
=References=<!-- References Section --><br />
<references/><br />
<br />
==Appendix== <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|700px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T16:39:33Z<p>Madabhushi: /* Voce Hardening */</p>
<hr />
<div>=Multi-Scale Modeling of Pure Vanadium=<br />
<br />
==Abstract==<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
==Introduction==<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the uncertainty in the generalized stacking fault energy which is propagated to the MD scale calculations of dislocation mobility, which is upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
==Methodology== <!-- Methodology Section --><br />
===Electronic Scale Simulations=== <!-- Methodology from Homework #1 --><br />
====Ab Initio calculations====<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
====MEAM parameter calibration====<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
===Molecular Dynamics Simulations=== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|800px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
===MDDP Simulations=== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
=== CPFEM Simulations=== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
=== Macroscale ISV Model Calibration=== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|1000px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
==Results== <!-- Results Section --><br />
<br />
===Electronic Scale Simulations===<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
===MEAM Parameter Calibraion=== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
===LAMMPS Simulations=== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
====Calculation of Dislocation Mobility====<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
====Sensitivity Analysis====<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
===MDDP Simulations=== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
====Voce Hardening====<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref name='ICME'></ref>. Using Eq. 3, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table XXX along with the fitted Voce equation and material parameters κ<sub>s</sub>,κ<sub>0</sub>, and h<sub>0</sub>. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ<sub>0</sub> increases with the number of FRS.<br />
<br />
=== CPFEM Simulations=== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
=== Macroscale ISV Model Calibration=== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
==Conclusion==<!-- Conclusion Section --><br />
==References==<!-- References Section --><br />
<references/><br />
<br />
==Appendix== <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
<br><br />
<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
<br><br />
[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix4.png|center|500px]]<br />
<br><br />
<div align="center">Figure A4. LAMMPS simulation results: Position vs time curves for various applies shear stresses for N =15708</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix5.png|center|700px]]<br />
<br><br />
<div align="center">Figure A5. CPFEM results for single crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br><br />
<br><br />
<br />
[[Image:Appendix6.PNG|center|700px]]<br />
<br><br />
<div align="center">Figure A6. CPFEM results for varying numbers of grains in tension with best fit parameters.</div><br />
<br />
[[Category: Research Paper]]<br />
[[Category: macroscale]]<br />
[[Category: mesoscale]]<br />
[[Category: microscale]]<br />
[[Category: nanoscale]]<br />
[[Category: Electronic Scale]]<br />
[[Category: metals]]</div>Madabhushihttps://icme.hpc.msstate.edu/mediawiki/index.php/Multi-Scale_Modeling_of_Pure_VanadiumMulti-Scale Modeling of Pure Vanadium2019-04-26T16:37:50Z<p>Madabhushi: </p>
<hr />
<div>=Multi-Scale Modeling of Pure Vanadium=<br />
<br />
==Abstract==<br />
<br />
Authors: Caleb O. Yenusah <sup>a,b</sup>, Abhijith Madabhushi <sup>a</sup>, William M. Furr <sup>a,b</sup>, Benmbarek Mouhsine <sup>a</sup><br />
<br><br />
a. Department of Mechanical Engineering, Mississippi State University, Mississippi State, MS, USA<br />
<br><br />
b. Center for Advanced Vehicular Systems, Mississippi State University, Mississippi State, MS, USA<br />
<br />
==Introduction==<br />
Integrated Computational Materials Engineering (ICME) sets a framework for the modeling of materials from the lowest length and time scales to the continuum scale. Bridges between the different scales are established in relation to the nature of the problem, or a continuum scale phenomenon of interest. In this case study, the plasticity of Vanadium is the continuum scale phenomenon of interest. To achieve this goal, Density Functional Theory (DFT) calculations were performed in Quantum Espresso<ref>P. Giannozzi et al., “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials,” J. Phys. Condens. Matter, vol. 21, no. 39, p. 395502, Sep. 2009.</ref> to obtain the energy verse volume curves for BCC, FCC, and HCP Vanadium, along with the coherence energy, elastic constants, generalized stacking fault energy (GSFE), and lattice constants for the stable BCC V phase. To upscale the information generated from the electronic scale to the atomic scale, A second nearest-neighbor (2NN) Modified Embedded Atom Method (MEAM) interatomic potential was generated from these results <ref name='MEAM'>Lee, Byeong-Joo, et al, "Second nearest-neighbor modified embedded atom method potentials for bcc transition metals," Physical Review B, vol. 64, no. 18, 2001.</ref>. This MEAM potential is used in the atomic scale in molecular dynamics (MD) calculations. The MD calculations can then be used to examine atomic scale phenomena like the movement of dislocations as a function of applied stress, temperature, and other boundary conditions. From the MD simulations, the atomic positions in the lattice under the influence of the dislocation are obtained, from which important variables like the dislocation velocities can be calculated; which in turn can be used to obtain the dislocation mobility. The dislocation mobility will be upscaled to the Mesoscale via Dislocation Dynamics (DD) calculations to study the stress-strain response of the material, especially the hardening behavior.<br />
<br />
This exercise is of utmost importance for capturing the critical stress-strain behavior post yielding known as plastic deformation. During the deformation of a material in the plastic regime, dislocation density increases, resulting in the interaction of dislocation, this interaction dictates the hardening behavior of a material. In this study, the Voce hardening rule parameters for a single crystal are determined from DD calculations, which are then used in a Finite Element Code (FEM) called CPFEM<ref>F. Roters, "Advanced material models for the crystal plasticity finite element method: development of a general CPFEM framework.No. RWTH-CONV-144,865th," in Material Science and Engineering, AACHEN, Germany, 2011</ref> to simulate hardening in a single crystal, followed by polycrystalline simulations for three different loading conditions (Tension, Compression, and Torsion) for a realistic quantification of the stress-strain response of the material to different stress states. This stress-strain behavior of the polycrystalline solid is then used to obtain a parameterized internal state variable model (ISV) for the material. The ISV model serves as a bridge to the continuum scale where large systems of engineering significance can be simulated. <br />
During the upscaling process, care is taken to capture the sensitivity of the computed values of the current scale to the values from the previous scale. To accomplish this, a sensitivity analysis is performed by creating three different test cases, starting from the MEAM potential creation. The best fit case is the resultant MEAM potential previously mentioned using the MPC calibration tool<ref name='MPC'>Carino, Ricolindo L., and Mark F. Horstemeyer. "Case Studies in Using MATLAB to Build Model Calibration Tools for Multiscale Modeling." Applications from Engineering with MATLAB Concepts. IntechOpen, 2016.</ref>. Information from an upper and lower bound potential is also upscaled to the MD simulations to observe the sensitivity. Similar care is taken at the mesoscale to capture the sensitivity of the values from the atomic scale. This is done to capture the uncertainties in the analyses like the uncertainty in the generalized stacking fault energy which is propagated to the MD scale calculations of dislocation mobility, which is upscaled to the mesoscale DD calculations of the hardening behavior, the CPFEM simulation of stress/strain response of single and polycrystalline, and finally, the parameterization of the ISV model. <br />
<br />
While the aim of this research endeavor is to obtain a working knowledge of the ICME design paradigm, future work will involve the improvement of the results produced in this work to accurately capture the continuum scale plasticity behavior of pure BCC Vanadium. The following sections describe the methodology involved in this work and the corresponding results obtained for the best fit, upper bound and lower bound cases.<br />
<br />
==Methodology== <!-- Methodology Section --><br />
===Electronic Scale Simulations=== <!-- Methodology from Homework #1 --><br />
====Ab Initio calculations====<br />
Ab initio calculations in this work were based on density functional theory (DFT) as implemented in the Quantum Espresso code. Projector augmented-wave (PAW) potential<ref>P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B, vol. 50, no. 24, pp. 17953–17979, Dec. 1994.</ref> and the Perdew-Burke-Ernzerhof generalized gradient approximation<ref>J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys. Rev. Lett., vol. 77, no. 18, pp. 3865–3868, Oct. 1996.</ref> were used. Convergence study was performed in two stages. First to determine the appropriate k-point mesh point needed, the plane wave energy cutoff was fixed at 653.07 eV and k-point mesh were varied. After the converged k-point mesh with respect to the energy cutoff is obtained, a second study was performed using the converged k-point mesh and different energy cutoff ensuring that all results are converged with respect to both parameters. K-point convergence study was also performed for lattice constant and bulk modulus. The results of this study can be found in the Appendix. Energy versus atomic spacing for bcc, fcc, and hcp structures was calculated using the converged energy cutoff and K-point mesh. This is needed for calibration of the MEAM parameters. The lattice constant, bulk modulus, and cohesive energy for the ground state bcc, V was calculated using Murnaghan equation of state (EOS). <br />
The generalized stacking fault energy (GSFE) values (E<sub>sf</sub>) were calculated using Eq. 1 for calibration of the MEAM parameters for a specific slip system. This ensures that plastic deformation is correctly reproduced by the MEAM potential developed. Due to the fact that vanadium belongs to bcc metal, the (110) plane is the most close-packed plane and the {110}<111> is the most preferable slip system family and should have the smallest GSFE curve. The GSFE curves for (110)[-111] , (110)[001] , and (110)[-110] were plotted. Since the former slip system had the smallest GSFE curve, the MEAM parameters were calibrated to the (110)[-111] slip plane and direction. <br />
<br><br />
[[Image:GSFE EQ.JPG|center|250px]]<br />
where E<sub>tot</sub> is the total energy of the structure with a stacking fault, N is the number of atoms in the system, ε is the total energy per atom in the bulk, and A is the unit cell area that is perpendicular to the stacking fault<ref>M. F. Horstemeyer, Integrated Computational Materials Engineering (ICME) for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science. John Wiley & Sons, 2012.</ref>.<br />
<br />
====MEAM parameter calibration====<br />
MEAM parameter calibration was performed using the MEAM Parameter Calibration software (MPC) developed by Mississippi State University<ref name='MPC'></ref>. MEAM parameters were calibrated to the energy verse atomic spacing and GSFE data calculated from DFT and the elastic moduli of bcc V found in the literature<ref>Alers, G. A. "Elastic moduli of vanadium." Physical Review 119, no. 5 (1960): 1532.</ref>. A comparison was made between the DFT calculated energy verse atomic spacing and GSFE curve and results from Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) utilizing the developed MEAM potential.<br />
<br />
===Molecular Dynamics Simulations=== <!-- Methodology form Homework #2 --><br />
The MD simulations were performed using Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). A “pad” type structure with BCC Vanadium (V) atoms was created for the purpose of the simulations as shown in the figure below. It was periodic in the X and Z, and non-periodic/shrink-wrapped in the Y direction. An Edge dislocation was introduced in the middle of the cell and constant shear stress was applied on the bottom and top face to propagate the dislocation in the X direction. A constant number of atoms, volume, and energy conditions were prescribed. The temperature of the system was fixed at 300K and equilibrated for 10000ps. The dislocation mobility for the edge dislocation as a function of applied stress and velocity was calculated from the MD simulation results. A convergence study was performed for applied shear stress for the different total number of atoms. The results from the MD simulations serve as the bridging parameters to the mesoscale DD calculations to capture the stress-strain behavior of the material. <br><br />
<br><br />
[[Image:OVITO.jpg|center|800px]]<br />
<div align="center">Figure 1. Atom positions within the Pad structure after the equilibration stage. </div><br />
<br />
<br><br />
<br />
===MDDP Simulations=== <!-- Methodology form Homework #2 & Homework #3 --><br />
In the next step, Multiscale Dislocation Dynamics Plasticity (MDDP)<ref>Zbib, Hussein M., Tomas Diaz de la Rubia, "A multiscale model of plasticity," International Journal of Plasticity, vol. 18, no. 9, pp. 1133-1163, 2002</ref> code was used to create a single Frank-Read source (SFRS) to perform the DD calculations. The code used two input files and an executable (BCCdata.exe) to generate the dislocation input file (DDinput). This input file was modified so that the Frank-Read pair created earlier is modified to a single Frank-Read source. Further modifications to the input file to account for the nearest neighbor atom information was corrected. The results of the MDDP calculations were later extracted and visualized using Tecplot. The process was repeated for a single crystal with multiple Frank-Read sources (MFRS). The material properties used in MDDP calculations are detailed in Table 1. Annealed initial state was assumed for the material with an initial dislocation density in the order of 10<sup>12</sup>. Preliminary studies were carried out to understand the effect of the number of Frank-Read sources on the initial dislocation density to determine the number of Frank-Read sources needed for a dislocation density in the order of 10<sup>12</sup>. The MFRS were randomly generated on different planes. For BCC vanadium, all glide planes are of the {110} family. A strain rate of 1.0E5 was used for all simulations. The material properties used in MDDP calculations are detailed in Table 1.<br />
<div align="center">Table 1: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Matl Props Table.PNG|center|800px]]<br />
<br />
All the values in the above table were obtained from the results of the calculations from the previous scales. The Voce Hardening law parameters were obtained from the results of the MDDP simulations which are upscaled to the mesoscale via the CPFEM simulations.<br />
<br />
=== CPFEM Simulations=== <!-- Methodology from Homework #3 & Homework #4 --><br />
Crystal plasticity finite element method (CPFEM) simulations were run for single crystals and 500 crystals in tension, compression, and shear. The Voce hardening parameters obtained from the MDDP results were used in these simulations, as well as the parameters in Table 2. C11, C12, and C44 are the single crystal elastic constants obtained from previous MPC calibrations, B is the dislocation drag coefficient previously obtained from Lammps simulations.<br />
<div align="center">Table 2: Material properties for BCC Vanadium used in MDDP Simulations </div><br />
[[Image:Elastic constants.JPG|center|600px]]<br />
These values were used in an Abaqus user material subroutine (UMAT) and applied to a single element in tension, compression, and shear. The slip mode examined was <-111> {110} for bcc Vanadium. To evaluate the effects of varying the number of crystals, the best fit case obtained from MDDP simulations was simulated in tension and compared.<br />
<br />
=== Macroscale ISV Model Calibration=== <!-- Methodology form Homework #4 --><br />
The results of the polycrystalline CPFEM simulations were used to generate the ISV model using the MSU ISV DMGfit 55p v1p1 tool<ref>Mississippi State University, "MSU ICME," Mississippi State University, [Online]. Available: https://icme.hpc.msstate.edu/mediawiki/index.php/DMGfit_55p_v1p1. [Accessed Monday April 2019].</ref>. This was done for the purpose of upscaling the results of the mesoscale onto the macroscale simulations. DMGfit was designed to be an interactive calibration tool to model damage and plasticity using the MSU ISV model. The model is a FORTRAN subroutine implemented in ABAQUS as a UMAT (user material subroutine). The purpose of the calibration tool is to determine ‘material constants’ that will be used as inputs in the finite element simulations. The DMG model itself is specified by 55 constants, which include the dataset and the fitted parameters as shown in the figure below. The upscaled results like bulk and shear modulus and other material properties serve as the input to the calibration tool along with the stress-strain data from CPFEM. The tool uses these inputs and subjects them to an optimization protocol called fminsearch. The relevant Best Fit, Upper Bound and the Lower Bound results of the previous scales like the tension, compression and torsion simulations from the polycrystalline CPFEM simulations are saved as data set files and are subjected to this process and three ISV models are obtained. <br />
The fitted ISV constants are added to an input deck to be used alongside the DMG UMAT to run a single element calculation in ABAQUS for the three loading conditions. The results obtained from the finite element calculation serve as verification of the plasticity-damage calibration.<br />
<br><br />
[[Image:DMGfit.jpg|center|1000px]]<br />
<div align="center">Figure 2. DMGfit 55 v1p1 calibration tool used to fit stress-strain behavior from CPFEM for BCC vanadium </div><br />
<br />
==Results== <!-- Results Section --><br />
<br />
===Electronic Scale Simulations===<br />
The Shear Modulus, the lattice constant, and the cohesive energy are obtained after the convergence study using the Universal Equations of State (UEOS). Out of the four EOS choices given, BIRCH1 and Keane models were not able to generate results. It seems like the code was unable to converge to a result. From the results generated by BIRCH1 and MURNAGHAN as shown in Table 3, the former yielded a lower lattice constant value but a higher bulk modulus and cohesive energy values. The value generated by MURNAGHAN was closer to Vanadium’s lattice constant as per literature<ref>Molecular dynamics simulation of vanadium using an interatomic potential fitted to finite temperature properties - ScienceDirect.” [Online]. Available: https://www.sciencedirect.com/science/article/pii/S002231150201019X. [Accessed: 09-Mar-2019]</ref> Therefore MURNAGHAN was used for the rest of the analysis performed in this study. <br />
<br> <br />
<!-- Results from Homework #1 --> <br />
<div align="center">Table 3. EOS study using the converged K-point and energy cut-off. </div> <br />
[[Image:DFT Table1.PNG|center|600px]]<br />
<br />
===MEAM Parameter Calibraion=== <br />
As a prelude to this study, the MEAM potential previously obtained was fine-tuned to accurately capture the behavior of the generalized stacking fault energy (GSFE). The coherence energy, bulk modulus, and lattice constants of the stable BCC V phase were well established in the literature from experiments and first principle calculations. However, the characteristics of the GSFE can have only been reported through first principle calculations <ref>Gui, Li-Jiang, and Yue-Lin Liu, "First-principles studying the properties of oxygen in vanadium: Thermodynamics and tensile/shear behavior," Computational Condensed Matter, vol. 7, pp. 7-13, 2016</ref><ref>Zhang, Xingming, et al., "The effects of interstitial impurities on the mechanical properties of vanadium alloys: A first-principles study," Journal of Alloys and Compounds, vol. 701, pp. 975-980, 2017</ref>. Therefore, any errors in it are quantified through a sensitivity analysis, since the GSFE is paramount to the plastic behavior of materials. This was performed using three different MEAM potentials, a good fit to the DFT calculations, another that underestimated the GSFE curve by 26% <ref name='MEAM'></ref>, and a third one that overestimated the GSFE curve by 22% (Figure 3). These MEAM potentials will hereon be referred to as Best fit, lower bound, and upper bound. It should be noted that the MEAM potential for V from Lee et al [1] underestimated the GSFE curve and this was use used as the lower bound. All the MEAM potential values are shown in Table 4. The values modified in the best fit potential in comparison with the original Lee et al. are A, β(0), β(2), β(3), t(1), t(2), t(3), the attraction and the repulsion parameters. <ref name='MEAM'></ref> MEAM potential to better fit the GSFE curve was therefore calculated from DFT. The energy versus atomic spacing for the three MEAM potentials is reported in Figure A1 in the Appendix along with predicted material properties which were compared with experimental results (Table A1).<br />
<div align="center">Table 4. Parameters for the second nearest-neighbor MEAM potential for V showing upper, lower bounds, and best fit to DFT. </div> <br />
[[Image:MEAM Image1.PNG|center|800px]]<br />
<br><br />
[[Image:DFT Image1.png|center|500px]]<br />
<div align="center">Figure 3. GSFE Curves for upper, lower bounds, and best fit MEAM to DFT for the [111] direction BCC Vanadium. </div><br />
<br />
===LAMMPS Simulations=== <!-- Results form Homework #2 --><br />
For the created structure type described in the earlier section, LAMMPS simulations were run to evaluate the dislocation velocity. The LAMMPS output files were then subjected to post-processing using the OVITO visualization software <ref>Alexander Stukowski, "Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool.," Modelling and Simulation in Materials Science and Engineering, vol. 18, no. 1, 2009. </ref>. The OVITO simulations give the position and time values for the edge dislocation during the simulation. The dislocation velocity is obtained by plotting the position value against time and taking its slope. To select appropriate dimensions for the Pad structure, a convergence study was run to see the behavior of dislocation velocity for applied shear stress of 1500 bar as a function of the number of atoms in the simulation cell. This study gave an optimal number of atoms to run the rest of the simulations. The results of the convergence study and the position vs time plot for the optimal number of atoms (15708) are shown in Figure A3 in the appendix. The value was selected with a view to balancing computational time and the accuracy of the results. A 1-5% range for convergence was considered during the study and the search for convergence was concluded after the last value for the number of atoms shown in Table 3 since a convergence of .457% was attained.<br />
<br><br />
A graph showing the variation in dislocation velocity with respect to the change in the applied shear stress was plotted (Figure 4) and the characteristics were compared with a similar graph from the ICME for Metals textbook (page no: 388, figure (9.7 a))<ref name='ICME'>Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for metals: using multiscale modeling to invigorate engineering design with science. John Wiley & Sons.</ref>. The figure in the textbook describes the similar effects for an edge location within FCC Aluminum. The velocity scale between the two curves is almost the same however, the velocity of the dislocation in BCC V is lower than FCC Al for the same values of applied stress. Thus, the slope of the FCC Al curve is higher than BCC V. Both curves predict a linear behavior for low applied shear values; however, the linear region in BCC V is between 0-150 MPa which higher than the FCC Al for which the linear region is between 0 – 100 MPa as shown in Figure 4. <br />
<br> <br />
[[Image:LAMMPS Image1.png|center|700px]]<br />
<div align="center">Figure 4: (a) Dislocation Velocity vs applied shear stress for edge dislocation in Vanadium vs (b) Figure 9.7a same figure for Al from ICME Textbook </div><br />
<br><br />
====Calculation of Dislocation Mobility====<br />
The dislocation mobility of a metal is a very important property that affects its strength and ductility<ref>Kang, Keonwook, Vasily V. Bulatov, and Wei Cai, "Singular orientations and faceted motion of dislocations in body-centered cubic crystals," Proceedings of the National Academy of Sciences, vol. 109, no. 38, pp. 15174-15178, 2012. </ref>. The drag-coefficient (B) is a parameter used to quantify the resistance to dislocation motion. Both quantities were evaluated from the velocity vs applied shear stress graph. The drag-coefficient (B) was found by taking the inverse of the slope of the curve in Figure 4 and multiplying it with the Burgers vector of the edge dislocation. Dislocation mobility, in turn, is the inverse of B. The obtained B and dislocation mobility values are shown in Table 5. <br />
====Sensitivity Analysis====<br />
After the evaluation of the dislocation mobility and its drag coefficient (B), a sensitivity analysis was conducted on the previously calibrated MEAM potential by comparing it with two other MEAM potentials. The lower bound potential is generated using the MEAM potential for Vanadium evaluated by Lee et al.<ref name='MEAM'></ref> It underestimated the maximum GSFE by 26 %. The upper bound was created using the existing Best fit potential and offsetting its maximum GSFE by 22% more than the best fit. The purpose of the sensitivity analysis is to also study the effect of the GSFE on the Mobility of the Edge dislocation. The LAMMPS simulations were performed for the other two MEAM potentials (upper and lower) using 150 MPa applied shear stress value. Figure 5 shows the position vs time curve for all the three cases and their corresponding velocity vs time curves. The slope for the lower bound potential in the position vs time curve was found to be greater than the best fit and the upper bound curve respectively, indicating higher dislocation velocities for the former. <br />
[[Image:LAMMPS Image2.png|center|700px]]<br />
<div align="center">Figure 5: (a) Position Vs time curve for Best Fit, Lower Bound and Upper Bound MEAM Potentials atoms, (b) Velocity vs Applied shear stress curve for Best fit, Lower Bound and Upper Bound MEAM potentials (applied shear = 150 MPa)</div><br />
<br><br />
<div align="center">Table 5. Dislocation Drag Coefficient and Dislocation Mobilities for the available MEAM Potentials</div><br />
[[Image:LAMMPS Image3.PNG|center|700px]]<br />
<br><br />
<br />
===MDDP Simulations=== <!-- Results form Homework #2 & Homework #3 --><br />
Figure 6(a-c) shows the dislocation motion under constant applied stress for the best fit MEAM. Figure 6a shows the initial SFRS, the propagation of the dislocation as the strain increases is shown in these figures.<br />
[[Image:MDDP Image1.png|center|700px]]<br />
<div align="center">Figure 6: MDDP Simulations for Single FRS for best fit MEAM at different frames (a) 0 (b) 27 (c) 91</div><br />
<br><br />
Multiple Frank-Read sources were modeled next to get the dislocation evolution and hardening behavior. Dislocation density evolution for different number of FRS is shown in Figure 7a. The initial dislocation density was 6.30E12, 1.30E13, 1.88E13, and 2.55E13 for 10, 20, 30, and 40 FRS respectively. The 10 FRS was shown for the subsequent calculation for the assumed anneal initial state of the material having dislocation density of the order of 1E12. From Figure 1, forest hardening region was assumed at 7E-8s for 10FRS, while the others have not yet reached a saturation point. The stress/strain response of the single crystal shown in Figure 7b. The stress/strain curve is identical for all the simulations with a yield point between 30MPa to 35MPa. The yield strain was about 3E-2m/m for all initial number of FRS.<br />
[[Image:MDDP Image2.png|center|800px]]<br />
<div align="center">Figure 7: (a) Dislocation density (DD) vs. time for different numbers of Frank-Read sources (b) Stress vs. strain curve for different numbers of Frank-Read sources </div><br />
====Voce Hardening====<br />
The Voce hardening equation can be written as:<br />
[[Image:Voce1.JPG|center|350px]]<br />
where 𝜅<sub>𝑠</sub>,𝜅<sub>0</sub>,𝑎𝑛𝑑 ℎ<sub>0</sub> are material properties obtained by correlating the equation with the hardening evolution predicted by DD. Because DD calculations start initially using a random distribution of source segments, the dislocation density calculated at the beginning of the simulation can be ignored and only the forest hardening region, which captures dislocation interactions would be used in the determination of the Voce hardening relationship. The dislocation hardening can be written using the classical Taylor relation:<br />
[[Image:Voce2.JPG|center|250px]]<br />
Here, ρ<sub>f</sub> is the forest dislocation density, b is the magnitude of the Burgers vector, μ shear modulus, and α is a constant representing an average of the junction strength over all existing configurations. α=0.35 in this work<ref></ref>[3]. Using Eq. 2, hardening (κ) as a function of time for various initial FRS is calculated and the results are shown in table XXX along with the fitted Voce equation and material parameters κ_s,κ_0, and h_0. A forest hardening region beginning at about 7E-8s can be observed for 10 FRS. For the others, a forest hardening region could not be identified, and extended simulation times are needed to identify the hardening region. But from a rough fit, κ_0 increases with the number of FRS<br />
<br />
=== CPFEM Simulations=== <!-- Results from Homework #3 & Homework #4 --><br />
[[Image:CPFEM Image1.png|center|800px]]<br />
<div align="center">Figure XXX: CPFEM results for 500 crystal simulations in tension, compression, and torsion for (a) the best fit case, (b) the lower bound, and (c) the upper bound case.</div><br />
<br />
=== Macroscale ISV Model Calibration=== <!-- Results form Homework #4 --><br />
[[Image:ISV Calib Image1.png|center|1000px]]<br />
<div align="center">Figure XXX: Verification results where best fit, lower bound, and upper bound columns are organized left to right.</div><br />
<br />
==Conclusion==<!-- Conclusion Section --><br />
==References==<!-- References Section --><br />
<references/><br />
<br />
==Appendix== <!-- Appendix Section --><br />
[[Image:EVcurve.jpg|center|1000px]]<br />
<br><br />
<div align="center">Figure A1. DFT results: MEAM Calibration: Energy vs. Atomic spacing for BCC, FCC, and HCP predicted by the three MEAM potentials (a) Upper bound (b) Best Fit (c) Lower bound.<br />
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<br><br />
<div align="center">Table A1. Comparison of the MEAM potentials to properties of V reported from experiments. All Experimental data are taken from Lee et al.[2] </div><br />
[[Image:TABLE A1.JPG|center|700px]]<br />
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[[Image:Appendix1.png|center|700px]]<br />
<br><br />
<div align="center">Figure A2. DFT results: Energy vs Atomic spacing curves for (a) various K-point meshes for constant energy cut-off value (b) various energy cut-off for constant converged K-point.</div><br />
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[[Image:Appendix3.PNG|center]]<br />
<br><br />
<div align="center">Figure A3. LAMMPS simulation results: Convergence graph showing convergence at N=15708</div><br />
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[[Image:Appendix4.png|center|500px]]