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An empirical relation for parameter mi in the Hoek–Brown criterion of anisotropic intact rocks with consideration of the minor principal stress and stress-to-weak-plane angle

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Abstract

The parameter mi accounts for the anisotropy of rock strength, and the accurate determination of mi is a primary requirement of the Hoek–Brown (H–B) strength criterion. In this study, a relation for mi as a bivariate, second-order, polynomial function of the minor principal stress (σ3) and the angle (β) between the major principal stress and weak plane is developed. First, the possible contribution of σ3 and β to mi is systematically investigated based on substantial uniaxial and triaxial compression test data of various rock types collected from the available literature. This investigation allows mi to be described in terms of σ3 and β. A self-defined fitting function is adopted to formulate the functional relationship among these metrics. Using this relation, the H–B strength criterion is modified, and its performance is evaluated in two examples. The examples illustrate the improved ability of the modified criterion to determine the strength of metamorphic and sedimentary rocks. The applicability of this criterion to additional rock types warrants further study.

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Abbreviations

σ 1 :

Major principal stress at failure

σ 1_average :

Average value of the peak strength at rock failure for all rock types

σ 1i_observed :

Observed peak strength at rock failure for a given minor principal stress at the ith data point

σ 1i_estimated :

Estimated peak strength at rock failure for a given minor principal stress at the ith data point

σ 3 :

Minor principal stresses at failure

σ ci :

Uniaxial compressive strength

σ cii :

Crack initiation stress

σ cβ :

Uniaxial compressive strength at angle β

σ t :

Uniaxial tensile strength

β :

Stress-to-weak-plane angle from the major principal stress direction

ϕ :

Internal friction angle

γ :

Empirical parameter

ξ m :

Value of β when mb is minimized

a :

Rock-type-dependent constant

A 1 :

Coefficient, which reflects the initial value of mi(β) under special conditions

A 2 :

Coefficient, which reflects the initial value of mi(β) when β is 0°

A 3 :

Coefficient, which reflects the variation rate with β

AAREP:

Average absolute relative error percentage

b :

Rock-type-dependent constant

B 1 :

Regression coefficient

C :

Σ3-Dependent constant

COA:

Coefficient of accordance

D :

Damage factor

GSI:

Geological strength index

k :

Slope of the curve between σ1 and σ3

k β :

β-Dependent anisotropic parameter

m b :

Rock mass material constant

m i :

Material parameter

m i(90) :

Value of mi(β) when β is 90°

n :

Number of repeated triaxial compression tests

N :

Number of datasets for all rock types

P 1 :

Coefficient, which reflects the initial value of mi(β) under special conditions

P 2 :

Value of β when mi(β) is minimized

P 3 :

Coefficient, which reflects the initial value of mi(β) when β is 0°

P 4 :

Coefficient, which reflects the initial decreasing rate of mi(β) under conditions of relatively high β

PE:

Percent error for each data point

s :

Rock mass material constant

t 1 :

Coefficient, which represents the increasing rate of mi(β) with the square of the orientation angle β

t 2 :

Coefficient, which represents the increasing rate of mi(β) with the orientation angle β

t 3 :

Coefficient, which represents the increasing rate of mi(β) with the confining pressure σ3

t 4 :

Coefficient, which represents the initial value when σ3 is 0 and β is 0°

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Acknowledgements

The work was funded by the National Key Research and Development Program of China (2017YFC1501305) and the China Postdoctoral Science Foundation (2018M642799), Science and Technology Research Project of Hubei Education Department (D2019038), Open Foundation of Top Disciplines in Yangtze University.

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Authors and Affiliations

  1. School of Geosciences, Yangtze University, Wuhan, 430100, Hubei, China

    Tao Wen

  2. Three Gorges Research Center for Geohazards, Ministry of Education, China University of Geosciences, Wuhan, 430074, Hubei, China

    Huiming Tang & Lei Huang

  3. Faculty of Engineering, China University of Geosciences, Wuhan, 430074, Hubei, China

    Huiming Tang, Asif Hamza & Yankun Wang

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  1. Tao Wen
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Contributions

Tao Wen contributed to investigation, methodology, project administration, validation, visualization, and writing of original draft. Huiming Tang contributed to conceptualization, data curation, funding acquisition, and supervision. Lei Huang contributed to data curation, supervision, review, and editing. Asif Hamza contributed to review and editing. Yankun Wang helped in conceptualization and supervision.

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Correspondence to Huiming Tang or Lei Huang.

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Appendix: Relation between k and ϕ

Appendix: Relation between k and ϕ

The slope k of the curve of σ1 versus σ3 can be obtained by the fitting curve and is expressed as

$$ k = \frac{{\partial \sigma_{1} }}{{\partial \sigma_{3} }} = 1 + \frac{{m_{i} }}{{\left( {m_{i} \frac{{\sigma_{3} }}{{\sigma_{ci} }} + 1} \right)^{0.5} }} $$
(23)

Conventionally, assuming that the cohesion and ϕ of the rock are constant during compression, the M–C criterion is regarded as a linear strength criterion and can be used to characterize the relationship between σ1 and σ3. The ϕ value of the rock is not constant during triaxial compression but gradually changes with σ3 [45]. Under a low minor principal stress, the rock exhibits mainly dilatancy and brittleness, and the microcracks inside the rock coalesce, fragmenting the rock, which leads to an increase in the volume [34, 35, 51, 52]. This phenomenon indicates that the rock has a relatively large ϕ. Under a high minor principal stress, the rock exhibits mainly ductility, and both microcrack opening and shear dilatancy are restricted by σ3, indicating that the rock has a relatively small ϕ [34, 35, 51, 52]. As σ3 increases, the rock shows ductile characteristics. With a further increase in σ3, the rock reaches a critical state, and the friction angle of the rock is basically zero.

The conventional linear M–C criterion for anisotropic intact rocks can be written as

$$ \sigma_{1} = \frac{1 + \sin \phi }{1 - \sin \phi }\sigma_{3} + \sigma_{c\beta } $$
(24)

where β is the stress-to-weak-plane angle observed from the major principal stress direction and σcβ is the uniaxial compressive strength at angle β.

According to the M–C criterion, the slope k of the curve between σ1 and σ3 can also be obtained by deriving Eq. (24) about σ3:

$$ k = \frac{1 + \sin \phi }{1 - \sin \phi } $$
(25)

Equation (25) can be used to solve for ϕ.

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Wen, T., Tang, H., Huang, L. et al. An empirical relation for parameter mi in the Hoek–Brown criterion of anisotropic intact rocks with consideration of the minor principal stress and stress-to-weak-plane angle. Acta Geotech. 16, 551–567 (2021). https://doi.org/10.1007/s11440-020-01039-y

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