Random phase field model for simulating mixed fracture modes in spatially variable rocks under impact loading

A novel rigorous random phase field model, capable of simulating mixed fracture modes in spatially heterogeneous rocks under impact loading, is proposed. By treating the critical energy release rate as a function dependent on spatial coordinates, a new energy functional is constructed. A governing equation distinct from that used in phase field models addressing dynamic fractures of homogeneous materials is derived using the variational principle. The driving force is then improved by introducing a compressive-shear driving force that incorporates the friction angle and cohesive strength, as well as tensile and tensile-shear driving forces that include the Lamé constants. The new governing equations can account for variations in the spatial distribution of the critical energy release rate and its gradient. The spatial distribution of fracture parameter is implemented by using the random field theory. The proposed model is validated using three numerical experiments, with the results demonstrating strong agreement with the corresponding experiments. The results indicate that the proposed model effectively simulates the mixed fracture modes observed in the specimens under impact loading more realistically compared with the homogeneous model. Compared with the conventional method that couple random fields and phase fields, the new model can well account for the spatial variability in the gradient of the critical energy release rate. The spatial variability of the critical energy release rate gradient significantly influences mixed fracture modes and strength.