Static bistability of spherical Caps

Matteo Taffetani*, Xin Jiang, Douglas P. Holmes, Dominic Vella

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

58 Citations (Scopus)


Depending on its geometry, a spherical shell may exist in one of two stable states without the application of any external force: There are two 'self-equilibrated' states, one natural and the other inside out (or 'everted'). Though this is familiar from everyday life-An umbrella is remarkably stable, yet a contact lens can be easily turned inside out-The precise shell geometries for which bistability is possible are not known. Here, we use experiments and finite-element simulations to determine the threshold between bistability and monostability for shells of different solid angle. We compare these results with the prediction from shallow shell theory, showing that, when appropriately modified, this offers a very good account of bistability even for relatively deep shells. We then investigate the robustness of this bistability against pointwise indentation. We find that indentation provides a continuous route for transition between the two states for shells whose geometry makes them close to the threshold. However, for thinner shells, indentation leads to asymmetrical buckling before snap-Through, while also making these shells more 'robust' to snap-Through. Our work sheds new light on the robustness of the 'mirror buckling' symmetry of spherical shell caps.

Original languageEnglish
Article number20170910
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2213
Publication statusPublished - May 2018

Bibliographical note

Funding Information:
Data accessibility. The datasets supporting this article may be found at Berw0. Authors’ contributions. D.V. and D.P.H. designed the research. M.T. performed the numerical analysis, D.V. and M.T. developed the shallow shell model. X.J. carried out the experiments. All authors discussed the results and contributed to the writing of the manuscript. Competing interests. We declare that we have no competing interests. Funding. The research leading to these results has received funding from the European Research Council under the European Union’s Horizon 2020 Programme/ERC grant agreement no. 637334 (D.V.). D.P.H. and X.J. are grateful for financial support from the NSF CMMI-1505125. Acknowledgements. We thank Mariano Galvagno for discussions and preliminary numerical investigations.

Publisher Copyright:
© 2018 The Author(s) Published by the Royal Society. All Rights Reserved.

Structured keywords

  • Engineering Mathematics Research Group


  • Buckling
  • Finite-element analysis
  • Shallow shell theory
  • Shell stability


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