The mesoscale, although used generically to mean many things that are "intermediate," means essentially crystal plasticity in the context of multiscale modeling of metals. Crystal plasticity essentially is a continuum theory, but 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 HCP crystals
- Visco Plasticity Self Consistent (VPSC) model: 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. 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
Phase-Field Modelling of Microstructural Evolution
A broad spectrum of moving boundary problems can be successfully handled with the diffuse-interface or phase-ﬁeld models. Such models are constructed by assuming that the free energy of a non-uniform system F, depends on (among other variables) the phase-ﬁeld variable , and its gradient . In the volume V:
F = (1)
Mathematically, the phase-ﬁeld formulation gives rise to two types of problems. The non-conserved phase-ﬁeld variables, such as those in solidiﬁcation and melting problems, are often assumed to evolve according to the ‘‘ﬁrst-order relaxation”, whereby the rate of change of the phase variable, , is proportional to the variational derivative of the free energy, mediated by the kinetic mobility B:
Such a formulation leads to a 2nd order, diffusion-type partial differential equation (PDE) for . However, if the phase- ﬁeld variable is subject to a conservation law, e.g.
then the rate is proportional to the divergence of the ﬂux, which in turn, is proportional to the gradient of a potential. This potential is equal to the variational derivative of the free energy:
Examples include diffusion-controlled solid-state phase transformations, and interfaces between immiscible ﬂuids . Typically, the 4th order evolution PDE (3) is coupled to a 2nd order PDEs (stress, heat, etc.). For example, in solid state phase transformations, the lattice continuity across the interfaces between mismatched phases produces elastic strains , so that the 2nd order elasticity PDEs are coupled to the phase ﬁeld Eq. (3). The problems in this class have been addressed in the past, with the numerical tools tailored for speciﬁc problems. However, a general numerical method for solving the coupled equations, applicable to a variety of geometries and boundary conditions, has remained a challenging problem. The ﬁnite difference method (FDM) was used by Cahn and Kobayashi for one-dimensional modeling of the rapid coarsening and buckling in coherently self-stressed thin plates. Johnson used FDM to study the growth of the intermediate phase in a thin ﬁlm diffusion couple. Leo et al.used a pseudo-spectral method for the phase-ﬁeld model of coarsening in a two dimensional, elastically stressed, binary alloys. Fourier transform techniques, and related Fourier-spectral methods , have been used to investigate the microstructure evolution (mostly coarsening) resulting from solid-state phase transformations. The drawbacks of such methods are seen when the problem requires modeling of irregular domains, nonlinear and history-dependent problems. A versatile method is needed, applicable to irregular domains, a variety of boundary conditions, and various forms of geometric and material nonlinearities. The ﬁnite element method (FEM) and the related mesh-free method are such methods. [Phasefield 1]
References pertaining to metals
- ↑ Mohsen Asle Zaeem, Sinisa Dj. Mesarovic, Finite element method for conserved phase fields: Stress-mediated diffusional phase transformation Original Research Article Journal of Computational Physics, Volume 229, Issue 24, 10 December 2010, Pages 9135-9149
Multilayer Thin Films
Fatigue Crack Incubation
Due to the vast differences between polymers and metals, it can be safely assumed that the mechanics that govern the incubation of fatigue cracks are also different. To help form a base-level understanding of which properties and conditions are most important for crack incubation in polymers, a design of experiments (DOE) methodology was employed to discern which of several chosen parameters, when varied, had the greatest impact on incubation. Briefly, the DOE algorithm allows a reduction in the number of numerical calculations required to isolate the effects of individual parameters on various outputs including fatigue life. For polymer fatigue, seven parameters were chosen, each with its own high and low boundary conditions. Normally, 2^7 or 128 separate simulations would be required to determine the individual effect of each parameter. However, implementing the DOE technique allows one to accomplish the same task with only 8 numerical calculations. The 7 parameters are a conglomeration of material properties and testing conditions. The parameters chosen were: matrix material, particle modulus, temperature, strain rate, stress state, strain amplitude, and mean strain.
References for Polymers
- ↑ DeVor RE, Chang T, Sutherland JW. Statistical quality design and control: contemporary concepts and methods, First edition. Macmillan, New York, 1992
Matlab tool box for texture MTEX
- MTEX 
 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.
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 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.