Phase field approximation of cohesive fracture models

TL;DR

This paper derives a cohesive fracture model as a limit of damage models, characterizing fracture energy through an optimal profile problem and connecting to classical fracture models like Dugdale's and Griffith's.

Contribution

It introduces a novel gamma-limit approach to derive cohesive fracture models from damage models with damage-dependent elastic coefficients.

Findings

Fracture energy is linear in small openings and finite at large openings.

The model recovers Dugdale's and Griffith's fracture models as special cases.

Surface energy density can exhibit power-law growth at small openings.

Abstract

We obtain a cohesive fracture model as a $\Gamma$-limit of scalar damage models in which the elastic coefficient is computed from the damage variable $v$ through a function $f_k$ of the form $f_k(v)=min\{1,\varepsilon_k^{1/2} f(v)\}$, with $f$ diverging for $v$ close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening $s$ at small values of $s$ and has a finite limit as $s\to\infty$. If the function $f$ is allowed to depend on the index $k$, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings.