Random phase field method for quasi-static and dynamic fracture propagation: Strict phase field equations based on variational principle

Abstract

Spatial variability in mechanical properties significantly affects fracture evolution in materials. A strict random phase field model suitable for spatially heterogeneous materials is developed, in which a gradient term of the critical energy release rate is introduced. Compared with the simple integration of traditional finite element models with random fields, which involves merely replacing deterministic mechanical properties in the governing equations, a more accurate coupling approach is adopted by incorporating a coordinate-dependent critical energy release rate into the energy functional, which is reformulated from existing models. The governing equation of the phase field is derived through variational principles. Unlike traditional random phase field models, the proposed model captures the spatial variation in the gradient of the critical energy release rate and effectively characterizes the directional rate of change in fracture toughness. The proposed model is implemented by using COMSOL Multiphysics and MATLAB, and validated through the rock fracture experiment. Quasi-static and dynamic fracture simulations reveal that in the presence of pronounced spatial heterogeneity, incorporating the critical energy release rate gradient can significantly alters fracture behavior in heterogeneous materials, often producing effects not captured by traditional random phase field models. Therefore, the rigorous random phase field model is indispensable for understanding and predicting the fracture behavior of spatially heterogeneous materials such as rocks.