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 Vo on the tension side of the hydrostatic axis, where Vo=(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 σII (σI≥σII≥σIII) and contains two material parameters, such as Vo and the internal friction angle ϕ.The Paul–Mohr–Coulomb failure criterion AσI+BσII+CσIII=1 is linear with three principal stresses, and it is formulated with three identifiable material constants, where A=(1−sinϕc)/(2Vosinϕc), B=(sinϕc−sinϕe)/(2Vosinϕcsinϕe), C=−(1+sinϕe)/(2Vosinϕe) and ϕc, ϕe are internal friction angles for compression (σII=σIII) and extension (σI=σII). 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.

► A linear failure surface is formulated with three principal stresses and three material parameters.
► The failure surface is a hexagonal pyramid with a common vertex.
► The material parameters are friction angles for compression and extension, and the vertex.
► The intermediate stress effect is included.
► The convexity of the failure surface on the π-plane can be approximated by a twelve-sided pyramid.