Metric-induced wrinkling of a thin elastic sheet

Peter Bella and Robert V. Kohn

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Submission date: 03. Apr. 2013
Pages: 25
published in: Journal of nonlinear science, 24 (2014) 6, p. 1147-1176 
DOI number (of the published article): 10.1007/s00332-014-9214-9
Bibtex
Keywords and phrases: thin films, energy scaling law, variable metric
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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.