To consider the effect of intermediate principal stress on rock failure, the generalized Hoek-Brown criterion was modified by incorporating a new term to quantitatively account for the nonlinearity of rocks in high stress states. The failure envelope of the modified generalized as the Hoek-Brown criterion in the principal stress space is a curved hexagonal pyramid which touches the three outer apices of the original Hoek-Brown criterion. In the (σ1 – σ3) – (σ3 + nσ2) plane, it has a nonlinear form of powerlaw curve. The comparison results with three failure criteria based on multi-axial test strength data of four rocks indicate that the new three-dimensional Hoek-Brown criterion achieved the least misfit to most of the test data, followed by the Mogi (1971) criterion, whereas the Hoek-Brown and the Drucker-Prager criterion have a poor performance compared to the former two criteria. The modified Hoek-Brown criterion is proved to be a good candidate for prediction multi-axia strength of rocks.


The famous Hoek-Brown strength criterion was developed specially for geomaterials by trial error and based on a wide range of triaxial test data. It has been applied successfully to a wide range of intact and fractured rock types during the past three decades. While, the Hoek-Brown criterion was originally proposed without considering the influence of the intermediate principal stress and could not predict the true triaxial strength of rocks very well. As a matter of fact, much more evidence has proved that the intermediate principal stress does affect the rock strength in many instances (Mogi 1976, 1971, 1972). Pan & Hudson (1988) attempted to extend the two-dimensional Hoek-Brown criterion to a three-dimensional version by establishing the relationship between the octahedral shear stress and the first stress invariant. To take into account the effect of intermediate principal stress, Singh et al. (1998) suggested a new strength criterion by replacing σ2 with average confining pressure for highly anisotropic rock material and jointed rock masses. Zhang and Zhu proposed a three-dimensional Hoek-Brown criterion for rock mass by combining the general Mogi criterion and the original Hoek-Brown criterion (Zhang and Zhu 2007, Zhang 2008). The Zhang Zhu criterion was then developed into a generalized version and modified utilizing three different Lode dependences to solve the problems with some stress paths (Zhang et al. 2013).

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