The Repulsion Property in Nonlinear Elasticity and a Numerical Scheme to Circumvent It

For problems in the Calculus of Variations that exhibit the Lavrentiev phenomenon, it is known that a \textit{repulsion property} may hold, that is, if one approximates the global minimizer in these problems by smooth functions, then the approximate energies will blow up. Thus, standard numerical schemes, like the finite element method, may fail when applied directly to these types of problems. In this paper we prove that a generalised repulsion property holds for variational problems in three dimensional elasticity that exhibit cavitation.<br/><br/>We propose a numerical scheme that circumvents the repulsion property, which utilizes an adaptation of the Modica and Mortola functional for phase transitions in liquids. In our scheme, the phase function is coupled, via the determinant of the deformation gradient, to the stored energy functional. We show that the corresponding approximations by this method satisfy the lower bound $\Gamma$--convergence property in the multi-dimensional, non--radial, case. The convergence to the actual cavitating minimizer is proved for a spherical body, in the case of radial deformations.