The Convergence of Regularized Minimizers for Cavitation Problems in Nonlinear Elasticity

Abstract

Consider a nonlinearly elastic body which occupies the region $\Omega \subset {\mathbb{R}} ^m$ ($m=2,3$) in its reference state and which is held in tension under prescribed boundary displacements on $\partial \Omega$. Let ${\bf x}_0 \in \Omega$ 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 ${\bf x}_0$ in the deformed body (this is the phenomenon of cavitation). For each $\epsilon >0$, define the regularized domains $\Omega _\epsilon = \Omega \backslash \overline{B_\epsilon ({\bf x}_0)}$ which contain a preexisting hole of radius $\epsilon >0$ centered on ${\bf x}_0$. Now consider the corresponding mixed displacement/traction problem on $\Omega _\epsilon$ in which the boundary $\partial \Omega$ is subject to the same boundary displacements and the deformed cavity surface (i.e., the image of $\partial B_\epsilon$) is required to be stress-free. It follows from variational arguments that there exists a weak solution ${\bf u} _\epsilon$ of this problem for each $\epsilon >0 $. In this paper we prove convergence of these regularized minimizers ${\bf u} _\epsilon$ in the limit as $\epsilon \rightarrow 0$. In particular, we show that if $\epsilon _n \to 0$, then, passing to a subsequence, ${\bf u} _{\epsilon _n} \rightarrow {\bf u}$, where ${\bf u}$ is a minimizer for the original pure displacement problem on $\Omega$. 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.