Failure criteria for rocks based on smooth approximations to Mohr–Coulomb and Hoek–Brown failure functions

Existing experimental evidence from true triaxial test (also referred to as poly-axial test) indicates that the influence of the intermediate principal stress (σ2) on rock strength is substantial. In order to consider this effect in rock engineering design, an adequate 3D failure criterion, which takes into account the effect of σ2, is required. Such a criterion should not only be a convex but also a smooth function of stress state; the latter to avoid singularities and to ensure the numerical stability. The most popular failure criteria used in rock mechanics, i.e., Mohr–Coulomb (M–C) and Hoek–Brown (H–B), both ignore the effect of σ2 and also have singular corners. Hence, in this paper, the M–C and H–B criteria are modified to their 3D versions based on geometric approximations of deviatoric sections that lead to smooth and convex surfaces for a wide range of strength parameters. The derived criteria inherit all the key properties of the original ones and are formulated in terms of stress invariants introduced by Nayak and Zienkiewicz. The latter provides an easy way to identify the strength parameters from non-linear multiple regression analysis using the poly-axial test data. In order to validate the proposed criteria, eight sets of poly-axial tests selected from literature are fitted. The results clearly show that the new criteria are quite accurate for all rock types considered. Finally, an extension of the proposed criteria to account for anisotropic rock conditions is discussed and an illustrative example is provided incorporating the functional forms of H–B and modified H–B criteria.