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Curvature-Driven Wrinkling of Thin Elastic Shells

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Abstract

How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on the wrinkling patterns of floating shells, we develop a rigorous method via \(\Gamma \)-convergence for answering this question to leading order in the shell’s thickness and other small parameters. The observed patterns involve “ordered” regions of well-defined wrinkles alongside “disordered” regions whose local features are less robust; as little to no tension is applied, the preference for order is not a priori clear. Rescaling by the energy of a typical pattern, we derive a limiting variational problem for the effective displacement of the shell. It asks, in a linearized way, to cover up a maximum area with a length-shortening map to the plane. Convex analysis yields a boundary value problem characterizing the accompanying patterns via their defect measures. Partial uniqueness and regularity theorems follow from the method of characteristics on the ordered part of the shell. In this way, we can deduce from the principle of minimum energy the leading order features of stamped elastic shells.

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Notes

  1. We picked up the term “weakly curved” from [37]. It indicates a family of shells also referred to as “shallow”, the deformations of which can be modeled using the Donnel–Mushtari–Vlasov theory [58, 81] or Marguerre’s theory of shallow shells [68]. Our stretching and bending terms become the ones from [68] under the substitution \(w \rightarrow w+p\), and the ones from [58, 81] under the further substitution \(u\rightarrow u-w\nabla p\).

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Acknowledgements

We thank Eleni Katifori, Joseph D. Paulsen, Yousra Timounay, and Desislava V. Todorova for sharing their experimental and numerical results on thin and ultrathin floating shells in advance of their publication. We thank Benny Davidovitch, Charles R. Doering, and Robert V. Kohn for helpful discussions.

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  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607, USA

    Ian Tobasco

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Ian Tobasco

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This work was supported by National Science Foundation Awards DMS-1812831, DMS-1813003, and DMS-2025000, and a University of Michigan Van Loo Postdoctoral Fellowship.

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Tobasco, I. Curvature-Driven Wrinkling of Thin Elastic Shells. Arch Rational Mech Anal 239, 1211–1325 (2021). https://doi.org/10.1007/s00205-020-01566-8

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