Phase-field modelling of cohesive fracture. Part I: $\Gamma$-convergence results

TL;DR

This paper establishes the mathematical foundation of a cohesive phase-field model by proving a $B3$-convergence result, unifying various models and extending previous results in fracture mechanics.

Contribution

It provides a rigorous $B3$-convergence proof for a broad class of cohesive phase-field energies, unifying different approaches and generalizing the Ambrosio-Tortorelli approximation.

Findings

Proved $B3$-convergence for cohesive phase-field energies

Unified multiple existing models within a single framework

Extended the Ambrosio-Tortorelli approximation to cohesive fracture

Abstract

The main aim of this three-part work is to provide a unified consistent framework for the phase-field modeling of cohesive fracture. In this first paper we establish the mathematical foundation of a cohesive phase-field model by proving a $\Gamma$-convergence result in a one-dimensional setting. Specifically, we consider a broad class of phase-field energies, encompassing different models present in the literature, thereby both extending the results in \cite{ContiFocardiIurlano2016} and providing an analytical validation of all the other approaches. Additionally, by modifying the functional scaling, we demonstrate that our formulation also generalizes the Ambrosio-Tortorelli approximation for brittle fracture, therefore laying the groundwork for a unified framework for variational fracture problems. The Part~II paper presents a systematic procedure for constructing phase-field models that reproduce prescribed cohesive laws, whereas the Part~III paper validates the theoretical results with applied examples.