Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T01:08:30.504Z Has data issue: false hasContentIssue false
  • English
  • Français

A field theory approach to stability of radial equilibria in nonlinear elasticity

Published online by Cambridge University Press:  24 October 2008

J. Sivaloganathan
J. Sivaloganathan
Affiliation:
School of Mathematics, University of Bath, Bath BA2 7AY
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper we study the stability of a class of singular radial solutions to the equilibrium equations of nonlinear elasticity, in which a hole forms at the centre of a ball of isotropic material held in a state of tension under prescribed boundary displacements. The existence of such cavitating solutions has been shown by Ball[1], Stuart [11] and Sivaloganathan[10]. Our methods involve elements of the field theory of the calculus of variations and provide a new unified interpretation of the phenomenon of cavitation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 3 , May 1986 , pp. 589 - 604
Copyright
Copyright © Cambridge Philosophical Society 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ball, J. M.. Discontinuous equilibria and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. London Ser. A 306 (1982), 557–611.Google Scholar
[2]Ball, J. M. and Marsden, J. E.. Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Ration. Mech. Anal. 86 (1984), 251–277.CrossRefGoogle Scholar
[3]Bolza, O.. Lectures on the Calculus of Variations (Dover Publications Inc., 1961).Google Scholar
[4]Cesari, L.. Optimisation Theory and Applications (Springer-Verlag, 1961).Google Scholar
[5]Gelfand, I. M. and Fomin, S. V.. Calculus of Variations (Prentice-Hall, 1963).Google Scholar
[6]Giaquinta, M.. Multiple Integrals in the Calculus of Variations, Annals of Maths. Studies 105 (Princeton University Press, 1983).Google Scholar
[7]Horgan, C. O. and Abeyaratne, R.. A bifurcation problem for a compressible nonlinearly elastic medium: growth of a microvoid. (To Appear in J. Elasticity).Google Scholar
[8]Morrey, C. B.. Multiple Integrals in the Calculus of Variations (Springer-Verlag, 1966).CrossRefGoogle Scholar
[9]Ogden, R. W.. Nonlinear elastic deformations (Ellis Horweed, Halstead Press, Whiley, 1984).Google Scholar
[10]Sivaloganathan, J.. Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Ration. Mech. Anal. (To appear).Google Scholar
[11]Stuart, C. A.. Radially symmetric cavitation for hyperelastic materials. Ann. Inst. H. Poincaé Anal. Non lineéaire 2 (1958), 33–66.CrossRefGoogle Scholar
[12]Truesdell, C. and Noll, W.. The nonlinear field theories of mechanics, Handbuch der Physik III/3 (Springer-Verlag, 1965).Google Scholar