Heterogeneous Thin Films of Martensitic Materials

:  We study the effective behavior of heterogeneous thin films with three competing length scales: the film thickness and the length scales of heterogeneity and material microstructure. We start with three-dimensional nonhomogeneous nonlinear elasticity enhanced with an interfacial energy of the van der Waals type, and derive the effective energy density as all length scales tend to zero with given limiting ratios. We do not require any a priori selection of asymptotic expansion or ansatz in deriving our results. Depending on the dominating length scale, the effective energy density can be identified by three procedures: averaging, homogenization and thin-film limit. We apply our theory to martensitic materials with multi-well energy density and use a model example to show that the “shape-memory behavior” can crucially depend on the ratios of these length scales. We comment on the effective conductivity of linear composites, and also on multilayers made of shape-memory and elastic materials.