Crack Occurrence in Bodies with Gradient Polyconvex Energies

In a set of infinitely many reference configurations differing from a chosen fit region B in the three-dimensional space and from each other only by possible crack paths, a set parameterized by special measures, namely curvature varifolds, energy minimality selects among possible configurations of a continuous body those that are compatible with assigned boundary conditions of Dirichlet-type. The use of varifolds allows us to consider both “material 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 curvature 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.