Effective fracture toughness for heterogeneous materials accounting for the worst-case scenario

We propose a formulation for the anisotropic effective fracture toughness of heterogeneous materials for the phase field description for cracks. As an approximation for the worst-case scenario among all possible translations of the representative volume element (RVE) of the microstructure, a particular translation of this RVE is first obtained for subsequent analysis. This translated RVE is then subjected to numerical experiments of tensile loading along given sampling directions from the pristine state to complete fracture. The dissipated energy normalized by an appropriate projected area is then formulated as the effective fracture toughness for the chosen direction. By construction, this formulation is able to account for possible tortuosity of the crack path, the irreversibility constraint, the non-interpenetration constraint, and the stress criterion for crack nucleation. Furthermore, it can predict multiple toughening effects: that due to tougher inclusions, that resulting from microscopic cracking tortuosity, and that arising from the contrast in the tensile strength when fracture toughness is uniform.