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A New Version of the Generalized Zhang–Zhu Strength Criterion and a Discussion on Its Smoothness and Convexity

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Abstract

The generalized Zhang–Zhu (GZZ) strength criterion has a high associated fitting accuracy for true triaxial test data. It makes direct use of the parameters of the Hoek–Brown criterion and has become one of the strength criteria recommended by the International Society for Rock Mechanics (ISRM). In this study, a new version of the GZZ criterion is developed by addressing the problems associated with the original version’s non-smoothness and non-convexity on the π plane. (i) The new criterion is smooth and derivable on the π plane with no numerical singularity in the strength calculation. It thus renders the application of complicated Lode dependence functions for modification of the non-smooth GZZ criterion unnecessary. (ii) The new criterion is non-convex only under the triaxial extension stress condition, which is not critical for those criteria developed from the Hoek–Brown strength criterion. The new criterion is transformed from incomplete convexity to full convexity with an increase in hydrostatic pressure I1. Additionally, there exists a critical transformation radius r0, which only relates to I1. The new criterion achieves full convexity when I1 is high and the quality of the rock mass is poor. (iii) Both the new criterion and the GZZ criterion have high fitting accuracy for rock strength data. Their fitting accuracy is much higher than that of the Pan–Hudson criterion. However, the fitted strength parameter m of the new criterion is closer to Hoek’s recommendation than that of the GZZ criterion, thus proving that the new criterion is applicable and reliable when used for rock strength calculation based on the Hoek–Brown criterion parameter system.

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Abbreviations

σ 1, σ 2, σ 3 :

Maximum, intermediate principal stress and minimum principal stress

σ i e :

Effective principal stress

I 1, I 2, I 3 :

First, second and third stress invariant

J 2, J 3 :

Second and third deviatoric stress invariant

θ σ :

Lode angle

σ c :

Unconfined compressive strength (UCS)

m i :

Material constant for intact rock

m b, s, a :

Strength parameters for rock masses

τ oct :

Octahedral shear stress

σ m , 2 :

Effective mean stress

(σ m , 2)new :

Effective mean stress of new GZZ

(σ m , 2)GZZ :

Effective mean stress of original GZZ

S ij :

The deviatoric stress tensor

ξ :

Hydrostatic stress axis

r(θ σ):

Radius of strength envelope to hydrostatic stress axis

r 0 :

Critical radius of the new criterion from non-full convex to full convex on the π plane

β :

Ratio of r(π/6) to r(− π/6)

mf :

Evaluation index of absolute error (SAE) method

R 2 :

Square of correlation coefficient

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Acknowledgements

The authors would like to acknowledge the financial support. This work was supported by the Special Fund for Basic Research on Scientific Instruments of the National Natural Science Foundation of China [4182780021]; Major Program of National Natural Science Foundation of China [41130751]; Emeishan-Hanyuan Highway Program [SRIG2019GG0004]; China Railway First Survey and Design Institute Group Co., Ltd [Grant no. 19-21-1].

Funding

This work was supported by the Special Fund for Basic Research on Scientific Instruments of the National Natural Science Foundation of China [4182780021]; Major Program of National Natural Science Foundation of China [41130751]; Emeishan-Hanyuan Highway Program [SRIG2019GG0004]; China Railway First Survey and Design Institute Group Co., Ltd [Grant no. 19-21-1].

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Correspondence to Wuqiang Cai.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Cai, W., Zhu, H., Liang, W. et al. A New Version of the Generalized Zhang–Zhu Strength Criterion and a Discussion on Its Smoothness and Convexity. Rock Mech Rock Eng 54, 4265–4281 (2021). https://doi.org/10.1007/s00603-021-02505-z

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  • DOI: https://doi.org/10.1007/s00603-021-02505-z

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