Unilateral gradient flow of the Ambrosio-Tortorelli functional by minimizing movements

TL;DR

This paper studies the evolution of fracture models using unilateral gradient flows of the Ambrosio-Tortorelli functional, revealing convergence to a heat equation outside evolving crack sets.

Contribution

It introduces a new analysis of unilateral gradient flows with irreversibility constraints, using maximal slope curves and implicit schemes, with asymptotic convergence results.

Findings

Convergence to a generalized heat equation outside crack sets

Development of a gradient flow framework with irreversibility constraints

Use of implicit Euler schemes for constructing evolutions

Abstract

Motivated by models of fracture mechanics, this paper is devoted to the analysis of unilateral gradient flows of the Ambrosio-Tortorelli functional, where unilaterality comes from an irreversibility constraint on the fracture density. In the spirit of gradient flows in metric spaces, such evolutions are defined in terms of curves of maximal unilateral slope, and are constructed by means of implicit Euler schemes. An asymptotic analysis in the Mumford-Shah regime is also carried out. It shows the convergence towards a generalized heat equation outside a time increasing crack set.