TY - JOUR

T1 - Metric-Induced Wrinkling of a Thin Elastic Sheet

AU - Bella, Peter

AU - Kohn, Robert V.

N1 - Funding Information:
Acknowledgments WearegratefultoStefanMüllerforpointingoutthattheassumptionuh(x,y)·e1=x is too rigid and for suggesting the ansatz we use to prove the h4/3 upper bound of Theorem 1. We also thank two anonymous referees for their insightful comments and suggestions. This work was begun while PB was a PhD student at the Courant Institute of Mathematical Sciences. Support from NSF Grant DMS-0807347 is gratefully acknowledged. Support is gratefully acknowledged from NSF Grants DMS-0807347 and OISE-0967140.
Publisher Copyright:
© 2014, Springer Science+Business Media New York.

PY - 2014/12

Y1 - 2014/12

N2 - 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 (For for certain classes of displacements. Our main result is that when deformations are subject to certain hypotheses, the minimum energy is of order (Formula Presented). We also show that when deformations are subject to more restrictive hypotheses, the minimum energy is strictly larger – of order (Formula Presented); 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 (Formula Presented) could be obtained by considering a larger class of deformations.

AB - 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 (For for certain classes of displacements. Our main result is that when deformations are subject to certain hypotheses, the minimum energy is of order (Formula Presented). We also show that when deformations are subject to more restrictive hypotheses, the minimum energy is strictly larger – of order (Formula Presented); 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 (Formula Presented) could be obtained by considering a larger class of deformations.

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U2 - 10.1007/s00332-014-9214-9

DO - 10.1007/s00332-014-9214-9

M3 - Article

AN - SCOPUS:84920255996

VL - 24

SP - 1147

EP - 1176

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

SN - 0938-8974

IS - 6

ER -