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

TL;DR

This paper investigates the transition from nonlinear to linearized Griffith fracture models under non-interpenetration constraints, establishing conditions for convergence and approximation of deformations with contact conditions.

Contribution

It characterizes the passage from nonlinear to linearized fracture theories under non-interpenetration constraints and provides approximation results for limiting displacements.

Findings

Sequences satisfying Ciarlet-Nečas condition converge to contact-satisfying limits

Counterexample shows convergence of energies is essential for the main result

Limiting displacements can be approximated by energy-convergent deformations

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.