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"Mesoscale" refers to an intermediate length scale that spans the range between the nanoscale and the macroscale. This length scale aids in linking microscale behavior to macroscale results.

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

Biaxial loading of a 9x9 mm Polycarbonate plate with 2 defects showing interacting stress fields. From this study.

MSU DMG v1.0 is an example of a plasticity-damage internal state variable model^{[1][2]}, 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.

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) here.

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.

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
- The TPISV model calibration tool is described by the TPGui User Manual.
- A Powerpoint tutorial for the calibration tool is TPgui-1.2-Tutorial.zip.

Crystal Plasticity Finite Element Model Tutorials are here.

A crystal plasticity model in the ABAQUS subroutine UMAT. Tutorial Video

**Input files using CPFEM for an aluminum simulation**

The ABAQUS input decks and a step-by-step Tutorial on how to use them to run CPFEM simulations can be downloaded from the cpfem decks repository (CAVS users only) , or can be viewed online by clicking on the name of each of the files below.

- CPFEM manual – Introduction to Crystal Plasticity Finite Element Method (CPFEM)
- umat_xtal.f - constitutive model - polycrystal average model
- texture.txti - initial orientation distribution
- fcc.sx - single crystal parameters
- test.xtali - control for the time step and deformation
- params_xtal.inc - number of slip systems
- numbers.inc - numerical constants
- An example of the single crystal and polycrystal texture evolution for aluminum is available here

**Input files using CPFEM for a magnesium simulation**

The ABAQUS input decks and a step-by-step Tutorial on how to use them to run CPFEM simulations can be downloaded (CAVS users only) here, or can be viewed online by clicking on the name of each of the files below.

- CPFEM manual – Introduction to Crystal Plasticity Finite Element Method (CPFEM)
- umat_xtal.f - constitutive model - polycrystal average model
- texture.txti - initial orientation distribution
- hcp.sx - single crystal parameters
- test.xtali - control for the time step and deformation
- params_xtal.inc - number of slip systems
- numbers.inc - numerical constants
- vert_hcp_121.01 - vertices parameters in Mg
- vert_hcp_121.03 - vertices parameters in Mg
- vert_hcp_121.05 - vertices parameters in Mg

**CPFEM in Calculix**

CPFEM can now also be run in Calculix, a free FEA software, using the same input files as for Abaqus. Follow the instructions here to get started.

Getting started in Matlab: MATLAB Tutorials

- How to use MATLAB: The Basics of MATLAB
- How to use MATLAB: The Basics of MATLAB 2
- How to Import Data from a Textfile
- How to Write Data to a Textfile
- How to make a stress-strain plot using MATLAB
- How to make a journal quality plot using MATLAB
- Example: How to make a journal quality plot with errorbars
- How to do basic image processing with MATLAB
- Design of Experiments with MATLAB: Part 1
- Design of Experiments with MATLAB: Part 2
- Design of Experiments with MATLAB: Part 3

3D reconstruction tool

- Dream.3D website
- DREAM.3D: A Digital Representation Environment for the Analysis of Microstructure in 3D (IMMI Journal)

Texture code that quantifies pole figures

By Fred Cox: LANL

- Tutorial

Constructs simulated 3D polycrystalline materials, where the input is typically grain size and shape data.

VCSG generates composite structure representative volume elements (RVEs) based on user input. VCSG is a graphical tool that uses Matlab and Python to generate composite RVEs for finite element analysis with randomly oriented and positioned filler. RVEs are built to be compatible with ABAQUS finite element software.

- VPSC Deformation Simulation of Polycrystalline FCC Aluminum
- VPSC7d_gui - A graphical user interface and calibration tool for the visco-plastic self-consistent (VPSC) model of plastic deformation of polycrystalline aggregates. The interface utilizes the model code, input files and output files described in Manual for Code Visco-Plastic Self-Consistent (VPSC), Version 7d, by C.N. Tomé and R.A. Lebensohn, Los Alamos National Laboratory, USA. VPSC7b_gui attempts to calculate model parameters that minimize the difference between experimental stress-strain data and the stress-strain curve generated by VPSC. If fitting multiple data sets simultaneously, VPSC7d_gui exploits multi-processing capabilities of the computational environment to generate the model stress-strain curves in parallel.

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.

Phase field method is a power computational tool to model the temporal and spatial evolution of microstructure in mesoscale region. Most materials have a complex microstructure that arise due to grain, grain boundaries, different phases, compositions, orientations and crystallography. The material properties can be improved if we could understand the evolution of the microstructure.

IMMI Journal article on OpenCalphad - a free thermodynamic software

Texture can be defined as a measure of the percentage of grains that are oriented with respect to a certain plane. The texture of a material is highly dependent upon the processing conditons. X-ray Diffraction is a viable technique that allows us to determine the initial and deformed bulk texture of a material being studied. The texture can be evaluated using Rigaku SmartLab X-ray Diffraction System, which is available on the campus of Mississippi State University at the Institute for Imaging & Analytical Technologies (I2AT).

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 the CAVS Technical Report. 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.

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]^{[3]}. 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 ^{[4]} 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.

- Multiscale Modeling of Concrete
- Reinforced Concrete

In metals, mesoscale implies crystal plasticity. Crystal plasticity is a continuum theory, but it has discrete quantities starting at the grain or crystal scale. Polycrystalline averaging starting from the grain scale could be compared to the polycrystalline internal state variable models that operate at the macroscale. It can reach down to the dislocation dynamics results to help determine the material constants for the work hardening rules. Two different crystal plasticity formulations are given here.

- Polycrystal formulations for FCC, BCC, and HCPVPSC model for polycrystalline averaged quantities

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 of interest include the texture (grain/crystal orientation, stress-strain behavior, hardening response, and twinning response if needed). Once the material model is calibrated and validated at the mesoscale, the results for the texture, yield surface, and hardening evolutions can be used in the macroscale internal state variable model and structural scale simulations.

Model Calibration for both the CP 1.0 and VPSC 1.0 models have fitting algorithms in which the material constants can be determined.

- CPfit 1.0
- VPSCfit 1.0

The following single element finite element input decks should be used to verify the material point simulator determinations: one element compression for 500 crystals aluminum polycrystal(ABAQUS-Standard) one element tension for 500 crystals aluminum polycrystal(ABAQUS-Standard) one element simple shear for 500 crystals aluminum polycrystal (ABAQUS-Standard)

See also: ABAQUS CPFEM

- Yield surface prediction of Aluminum on rolling
- Visco-Plastic Self-Consistent (VPSC) Deformation Simulation of Polycrystalline FCC Aluminum
- One element deformation of Aluminum

- A channel die compression simulation on Mg AM30
- Twinning and double twinning upon compression of prismatic textures in an AM30 magnesium alloy
- One element deformation of Magnesium

- A mesomechanics parametric finite element study of damage growth and coalescence in polymers using an Elastoviscoelastic-Viscoplastic internal state variable model

- A mesomechanics parametric finite element study of damage growth and coalescence in polymers using an Elastoviscoelastic-Viscoplastic internal state variable model

- ↑ 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
- ↑ 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
- ↑ 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.
- ↑ 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 Sciencie,USA(1996)

- E.B. Marin, On the formulation of a crystal plasticity model, Sandia National Laboratories, CA, 2006, SAND2006-4170.
- S. Groh, E.B. Marin, M.F. Horstemeyer, et al., Multiscale modeling of the plasticity in an aluminum single crystal, International Journal of Plasticity, 25(2009), 1456-1473.
- E.B. Marin, P.R. Dawson, On modeling the elasto-viscoplastic response of metals using polycrystal plasticity, Computer Methods in Applied Mechanics and Engineering, 165(1998), 1-21.
- E.B. Marin, P.R. Dawson, J.T. Jenkins, Aggregate size effect on the predicted plastic response of hexagonal close-packed polycrystals, Modelling Simulation Materials Science Engineering, 3(1995), 845-864.
- F. Roters, Application of crystal plasticity FEM from single crystal to bulk polycrystal, Computational Materials Science, 32(2005), 509-517.
- F. Roters, P. Eisenlohr, L. Hantcherli, et al., Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications, Acta Materialia, 58(2010), 1152-1211.
- D. Peirce, R.J. Asaro, A. Needleman, Analysis of nonuniform and localized deformation in FCC single crystals, Acta Metallurgica, 30(1982), 1087-1119.
- D. Peirce, R.J. Asaro, A. Needleman, Material rate dependence and localized deformation in crystalline solids, Acta Metallurgica, 31(1983), 1951-1976.