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Exact solutions for the wrinkle patterns of confined elastic shells

Abstract

Complex textured surfaces occur in nature and industry, from fingerprints to lithography-based micropatterns. Wrinkling by confinement to an incompatible substrate is an attractive way of generating reconfigurable patterned topographies, but controlling the often asymmetric and apparently stochastic wrinkles that result remains an elusive goal. Here, we describe a new approach to understanding the wrinkles of confined elastic shells, using a Lagrange multiplier in place of stress. Our theory reveals a simple set of geometric rules predicting the emergence and layout of orderly wrinkles, and explaining a surprisingly generic co-existence of ordered and disordered wrinkle domains. The results agree with numerous test cases across simulation and experiment and represent an elementary geometric toolkit for designing complex wrinkle patterns.

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Fig. 1: Wrinkling of confined shells.
Fig. 2: Simple rules for wrinkles.
Fig. 3: Floating shells.
Fig. 4: Variable-curvature shells.
Fig. 5: Deducing the simple rules.
Fig. 6: Open questions.

Data availability

The parameters for the shells in Figs. 14 are presented in Supplementary Tables S1S3. Dimensionless parameter ranges for the experiments and simulations are given in Methods. Individual parameters for all experiments are also provided as a Supplementary Datafile. All other data that support the findings of this study are available from the corresponding authors upon reasonable request.

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Acknowledgements

We thank B. Davidovitch, V. Démery, C. R. Doering, G. Francfort, S. Hilgenfeldt, R. D. James, R. V. Kohn, N. Menon, P. Plucinsky, D. Vella and A. Waas for helpful discussions. This work was supported by NSF awards DMS-1812831 and DMS-2025000 (I.T.); NSF award DMR-CAREER-1654102 (Y.T., J.D.P.); NSF award PHY-CAREER-1554887, University of Pennsylvania MRSEC award DMR-1720530 and CEMB award CMMI-1548571, and a Simons Foundation award 568888 (D.T. and E.K.).

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Contributions

I.T., E.K. and J.D.P. conceived and designed the research. I.T. developed and implemented the theory. D.T. and E.K. conducted and analysed the simulations. Y.T., G.C.L. and J.D.P. conducted and analysed the experiments. I.T., Y.T., D.T., E.K. and J.D.P. wrote the manuscript.

Corresponding authors

Correspondence to Ian Tobasco, Joseph D. Paulsen or Eleni Katifori.

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Nature Physics thanks Enrique Cerda and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary information

Supplementary information

Supplementary Tables S1–S3, Figs. S1–S3 and text.

Supplementary data

Supplementary datafile containing individual parameters for all experiments.

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Tobasco, I., Timounay, Y., Todorova, D. et al. Exact solutions for the wrinkle patterns of confined elastic shells. Nat. Phys. 18, 1099–1104 (2022). https://doi.org/10.1038/s41567-022-01672-2

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