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-Ne\v{c}as condition in SBV^2 and for which a convergence of the energies is ensured, are shown to admit asymptotic representations in GSBD^2 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-Ne\v{c}as condition. The proof relies on a piecewise Korn-Poincar\'e inequality in GSBD2, on a careful blow-up analysis around jump points, as well as on a refined GSBD^2-density result guaranteeing enhanced contact conditions for the approximants.