Phase-field model for 2D cohesive-frictional shear fracture: An energetic formulation

This paper presents a phase -field model for cohesive-frictional shear fracture. The model is derived based on two energetic principles: the energy conservation law and a variational inequality of virtual work that serves as a stability condition. We show that all the governing equations for frictional fracture can be obtained from the above two principles, including the equilibrium condition, the phase -field evolution law and, most importantly, the yield function and flow rule for the plasticity-like frictional slip. The information of crack direction is naturally included in the flow rule. Theoretical and numerical results show that the proposed model is faithfully consistent with the Mohr-Coulomb theory of frictional failure in terms of the crack nucleation stress and direction. In addition, the presented phase -field model has an explicit frictional cohesive law.