Phase field approximation of cohesive fracture models

Source abstract

We obtain a cohesive fracture model as Γ-limit, as \varepsilon \rightarrow 0 , of scalar damage models in which the elastic coefficient is computed from the damage variable v through a function f_{\varepsilon } of the form f_{\varepsilon }(v) = \mathrm{\min }⁡\{1,\varepsilon ^{\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\rightarrow \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.