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Damage characteristics analysis and constitutive model establishment for deep rock considering pre-static loads and frequent dynamic disturbance

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Abstract

In order to characterize the mechanical behaviors of deep rock mass in complex geological environments, a statistical damage constitutive model was established according to the element combination method. In this model, the effects of pre-static loads and frequent dynamic disturbance on the mechanical and statistical parameters of rock were considered and a new function was introduced to deduce the damage evolution equation. This model was then verified against relevant test and literature data. The results show that the damage evolution rate curve gradually moves to the right along the strain axis and the peak value of the curve decreases first and then increases with the increase of parameter k related to the disturbance number. When k is large, the damage evolution rate curve appears a 'platform'. This platform may correspond to the stable development stage of damage. Besides, pre-static loads can improve the impact resistance of rock. The energy release in the rebound stage under pre-static loads is higher than that without pre-static loads when the disturbance number is constant. The energy release and impact resistance of rock decrease with the increase of disturbance number if the pre-static loads remain unchanged. The proposed model can better describe the mechanical properties and stress–strain behaviors of rock under the combination of pre-static loads and frequent dynamic disturbance. For the rebound type curve,  its predictions are very consistent with test data in the pre-peak stage; For the strain-softening type curve, it does not only  well with test data in the pre-peak stage but also has high fitting accuracy in the post-peak stage. This study can provide beneficial reference for the evaluation of the safety and stability of deep rock engineering.

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Data availability

All data used during the study appear in the submitted article.

Abbreviations

x, p(x):

Independent variable and probability density function of Weibull distribution

N, N f :

Total number of representative volume elements and the number of destroyed representative volume elements, respectively

m, F 0 :

Weibull distribution parameters

σ 1, σ 2, σ 3 :

First, second, third principal stress, respectively

σ 1 *, σ 2 *, σ 3 * :

First, second, third principal effective stress, respectively

σ, σ * :

Stress and effective stress

ε, ε 1 :

Strain and first principal strain

I 1 :

First invariant of stress tensor

J 2 * :

Second invariant of effective deviatoric stress tensor

φ 0, α 0 :

Internal friction angle and the parameter related to internal friction angle when rock yields

E 0, E, :

Initial elastic modulus, average elastic modulus

E 11, E d :

Elastic modulus of the elastic element H1, elastic modulus of the dynamic stress–strain curve

ε 0 :

Weibull distribution scale parameter

Y, k :

Introduced correction function and parameter of the function related to frequent dynamic disturbance

Q, T :

Calculation parameters of  rock constitutive equation

D, D ’ :

Damage variable and damage evolution rate, respectively

σ 11, σ 12, σ 2, σ a :

Stress of the elastic element H1, viscosity element, elastic element H2, damaged element Da, respectively

ε 11, ε 12, ε 2, ε a :

Strain of the elastic element H1, viscosity element, elastic element H2, damaged element Da, respectively

μ :

Poisson’s ratio

\(\dot{\sigma }_{11}\) :

Stress rate of the elastic element H1

\(\dot{\varepsilon }_{1} ,\dot{\varepsilon }_{12}\) :

Strain rate of the Maxwell body and viscosity element, respectively

η, t :

Viscosity coefficient and time, respectively

ε rt, ε r 0 :

Starting and ending strain of the rebound stage of stress–strain curve, respectively

σ r, ε r :

Stress and strain of the rebound stage of stress–strain curve, respectively

W r :

Energy released by  sample after disturbance

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Acknowledgements

This study was financially supported by National Natural Science Foundation of China (12272119, U1965101), which is gratefully appreciated.

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Authors and Affiliations

  1. School of Civil and Hydraulic Engineering, Hefei University of Technology, Hefei, 230009, China

    Chenchen Feng, Zhiliang Wang, Songyu Li & XuTao Wu

  2. Anhui Key Laboratory of Civil Engineering Structures and Materials, Hefei University of Technology, Hefei, 230009, China

    Zhiliang Wang

  3. School of Mechanics and Civil Engineering, China University of Mining and Technology, Jiangsu, 221116, China

    Jianguo Wang

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  1. Chenchen Feng
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  3. Jianguo Wang
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Contributions

Chenchen Feng and Zhiliang Wang wrote the main manuscript text and prepared the figures; Jianguo Wang, Songyu Li and Xutao Wu prepared the tables and check the grammar of manuscript. All authors reviewed the manuscript.

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Correspondence to Zhiliang Wang.

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Feng, C., Wang, Z., Wang, J. et al. Damage characteristics analysis and constitutive model establishment for deep rock considering pre-static loads and frequent dynamic disturbance. Environ Earth Sci 82, 559 (2023). https://doi.org/10.1007/s12665-023-11220-7

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