Journal of Elasticity

, Volume 126, Issue 2, pp 135–154 | Cite as

On the Role of In-Plane Compliance in Edge Wrinkling

  • Ciprian D. Coman
  • Andrew P. Bassom


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.


Thin films Wrinkling Föppl-von Kármán plate equations Asymptotic methods 

Mathematics Subject Classification




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


  1. 1.
    Bainov, D., Simeonov, P.: Integral Inequalities and Applications. Springer, New York (2013) MATHGoogle Scholar
  2. 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. 3.
    Cerda, E., Mahadevan, L.: Geometry and physics of wrinkling. Phys. Rev. Lett. 90, 074302 (2003) ADSCrossRefGoogle Scholar
  4. 4.
    Chia, C.Y.: Nonlinear Analysis of Plates. McGraw-Hill, New York (1980) Google Scholar
  5. 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. 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. 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. 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. 9.
    Dickey, R.W.: The plane circular elastic surface under normal pressure. Arch. Ration. Mech. Anal. 26, 219–236 (1967) CrossRefMATHMathSciNetGoogle Scholar
  10. 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. 11.
    Knops, R.J., Payne, L.E.: Uniqueness Theorems in Linear Elasticity. Springer, Berlin (1971) CrossRefMATHGoogle Scholar
  12. 12.
    Langhaar, H.L., Stippes, M.C.: Location of extreme stresses. J. Elast. 6, 83–87 (1976) CrossRefMATHGoogle Scholar
  13. 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. 14.
    Ng, T.T.: Edge effects in pressurized membranes. J. Eng. Mech. 128, 1100–1104 (2002) CrossRefGoogle Scholar
  15. 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. 16.
    Reismann, H.: Elastic Plates: Theory and Application. Wiley, New York (1988) MATHGoogle Scholar
  17. 17.
    Puntel, E., Deseri, L., Fried, E.: Wrinkling of a stretched thin sheet. J. Elast. 105, 137–170 (2011) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Shilkrut, D.I.: Stability of Nonlinear Shells. Elsevier, Amsterdam (2002) MATHGoogle Scholar
  19. 19.
    Stein, M., Hedgepeth, J.M.: Analysis of partly wrinkled membranes. Technical Note, NASA-TN D-813 (1961) Google Scholar
  20. 20.
    Stroebel, G.J., Warner, W.H.: Stability and secondary bifurcations for von Kármán plates. J. Elast. 3, 185–202 (1973) CrossRefGoogle Scholar
  21. 21.
    Timoshenko, S.P., Gere, J.M.: Theory of Plates and Shells, 2nd edn. McGraw-Hill, New York (1961) Google Scholar
  22. 22.
    Ventsel, E., Krauthammer, T.: Thin Plates and Shells: Theory, Analysis, and Applications. Dekker, New York (2001) CrossRefGoogle Scholar
  23. 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. 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

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Materials DivisionNational Physical LaboratoryTeddingtonUK
  2. 2.School of Physical SciencesUniversity of TasmaniaHobartAustralia

Personalised recommendations