Identification of fracture models based on phase field for crack propagation in heterogeneous lattices in a context of non-separated scales

The construction of a homogeneous medium equivalent to a heterogeneous one under quasi-brittle fracture is investigated in the case of non-separated scales. At the microscale, the phase field method to fracture is employed. At the scale of the homogeneous medium, another phase field model either isotropic or anisotropic, depending on the microscale crack length and on the underlying microstructure, is assumed. The coefficients of the unknown phase field model for the homogeneous model are identified through the mechanical response of a sample subjected to fracture whose microstructure is fully described and estimated numerically. We show that the identified models can reproduce both the mechanical force response as well as overall crack paths with good accuracy in other geometrical configurations than the one used to identify the homogeneous model. Several numerical examples, involving cracking in regular lattices of both hard particles and pores, are presented to show the potential of the technique.