Paul-Mohr-Coulomb failure surface of rock in the brittle regime

The Paul-Mohr-Coulomb failure criterion includes the intermediate principal stress σ_II and friction angles at the limiting stress states of σ_II = σ_III and σ_II = σ_I, where σ_I and σ_III are major and minor principal stresses. Conventional triaxial compression (σ_II = σ_III), extension (σ_II = σ_I), and plane strain (σ_I ≠ σ_II ≠ σ_III) experiments were performed on dry rock. The failure data were plotted in principal stress space, and material parameters were determined in the context of two internal friction angles and the theoretical uniform triaxial (all-around equal) tensile strength. Assuming isotropy, the triaxial compression and extension results were used to construct a six-sided pyramidal failure surface, and the extension friction angle was larger than the compression friction angle, a sufficient but not necessary condition of the intermediate stress effect. To capture the behavior of the rock in multiaxial loading, the Paul-Mohr-Coulomb criterion was extended to form a 12-sided pyramid with best fit planes.