arXiv:2102.06641v1 [math.AP] 12 Feb 2021 Crack occurrence in bodies with gradient polyconvex energies Martin Kruˇ z´ ık*, Paolo Maria Mariano**, Domenico Mucci*** *Czech Academy of Sciences, Institute of Information Theory and Automation Pod Vod´ arenskou v˘ eˇ z´ ı 4, CZ-182 00 Prague 8, Czechia e-mail: kruzik@utia.cz **DICEA, Universit` a di Firenze via Santa Marta 3, I-50139 Firenze, Italy e-mail: paolomaria.mariano@unifi.it, paolo.mariano@unifi.it ***DSMFI, Universit` a di Parma Parco Area delle Scienze 53/A, I-43134 Parma, Italy e-mail: domenico.mucci@unipr.it Abstract Energy minimality selects among possible configurations of a continuous body with and without cracks those compatible with assigned boundary conditions of Dirichlet- type. 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 en- ergy. 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. Key words: Fracture, Varifolds, Ground States, Shells, Microstructures, Calculus of Variations 1 Introduction Deformation-induced material effects involving interactions beyond those of first-neighbor-type can be accounted for by considering, among the fields defining states, higher-order deformation gradients. In short, we can say that these effects emerge from latent microstructures, intending those which do not strictly require to be represented by independent (observable) variables accounting for small-spatial-scale degrees of freedom. Rather they are such