Homogenization of Layered Materials with Stiff Components

Fabian Christowiak
2018
In the context of homogenization of elastoplastic materials, we study the effects of a stiff component on the macroscopic behavior of a material of fine layered structure. To model these materials a variational approach is chosen in accordance to the assumptions of finite elasticity. In view of the macroscopic material response, the elasticity of the individual thin layer stands in contrast to the stiffness of the components, leading to the question of optimal scaling relations between layer
more » ... ckness and stiffness. We answer this question by identifying two regimes. For sufficiently strong stiffness, we provide a new type of asymptotic rigidity theorem, which enables us to give a full characterization of the rather restricted class of macroscopic material responses. In particular, we show that in two dimensions, if volume is preserved locally, this class comprises only globally rotated shear deformations. In contrast, we illustrate with explicit examples that for small elastic constants bending and wrinkling of layers leads to much broader possibilities for deformations. This characterization result allows to determine homogenization formulas for a manifold of variational models for finite elastoplastic materials via Gamma-convergence. In this work, we provide two homogenization results, one for elastic materials and one in the context of crystal plasticity. Firstly, assuming that the elastic softer layers are described by an energy density whose polyconvex envelope coincides with its quasiconvex envelope, we establish an explicit homogenization formula and discuss its relations to cell and multicell formulas. Secondly, we consider a stiff material in two dimensions with one active slip system in every other layer. Here, the homogenization formula strongly depends on the orientation of the slip system. In particular, the intuition that a slip system orthogonal to the stiff layers should be blocked, while for a parallel orientation it should be unhindered, is rigorously confirmed. Due to the distinct differential [...]
doi:10.5283/epub.37697 fatcat:reyaoaipybfmrgsnkrwpaxiepi