Phase field approximation of cohesive fracture models

We obtain a cohesive fracture model as Γ-limit, as ε→0ε→0, of scalar damage models in which the elastic coefficient is computed from the damage variable v   through a function fεfε of the form fε(v)=min⁡{1,ε12f(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→∞s→∞. 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.