# Practical Estimates of Tensile Strength and Hoek–Brown Strength Parameter *m* _{i} of Brittle Rocks

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## Abstract

By applying the Griffith stress criterion of brittle failure, one can find that the uniaxial compressive strength (σ_{c}) of rocks is eight times the value of the uniaxial tensile strength (σ_{t}). The Griffith strength ratio is smaller than what is normally measured for rocks, even with the consideration of crack closure. The reason is that Griffith’s theories address only the initiation of failure. Under tensile conditions, the crack propagation is unstable so that the tensile crack propagation stress (σ_{cd})_{t} and the peak tensile strength σ_{t} are almost identical to the tensile crack initiation stress (σ_{ci})_{t}. On the other hand, the crack growth after crack initiation is stable under a predominantly compressive condition. Additional loading is required in compression to bring the stress from the crack initiation stress σ_{ci} to the peak strength σ_{c}. It is proposed to estimate the tensile strength of strong brittle rocks from the strength ratio of \( R = {\frac{{\sigma_{\text{c}} }}{{\left| {\sigma_{\text{t}} } \right|}}} = 8{\frac{{\sigma_{\text{c}} }}{{\sigma_{\text{ci}} }}}. \) The term \( {\frac{{\sigma_{\text{c}} }}{{\sigma_{\text{ci}} }}} \) accounts for the difference of crack growth or propagation in tension and compression in uniaxial compression tests. \( {\frac{{\sigma_{c} }}{{\sigma_{ci} }}} \) depends on rock heterogeneity and is larger for coarse grained rocks than for fine grained rocks. σ_{ci} can be obtained from volumetric strain measurement or acoustic emission (AE) monitoring. With the strength ratio *R* determined, the tensile strength can be indirectly obtained from \( \left| {\sigma_{\text{t}} } \right| = {\frac{{\sigma_{\text{c}} }}{R}} = {\frac{{\sigma_{\text{ci}} }}{8}}. \) It is found that the predicted tensile strengths using this method are in good agreement with test data. Finally, a practical estimate of the Hoek–Brown strength parameter *m* _{i} is presented and a bi-segmental or multi-segmental representation of the Hoek–Brown strength envelope is suggested for some brittle rocks. In this fashion, the rock strength parameters like σ_{t} and *m* _{i}, which require specialty tests such as direct tensile (or Brazilian) and triaxial compression tests for their determination, can be reasonably estimated from uniaxial compression tests.

## Keywords

Griffith’s theory Uniaxial compressive strength Tensile strength Crack initiation Crack propagation/damage AE Volumetric strain## Notes

### Acknowledgments

The author would like to thank Mr. S.J. Kim, Mr. G. Maybee, and Mr. D. Marr for conducting the laboratory tests at the Geomechanics Research Centre, MIRARCO-Mining Innovation, Laurentian University, Canada. The author also would like to thank Drs. D. McCreath and E. Hoek for reviewing the manuscript, and Drs. P. Kaiser and D. Martin for many helpful discussions on brittle failure of rocks.

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