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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
35/2013

Metric-induced wrinkling of a thin elastic sheet

Peter Bella and Robert V. Kohn

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 $h$. 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.

Received:
Apr 3, 2013
Published:
Apr 5, 2013
Keywords:
thin films, energy scaling law, variable metric

Related publications

inJournal
2014 Repository Open Access
Peter Bella and Robert V. Kohn

Metric-induced wrinkling of a thin elastic sheet

In: Journal of nonlinear science, 24 (2014) 6, pp. 1147-1176