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Exact solutions for the wrinkle patterns of confined elastic shells

Nature Physics volume 18, pages 1099–1104 (2022)Cite this article

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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.

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Data availability

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.

References

  1. Sharon, E., Roman, B., Marder, M., Shin, G.-S. & Swinney, H. L. Buckling cascades in free sheets. Nature 419, 579–579 (2002).

    Article  ADS  Google Scholar 

  2. Cerda, E. & Mahadevan, L. Geometry and physics of wrinkling. Phys. Rev. Lett. 90, 074302 (2003).

    Article  ADS  Google Scholar 

  3. Audoly, B. & Pomeau, Y. Elasticity and Geometry: from Hair Curls to the Non-linear Response of Shells (Oxford Univ. Press, 2010)

  4. Shyer, A. E. et al. Villification: how the gut gets its villi. Science 342, 212–218 (2013).

    Article  ADS  Google Scholar 

  5. Gemmer, J., Sharon, E., Shearman, T. & Venkataramani, S. C. Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets. EPL 114, 24003 (2016).

    Article  ADS  Google Scholar 

  6. Xu, F., Fu, C. & Yang, Y. Water affects morphogenesis of growing aquatic plant leaves. Phys. Rev. Lett. 124, 038003 (2020).

    Article  ADS  Google Scholar 

  7. Fei, C. et al. Nonuniform growth and surface friction determine bacterial biofilm morphology on soft substrates. Proc. Natl Acad. Sci. USA 117, 7622–7632 (2020).

    Article  ADS  Google Scholar 

  8. Hure, J., Roman, B. & Bico, J. Stamping and wrinkling of elastic plates. Phys. Rev. Lett. 109, 054302 (2012).

    Article  ADS  Google Scholar 

  9. King, H., Schroll, R. D., Davidovitch, B. & Menon, N. Elastic sheet on a liquid drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities. Proc. Natl Acad. Sci. USA 109, 9716–9720 (2012).

    Article  ADS  Google Scholar 

  10. Paulsen, J. D. Wrapping liquids, solids, and gases in thin sheets. Annu. Rev. Condens. Matter Phys. 10, 431–450 (2019).

    Article  ADS  Google Scholar 

  11. Vella, D. Buffering by buckling as a route for elastic deformation. Nat. Rev. Phys. 1, 425–436 (2019).

    Article  Google Scholar 

  12. Timounay, Y. et al. Sculpting liquids with ultrathin shells. Phys. Rev. Lett. 127, 108002 (2021).

    Article  ADS  Google Scholar 

  13. Breid, D. & Crosby, A. J. Curvature-controlled wrinkle morphologies. Soft Matter 9, 3624–3630 (2013).

    Article  ADS  Google Scholar 

  14. Stoop, N., Lagrange, R., Terwagne, D., Reis, P. M. & Dunkel, J. Curvature-induced symmetry breaking determines elastic surface patterns. Nat. Mater. 14, 337–342 (2015).

    Article  ADS  Google Scholar 

  15. Reis, P. M. A perspective on the revival of structural (in)stability with novel opportunities for function: from buckliphobia to buckliphilia. J. Appl. Mech. 82, 111001 (2015).

    Article  ADS  Google Scholar 

  16. Aharoni, H. et al. The smectic order of wrinkles. Nat. Commun. 8, 15809 (2017).

    Article  ADS  Google Scholar 

  17. Bella, P. & Kohn, R. V. Wrinkling of a thin circular sheet bonded to a spherical substrate. Philos. Trans. R. Soc. A 375, 20160157 (2017).

    Article  ADS  MATH  Google Scholar 

  18. Zhang, X., Mather, P. T., Bowick, M. J. & Zhang, T. Non-uniform curvature and anisotropic deformation control wrinkling patterns on tori. Soft Matter 15, 5204–5210 (2019).

    Article  ADS  Google Scholar 

  19. Davidovitch, B., Sun, Y. & Grason, G. M. Geometrically incompatible confinement of solids. Proc. Natl Acad. Sci. USA 116, 1483–1488 (2019).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Tovkach, O. et al. Mesoscale structure of wrinkle patterns and defect-proliferated liquid crystalline phases. Proc. Natl Acad. Sci. USA 117, 3938–3943 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  21. Pretzl, M. et al. A lithography-free pathway for chemical microstructuring of macromolecules from aqueous solution based on wrinkling. Langmuir 24, 12748–12753 (2008).

    Article  Google Scholar 

  22. Yang, S., Khare, K. & Lin, P.-C. Harnessing surface wrinkle patterns in soft matter. Adv. Funct. Mater. 20, 2550–2564 (2010).

    Article  Google Scholar 

  23. Chen, C.-M. & Yang, S. Wrinkling instabilities in polymer films and their applications. Polym. Int. 61, 1041–1047 (2012).

    Article  Google Scholar 

  24. Li, Z. et al. Harnessing surface wrinkling–cracking patterns for tunable optical transmittance. Adv. Opt. Mater. 5, 1–7 (2017).

    Article  ADS  Google Scholar 

  25. Wagner, H. Ebene blechwandträger mit sehr dünnem stegblech. Z. Flugtech. Motorluftshiffahrt 20, 200 (1929).

    Google Scholar 

  26. Pipkin, A. C. The relaxed energy density for isotropic elastic membranes. IMA J. Appl. Math. 36, 85–99 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  27. Steigmann, D. J. Tension-field theory. Proc. R. Soc. Lond. Ser. A 429, 141–173 (1990).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Davidovitch, B., Schroll, R. D., Vella, D., Adda-Bedia, M. & Cerda, E. A. Prototypical model for tensional wrinkling in thin sheets. Proc. Natl Acad. Sci. USA 108, 18227–18232 (2011).

    Article  ADS  MATH  Google Scholar 

  29. Bella, P. & Kohn, R. V. Wrinkles as the result of compressive stresses in an annular thin film. Commun. Pure Appl. Math. 67, 693–747 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  30. Hohlfeld, E. & Davidovitch, B. Sheet on a deformable sphere: wrinkle patterns suppress curvature-induced delamination. Phys. Rev. E 91, 012407 (2015).

    Article  ADS  Google Scholar 

  31. Vella, D., Huang, J., Menon, N., Russell, T. P. & Davidovitch, B. Indentation of ultrathin elastic films and the emergence of asymptotic isometry. Phys. Rev. Lett. 114, 014301 (2015).

    Article  ADS  Google Scholar 

  32. Taffetani, M. & Vella, D. Regimes of wrinkling in pressurized elastic shells. Philos. Trans. R. Soc. A 375, 20160330 (2017).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Amar, MartineBen & Jia, F. Anisotropic growth shapes intestinal tissues during embryogenesis. Proc. Natl Acad. Sci. USA 110, 10525–10530 (2013).

    Article  Google Scholar 

  34. van Rees, W. M., Vouga, E. & Mahadevan, L. Growth patterns for shape-shifting elastic bilayers. Proc. Natl Acad. Sci. USA 114, 11597–11602 (2017).

    Article  ADS  Google Scholar 

  35. Tobasco, I. Curvature-driven wrinkling of thin elastic shells. Arch. Ration. Mech. Anal. 239, 1211–1325 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  36. Ciarlet, P. G. Mathematical Elasticity. Vol. II, Studies in Mathematics and its Applications, Vol. 27 (North-Holland, 1997).

  37. Maxwell, J. C. XLV. On reciprocal figures and diagrams of forces. Philos. Mag. 27, 250–261 (1864).

    Article  Google Scholar 

  38. S. P., Timoshenko, History of Strength of Materials. With a Brief Account of the History of Theory of Elasticity and Theory of Structures (McGraw-Hill Book Company, 1953).

  39. Prager, W. On ideal locking materials. Trans. Soc. Rheol. 1, 169–175 (1957).

    Article  MATH  Google Scholar 

  40. Fung, Y. C. Elasticity of soft tissues in simple elongation. Am. J. Physiol. 213, 1532–1544 (1967).

    Article  Google Scholar 

  41. Prager, W. On the formulation of constitutive equations for living soft tissues. Q. Appl. Math. 27, 128–132 (1969).

    Article  Google Scholar 

  42. Elettro, H., Neukirch, S., Vollrath, F. & Antkowiak, A. In-drop capillary spooling of spider capture thread inspires hybrid fibers with mixed solid–liquid mechanical properties. Proc. Natl Acad. Sci. USA 113, 6143–6147 (2016).

    Article  ADS  Google Scholar 

  43. Grandgeorge, P. et al. Capillarity-induced folds fuel extreme shape changes in thin wicked membranes. Science 360, 296–299 (2018).

    Article  ADS  Google Scholar 

  44. Timounay, Y. et al. Crumples as a generic stress-focusing instability in confined sheets. Phys. Rev. X 10, 021008 (2020).

    Google Scholar 

  45. Pocivavsek, L. et al. Stress and fold localization in thin elastic membranes. Science 320, 912–916 (2008).

    Article  ADS  Google Scholar 

  46. Brau, F., Damman, P., Diamant, H. & Witten, T. A. Wrinkle to fold transition: influence of the substrate response. Soft Matter 9, 8177–8186 (2013).

    Article  ADS  Google Scholar 

  47. Paulsen, J. D. et al. Geometry-driven folding of a floating annular sheet. Phys. Rev. Lett. 118, 048004 (2017).

    Article  ADS  Google Scholar 

Download references

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.).

Author information

Authors and Affiliations

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

    Ian Tobasco

  2. Department of Physics, Syracuse University, Syracuse, NY, USA

    Yousra Timounay, Graham C. Leggat & Joseph D. Paulsen

  3. BioInspired Syracuse: Institute for Material and Living Systems, Syracuse University, Syracuse, NY, USA

    Yousra Timounay & Joseph D. Paulsen

  4. Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, USA

    Desislava Todorova & Eleni Katifori

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  1. Ian Tobasco
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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.

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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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