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An Elasto-Plastic Damage Model for Rocks Based on a New Nonlinear Strength Criterion

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Abstract

The strength and deformation characteristics of rocks are the most important mechanical properties for rock engineering constructions. A new nonlinear strength criterion is developed for rocks by combining the Hoek–Brown (HB) criterion and the nonlinear unified strength criterion (NUSC). The proposed criterion takes account of the intermediate principal stress effect against HB criterion, as well as being nonlinear in the meridian plane against NUSC. Only three parameters are required to be determined by experiments, including the two HB parameters σc and m i . The failure surface of the proposed criterion is continuous, smooth and convex. The proposed criterion fits the true triaxial test data well and performs better than the other three existing criteria. Then, by introducing the Geological Strength Index, the proposed criterion is extended to rock masses and predicts the test data well. Finally, based on the proposed criterion, a triaxial elasto-plastic damage model for intact rock is developed. The plastic part is based on the effective stress, whose yield function is developed by the proposed criterion. For the damage part, the evolution function is assumed to have an exponential form. The performance of the constitutive model shows good agreement with the results of experimental tests.

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Abbreviations

m i :

Strength parameters of the original HB criterion

m b , s, a :

Strength parameters of the generalized HB criterion

σ c :

Uniaxial compression strength of intact rock

σ t :

Uniaxial tensile strength of intact rock

GSI:

Geological Strength Index of rock masses

D :

Disturbance of rock masses

\(\sigma_{ij}\) :

Stress tensor in normal stress space

\(\bar{\sigma }_{ij}\) :

Stress tensor in transformed stress space

\(\sigma_{i} \;(i = 1,2,3)\) :

Principal stress at failure in normal stress space

\(\sigma_{{ 1 {\text{c}}}}\), \(\sigma_{{ 3 {\text{c}}}}\) :

The first and third principal stresses under triaxial compression condition

\(\sigma_{{ 1 {\text{t}}}}\), \(\sigma_{{ 3 {\text{t}}}}\) :

The first and third principal stresses under triaxial extension condition

\(\hat{\sigma }_{i} \;(i = 1,2,3)\) :

Principal stress at failure in characteristic stress space

p, q :

Mean stress and generalized shear stress in normal stress space

p c, q c :

Mean stress and generalized shear stress under triaxial compression condition

p t, q t :

Mean stress and generalized shear stress under triaxial extension condition

\(\hat{p}\), \(\hat{q}\) :

Mean stress and generalized shear stress in characteristic stress space

\(\bar{p}\), \(\bar{q}\) :

Mean stress and generalized shear stress in transformed stress space

p r :

Reference stress

M β :

Ratio of \(\hat{q}\) to \(\hat{p}\)

M f :

Ratio of \(\bar{q}\) to \(\bar{p}\) under triaxial compression condition

β :

Strength parameter of NUSC

M t :

Ratio of \(q_{\text{t}}\) to \(q_{\text{c}}\)

F :

Fitness function

n :

Number of test series with a certain σ3 for a specific rock

m :

Number of test series for a specific rock

DC:

Determination coefficient

\(q_{i}^{\text{test}}\) :

Experimental value of the generalized shear stress

\(q_{i}^{\text{calc}}\) :

Predicted value of the generalized shear stress

\(\bar{q}_{\text{test}}\) :

Mean value of \(q_{i}^{\text{test}}\)

φ c :

Friction angle of rock

κ :

Hardening function

κ o :

Initial hardening rate

σ, \(\tilde{\varvec{\sigma }}\) :

Nominal and effective stress tensor

ε, ε p :

Total and plastic strain tensor

D e :

Elastic stiffness tensor

\(\varepsilon_{ij}^{p}\) :

Plastic strain component

\(\varepsilon_{\text{v}}^{p}\) :

Plastic volume strain

γ p :

Equivalent plastic shear strain

γ pf :

Equivalent plastic shear strain corresponding to the peak strength

A :

Equivalent plastic shear strain for the mean strength of σc/3

σ o :

Three-dimensional tensile strength

f d :

Damage loading function

κ d :

Damage-driving variable

g d :

Damage function

χ 1 :

Softening ductility measure

d :

Damage variable

b 1 :

Parameter controlling the slope of the softening curve

b 2 :

Parameter controlling the values of χ1

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Acknowledgements

This research is supported by the National Basic Research Program of China (2015CB057902), the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (51421005), the Beijing Municipal Natural Science Foundation (8164049) and the National Natural Science of China (51608015; 51678015). This support is gratefully acknowledged.

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Authors and Affiliations

  1. School of Civil and Environmental Engineering, University of Science and Technology Beijing, Beijing, 100083, China

    Jingqi Huang

  2. Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing, 100124, China

    Mi Zhao, Xiuli Du & Chao Ma

  3. Department of Civil Engineering, Tsinghua University, Beijing, 100084, China

    Jingqi Huang & Jingbo Liu

  4. State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources and Hydropower, Sichuan University, Chengdu, 610065, Sichuan, China

    Feng Dai

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  1. Jingqi Huang
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Correspondence to Mi Zhao.

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Huang, J., Zhao, M., Du, X. et al. An Elasto-Plastic Damage Model for Rocks Based on a New Nonlinear Strength Criterion. Rock Mech Rock Eng 51, 1413–1429 (2018). https://doi.org/10.1007/s00603-018-1417-1

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