Singular Equilibria in Nonlinear Elasticity

Jay Sivaloganathan Department of Mathematical Sciences University of Bath Bath, BA2 7AY UK

Abstract

This talk centres on discontinuous weak solutions to the equilibrium equations of nonlinear elasticity which may be interpreted as fractures forming in an initially perfect material. The approach is motivated by the fundamental work in [1] and the subsequent development in [4].

I will first give a brief introduction to the variational approach to nonlinear elasticity and review some necessary conditions for a minimiser. I will then present results on the existence and properties of singular equilibria using the distributional determinant and degree theory (see [4], [2], [5]) to characterise the set of admissible deformations. Finally, I will indicate applications of these ideas to modelling flaws in a material and propose a possible mechanism for the initiation of cracks (see [3]). (Much of the work presented is joint with S.J. Spector (S. Illinois).)

References

[1] J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. London, A306, (1982), 557–611.

[2] J. Sivaloganathan and S. J. Spector, On the existence of minimisers with prescribed singular points in nonlinear elasticity. J. Elasticity 59, (2000), 83–113.

[3] J. Sivaloganathan and S. J. Spector, On cavitation, configurational forces and implications for fracture in a nonlinearly elastic material. J. Elasticity 67, (2002), 25–49.

[4] S. Muller and S.J. Spector, An existence theory for nonlinear elasticity that allows for Cavitation. Arch. Rational Mech. Anal., 131, (1995), 1–66.

[5] J. Sivaloganathan, S.J. Spector and V. Tilakraj, The convergence of regularised minimisers for cavitation problems in nonlinear elasticity. (Preprint 2003).