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This paper is motivated by the complex blister patterns sometimes seen inthin elastic films on thick, compliant substrates. These patterns are ofteninduced by an elastic misfit which compresses the film. Blistering permits thefilm to expand locally, reducing the elastic energy of the system. It isnatural to ask: what is the minimum elastic energy achievable by blistering ona fixed area fraction of the substrate? This is a variational problem involvingboth the {\it elastic deformation} of the film and substrate and the {\itgeometry} of the blistered region. It involves three small parameters: the {\itnondimensionalized thickness} of the film, the {\it compliance ratio} of thefilm/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 elasticenergy of the film, and a local approximation for the elastic energy of thesubstrate.  For a 1D version of the problem, we obtain "matching" upper and lower boundson the minimum energy, in the sense that both bounds have the same scalingbehavior 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 theirscaling with respect to the nondimensionalized thickness, not in the dependenceon the compliance ratio and the mismatch strain. The upper bound considers a 2Dlattice 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 acertain parameter regime, the elastic energy of this lattice is significantlylower than that of a few large blisters.

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