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A statistical damage constitutive model for rock based on modified Mohr–Coulomb strength criterion

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Abstract

The study on strength criterion and constitutive relationship for rock in the deep strata is an essential and paramount requirement for the analysis and design of deep underground engineering. Traditional strength criteria were generally proposed based on rock mechanics tests under low confining pressure and overestimated the strength of rock under high pressure. This is a significant safety risk for the construction of deep underground projects. Hence, a modified Mohr–Coulomb strength criterion was proposed by employing the critical state concept for rock and rock critical state coefficient. Under the assumption that the meso-element strength of rock follows the two-parameter Weibull distribution and meso-element failure follows the modified Mohr–Coulomb strength criterion, a new statistical damage constitutive model for rock in the deep strata was proposed. The modified Mohr–Coulomb strength criterion and the statistical damage constitutive model were verified with the results from the conventional triaxial compression test of quartz sandstone. The results show that the proposed damage constitutive model can well grasp the nonlinear strength properties and simulate the stress–strain relationships under different confining pressures, and its parameters have definite physical meaning. Moreover, the parameters of the proposed model, obtained from triaxial strength tests performed at low confining pressure (the confining pressure is less than 20 MPa), can be directly used to accurately predict the triaxial strength and the stress–strain relationships of the tested red sandstone in this study at higher confining pressure.

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Abbreviations

M-C:

Mohr–Coulomb strength criterion

D-P:

Drucker–Prager strength criterion

H-B:

Hoek–Brown strength criterion

τ ψ :

Shear stress on rock failure plane (MPa)

σ ψ :

Normal stress on rock failure plane (MPa)

ψ :

Rock fracture angle (°)

σ 1, σ 3 :

Major and minor principle stress (MPa)

β :

The inclination angle of the tangent line of the strength envelope (°)

p :

The extreme value of deviatoric stress (MPa)

σ rc :

Critical effective confining pressure (MPa)

n :

The critical state coefficient

c :

Cohesion of rock material (MPa)

φ :

Internal friction angle of rock material (°)

M :

The intermediate variable

c 0 :

Cohesion of modified M-C strength criterion (MPa)

φ 0 :

Internal friction angle of modified M-C strength criterion (°)

σ c :

Uniaxial compressive strength of rock (MPa)

(σ 1cσ 3c):

Axial deviatoric stress on critical failure state (MPa)

\({\sigma}_i^{\prime }\) :

Effective stress (MPa)

D :

Damage variable

A, A 1, A 2 :

Total action area; undamaged portion; damaged portion

\({\sigma}_1^{\mathrm{r}}\) :

Axial residual stress (MPa)

E :

Elastic modulus (MPa)

μ :

Poisson’s ratio

F :

Mesoscopic element strength

m, F 0 :

The distribution parameters

σ p :

The peak stress on the theoretical curves (σ1ε1 curves) (MPa)

ε p :

The peak point on the theoretical curves (σ1ε1 curves)

(σ 1σ 3)dp :

The peak deviatoric stress on test curves (MPa)

ε dp :

The peak point strain on test curves

(σ 1σ 3)r :

The residual stress on test curves (MPa)

ε c :

The initial strain of the rock under hydrostatic pressure

d :

The square sum of the fitting deviations

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Acknowledgements

This study was financially supported by “The National Natural Science Foundation of China (No. 41772308)”. We would like to express our sincere appreciation to the anonymous reviewers for many valuable suggestions.

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Correspondence to Yawu Zeng.

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Responsible Editor: Zeynal Abiddin Erguler

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Peng, Z., Zeng, Y., Ye, Y. et al. A statistical damage constitutive model for rock based on modified Mohr–Coulomb strength criterion. Arab J Geosci 14, 2679 (2021). https://doi.org/10.1007/s12517-021-08862-x

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