Convergence of critical points for a phase-field approximation of 1D cohesive fracture energies

TL;DR

This paper studies the convergence of critical points in phase-field models for 1D cohesive fracture, showing they approximate and are approximated by critical points of the limiting fracture energy functional.

Contribution

It establishes the asymptotic behavior of critical points in phase-field approximations of cohesive fracture energies in one dimension, linking them to the limit functional.

Findings

Critical points of phase-field energies converge to a class of critical points of the limit functional.

Each critical point of the limit functional can be approximated by phase-field critical points.

Results are specific to the one-dimensional setting.

Abstract

Variational models for cohesive fracture are based on the idea that the fracture energy is released gradually as the crack opening grows. Recently, [Conti, Focardi, and Iurlano, Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire, 2016] proposed a variational approximation via $\Gamma$-convergence of a class of cohesive fracture energies by phase-field energies of Ambrosio-Tortorelli type, which may be also used as regularization for numerical simulations. In this paper we address the question of the asymptotic behaviour of critical points of the phase-field energies in the one-dimensional setting: we show that they converge to a selected class of critical points of the limit functional. Conversely, each critical point in this class can be approximated by a family of critical points of the phase-field functionals.