Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Roberto Alessi, Francesco Colasanto, Matteo Focardi

TL;DR

This work develops a rigorous $Γ$-convergence theory for a broad class of cohesive phase-field energies in one dimension, unifying prior cohesive-field models and revealing how a suitable scaling recovers the Ambrosio–Tortorelli brittle-fracture approximation. By introducing a general phase-field functional with adaptable bulk and surface terms, the authors derive compactness, lower-bound, and asymptotic results that connect diffuse regularizations to sharp-interface cohesive energies. The framework accommodates different choices of degradation and dissipation functions, yielding explicit surface energy densities and bulk energies that can reproduce prescribed cohesive laws. The results provide a solid mathematical basis for designing phase-field models of cohesive fracture and lay groundwork for the subsequent Parts II and III, which construct models for target cohesive laws and validate the theory with applications. Overall, the paper advances variational fracture theory by unifying brittle and cohesive phase-field approaches under a common $Γ$-convergence paradigm.

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 $Γ$-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.

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

TL;DR

This work develops a rigorous -convergence theory for a broad class of cohesive phase-field energies in one dimension, unifying prior cohesive-field models and revealing how a suitable scaling recovers the Ambrosio–Tortorelli brittle-fracture approximation. By introducing a general phase-field functional with adaptable bulk and surface terms, the authors derive compactness, lower-bound, and asymptotic results that connect diffuse regularizations to sharp-interface cohesive energies. The framework accommodates different choices of degradation and dissipation functions, yielding explicit surface energy densities and bulk energies that can reproduce prescribed cohesive laws. The results provide a solid mathematical basis for designing phase-field models of cohesive fracture and lay groundwork for the subsequent Parts II and III, which construct models for target cohesive laws and validate the theory with applications. Overall, the paper advances variational fracture theory by unifying brittle and cohesive phase-field approaches under a common -convergence paradigm.

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 -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.

Paper Structure

This paper contains 15 sections, 20 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. A general phase-field model
  4. Contents of this Paper
  5. Technical results
  6. Preliminaries
  7. Technical results for the diffuse part
  8. Properties of the surface energy densities
  9. Case ...
  10. Case ...
  11. ...-convergence and compactness
  12. Case ...
  13. Case ...
  14. Dirichlet boundary values problem
  15. Corollaries of Theorem \ref{['t:finale']}

Key Result

Theorem 1.1

Assume (Hp $1$)-(Hp $4$), and e:FailureS def hold with $\overline{\varsigma}\in(0,\infty]$. Let $\mathcal{F}_\varepsilon$ be the functional defined in functeps. Then, for all $(u,v)\in L^1(\Omega,\mathbb{R}^2)$ where, $F_{\,\overline{\varsigma}}:L^1(\Omega,\mathbb{R}^2)\to[0,\infty]$ is defined by

Figures (1)

  • Figure 1: Paradigmatic fracture problem: qualitative trends of the surface energy density $g$ and associated cohesive stress $\IfNoValueTF{-NoValue-}{\sigma}{\sigma\ar{-NoValue-}}$ with respect to the crack opening (displacement jump) $\left[ u \right]$ for brittle and cohesive fracture models.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 27 more