Linear failure criteria with three principal stresses

Any failure criterion can be represented as a surface in principal stress space σ₁, σ₂, σ₃ (with no order implied), and the shape of the surface depends on the functional form of the criterion. For isotropic rock that exhibits a pressure dependence on strength, the simplest failure criterion is a linear function, and the failure surface is a hexagonal pyramid with a common vertex V_o on the tension side of the hydrostatic axis, where V_o=(theoretical) uniform triaxial tensile strength. An example of a pyramidal failure surface is the popular Mohr-Coulomb criterion, which is independent of the intermediate principal stress σ_II (σ_I≥σ_II≥σ_III) and contains two material parameters, such as V_o and the internal friction angle φ. The Paul-Mohr-Coulomb failure criterion Aσ_I+Bσ_II+Cσ_III=1 is linear with three principal stresses, and it is formulated with three identifiable material constants, where A=(1-sinφ_c)/(2V_osinφ_c), B=(sinφ_c-sinφ_e)/(2V_osinφ_csinφ_e), C=-(1+sinφ_e)/(2V_osinφ_e) and φ_c, φ_e are internal friction angles for compression (σ_II=σ_III) and extension (σ_I=σ_II). The convex nature of the failure surface at constant mean stress can be approximated by additional planes with appropriate material parameters. To demonstrate the utility of the linear failure criterion, a series of conventional triaxial compression and extension experiments were performed on an isotropic rock. The results were processed using the developed data fitting techniques, and the material parameters for the six-sided pyramidal failure surface were determined. A multi-axial experiment was also performed to evaluate the convexity of the failure surface, and a twelve-sided pyramid was constructed and the appropriate equations were derived.