# On the Role of In-Plane Compliance in Edge Wrinkling

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## Abstract

Bifurcations of a thin circular elastic plate subjected to uniform normal pressure are investigated by taking into account the in-plane compliance of the edge restraint. This effect amounts to introducing a Hookean spring relating the radial components of the membrane stress tensor and the corresponding in-plane displacement fields. The addition of this new feature gives rise to an adaptive radial stretching of our configuration, which is intimately linked to the strength of the applied pressure. The Föppl-von Kármán nonlinear plate theory, in conjunction with singular perturbation arguments, help us to establish the nature of the localised wrinkling observed in numerical simulations. Asymptotic analysis of the problem provides some simple qualitative predictions for the dependence of the critical load on a number of key dimensionless parameters.

## Keywords

Thin films Wrinkling Föppl-von Kármán plate equations Asymptotic methods## Mathematics Subject Classification

74G60## Notes

### Acknowledgements

The referees are thanked for several useful comments that led to improvements in the presentation of this work.

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