Crack occurrence in bodies with gradient polyconvex energies

Abstract

Energy minimality selects among possible configurations of a continuous body with and without cracks those compatible with assigned boundary conditions of Dirichlettype. Crack paths are described in terms of curvature varifolds so that we consider both "phase" (cracked or non-cracked) and crack orientation. The energy considered is gradient polyconvex: it accounts for relative variations of second-neighbor surfaces and pressure-confinement effects. We prove existence of minimizers for such an energy. They are pairs of deformations and varifolds. The former ones are taken to be SBV maps satisfying an impenetrability condition. Their jump set is constrained to be in the varifold support.