Any failure criterion can be represented as a surface in principal stress space, and the shape of the surface depends on the functional form of the criterion. For isotropic materials that exhibit 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. An example of a pyramidal failure surface for rock is the popular Mohr-Coulomb criterion, which is independent of the intermediate principal stress and thus contains two material parameters.
A linear failure criteria in three principal stress is formulated with three material constants: internal friction angles for (i) compression c and (ii) extension e, and (iii) a common vertex Vo. Nonlinearity on the failure surface can be approximated by additional planes with appropriate material parameters (i), (ii), and (iii). To demonstrate the utility of the linear failure criterion, a series of conventional triaxial compression (II = III) and extension (I = II) 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. Multi-axial (I ≠ II ≠ III) experiments were also performed to evaluate the nonlinearity, and a twelve-sided pyramid was constructed and the appropriate equations were derived.