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A Unified Constitutive Model for Rock Based on Newly Modified GZZ Criterion

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Abstract

This paper proposes a unified constitutive model for rock based on the newly modified generalized Zhang-Zhu (GZZ) criterion. The constitutive model adopts a non-associated plastic flow rule and a continuous potential function that takes the three effective principal stresses into account. To reflect strain-softening, strain-hardening, and elastic-perfectly plastic behavior of rock in a unified way, a general expression is proposed to model the post-failure behavior of rock using the deviatoric plastic shear strain as the fundamental variable. The proposed constitutive model has been successfully implemented in a 3D finite-difference code and validated using it to simulate the true triaxial test of two types of rocks and comparing the simulation results with the experimental data. Finally, a 3D numerical model based on the proposed constitutive model is constructed to simulate a highway rock tunnel during construction. The results show that the predicted displacements of the rock tunnel are in good agreement with the field measurements.

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Fig. 1

(After Clausen and Damkilde 2008)

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Abbreviations

E rm :

Young’s modulus of rock mass

\(I_{1}^{*} ,I_{2}^{*} ,I_{3}^{*}\) :

Transformed first, second and third stress invariants

J 2, J 3 :

Second and third deviatoric stress invariants

m b, s, a :

Material constant for rock masses defined in Hoek–Brown criterion

m bi, m br :

Initial and residual value of \({m}_{b}\)

m d :

Material constant defining potential function

m di, m dr :

Initial and residual value of \({m}_{d}\)

m i :

Material constant for the intact rock

\(p^{\prime}\) :

Mean effective stress

s i, s r :

Initial and residual value of \(s\)

\(\varepsilon_{f}\) :

A parameter controlling the softening/hardening of the yield function

\(\varepsilon_{g}\) :

A parameter controlling the evolution of the potential function

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

Plastic deviatoric shear strain

\(\varepsilon_{x} ,\varepsilon_{y} ,\varepsilon_{z} ,\varepsilon_{xy} ,\varepsilon_{xz} ,\varepsilon_{yz}\) :

Six basic strain components

\(\theta_{\sigma }\) :

Lode’s angle

\(\sigma_{1}^{{\prime}} ,\sigma_{2}^{{\prime}} ,\sigma_{3}^{{\prime}}\) :

Maximum, intermediate, and minimum effective principal stresses

\(\sigma_{c}\) :

Unconfined compressive strength of intact rock

\(\sigma_{m,2}^{{\prime}}\) :

Effective mean stress

\(\sigma^{\prime}_{x} ,\sigma^{\prime}_{y} ,\sigma^{\prime}_{z} ,\tau_{xy} ,\tau_{xz} ,\tau_{yz}\) :

Six basic stress components

\(\tau_{{{\text{oct}}}}\) :

Octahedral shear stress

D :

Disturbance factor reflecting the level of blast damage and stress relaxation to rock mass

E, v :

Young’s modulus and Poisson’s ratio

GSI:

Geological strength index

K, G :

Bulk modulus and shear modulus

f, g :

Yield function and potential function

λ :

Plastic multiplier for flow rule

\(\varepsilon\) :

\(\left[ {\varepsilon_{x} ,\varepsilon_{y} ,\varepsilon_{z} ,2\varepsilon_{xy} ,2\varepsilon_{xz} ,2\varepsilon_{yz} } \right]\)

\(\sigma\) :

\(\left[ {\sigma_{x}^{{\prime}} ,\sigma_{y}^{{\prime}} ,\sigma_{z}^{{\prime}} ,\tau_{xy} ,\tau_{xz} ,\tau_{yz} } \right]\)

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Authors and Affiliations

  1. Department of Civil and Architectural Engineering and Engineering Mechanics, The University of Arizona, Tucson, AZ, 85716, USA

    Haohua Chen & Lianyang Zhang

  2. Department of Geotechnical Engineering, Tongji University, Shanghai, 200092, China

    Hehua Zhu

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  1. Haohua Chen
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  2. Hehua Zhu
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  3. Lianyang Zhang
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Correspondence to Lianyang Zhang.

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Chen, H., Zhu, H. & Zhang, L. A Unified Constitutive Model for Rock Based on Newly Modified GZZ Criterion. Rock Mech Rock Eng 54, 921–935 (2021). https://doi.org/10.1007/s00603-020-02293-y

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