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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The parameters for the shells in Figs. 1–4 are presented in Supplementary Tables S1–S3. 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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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.).
The authors declare no competing interests.
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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