Wrinkling as a result of relaxation of compressive stresses in thin films

I will talk about two (and, if time permits, three) problems I worked on recently - all of them were a joint work with Prof. Robert V. Kohn.

In the first one we study the wrinkling of a thin elastic sheet caused by a prescribed non-Euclidean metric. This is a model problem for the patterns seen, for example, in torn plastic sheets and the leaves of plants. Following the lead of other authors we adopt a variational viewpoint, according to which the wrinkling is driven by minimization of an elastic energy subject to appropriate constraints and boundary conditions. We prove an upper bound and a matching lower bound with the conjectured scaling for the minimum of the energy. We also discuss the minimization of the same functional over a more restricted class, again proving matching upper and lower bounds.

The second problem involves an elastic sheet loaded in tension which wrinkles and the length scale of the wrinkles tends to zero with vanishing thickness of the sheet. Our main achievement is identification of the scaling law of the minimum energy as the thickness of the sheet tends to zero. This requires proving an upper bound and a lower bound that scale the same way. We prove both bounds first in a simplified Kirchhoff-Love setting and then in the nonlinear three-dimensional setting. To obtain the optimal upper bound, we need to adjust a naive construction (one family of wrinkles superimposed on a planar deformation) by introducing a cascade of wrinkles. The lower bound is more subtle, since it must be ansatz-free.