Voronoi Polygon–Based Crack Density Function for Evaluating Mesoscopic Damage in Shale under Biaxial Compression

The accurate evaluation of mesoscopic damage in granular materials, particularly shale, is crucial for understanding their mechanical behavior and ensuring the stability and safety of engineering structures. However, existing methods for quantifying damage in granular materials often have limitations in accurately assessing the degree of mesoscopic damage. This study proposes a new crack density function based on Voronoi polygons to assess mesoscopic damage in granular materials, with a focus on shale under biaxial compression. Using PFC2D numerical simulations, we systematically investigate the effects of bedding angle and confining pressure on the mesoscopic damage evolution, stress-strain relationships, and crack density-strain relationships. The results reveal four distinct stages of crack density evolution during biaxial compression: crack initiation, propagation, penetration, and equilibrium. Notably, the peak stress occurs after the onset of crack penetration. We find that a crack density of approximately 4% indicates the critical state of the specimen, beyond which failure ensues rapidly. Additionally, increasing confining pressure inhibits crack growth, prolongs the time to reach the equilibrium stage, and increases the crack density in the stable state. This study offers valuable insights into the crack density growth process and its influencing factors in shale brittle materials. The proposed crack density function based on Voronoi polygons provides a novel tool for evaluating mesoscopic damage in other granular materials, thereby enhancing the theoretical foundations for the application of shalelike brittle materials and contributing to engineering safety.