Crack occurrence in bodies with gradient polyconvex energies

TL;DR

This paper investigates crack formation in elastic bodies using a gradient polyconvex energy model, proving the existence of minimizers that include deformation maps and crack configurations represented by varifolds.

Contribution

It introduces a novel variational framework combining gradient polyconvex energies with curvature varifolds to model cracks and proves the existence of minimizers within this setting.

Findings

Existence of minimizers for the combined deformation and crack configuration.

Deformations are modeled as SBV maps with impenetrability conditions.

Crack paths are characterized using curvature varifolds.

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 \textquotedblleft 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 the 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.