View source for Homework 3 and 4 Contribution - Group 1

===Overview===

Crystal plasticity models became popular during 1980s as a tool to study deformation and texture behavior of metals during material processing<ref>P. Dawson, On modeling of mechanical property changes during flat rolling of aluminum, ''International Journal of Solids and Structures'', 23 (7)(1987) 947–968.</ref> and shear localization<ref>D. Peirce, R. Asaro, A. Needleman, An analysis of nonuniform and localized deformation in ductile single crystals, ''Acta Metallurgica'', 30 (6)(1982) 1087–1119.</ref><ref>M. Rashid, S. Newmat-Nasser, Modeling very large plastic flows at very large strain rates for large-scale computation, ''Computers & Structures'', 37 (2) (1990) 119–132.</ref>. The basic elements of the theory comprise (i) kinetics including self and latent hardening components<ref>U. Krocks, C. Tome, H. Wenk, Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Effect on Materials Properties, 2000.</ref>; (ii) kinematics in which the concept of the plastic spin plays an important role; and (iii) intergranular constraint laws to govern interactions among crystals or grains. The theory is commonly acknowledged for providing realistic prediction/correlation of texture development and stress-strain behavior at large strains as it joins continuum theory with discretized crystal activity<ref>M. Horstemeyer, Integrated Computation Materials Engineering (ICME) for Metals, John Wiley & Sons, Inc., 2012.</ref>.

===Crystal Plasticity===

The crystal plasticity simulations for the one, twenty, fifty-five grains were successfully ran in the tension, compression, and shear loading conditions.  Four hardening cases taken from Groh<ref>S. Groh, E.B. Marin, M.F. Horstemeyer, and H.M. Zbib. Multiscale modeling of the plasticity in an aluminum single crystal. ''International Journal of Plasticity'', 25(8):1456-1473, 2009.</ref> were studied for each loading condition. For all loading conditions the lowest saturation strength variable, κ<sub>s</sub> = 46.75 MPa created the curves with the higher ultimate stress, with κ<sub>s</sub> = 47.75, κ<sub>s</sub> = 48.84, and κ<sub>s</sub> = 49.10 MPa following, respectively.  Figure 1 illustrates the stress-strain relationships for the tension loading condition for the various grain numbers and hardening constant cases.


[[File:icme_gp1_cp.png|center|thumb|600px|Figure 1: Stress-strain relations from the tension loading condition at grain densities of (a) 1 grain, (b) 20 grains, and (c) 55 grains.]]


The representation of orientations within the crystal structures are usually given by Euler angles. The angles are based off of an orientation in three-dimensional Euclidean space and represent the rotations from the reference to the current configuration. For this study, the Euler angles were randomly generated and then graphically represented using pole figures. The pole figures (Figures 2) show the crystallographic preferred orientations (texture) on the face centered cubic (111) plane via the stereographic projection method.


[[File:icme_gp1_polefigures.png|center|thumb|600px|Figure 2: Pole figures comparing texture for (a) tension, (b) compression, (c)shear. The difference between the texture developments in the tension, shear, and compression loading conditions can be noticed.]]

===Macroscale Internal State Variable Plasticity===

In order to quantify the effects of damage, stress-strain data from the above crystal plasticity investigation was used to fit the constants from the [[Material_Models#MSU_ISV_Plasticity-Damage_Model|Internal State Variable Plasticity-Damage Model DMG-fit tool]].  Table 1 shows the non-zero constants used to fit the tension data.  Figure 3 shows the fit.


{| border="1" class="wikitable" style="margin: 1em auto 1em auto;"
|+ '''Table 1: Non-zero DMGfit material parameters'''
| align="center" width="75px" style="background:#f0f0f0;"|'''Parameter'''
| align="center" width="50px" style="background:#f0f0f0;"|'''G'''
| align="center" width="50px" style="background:#f0f0f0;"|'''Bulk'''
| align="center" width="50px" style="background:#f0f0f0;"|'''Kic'''
| align="center" width="50px" style="background:#f0f0f0;"|'''Tint'''
| align="center" width="50px" style="background:#f0f0f0;"|'''Tmelt'''
| align="center" width="50px" style="background:#f0f0f0;"|'''heat'''
| align="center" width="50px" style="background:#f0f0f0;"|'''nv'''
| align="center" width="50px" style="background:#f0f0f0;"|'''fn'''
| align="center" width="50px" style="background:#f0f0f0;"|'''dcs'''
| align="center" width="50px" style="background:#f0f0f0;"|'''dcs0'''
| align="center" width="50px" style="background:#f0f0f0;"|'''C1'''
| align="center" width="50px" style="background:#f0f0f0;"|'''C3'''
| align="center" width="50px" style="background:#f0f0f0;"|'''C5'''
| align="center" width="50px" style="background:#f0f0f0;"|'''C9'''
| align="center" width="50px" style="background:#f0f0f0;"|'''C13'''
| align="center" width="50px" style="background:#f0f0f0;"|'''CAcon'''

|-
| ||align="center"|25563.9||align="center"|66666.7||align="center"|1000||align="center"|297||align="center"|933.52||align="center"|0.370521||align="center"|0.3||align="center"|0.001||align="center"|30||align="center"|30||align="center"|2.1||align="center"|10||align="center"|500||align="center"|120.339||align="center"|0.166||align="center"|0.5
|}


[[File:icme_gp1_DMGfit.png|center|thumb|600px|Figure 3: Stress-strain comparison between results gained from crystal plasticity calculations and plasticity-damage calculations.]]


Following the crystal plasticity simulations and DMGfit routine, a one element finite element simulation in ABAQUS with the ISV-Damage UMAT was ran using the constants in Table 1.  Figure 4 serves to validate the plasticity-damage model as the stress-strain response shows good agreement.


[[File:icme_gp1_validation.png|center|thumb|600px|Figure 4: Comparison between stress-strain results from crystal plasticity finite element simulations and plasticity damage finite element simulations.]]

===References===
<references/>