Phase field method of multi-mode fracture propagation in transversely isotropic brittle rock

A phase field model (PFM) for simulating multi-mode brittle fractures in transversely isotropic rock is proposed. In the model, we employ a novel decomposition form of strain energy consisting of tensile, tensile–shear and compressive–shear components to improve the fracture driving force, and the positive–negative decomposition forms of elastic energy are considered as an approximate extension of the spectral decomposition method for transversely isotropic materials. We also introduce the tensor form of the critical energy release rate and Lamé constants, which breaks through the limitations of traditional PFMs that cannot well reproduce fracture anisotropy and multi-mode brittle fracture in transversely isotropic rock, i.e. the tensile, shear, and mixed-mode fractures. A new mixed phase field model is established, which is solved in COMSOL by employing a staggered scheme to solve the phase field, displacement field, and historical field independently. Subsequently, the accuracy and effectiveness of the proposed model are comprehensively verified by a series of 2D and 3D numerical examples. The results show that the proposed model agrees well with the existing model in capturing tensile and tensile–shear fractures. Compressive–shear fractures can be additionally captured well in the compressive state.