Non-interpenetration conditions in the passage from nonlinear to linearized Griffith fracture

Abstract

We characterize the passage from nonlinear to linearized Griffith-fracture theories under non-interpenetration constraints. In particular, sequences of deformations satisfying a Ciarlet-Nečas condition in SBV² and for which a convergence of the energies is ensured, are shown to admit asymptotic representations in GSBD² satisfying a suitable contact condition. With an explicit counterexample, we prove that this result fails if convergence of the energies does not hold. We further prove that each limiting displacement satisfying the contact condition can be approximated by an energy-convergent sequence of deformations fulfilling a Ciarlet-Nečas condition. The proof relies on a piecewise Korn-Poincaré inequality in GSBD², on a careful blow-up analysis around jump points, as well as on a refined GSBD²-density result guaranteeing enhanced contact conditions for the approximants.