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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.
    Sivaloganathan, Jeyabal and Spector, Scott J. 2010. On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity II: Compressible Materials. Archive for Rational Mechanics and Analysis, Vol. 196, Issue. 2, p. 395.
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    Sivaloganathan, J. 1992. Singular minimisers in the Calculus of Variations: a degenerate form of cavitation. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Vol. 9, Issue. 6, p. 657.
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    Ball, J. M. and Knowles, G. 1987. A numerical method for detecting singular minimizers. Numerische Mathematik, Vol. 51, Issue. 2, p. 181.
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A field theory approach to stability of radial equilibria in nonlinear elasticity

  • J. Sivaloganathan (a1)
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.

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COPYRIGHT: © Cambridge Philosophical Society 1986
References
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[1]Ball, J. M.. Discontinuous equilibria and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. London Ser. A 306 (1982), 557–611.
[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.
[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), 33–66.
[12]Truesdell, C. and Noll, W.. The nonlinear field theories of mechanics, Handbuch der Physik III/3 (Springer-Verlag, 1965).
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
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