A generalized higher-order phase-field model for brittle fracture in anisotropic rocks

In this work, we formulate a generalized higher-order phase-field model within the rotated anisotropic framework for simulating brittle phenomena in anisotropic rocks. Our phase-field model accounts for both the anisotropic critical fracture energy release rate and the anisotropic degradation in stiffness. The innovative aspects of this model include (i) a fourth-order structural tensor enabling simulations of strongly anisotropic fractures with arbitrary, non-orthogonal symmetry axes for capturing the complexity of natural geological media; (ii) a volumetric–deviatoric coupling energy density for transitions from anisotropic responses in the undamaged state to isotropic responses in the damaged state; (iii) a patch-based Hessian recovery algorithm ensuring stable solutions of the higher-order PDEs to reduce the computational cost; and (iv) stochastic perturbations integrated into the anisotropic crack surface density function to capture microstructural heterogeneity. Several numerical benchmark examples are provided. The numerical results are compared with some laboratory experiments on brittle fracture in anisotropic rocks.