Acta Mechanica

, Volume 224, Issue 2, pp 399–423 | Cite as

Some applications of the WKB method to the wrinkling of bi-annular plates in tension

  • Ciprian D. ComanEmail author


An application of WKB methods is proposed here for a stretched annular thin plate with piecewise-constant mechanical properties (also known as a bi-annular plate). Unlike the classical scenario involving only a simple annular such plate, in certain cases the neutral stability curve fails to be convex and the critical eigenmodes behave rather differently as the plate becomes progressively thinner (equivalent to \({\mu \to \infty}\) in our notations). On one side of this curve, the corresponding eigenmodes are localised near the inner rim of the annulus, while in the remaining part these functions are concentrated along the interface separating the two annular sub-regions. By using the asymptotic reduction technique proposed by Coman and Haughton in (Acta Mech 185:179–200, 2006), the original fourth-order three-point boundary-value problem is formally reduced to a pair of second-order differential equations coupled through a set of matching conditions at the interface. It is shown that for \({\mu \gg 1}\) the critical eigenvalues for both cases mentioned above can be approximated by solving a couple of simple transcendental equations and that the results predicted compare well with the direct numerical simulations of the original problem.


Direct Numerical Simulation Mode Number Annular Plate Neutral Stability Curve Annular Domain 
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  1. 1.
    Frostig Y., Simites G.J.: Buckling of multi-annular plates. Comput. Struct. 24, 443–454 (1986)zbMATHCrossRefGoogle Scholar
  2. 2.
    Frostig Y., Simites G.J.: Buckling of ring-stiffened multi-annular plates. Comput. Struct. 29, 519–526 (1988)CrossRefGoogle Scholar
  3. 3.
    Greenberg J.B., Stavsky Y.: Axisymmetric vibrations of concentric dissimilar isotropic composite plates. Compos. Part B 30, 553–567 (1999)CrossRefGoogle Scholar
  4. 4.
    Greenberg J.B., Stavsky Y.: Axisymmetric vibrations of concentric dissimilar orthotropic composite annular plates. J. Sound Vib. 254, 849–865 (2002)CrossRefGoogle Scholar
  5. 5.
    Coman C.D., Haughton D.M.: Localised wrinkling instabilities in radially stretched annular thin films. Acta Mech. 185, 179–200 (2006)zbMATHCrossRefGoogle Scholar
  6. 6.
    Coman C.D., Bassom A.P.: On the wrinkling of a pre-stressed annular thin film in tension. J. Mech. Phys. Solids 55, 1601–1617 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Coman, C.D., Bassom, A.P.: Erratum to “On the wrinkling of a pre-stressed annular thin film in tension [Journal of the Mechanics and Physics of Solids (2007) 1601]. J. Mech. Phys. Solids 60, 1240–1240 (2012)Google Scholar
  8. 8.
    Coman, C.D., Liu, X.: Buckling-resistant thin annular plates in tension. Int. J. Solids Struct. (submitted)Google Scholar
  9. 9.
    Géminard J.C., Bernal R., Melo F.: Wrinkle formations in axis-symmetrically stretched membranes. Eur. J. Phys. E 15, 117–126 (2004)CrossRefGoogle Scholar
  10. 10.
    Brush D.O., Almroth Bo.O.: Buckling of Bars, Plates and Shells. McGraw-Hill, New York (1975)zbMATHGoogle Scholar
  11. 11.
    Timoshenko S.P., Gere J.M.: Theory of Elastic Stability. McGraw Hill, New York (1961)Google Scholar
  12. 12.
    Ventsel E., Krauthammer T.: Thin Plates and Shells: Theory and Applications. Marcel Dekker, New York (2001)CrossRefGoogle Scholar
  13. 13.
    Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang Y.: AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (1997)Google Scholar
  14. 14.
    Coman C.D.: Edge-buckling in stretched thin films under in-plane bending. Z. Angew. Math. Phys 58, 510–525 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Coman C.D., Bassom A.P.: Higher-order asymptotics for edge-buckling of pre-stressed thin plates under in-plane bending. J. Eng. Math. 63, 327–338 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Coman C.D., Bassom A.P.: Wrinkling of pre-stressed annular thin films under azimuthal shearing. Math. Mech. Solids 55, 513–531 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Coman C.D.: Elastic instabilities caused by stress concentration. Int. J. Eng. Sci. 46, 877–890 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Coman C.D., Bassom A.P.: On a class of buckling problems in a singularly perturbed domain Quart. J. Mech. Appl. Math. 62, 89–103 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2012

Authors and Affiliations

  1. 1.Schlumberger Gould ResearchCambridgeEngland

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