Non-ordinary state-based peridynamics model for rock crack propagation: a combined stress-energy fracture method

Non-ordinary state-based peridynamics (NOSB-PD) has been widely applied to rock fracture modeling due to its compatibility with continuum mechanics. However, the classical formulation frequently produces zero-energy modes and nonphysical displacement oscillations as a result of insufficient local deformation compatibility. Moreover, the direct use of classical shear failure criteria often leads to inconsistencies between simulated and experimentally observed shear crack paths. To address these issues, this study employs a bond-level deformation gradient constructed from the averaged deformation gradient at the bond scale, which restores local compatibility and effectively suppresses zero-energy modes. In addition, a triple shear energy criterion is incorporated to distinguish different shear-related mechanical states, thereby improving the physical consistency of shear crack identification and the accuracy of crack path prediction. The improved model is examined through three benchmark numerical examples and simulations of fractured rock. The results demonstrate that it can accurately reproduce crack initiation, propagation, and coalescence under complex loading conditions, while maintaining numerical stability and requiring minimal parameter calibration. Compared with existing NOSB-PD implementations, the approach offers a clearer mathematical structure and enhanced predictive capability, providing a reliable and applicable numerical tool for crack evolution analysis in rock materials.