A theoretical derivation of the Hoek–Brown failure criterion for rock materials

This study uses a three-dimensional crack model to theoretically derive the Hoek–Brown rock failure criterion based on the linear elastic fracture theory. Specifically, we argue that a failure characteristic factor needs to exceed a critical value when macro-failure occurs. This factor is a product of the micro-failure orientation angle (characterizing the density and orientation of damaged micro-cracks) and the changing rate of the angle with respect to the major principal stress (characterizing the microscopic stability of damaged cracks). We further demonstrate that the factor mathematically leads to the empirical Hoek–Brown rock failure criterion. Thus, the proposed factor is able to successfully relate the evolution of microscopic damaged crack characteristics to macro-failure. Based on this theoretical development, we also propose a quantitative relationship between the brittle–ductile transition point and confining pressure, which is consistent with experimental observations.