Phase field modeling of mixed-mode crack in rocks incorporating heterogeneity and frictional damage.

• A crack growth model with mixed-mode and frictional damage is implemented into the phase field method to comprehensively investigate crack nucleation, growth, and interaction in heterogeneous rock. • A gradual shift in wing crack initiation from the middle to the end tip of the pre-existing macrocrack, accompanied by a systematic decrease in wing crack length as an increase in macrocrack inclination angle. • Significant increase in the peak strength and Young's modulus of macrocracked samples with increasing macrocrack inclination and homogeneity index. Heterogeneous rocks often exhibit a complex failure pattern of distributed tensile and shear cracks (mixed-mode cracks). In the present paper, we propose a modified phase field modeling (PFM) approach to simulate mixed-mode cracking in rocks, taking into account the effects of material heterogeneity and frictional damage. The proposed model combines the phase field method with a heterogeneous material representation and incorporates a frictional damage to capture the complex behavior of cracks in rocks. A variational formulation is derived based on the Mohr-Coulomb criterion to describe the evolution of the crack phase field. The proposed PFM-based mixed-mode crack growth model is validated against laboratory uniaxial compressive strength (UCS) tests on sandstone samples that contain a single macroscopic crack using digital image correlation (DIC). We performed a systematic characterization of the cracking and failure process as a function of macrocrack inclination angle, and found that the simulations well match the laboratory results. The results show that the length of wing cracks gradually decreases, and the nature of the damage gradually changes from tensile to shear as a function of macrocrack inclination angle. Based on this validation, we further developed a PFM model that considers multiple macrocracks. In these multi-crack simulations, we found that the crack initiation stress and peak stress decrease and then increase as the ligament angle increases. The proposed PFM-based mixed-mode crack growth model allows for the transition from single to multi-crack simulations and enables a more accurate assessment of the cracking evolution in rock. [ABSTRACT FROM AUTHOR]