A novel constitutive modelling approach measured under simulated freeze–thaw cycles for the rock failure


In this study, we derived a computational modelling relation between the model parameters and characteristic parameters of rock deformation and failure via total differentials. We construct a damage model of rock under simulated freeze–thaw cycles and loading based on continuum damage mechanics theory that considers the influence of confining pressure and the random characteristics of rock material defects. This model reflects the variation in regulation between the internal mechanism of freeze–thaw damage and selected physical variables, making it more adaptable. We further analyze the evolution of microdamage and induced material mechanical properties of the rock using our proposed model, producing a total damage evolution curve under freeze–thaw cycles and loading that reflects the closure, initiation, propagation and coalescence of internal microcracks, as well as the subsequent appearance of macrocracks and rock failure. As the number of freeze–thaw cycles increases, rock damage intensifies, as demonstrated by the material’s deteriorating micromechanical properties. However, in later stages of deformation, both the strain and plasticity of the rock increase. With increasing confining pressure, rock damage and the damage accumulation rate, peak damage evolution ratio and descending segment after the peak decrease which manifest in the enhanced resistance of the rock to failure and increased macroscopic plastic deformation. Finally, we perform triaxial compression tests of rock under freeze–thaw cycles to validate our model. The macroscopic rock deformation and failure predicted by our model’s damage characteristics analysis are consistent with our experimental result.

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σ :

Nominal stress vector

C :

Elastic matrix

ε :

Nominal strain vector

D :


F :

Infinitesimal intensity

D :

Loading damage

E n :

Elastic modulus after n freeze–thaw cycles

D m :

Total damage

μ :

Poisson’s ratio

I 1 :

First invariant of the stress tensor

J 2 :

Second invariant of stress deviation

α 0 :

Parameter related to rock mechanics characteristics

σ *1 :

Effective axial stress

σ *3 :

Effective lateral stress

σ 1 :

Nominal axial stress

σ 3 :

Lateral nominal stress

φ :

Internal friction angle

σ cn :

Peak stress at an appropriate confining pressure

ε cn :

Corresponding strain


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The authors are grateful for financial support from the National Natural Science Foundation of China (Nos. 11172232, 51774231, 11872299, 41772333); The Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China (2018JQ4026); Future Scientists Program of “Double First Rate” of China University of Mining and Technology (2019WLKXJ076).

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Correspondence to Chao Yuan.

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Zhang, H., Yuan, C., Yang, G. et al. A novel constitutive modelling approach measured under simulated freeze–thaw cycles for the rock failure. Engineering with Computers 37, 779–792 (2021). https://doi.org/10.1007/s00366-019-00856-4

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  • Freeze–thaw cycle
  • Rock failure
  • Constitutive model
  • Confining pressure