Science at the triple point between mathematics, mechanics and materials science

Publication 40

Metric-Induced Wrinkling of a Thin Elastic Sheet

Authors:

Peter Bella
Max Planck Institute for Mathematics in the Sciences

Robert V Kohn
Courant Institute of Mathematical Sciences
New York University

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 the deformations are subject to certain hypotheses, the minimum energy is of order $h^{4/3}$. We also show that when the 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.
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bella-kohn-metric-induced.pdf

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