Abstract
How to accurately characterize the nonlinear characteristics of rock mass creep has long been a research hotspot. The traditional Kelvin model cannot accurately characterize the nonlinear variation process in the decay creep stage and the dual characteristics of power function and exponential function in the accelerated creep stage, so it needs to be modified. Based on nonlinear rheological theory and damage theory, this paper establishes a damage creep model that simultaneously considers instantaneous elastic strain, nonlinear viscoelastic strain, viscous strain, and nonlinear viscoplastic strain. The differential damage constitutive equations under one-dimensional and three-dimensional stress states are derived, the method for determining model parameters is presented, and the model is verified based on the triaxial compression creep test results of sandstone. Meanwhile, a sensitivity analysis is performed on the parameters of the modified model. The results show that the modified model can accurately reflect the entire creep process of sandstone under different confining pressures. In uniaxial compression creep tests, the peak strain is most sensitive to changes in the parameters η₂ and η₄. Under high confining pressure, the peak strain is most sensitive to changes in the parameters β, n, and η₄.
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Abbreviations
- E 1 :
-
The elastic modulus of the elastic body
- E 2 :
-
The elastic modulus of the nonlinear Kelvin body
- η 1 :
-
Viscosity coefficient of the nonlinear Kelvin body
- η 3 :
-
Viscosity coefficient of the viscous body
- E 3 :
-
Elastic modulus of the nonlinear damage viscoplastic body
- η 3 :
-
Viscosity coefficient of the nonlinear damage viscoplastic body
- η 4 :
-
Viscosity coefficient of the nonlinear damage viscoplastic body
- D :
-
Damage variable
- σ s :
-
Yield strength of rock
- σ e, ε e :
-
Stress and strain in the elastic deformation stage
- σ ve, ε ve :
-
Stress and strain in the decay creep stage
- σ v, ε v :
-
Stress and strain in the steady creep stage
- σ vp, ε vp :
-
Stress and strain in the accelerated creep stage
- σ, ε :
-
Total stress and total strain
- β :
-
Time-dependent constant in the nonlinear Kelvin body
- C , n :
-
Material parameters related to damage variable D
- σ ij, ε ij :
-
Stress tensor and strain tensor
- S ij, e ij :
-
Deviatoric stress tensor and deviatoric strain tensor
- σ m, ε m :
-
Spherical stress and spherical strain
- δ ij :
-
Kronecker tensor
- t f :
-
Time to creep failure
- \(\varepsilon_{ij}\) :
-
Total strain under three-dimensional stress state
- \(\varepsilon_{ij}^{e}\), \(\varepsilon_{ij}^{ve}\), \(\varepsilon_{ij}^{v}\), \(\varepsilon_{ij}^{vp}\) :
-
Strain at each stage under three-dimensional stress state
- G 1, K 1 :
-
Shear modulus and bulk modulus of the elastic body
- G 2 :
-
Shear modulus of the nonlinear Kelvin body
- F :
-
Rock yield function
- F 0 :
-
Initial value of rock yield function
- K :
-
Specified constants
- Q :
-
Plastic potential function
- J 2 :
-
Second invariant of the stress deviator
- I 1 :
-
First invariant of the stress tensor
- α, k :
-
Material parameters
- \(\phi ,c\) :
-
The internal friction angle and cohesion of rock
- RMSE :
-
Root Mean Square Error
- R 2 :
-
Coefficient of determination
- \(\delta \varepsilon\) :
-
Strain rate
Acknowledgements
This study was supported by the Guizhou Science and Technology Support Program Project (Qian Ke He Zhi Cheng [2026] General No. 305), Guizhou Science and Technology Plan Project (Qian Ke He Ji Chu QN [2025] No. 223), Scientific Research Startup Project for High-Level Talents, Guizhou Institute of Technology (2025GCC023), and Science and Technology Program of Guizhou Province (Qiankehe Platform KXJZ [2024]020). All of the support provided is gratefully acknowledged.
Funding
The Guizhou Science and Technology Support Program Project (Qian Ke He Zhi Cheng [2026] General No. 305), Guizhou Science and Technology Plan Project (Qian Ke He Ji Chu QN [2025] No. 223), Scientific Research Startup Project for High-Level Talents, Guizhou Institute of Technology (2025GCC023), and Science and Technology Program of Guizhou Province (Qiankehe Platform KXJZ [2024] 020).
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Tian, X., Jia, B., Sun, D. et al. Research on improved nonlinear viscoelastic-plastic damage model of rock and parameter identification. Sci Rep (2026). https://doi.org/10.1038/s41598-026-53436-0
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DOI: https://doi.org/10.1038/s41598-026-53436-0
