The Convergence of Regularized Minimizers for Cavitation Problems in Nonlinear Elasticity

Abstract. Consider a nonlinearly elastic body which occupies the region Ω ⊂ Rm (m = 2, 3) in its reference state and which is held in tension under prescribed boundary displacements on ∂Ω. Let x0 ∈ Ω be any fixed point in the body. It is known from variational arguments that, for sufficiently large boundary displacements, there may exist discontinuous weak solutions of the equilibrium equations corresponding to a hole forming at x0 in the deformed body (this is the phenomenon of cavitation). For each > 0, define the regularized domains Ω = Ω\B (x0) which contain a preexisting hole of radius > 0 centered on x0. Now consider the corresponding mixed displacement/traction problem on Ω in which the boundary ∂Ω is subject to the same boundary displacements and the deformed cavity surface (i.e., the image of ∂B ) is required to be stress-free. It follows from variational arguments that there exists a weak solution u of this problem for each > 0. In this paper we prove convergence of these regularized minimizers u in the limit as → 0. In particular, we show that if n → 0, then, passing to a subsequence, u n → u, where u is a minimizer for the original pure displacement problem on Ω. Finally, we study the effect on cavitation of regularizing the variational problem by introducing a surface energy term which penalizes the formation and growth of cavities.