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The Macroscale is a continuum point, where one develops the constitutive model for the 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).

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.

Finally, to garner more information about the information bridges between length scales go to the MSU Education page.

Material Models

Human Head/Brain Models

Several options for the modeling of the head and brain exist. Samaka and Tarlochan [2013][1] and Caroline and Remy [2009],[2] provide an overview of these models and the softwares used to implement the models:

  • EEVC WG17 Adult Headform Finite Element Model [3]
  • Wayne State University (WSU) Head (Brain) Injury Model (WSUHIM) [4]
  • TNO Head FE Model [5] [8]
  • Head Brain Model [6] [9]
  • University of Louis Pasteur Finite Element Model (ULP FEM)of the Human Head [7]
  • Politecnico di Torino University Finite Element Model of the Human Head [8] [10]
  • Harvard Medical School Finite Element Model [9]
  • University College Dublin Brain Trauma Model (UCDBTM) [10] [11]; [12]
  • Strasbourg University Finite Element Head Model (SUFEHM) [2] [13]; [14]

Internal State Variable Plasticity-Damage (MSU ISV-DMG) Model

The Mississippi State University Internal State Variable (ISV) plasticity-damage model (DMG) production version 1.0 is being released along with its model calibration tool (DMGfit). The model equations and material model fits are explained in 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.

Johnson-Cook Flow Stress (JC) Model

The Johnson-Cook (JC) constitutive model is an empirically based flow model originally intended for the prediction of inelastic deformation in solid materials [11]. 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[12].

Mechanical Threshold Stress (MTS) Model

The Mechanical Threshold Stress (MTS) Model is a flow stress model that considers the effects of dislocation motion and interaction on macroscale deformation[13]. 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.

MultiStage Fatigue (MSF) Model

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.

ThermoPlastic Internal State Variable (TP-ISV) Model

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][14]. 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 [15] 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 * 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 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.

WARP3D - Open Source Code for 3D Nonlinear Analysis of Solids

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.

Zerilli-Armstrong Flow Stress (ZA) Model

The Zerilli-Armstrong (ZA) model is a flow stress model based upon dislocation mechanics [16]. 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.


DMG v1.0

MSU DMG v1.0 is an example of a plasticity-damage internal state variable model[17] [18], 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 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.

A video tutorial for calibrating DMG is found here.

An in depth written tutorial of DMGfit can be found here.

A guide to setting up the DMG UMAT in Calculix can be found here.

The following single element finite element input decks should be used to verify the material point simulator (DMGfit) determinations:

  • one element compression for aluminum A356 (ABAQUS-Implicit)here
  • one element compression for aluminum A356 (ABAQUS-Explicit)here
  • one element tension for aluminum A356 (ABAQUS-Implicit) here
  • one element tension for aluminum A356 (ABAQUS-Explicit)here
  • one element simple shear for aluminum A356 (ABAQUS-Implicit)here
  • one element simple shear for aluminum A356 (ABAQUS-Explicit)here

One element explicit compression A356 input decks Model Validation simulations include the following: notch tensile specimen (1/8 space mesh) for aluminum A356 (ABAQUS-Implicit)

The ABAQUS input decks and instruction on how to one element simulations can be downloaded ('Download GNU tarball') here, or can be viewed online by clicking 'view' for each of the files.


The macroscale tools also include fatigue analysis tools. One example is the 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.

Thermoplastic Internal State Variable (TPISV) Model

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.

  • One element tension for Polycarbonate (ABAQUS-Explicit) - here

To learn how to use the parameters fitting routine, you can refer to the documentation (TPGui User Manual; TPGui Tutorial).

A stand-alone TP tool is available from the online code repository. Please refer to the documentation (online help and tutorial) to learn how to use this tool.

Macroscale Research



Constitutive Models for ferroelectric materials under combined mechanical and electrical loading














Constitutive Models for ferroelectric materials under combined mechanical and electrical loading




  1. H. Samaka and F. Tarlochan, “Finite Element (FE) Human Head Models/Literature Review”, Internat. J. Sci. Tech. Res., 2(7), 17-31., 2013.[1]
  2. 2.0 2.1 D. Caroline and W. Remy, "Head Injury Prediction Tool for Protective Systems Optimisation”, 7th European LS-Dyna Users Conference, 2009.[2]
  3. 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.
  4. 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.
  5. [3]
  6. 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. [4]
  7. 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
  8. G. Belingardi,G. Chiandussi, and I. Gaviglio, "Development and Validation of a New Finite Element Model of Human Head", 1-9., [5]
  9. 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.[6]
  10. 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, [7]
  11. 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.
  12. 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.
  13. 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.
  14. 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.ISV_MODEL_POLYMER_PAPER_ACTAMECHANICA
  15. 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)
  16. Zerilli, F.J., Armstrong, R.W., “Dislocation-mechanics-based constitutive relations for material dynamcis calculations.” Journal of Applied Physics 61.5 (1987): 1816-1825.
  17. 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
  18. 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


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Pages in category "Macroscale"

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Material Models