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

- 199 Downloads
- 3 Citations

## 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.

## References

- 1.Bainov, D., Simeonov, P.: Integral Inequalities and Applications. Springer, New York (2013) MATHGoogle Scholar
- 2.Bennati, S., Degl’innocenti, S., Padovani, C.: On the sign of the internal stresses in a problem of plane elasticity. J. Elast.
**32**, 155–174 (1993) CrossRefMATHMathSciNetGoogle Scholar - 3.Cerda, E., Mahadevan, L.: Geometry and physics of wrinkling. Phys. Rev. Lett.
**90**, 074302 (2003) ADSCrossRefGoogle Scholar - 4.Chia, C.Y.: Nonlinear Analysis of Plates. McGraw-Hill, New York (1980) Google Scholar
- 5.Coman, C.D.: Asymmetric bifurcations in a pressurised circular thin plate under initial tension. Mech. Res. Commun.
**47**, 11–17 (2013) CrossRefGoogle Scholar - 6.Coman, C.D., Matthews, M.T., Bassom, A.P.: Asymptotic phenomena in pressurised thin films. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci.
**471**, 20150471 (2015) ADSCrossRefGoogle Scholar - 7.Coman, C.D., Bassom, A.P.: On the nonlinear membrane approximation and edge-wrinkling. Int. J. Solids Struct.
**82**, 85–94 (2016) CrossRefGoogle Scholar - 8.Croll, J.G.A.: A tension field solution for non-linear circular plates. In: Dawe, D.J., Horsington, R.W., Kamtekar, A.G., Little, G.H. (eds.) Aspects of the Analysis of Plate Structures. Oxford University Press, New York (1985) Google Scholar
- 9.Dickey, R.W.: The plane circular elastic surface under normal pressure. Arch. Ration. Mech. Anal.
**26**, 219–236 (1967) CrossRefMATHMathSciNetGoogle Scholar - 10.Feng, L., Li, X., Shi, T.: Nonlinear large deflection of thin film overhung on compliant substrate using shaft-loaded blister test. J. Appl. Mech.
**82**, 091001 (2015) ADSCrossRefGoogle Scholar - 11.Knops, R.J., Payne, L.E.: Uniqueness Theorems in Linear Elasticity. Springer, Berlin (1971) CrossRefMATHGoogle Scholar
- 12.Langhaar, H.L., Stippes, M.C.: Location of extreme stresses. J. Elast.
**6**, 83–87 (1976) CrossRefMATHGoogle Scholar - 13.Mikulas, M.M.: Behaviour of a flat stretched membrane wrinkled by the rotation of an attached hub. Technical Report, NASA-TN D-2456 (1964) Google Scholar
- 14.Ng, T.T.: Edge effects in pressurized membranes. J. Eng. Mech.
**128**, 1100–1104 (2002) CrossRefGoogle Scholar - 15.Panov, D.Y., Feodosiev, V.I.: On the equilibrium and loss of stability of shallow shells in the case of large displacement. Prikl. Mat. Meh.
**20**, 389–406 (1948) Google Scholar - 16.Reismann, H.: Elastic Plates: Theory and Application. Wiley, New York (1988) MATHGoogle Scholar
- 17.Puntel, E., Deseri, L., Fried, E.: Wrinkling of a stretched thin sheet. J. Elast.
**105**, 137–170 (2011) CrossRefMATHMathSciNetGoogle Scholar - 18.Shilkrut, D.I.: Stability of Nonlinear Shells. Elsevier, Amsterdam (2002) MATHGoogle Scholar
- 19.Stein, M., Hedgepeth, J.M.: Analysis of partly wrinkled membranes. Technical Note, NASA-TN D-813 (1961) Google Scholar
- 20.Stroebel, G.J., Warner, W.H.: Stability and secondary bifurcations for von Kármán plates. J. Elast.
**3**, 185–202 (1973) CrossRefGoogle Scholar - 21.Timoshenko, S.P., Gere, J.M.: Theory of Plates and Shells, 2nd edn. McGraw-Hill, New York (1961) Google Scholar
- 22.Ventsel, E., Krauthammer, T.: Thin Plates and Shells: Theory, Analysis, and Applications. Dekker, New York (2001) CrossRefGoogle Scholar
- 23.Williams, J.G.: Energy release rates for the peeling of flexible membranes and the analysis of blister tests. Int. J. Fract.
**87**, 265–288 (1997) CrossRefGoogle Scholar - 24.Zhao, M.H., Zhou, J., Yang, F., Liu, T., Zhang, T.Y.: Effects of substrate compliance on circular buckle delamination of thin films. Eng. Fract. Mech.
**74**, 2334–2351 (2007) CrossRefGoogle Scholar