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The finite element method is a powerful complement to polycrystal plasticity theory for modeling the non-uniform deformation of crystalline solids. Polycrystal plasticity provides a micro-mechanical model for slip-dominated plastic flow and serves as a constitutive theory for deformation simulations (Kocks et al. 1998). The finite element method offers a numerical means to solve partial differential equations, such as the field equations of elasticity or plasticity (Zienkiewicz et al. 1989). The two can be combined in different ways depending on the goals of a modeling effort.
Finite elements and polycrystal plasticity may be applied to the detailed modeling of a collection of grains that represent a sample of the material. In this case, there are one or more finite elements discretizing each grain and balance laws for momentum and mass are applied at the level of individual crystals. A second combination of finite elements and polycrystal plasticity is to embed polycrystal theory within a finite element formulation for physical systems that are far larger than the dimension of a grain. Polycrystal plasticity serves as a constitutive theory in essentially the same way as continuum elastoplasticity models. Balance laws are applied at the larger continuum scale. We refer to these as small-scale and large-scale applications, respectively. Care must be exercised in assuring consistency between the macroscopic material element volume and the polycrystal dimensions.
With respect to characterizing these applications, it is useful to define a geometric parameter, ζ , as the relative sizes of a finite element and a crystal. Allowing h to be a characteristic dimension of an element and d to be the representative grain size the parameter ζ simply is h/d. Here, large ζ implies large numbers of crystals in each element; small ζ implies many elements within each crystal. For small-scale applications, there is a …