Infinite energy cavitating solutions: a variational approach

TL;DR

This paper introduces a variational approach to cavitation in compressible elasticity, defining a modified energy functional that admits finite energy cavitating solutions and analyzing their properties.

Contribution

It develops a new modified energy functional differing by a null Lagrangian, enabling the existence of finite energy cavitating solutions and establishing their variational and boundary conditions.

Findings

Existence of radial energy minimizers with cavitation

Modified boundary conditions for cavitating solutions

Numerical scheme convergence for singular cavitating solutions

Abstract

We study the phenomenon of cavitation for the displacement boundary value problem of radial, isotropic compressible elasticity for a class of stored energy functions of the form $W(F) + h(\det F)$, where $W$ grows like $||F||^n$, and $n$ is the space dimension. In this case it follows (from a result of Vodopyanov, Goldshtein and Reshetnyak) that discontinuous deformations must have infinite energy. After characterizing the rate at which this energy blows up, we introduce a modified energy functional which differs from the original by a null lagrangian, and for which cavitating energy minimizers with finite energy exist. In particular, the Euler--Lagrange equations for the modified energy functional are identical to those for the original problem except for the boundary condition at the inner cavity. This new boundary condition states that a certain modified radial Cauchy stress function has to vanish at the inner cavity. This condition corresponds to the radial Cauchy stress for the original functional diverging to $-\infty$ at the cavity surface. Many previously known variational results for finite energy cavitating solutions now follow for the modified functional, such as the existence of radial energy minimizers, satisfaction of the Euler-Lagrange equations for such minimizers, and the existence of a critical boundary displacement for cavitation. We also discuss a numerical scheme for computing these singular cavitating solutions using regular solutions for punctured balls. We show the convergence of this numerical scheme and give some numerical examples including one for the incompressible limit case. Our approach is motivated in part by the use of the renormalized energy for Ginzberg-Landau vortices.