A novel and simple variationally-consistent phase-field cohesive zone model for mixed-mode fracture

In rock -like materials, mode I and II fracture toughness are distinctive while the compressive strength is much greater than the tensile strength. Besides, mixed -mode fractures instead of pure mode I or II fractures occur under complicated load cases in those materials. Therefore, to capture the mixed -mode cracking process, we developed a mixed -mode phase -field cohesive zone model based on the unified phase -field theory proposed by Wu. The model was built based on a strain energy decomposition scheme and the softening behavior of each mode is controlled by an independent degradation function. Thus, independent values of strength and toughness can be given for different failure modes simultaneously. A simple mixture rule of fracture toughness was introduced to describe the transition between different failure modes. The model can keep variational consistency perfectly and governing equations for displacement- and phase -field, which was never reported in other works. Several numerical examples verified the effectiveness and flexibility of this work. The present work showed a simple but effective way to simulate mixed -mode fracture.