Metric-Induced Wrinkling of a Thin Elastic Sheet

Abstract

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 begin with a broad introduction, including a discussion of key examples (some well-known, others apparently new) that demonstrate the overall character of the problem. We then focus on how the minimum energy scales with respect to the sheet thickness $h$ for certain classes of displacements. Our main result is that when deformations are subject to certain hypotheses, the minimum energy is of order $h^{4/3}$. We also show that when deformations are subject to more restrictive hypotheses, the minimum energy is strictly larger – of order $h$; it follows that energy minimization in the more restricted class is not a good model for the applications that motivate this work. Our results do not explain the cascade of wrinkles seen in some experimental and numerical studies, and they leave open the possibility that an energy scaling law better than $h^{4/3}$ could be obtained by considering a larger class of deformations.