Self-similar folding patterns and energy scaling in compressed elastic sheets

Thin elastic sheets under isotropic compression, such as for example blisters formed by thin films which debonded from the substrate, can exhibit remarkably complex folding patterns. We discuss the scaling of the elastic energy with respect to the film thickness, and show that in certain regimes the optimal energy scaling can be reached by self-similar folding patterns that refine towards the boundary, in agreement with experimental observations. We then extend the analysis to anisotropic compression, and discuss a simplified scalar model which suggests the presence of a transition between a regime where the deformation is governed by global properties of the domain and another one where the direction of maximal compression dominates and the scale of the folds is mainly determined by the distance to the boundary in the direction of the folds themselves.