Skip to main content

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

Abstract

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Abbreviations

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 :

Time

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

References

  • Baud P, Wong TF, Zhu W (2014) Effects of porosity and crack density on the compressive strength of rocks. Int J Rock Mech Min Sci 67:202–211

    Article  Google Scholar 

  • Cao WG, Zhang C, He M, Liu T (2016) Statistical damage simulation method of strain softening deformation process for rocks considering characteristics of void compaction stage. China J Rock Mech Rock Eng 38:1754–1761

    Google Scholar 

  • Chai SB, Li JC, Zhang QB, Li HB, Li NN (2016) Stress wave propagation across a rock mass with two non-parallel joints. Rock Mech Rock Eng 49(10):4023–4032

    Article  Google Scholar 

  • Chai SB, Li JC, Rong LF, Li NN (2017) Theoretical study for induced seismic wave propagation across rock masses during underground exploitation. Geomech Geophys Geo 3(2):95–105

    Article  Google Scholar 

  • Chen Y, Man CS, Tanuma K, Kube CM (2018) Monitoring near-surface depth profile of residual stress in weakly anisotropic media by Rayleigh-wave dispersion. Wave Motion 77:119–138

    Article  Google Scholar 

  • Cheng Y, Song ZP, Jin JF, Wang JB, Wang T (2019) Experimental study on stress wave attenuation and energy dissipation of sandstone under full deformation condition. Arab J Geosc 12(23):1–14

    Article  Google Scholar 

  • Du K, Li X, Tao M, Wang XF (2020a) Experimental study on acoustic emission (AE) characteristics and crack classification during rock fracture in several basic lab tests. Int J Rock Mech Min 133:104411

    Article  Google Scholar 

  • Du K, Yang CZ, Su R, Tao M, Wang XF (2020b) Failure properties of cubic granite, marble, and sandstone specimens under true triaxial stress. Int J Rock Mech Min 130:104309

    Article  Google Scholar 

  • Du K, Sun Y, Zhou J, Wang XF, Tao M, Yang CZ, Khandelwal M (2021) Low amplitude fatigue performance of sandstone, marble, and granite under high static stress. Geomech Geophys Geo 7(3):1–21

    Google Scholar 

  • Fan LF, Sun HY (2015) Seismic wave propagation through an in-situ stressed rock mass. J Appl Geophys 121:13–20

    Article  Google Scholar 

  • Fan LF, Wang M, Wu ZJ (2021) Effect of nonlinear deformational macrojoint on stress wave propagation through a double-scale discontinuous rock mass. Rock Mech Rock Eng 54(3):1077–1090

    Article  Google Scholar 

  • Han DH, Nur A, Morgan D (1986) Effects of porosity and clay content on wave velocities in sandstones. Geophysics 51:2093–2107

    Article  Google Scholar 

  • Han B, Xie SY, Shao JF (2016) Experimental investigation on mechanical behavior and permeability evolution of a porous limestone under compression. Rock Mech Rock Eng 49(9):3425–3435

    Article  Google Scholar 

  • Hu JN, Fu LY, Wei W, Zhang Y (2018) Stress-associated intrinsic and scattering attenuation from laboratory ultrasonic measurements on shales. Pure Appl. Geophys 175(3):929–962

    Article  Google Scholar 

  • Jiang JQ, Su GS, Liu YX, Zhao GF, Yan XY (2021) Effect of the propagation direction of the weak dynamic disturbance on rock failure: an experimental study. B Eng Geol Environ 80(2):1507–1521

    Article  Google Scholar 

  • Jin JF, Yuan W, Wu Y, Guo ZQ (2020) Effects of axial static stress on stress wave propagation in rock considering porosity compaction and damage evolution. J Cent South Univ 27(2):592–607

    Article  Google Scholar 

  • Li JC, Ma GW, Zhao J (2010) An equivalent viscoelastic model for rock mass with parallel joints. J Geophys Res-Sol Ea 115(B3)

  • Li JC, Ma GW, Zhao J (2011) Equivalent medium model with virtual wave source method for wave propagation analysis in jointed rock masses. Adv Rock Dynamics Ap

  • Li JC, Li HB, Zhao J (2015) An improved equivalent viscoelastic medium method for wave propagation across layered rock masses. Int J Rock Mech Min 73:62–69

    Article  Google Scholar 

  • Liu CL, Ahrens TJ (1997) Stress wave attenuation in shock-damaged rock. J Geophys Res 102(B3):5243–5250

    Article  Google Scholar 

  • Liu TT, Li JC, Li HB, Li XP, Zheng Y, Liu H (2017) Experimental study of s-wave propagation through a filled rock joint. Rock Mech Rock Eng 50(10):2645–2657

    Article  Google Scholar 

  • Ma GW, Fan LF, Li JC (2013) Evaluation of equivalent medium methods for stress wave propagation in jointed rock mass. Int J Numer Anal Met 37(7):701–715

    Article  Google Scholar 

  • Majstorović J, Belinić T, Namjesnik D, Dasović I, Herak D, Herak M (2017) Intrinsic and scattering attenuation of high-frequency S-waves in the central part of the External Dinarides. Phys Earth Planet In 270:73–83

    Article  Google Scholar 

  • Mindlin RD (1960) Waves and vibrations in isotropic, elastic plates. Structure Mechanics 199-232

  • Mogilevskaya SG, Lecampion B (2018) A lined hole in a viscoelastic rock under biaxial far-field stress. Int J Rock Mech Min 106:350–363

    Article  Google Scholar 

  • Niu LL, Zhu WC, Li SH, Guan K (2018) Determining the viscosity coefficient for viscoelastic wave propagation in rock bars. Rock Mech Rock Eng 51(5):1347–1359

    Article  Google Scholar 

  • Niu LL, Zhu WC, Li S, Liu XG (2020) Spalling of a one-dimensional viscoelastic bar induced by stress wave propagation. Int J Rock Mech Min Sci 131:104317

    Article  Google Scholar 

  • Proskuryakov NM, Livenskii VS, Kuznetsov HF (1975) Study of the velocity of propagation of elastic waves in relation to stress in salt rocks under uniaxial compression. J Min Sci 11(1):68–69

    Google Scholar 

  • Schenk V (1971) Attenuation coefficients of the maximum amplitude and the spectral amplitude of stress waves in non-elastic zones of explosive sources. Pure Appl Geophys 90(1):61–69

    Article  Google Scholar 

  • Schoenberg M (1980) Elastic wave behavior across linear slip interfaces. J Acoust Soc Am 68(5):1516–1521

    Article  Google Scholar 

  • Shkuratnik VL, Nikolenko PV, Koshelev AE (2016) Stress dependence of elastic P-wave velocity and amplitude in coal specimens under varied loading conditions. J Min Sci 52(5):873–877

    Article  Google Scholar 

  • Sun Q, Zhu SY (2014) Wave velocity and stress/strain in rock brittle failure. Environ Earth Sci 72(3):861–866

    Article  Google Scholar 

  • Wang LL (2007) Foundations of stress waves. Elsevier, Amsterdam

    Google Scholar 

  • Wang EY, He XQ (2000) An experimental study of the electromagnetic emission during the deformation and fracture of coal or rock. Chin J Geophs 43(1):134–140

    Article  Google Scholar 

  • Wang HT, He MM, Pang F, Chen YS, Zhang ZQ (2021) Energy dissipation-based method for brittleness evolution and yield strength determination of rock. J Pet Sci Eng 200:108376

    Article  Google Scholar 

  • Xie SY, Shao JF (2015) An experimental study and constitutive modeling of saturated porous rocks. Rock Mech Rock Eng 48(1):223–234

    Article  Google Scholar 

  • Yan ZL, Dai F, Liu Y, Du HB, Luo J (2020) Dynamic strength and cracking behaviors of single-flawed rock subjected to coupled static–dynamic compression. Rock Mech Rock Eng 53:4289–4298

    Article  Google Scholar 

  • Ye ZY, Li XB, Zhou ZL, Yin SB, Liu XL (2009) Static-dynamic coupling strength and deformation characteristics of rock under triaxial compression. Roc Soi Mech 30(7):1981–1986

    Google Scholar 

  • Yin ZQ, Li XB, Yin TB, Jin Jiefang DK (2012) Critical failure characteristics of high stress rock induced by impact disturbance under confining pressure unloading. Chin J Rock Mech Eng 31(7):1355–1362

    Google Scholar 

  • Yuan W, Wang X, Wang XB (2020) Numerical investigation on effect of confining pressure on the dynamic deformation of sandstone. Eur J Environ Civ 1-18

  • Zhang JZ, Zhou XP, Peng Y (2019a) Viscoplastic deformation analysis of rock tunnels based on fractional derivatives. Tunn Undergr Space Technol 85:209–219

    Article  Google Scholar 

  • Zhang SH, Wu SC, Duan K (2019b) Study on the deformation and strength characteristics of hard rock under true triaxial stress state using bonded-particle model. Comput Geotech 112:1–16

    Article  Google Scholar 

  • Zhao J, Zhao XB, Cai JG (2006) A further study of P-wave attenuation across parallel fractures with linear deformational behaviour. Int J Rock Mech Min 43(5):776–788

    Article  Google Scholar 

  • Zhu JB, Zhai TQ, Liao ZY, Yang SQ, Liu XL, Zhou T (2020a) Low-amplitude wave propagation and attenuation through damaged roc-k and a classification scheme for rock fracturing degree. Rock Mechanics and Rock Engineering 53(9):3983–4000

    Article  Google Scholar 

  • Zhu SJ, Zhou FB, Kang JH, Wang YP, Li HJ, Li GH (2020b) Laboratory characterization of coal P-wave velocity variation during adsorption of methane under tri-axial stress condition. Fuel 272:117698

    Article  Google Scholar 

Download references

Funding

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jiefang Jin.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Additional information

Responsible Editor: Zeynal Abiddin Erguler

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1007/s12517-022-10461-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12517-022-10461-3

Keywords

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