Blister patterns and energy minimization in compressed thin films on compliant substrates

This paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick, compliant substrates. These patterns are often induced by an elastic misfit which compresses the film. Blistering permits the film to expand locally, reducing the elastic energy of the system. It is natural to ask: what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate? This is a variational problem involving both the {\it elastic deformation} of the film and substrate and the {\it geometry} of the blistered region. It involves three small parameters: the {\it nondimensionalized thickness} of the film, the {\it compliance ratio} of the film/substrate pair and the {\it mismatch strain}. In formulating the problem, we use a small-slope (F\"oppl-von K\'arm\'an) approximation for the elastic energy of the film, and a local approximation for the elastic energy of the substrate. For a 1D version of the problem, we obtain "matching" upper and lower bounds on the minimum energy, in the sense that both bounds have the same scaling behavior with respect to the small parameters. For a 2D version of the problem, our results are less complete. Our upper and lower bounds only "match" in their scaling with respect to the nondimensionalized thickness, not in the dependence on the compliance ratio and the mismatch strain. The upper bound considers a 2D lattice of blisters, and uses ideas from the literature on the folding or "crumpling" of a confined elastic sheet. Our main 2D result is that in a certain parameter regime, the elastic energy of this lattice is significantly lower than that of a few large blisters.