Heterogeneous thin films of martensitic materials

We study the effective behavior of heterogeneous thin films with three comp eting length scales: the film thickness and the length scales of heterogene ity and material microstructure. We start with three-dimensional nonhomogen eous nonlinear elasticity enhanced with an interfacial energy of the van de r Waals type, and derive the effective energy density as all length scales tend to zero with given limiting ratios. We do not require any apriori sele ction of asymptotic expansion or ansatz in deriving our results. Depending on the dominating length scale, the effective energy density can be identif ied by three procedures: averaging, homogenization and thin-him limit. We a pply 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 c onductivity of linear composites, and also on multilayers made of shape-mem ory and elastic materials.