Phase field approximation of cohesive fracture models

Abstract

We obtain a cohesive fracture model as Γ-limit, as $\epsilon \to 0$, of scalar damage models in which the elastic coefficient is computed from the damage variable v through a function $f_{\epsilon }$ of the form $f_{\epsilon }(v)=\min \{1,{\epsilon }^{\frac{1}{2}}f(v)\}$, 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 in addition the function f is allowed to depend on the parameter ε, 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.