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An equivalent medium model of stress wave propagation through a three-dimensional geo-stressed rock


Applying the equivalent medium method, a dynamic constitutive equation of rock with three-dimensional geo-stress is constructed by modifying the Kelvin-Voigt model, and a theoretical model of stress wave propagation through a three-dimensional geo-stressed rock is proposed. Based on the theory of one-dimensional stress wave propagation, the wave equation of the theoretical model is derived, and the analytical formulas of the stress wave propagation velocity, spatial attenuation coefficient and response frequency are obtained by using harmonic solution. Based on stress wave propagation experimental, the proposed theoretical model is verified by comparing the experimental and theoretical results. Based on the validated theoretical model, the effects of three-dimensional geo-stress on stress wave propagation velocity, spatial attenuation coefficient and response frequency are studied by using the parametric study. The results show that the proposed model of stress wave propagation can effectively study the propagation of stress wave in three-dimensional geo-stressed rock. Three-dimensional geo-stress varies the level of a rock porosity and damage, which makes the rock have different equivalent modulus, and then affects the stress wave propagation characteristics. Moreover, the initial porosity, initial elastic modulus, viscosity coefficient of a rock and vibration frequency have significant influence on the stress wave propagation velocity, spatial attenuation coefficient and response frequency.

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σ, ε :

Total axial stress and strain of the rock, respectively

σ D, ε D :

Stress and strain of the damage body, respectively

σ η, ε η :

Stress and strain of the viscous body, respectively

σ s, σ c, σ d :

Axial static stress, confining pressure and dynamic stress, respectively

R s, R p, η v :

Skeleton body, void body and viscous body of the rock, respectively

\({\varepsilon}_D^V\), \({\varepsilon}_D^R\) :

Strain of the micro-element rock void body and skeleton body, respectively

E 1, E 2 :

Initial elastic modulus of void body and skeleton body of the rock, respectively

μ 1, μ 2 :

Poisson’s ratios of void body and skeleton body of the rock, respectively

γ 0 :

Initial porosity of the rock

η :

Viscosity coefficient of the rock

β :

Reciprocal of equivalent modulus of the rock

x :

Propagation distance

t :


ω q, ω w :

Vibration and response frequencies, respectively

k t, k s :

Time and spatial wavenumbers, respectively

α s :

Spatial attenuation coefficient, respectively

C q :

Stress wave propagation velocity


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The study has been supported by the Projects (51664017, 51964015) supported by the National Natural Science Foundation of China and the Project (JXUSTQJBJ2017007) supported by the Program of Qingjiang Excellent Young Talents of Jiangxi University of Science and Technology, China.

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Correspondence to Jiefang Jin.

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Responsible Editor: Zeynal Abiddin Erguler

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Jin, J., Xu, H., Guo, Z. et al. An equivalent medium model of stress wave propagation through a three-dimensional geo-stressed rock. Arab J Geosci 15, 1236 (2022).

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  • Three-dimensional geo-stress
  • Theoretical model of stress wave propagation
  • Stress wave propagation velocity
  • Spatial attenuation coefficient
  • Response frequency