Approximation of a Brittle Fracture Energy with a Constraint of Non-Interpenetration

TL;DR

This paper extends phase field approximation methods for brittle fracture energy to include a non-interpenetration constraint, ensuring physically realistic crack modeling by preventing interpenetration of crack faces.

Contribution

It introduces a variational approximation of brittle fracture energy that incorporates a non-interpenetration constraint within a phase field framework.

Findings

The approximation converges in the sense of Γ-convergence.

The non-interpenetration constraint is effectively incorporated into the phase field model.

The approach ensures physically realistic crack propagation without interpenetration.

Abstract

Linear fracture mechanics (or at least the initiation part of that theory) can be framed in a variational context as a minimization problem over a SBD type space. The corresponding functional can in turn be approximated in the sense of $\Gamma$-convergence by a sequence of functionals involving a phase field as well as the displacement field. We show that a similar approximation persists if additionally imposing a non-interpenetration constraint in the minimization, namely that only nonnegative normal jumps should be permissible. 2010 Mathematics subject classification: 26A45