A Simple Three-dimensional Failure Criterion for Rocks Based on the Hoek–Brown Criterion

Abstract

A simple three-dimensional (3D) failure criterion for rocks is proposed in this study. This new failure criterion inherits all the features of the Hoek–Brown (HB) criterion in characterizing the rock strength in triaxial compression, and also accounts for the influence of the intermediate principal stress. The failure envelope surface has non-circular convex sections in the deviatoric stress plane, and is smooth, except in triaxial compression. In particular, the failure function of the proposed criterion has a similar simple expression as that of the HB criterion in terms of the principal stresses. The material parameters can be calibrated from tests in conventional triaxial compression, and predictions using this new criterion generally compare well with polyaxial testing data for a variety of rocks. Comparison of the new 3D failure criterion and two existing criteria demonstrates that the new failure criterion performs better in characterizing the rock strength. A unified expression for the 3D failure criteria is further provided, retaining the features of the classical criteria and recovering several existing ones as specific cases.

Appendix: Unified Expression of the 3D Failure Criterion for Intact Rocks

For intact rocks, a = 0.5, the unified expression given by Eq. (21) can be written as

$$3\frac{{J^{\prime}_{2} }}{{\sigma_{\text{ci}}^{2} }} + \frac{{m_{b} }}{\sqrt 3 }A\left( \theta \right)\frac{{\sqrt {J^{\prime}_{2} } }}{{\sigma_{\text{ci}} }} - s - \frac{{m_{b} I_{1}^{'} }}{{3\sigma_{\text{ci}} }} = 0$$

(43)

Then $\sqrt {J^{\prime}_{2} } /\sigma_{\text{ci}}$ can be explicitly derived from Eq. (43)

$$\frac{{\sqrt {J^{\prime}_{2} } }}{{\sigma_{\text{ci}} }} = \frac{{ - b + \sqrt {b^{2} - 4ac} }}{2a} = \frac{1}{6}\left[ { - \frac{{m_{b} }}{\sqrt 3 }A\left( \theta \right) + \sqrt {\frac{{m_{b}^{2} }}{3}A^{2} \left( \theta \right) + 12\left( {m_{b} \frac{{\sigma^{\prime}_{m} }}{{\sigma_{\text{ci}} }} + s} \right)} } \right]$$

(44)

Or in terms of Haigh–Westergaard coordinates

$$\frac{\rho }{{\sqrt 2 \sigma_{\text{ci}} }} = \frac{{ - b + \sqrt {b^{2} - 4ac} }}{2a} = \frac{1}{6}\left[ { - \frac{{m_{b} }}{\sqrt 3 }A\left( \theta \right) + \sqrt {\frac{{m_{b}^{2} }}{3}A^{2} \left( \theta \right) + 12\left( {\frac{{\xi m_{b} }}{{\sqrt 3 \sigma_{\text{ci}} }} + s} \right)} } \right]$$

(45)

Finally, we obtain the following invariant representation for the unified expression

$$\rho = \frac{{\sqrt 2 \sigma_{ci} m_{b} }}{6}\left[ { - \frac{A\left( \theta \right)}{\sqrt 3 } + \sqrt {\frac{{A^{2} \left( \theta \right)}}{3} + 12\left( {\frac{\xi }{{\sqrt 3 m_{b} \sigma_{ci} }} + \frac{s}{{m_{b}^{2} }}} \right)} } \right]$$

(46)

The invariant representation for the new failure criterion (see Eq. (29)) is given by

$$\rho = \sqrt 2 \sigma_{\text{ci}} m_{b} \left[ { - \frac{\sqrt 3 }{9}\cos \left( {\frac{\pi }{3} - \theta } \right) + \sqrt {\frac{1}{27}\cos^{2} \left( {\frac{\pi }{3} - \theta } \right) + \frac{1}{3}\left( {\frac{s}{{m_{b}^{2} }} + \frac{\xi }{{\sqrt 3 \sigma_{\text{ci}} m_{b} }}} \right)} } \right]$$

(47)

Rearranging Eq. (47) gives

$$\rho = \frac{{\sqrt 2 \sigma_{\text{ci}} m_{b} }}{6}\left[ { - \frac{2}{\sqrt 3 }\cos \left( {\frac{\pi }{3} - \theta } \right) + \sqrt {\frac{4}{3}\cos^{2} \left( {\frac{\pi }{3} - \theta } \right) + 12\left( {\frac{s}{{m_{b}^{2} }} + \frac{\xi }{{\sqrt 3 \sigma_{\text{ci}} m_{b} }}} \right)} } \right]$$

(48)

which can be also derived from Eq. (46) by setting A(θ) = 2cos(π/3−θ).