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Modified Hoek–Brown failure criterion for anisotropic intact rock under high confining pressures

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Abstract

The extensively used modified anisotropic Hoek–Brown (H-B) failure criterion often overestimates the rock strength under a confining pressure (σ3) exceeding the uniaxial compressive strength (UCS) of rock. In this study, to rectify this problem, the error term (B1σ32) is added to the modified anisotropic H-B strength criterion. The coefficient (B1) in the error term is determined by introducing the anisotropic parameter (kα), and the rock critical confining pressure is adopted for the separation of brittle and ductile rock behaviors. The relationship between the anisotropic parameter (kα) and joint surface dip angle (α) is also established. A new failure criterion that considers rock anisotropy for predicting the rock strength under a high confining pressure (M-HB-HC) is proposed herein. Analyses reveal that the critical coefficient (n) is dependent of the angle (α) and parameter (kα). The strength predictions for different types of rocks using four failure criteria were compared under four confining pressure conditions. The comparison shows that the M-HB-HC failure criterion has better accuracy than other failure criteria; it can describe the transformation from brittle to ductile behaviors of rock as confining pressures increase. Furthermore, we used discrete element numerical software (3DEC) to conduct nine groups of numerical simulations of triaxial confining pressure tests to verify the accuracy of the criterion. The results of the numerical simulation are in good agreement with the experimental data. The M-HB-HC failure criterion is more suitable for rocks with one set of joint surfaces.

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Data availability

All data, models, or codes generated or used during the study are available from the corresponding author upon request.

Abbreviations

β (°):

Angle between the loading direction and plane of anisotropy

β m (°):

Angle describing minimum uniaxial compressive strength

α (°):

Joint surface dip angle

α β :

Reduction factor of strength associated with rock anisotropy

k β :

Anisotropic parameter considering angle β

k α :

Anisotropic parameter considering angle α

σ ci (MPa):

Uniaxial compressive strength

σ ci ,β (MPa):

UCS, which depends on angle β

σ ci ,α (MPa):

UCS, which is dependent on the joint surface dip angle α

σ 1 j (MPa):

Maximum principal stress limit value of joint surface friction force when β is equal to \(\frac{\pi }{4}-\frac{{\varphi }_{j}}{2}\)

σ ccp (MPa):

Critical confining pressure

σ ci ,αmax (MPa):

Maximum uniaxial compressive strength considering joint surface dip angle α

σ ci ,αmin (MPa):

Minimum uniaxial compressive strength considering joint surface dip angle α

σ 1 (MPa):

Major principal stress

σ 1-σ 3 (MPa):

Deviatoric stress

σ 3 (MPa):

Minor principal stress

σ cr :

Critical confining pressure in Zhang’s new I-HB-MW failure criterion

c j :

Cohesive force on the joint surface

μ j :

Friction coefficient of the joint surface

φ j (°):

Internal friction angle on the joint surface

B 1 σ 3 2 :

Error term

B 1 :

Coefficient in the error term and a constant related to rock type

ω :

Relative error (%)

n :

Critical coefficient of the rock

s :

Rock mass material constant of Hoek–Brown failure criterion

s an :

Rock mass material constant of Hoek–Brown failure criterion

s θ :

Rock mass material constants in Lee’s failure criterion

m i :

Dimensionless empirical parameter of Hoek–Brown failure criterion

m p :

Dimensionless empirical parameter with angle β of Saroglou’s modified Hoek–Brown failure criterion

m b,an :

Reduced value of mi parameter for anisotropic rock mass

m θ :

Rock mass material constant in Lee’s failure criterion

ζ :

Rock mass material constant of Hoek–Brown failure criterion

ζ m :

Value of angle β at minimum mp

ζ s :

Value of angle β at the minimum s

r :

Strength reduction factor defined by Saeidi et al. (2013)

A :

Parameter related to the rock shear shrinkage failure

R c :

Index of anisotropic rocks strength degree

a m 1;2 :

Coefficient in Lee’s failure criterion

a s 1;2 :

Coefficient in Lee’s failure criterion

Ωs 0 :

Second-order tensors in Lee’s failure criterion

Ωm 0 :

Second-order tensors in Lee’s failure criterion

a,b :

Experiment constants in Chen’s deviatoric stress–strain condition

d :

Exponent in the error term (B1σ3d)

θ :

Exponent in the expression of mp

ξ :

Exponent in the expression of s

A2, A 3 :

Material constants in the expression of θ

p2, p 3 :

Material constants in the expression of ξ

Q, W :

Constants in the expression of σci,β

Q 1, Q 2, W 1, and W 2 :

Constantsin McLamore and Gray’sexpression

t, o :

Positive integers exponentsin McLamore and Gray’sexpression

β t :

Angle corresponding to the minimum uniaxial compressive strength

τ :

Shear stress

c :

m p k α

d :

c+(c2+54)0.5

f :

4d(cd+27)0.5/(27)0.5σci,α

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Funding

This study was supported by the National Natural Science Foundation of China (Nos. 51774018, 12172036), BUCEA Doctor Graduate Scientific Research Ability Improvement Project (Nos. DG2021003), BUCEA Post-Graduate Innovation Project (Nos. PG2021007), and Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT,IRT_17R06).

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Authors and Affiliations

  1. Beijing International Cooperation Base for Transportation Infrastructure Construction, Beijing Advanced Innovation Center for Future Urban Design, Beijing University of Civil Engineering and Architecture, Beijing, 100044, China

    Zefan Wang, Chengzhi Qi, Liren Ban, Hang Yu & Haonan Wang

  2. School of Chinese Language and Literature, Beijing Foreign Studies University, Beijing, 100089, China

    Ziming Fu

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  1. Zefan Wang
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  2. Chengzhi Qi
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  4. Hang Yu
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Correspondence to Chengzhi Qi.

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Wang, Z., Qi, C., Ban, L. et al. Modified Hoek–Brown failure criterion for anisotropic intact rock under high confining pressures. Bull Eng Geol Environ 81, 333 (2022). https://doi.org/10.1007/s10064-022-02831-8

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