A two-scale model for frictional cracks in 3D fractured brittle media with the extended finite element method

Stress concentration induced by the heterogeneity in brittle geomaterials is generally considered as the driving force in the evolution of the microstructure (such as the crack and pore microstructure). Specifically, modeling heterogeneity is key to properly predicting the nucleation, coalescence and propagation of micro-cracks in brittle solids. In this paper, we propose a two-scale model for frictional cracks in fractured brittle media. The major crack in the study domain is modeled at a macro level, while the micro-cracks are modeled at a finer scale. The macro-scale behavior is described by a standard boundary value problem. The finer-scale problem is modeled using the notion of representative elementary volume (REV) consisting of a solid volume with distributed micro-cracks. Periodic boundary condition and small strain formulation are assumed in the finer-scale analysis. The scale bridging mechanism is borrowed from the standard homogenization technique. The proposed model is implemented with the extended finite element method. The macro stress at each Gauss point in the finite element formulation is computed as the volume average of finer-scale stresses in each corresponding REV. The macro tangent operator is computed using a perturbation method. For 3D problems, six independent linear perturbation analyses are carried out for each numerical integration point. Our numerical examples capture the nucleation and coalescence of micro-cracks, which can be used to infer the potential propagation direction of the major crack.