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

  • J. Sivaloganathan (a1)

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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.

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[1]Ball, J. M.. Discontinuous equilibria and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. London Ser. A 306 (1982), 557611.
[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), 251277.
[3]Bolza, O.. Lectures on the Calculus of Variations (Dover Publications Inc., 1961).
[4]Cesari, L.. Optimisation Theory and Applications (Springer-Verlag, 1961).
[5]Gelfand, I. M. and Fomin, S. V.. Calculus of Variations (Prentice-Hall, 1963).
[6]Giaquinta, M.. Multiple Integrals in the Calculus of Variations, Annals of Maths. Studies 105 (Princeton University Press, 1983).
[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).
[8]Morrey, C. B.. Multiple Integrals in the Calculus of Variations (Springer-Verlag, 1966).
[9]Ogden, R. W.. Nonlinear elastic deformations (Ellis Horweed, Halstead Press, Whiley, 1984).
[10]Sivaloganathan, J.. Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Ration. Mech. Anal. (To appear).
[11]Stuart, C. A.. Radially symmetric cavitation for hyperelastic materials. Ann. Inst. H. Poincaé Anal. Non lineéaire 2 (1958), 3366.
[12]Truesdell, C. and Noll, W.. The nonlinear field theories of mechanics, Handbuch der Physik III/3 (Springer-Verlag, 1965).

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

  • J. Sivaloganathan (a1)

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