A phase‐field porous media fracture model based on homogenization theory

Abstract

AbstractA novel regularized fracture model for crack propagation in porous media is proposed. Our model is obtained through formal asymptotic expansions. We start with a regularized quasi‐static fracture model posed in a periodically perforated domain obtained by periodic extension of a rescaled unit cell with a hole. This setup allows us to write two separated minimality conditions for the primary (displacement) and secondary (or phase field) variables plus a balance of energy relation. Then we apply the usual asymptotic expansion matching to deduce limit relations when the rescaling parameter of the unit cells vanishes. By introducing cell problems solutions and a homogenized tensor, we can recast the obtained relations into a novel model for crack propagation in porous media. The proposed model can be interpreted as a regularized quasi‐static fracture model for porous media. This model yields two separated (homogenized) minimality conditions for the primary and secondary variables and a balance of a homogenized energy relation.