MiS Preprint Repository

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 $h$. Our main result is that when the deformations are subject to certain (physically reasonable) hypotheses, the minimum energy is of order $h^{4/3}$. We also show that when the deformations are subject to a more restrictive hypothesis, 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.