Variational modeling of multilayer films with coherent and incoherent interlayer interfaces

Abstract

A novel variational model is proposed to address design control for composite multilayered metamaterials self-assembled via vapor deposition. The model is formulated within the framework of continuum mechanics, with the reference configuration corresponding to the equilibrium lattice of the substrate material. To account for the potential mismatch with the free-standing equilibrium lattices of each layer’s material, following the literature on Stress-Driven Rearrangement Instabilities, a nonzero mismatch strain varying across layers is considered. Moreover, building on the results of [47], the model allows for the treatment of the interplay between coherent and incoherent regions, which can coexist at each interlayer interface, as both elastic and surface effects—and their competition—are taken into account. The surface of each film layer is assumed to satisfy the“exterior graph condition” introduced in [47], which allows bulk cracks to be of non-graph type. By applying the direct method of calculus of variations under a constraint on the number of connected components of the cracks that are not connected to the surface of the film layers, the existence of energy minimizers is established in two dimensions. As a byproduct of the analysis, advancements are also made in the state of the art in the variational modeling of single-layered films by allowing the substrate surface to be free and including the possibility of delamination from the substrate.