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Evolution models of the strength parameters and shear dilation angle of rocks considering the plastic internal variable defined by a confining pressure function

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Abstract

Based on the triaxial test results of 30 types of rocks, by analysing the confining pressure function and defining a new plastic internal variable, the evolution laws of the strength parameters and shear dilation angle with a defined plastic internal variable are studied, and the corresponding evolution models are established. First, the complete stress-strain curves of 30 types of rocks are collected from published literature; from these curves, the critical equivalent plastic strains under different confining pressures are extracted. With the confining pressure and critical equivalent plastic strain data of the 30 types of rocks, fitting is performed for 23 different functions. The results demonstrate that the three-parameter allometric power-type function is the best to serve as the confining pressure function to define the plastic internal variable. Second, the strength and plastic strain data of the 30 types of rocks are extracted and transformed into the strength and plastic internal variable data. By analysing the evolution laws of the strength parameters considering the plastic internal variable, the Gaussian function is adopted to uniformly characterise the variation in the strength parameters with the plastic internal variable. Third, the shear dilation angle, confining pressure and plastic strain data of the 30 types of rocks are extracted and transformed into shear dilation angle, confining pressure and plastic internal variable data. By analysing the evolution law of the shear dilation angle considering the confining pressure and the plastic internal variable, a negative exponential function is adopted to uniformly characterise the nonlinear evolution of the shear dilation angle. Finally, the proposed evolution models of the strength parameters and the shear dilation angle are integrated into ABAQUS. By comparing the simulated complete stress-strain curves with the experimental curves of the different rock types, it is verified that the proposed models can be used to correctly simulate the nonlinear deformation and failure of different rock types. This research overcomes the shortcomings of the existing models and has wide application prospective.

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Abbreviations

κ :

Plastic internal variable

ε 1y and ε 3y :

Axial and lateral strains at the initial yield point

ε 1r and ε 3r :

Axial and lateral strains at the starting point of the residual stage

\( {\varepsilon}_{1\mathrm{r}}^{\mathrm{p}} \) and \( {\varepsilon}_{3\mathrm{r}}^{\mathrm{p}} \) :

Axial and lateral critical plastic strains

\( {\overline{e}}^{\mathrm{p}} \) and \( {\overline{e}}_{\mathrm{r}}^{\mathrm{p}} \) :

Equivalent plastic strain and critical equivalent plastic strain

f(σ 3/σ c):

Confining pressure function

σ 3 and σ c :

Confining pressure and unit stress

A1, A2 and A3 :

Parameters in the three-parameter allometric power-type confining pressure function

\( \Delta {\varepsilon}_1^{\mathrm{p}} \) and \( \Delta {\varepsilon}_{\mathrm{v}}^{\mathrm{p}} \) :

Axial and volumetric plastic strain increments

E :

Young’s modulus

ν :

Poisson’s ratio

c :

(Cohesion)

φ :

Internal friction angle

ψ :

Shear dilation angle

c max, c min, κ c and ξ c :

Parameters in the evolution function of cohesion (Eq. (7))

φ max, φ min, κ φ and ξ φ :

Parameters in the evolution function of the internal friction angle (Eq. (7))

α 1, α 2, β 1 and β 2 :

Parameters in the evolution function of the shear dilation angle (Eq. (9))

Δλ n :

Plastic multiplier determined by the plastic consistency condition

D :

Elastic matrix

σ :

Stress vector

\( {\boldsymbol{\sigma}}_{\mathrm{n}}^{\mathrm{trial}} \) :

Trial stress vector

Δε n :

Strain increment vector

e p and \( {\boldsymbol{e}}_{\mathrm{r}}^{\mathrm{p}} \) :

Deviatoric plastic strain vector and critical deviatoric plastic strain vector

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Acknowledgements

No conflict of interest exits in the submission of this manuscript that is approved by all authors for publication. We declare that the work described is an original research that has not been published previously. All the authors listed have approved the manuscript that is enclosed.

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Authors and Affiliations

  1. State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, 430072, China

    Junchao Jin, Chengxue She & Pengyang Shang

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  1. Junchao Jin
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  2. Chengxue She
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Corresponding author

Correspondence to Chengxue She.

Appendix

Appendix

In Fig. 13, the values of the strength parameters for different plastic internal variables of rocks - are plotted, and the fitting curves are also drawn via Eq. (7) with the parameter values listed in Table 8.

Fig. 13
figure 13

Strength parameter values for different κ values (open squares represent the cohesion, solid circles represent the internal friction angle) of rocks - , and the curves fitted by Eq. (7). ⑮

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Table 8 Parameter values of Eq. (7) fitted by the strength parameter data of rocks -
Full size table

In Fig. 14, the shear dilation angle values assumed for plastic internal variables under different confining pressures of rocks - are displayed, and the fitting curves by Eq. (9) are also given with the parameter values listed in Table 9.

Fig. 14
figure 14

Calculated shear dilation angles (symbols) of rocks - , and the curves fitted by Eq. (9)

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Table 9 Parameter values of Eq. (9) fitted by the shear dilation angle data of rocks -
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Jin, J., She, C. & Shang, P. Evolution models of the strength parameters and shear dilation angle of rocks considering the plastic internal variable defined by a confining pressure function. Bull Eng Geol Environ 80, 2925–2953 (2021). https://doi.org/10.1007/s10064-020-02040-1

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  • DOI: https://doi.org/10.1007/s10064-020-02040-1

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