Integral form of mixed-mode crack opening in the phase field method

Phase-field modelling of cracks has gained popularity in the fracturing analysis recently. Originally developed for brittle fracture, the method has now been extended to cohesive fracture. In the latter case, the crack opening displacement is an essential variable, due to the dependence on it of the interface tractions. But also in brittle fracture, the crack opening displacement is crucial in certain applications, for instance for transport of fluids in the cracks. Herein, we derive the complete formulation for mixed-mode crack opening within the framework of the phase-field model, given for brittle as well as for cohesive fracture. The crack opening displacement is associated with a line integral that is perpendicular to the crack. Different factors and matrices apply in the integration for both fracture models and crack directions. These derivations have been validated analytically through an edge-cracked problem and numerically through curved crack scenarios.