Elastic Wave Propagation through a Layered Rock Mass with Adhesive Bedding Planes
Publication: International Journal of Geomechanics
Volume 23, Issue 12
Abstract
Wave propagation in a layered rock mass is a common problem in geotechnical engineering. The bedding planes in a layered rock mass are generally in situ stressed and adhesively bonded. This study extended the time-domain recursive method to analyze the wave propagation in a layered rock mass with adhesive bedding planes. The maximum stress criterion was used to indicate the adhesive failure of bedding planes. The Bandis–Barton and Coulomb slip models were used to characterize the normal and tangential behaviors of bedding planes after the adhesive bond fails. Based on the backward differentiation formula, an analytical solution reflecting four possible states of the bedding plane and the in situ stresses in rock mass was established. Subsequently, the solution was verified for various conditions. Besides, parametric studies were carried out to assess the influences of adhesive properties of bedding planes and in situ stresses on wave transmission. The transmission coefficient of seismic waves increases linearly as the adhesive strength of the bedding plane increases. The in situ normal stress could facilitate wave transmission across the bedding plane, while the in situ shear stress causes the direction-dependency of transmitted waves. Furthermore, the impacts of bedding plane adhesion and in situ stresses on wave propagation were influenced by the amplitude, frequency, and impinging angle of incident waves. The welded model that ignores the adhesion failure of the bedding plane and the unbonded models that neglect the interface adhesion could overestimate or underestimate the transmitted wave across a bedding plane.
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Data Availability Statement
All data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request, including the data for all figures.
Acknowledgments
This research was funded by the National Natural Science Foundation of China (Grant Numbers: 52079134 and 51991393).
Notation
The following symbols are used in this paper:
- AP
- amplitude of incident P wave;
- As
- amplitude of incident S wave;
- cp
- propagation velocity of P wave;
- cs
- propagation velocity of S wave;
- dmax
- the maximum closure of bedding plane;
- Em
- Young’s modulus of the mth rock;
- f
- frequency of incident wave;
- hm
- thickness of the mth rock layer;
- kn
- normal stiffness of bedding plane;
- kn0
- initial normal stiffness of bedding plane;
- tangential stiffness of bedding plane;
- Tpc
- transmission coefficient of P wave;
- Tsc
- transmission coefficient of S wave;
- tn
- adhesive tensile strength of bedding plane;
- ts
- adhesive shear strength of bedding plane;
- uc
- relative displacement at which contact surfaces begin to slide;
- un
- bedding plane closure;
- ux
- displacement in x-direction;
- uz
- displacement in z-direction;
- relative displacement between two contact surfaces;
- vlp
- particle velocity induced by downward-running P wave;
- vls
- particle velocity induced by downward-running S wave;
- vrp
- particle velocity induced by upward-running P wave;
- vrs
- particle velocity induced by upward-running S wave;
- vx
- particle velocity in x-direction;
- vz
- particle velocity in z-direction;
- zp
- impedance of P wave;
- zs
- impedance of S wave;
- α
- incident angle of P wave;
- β
- incident angle of S wave;
- νm
- Poisson’s ratio of the mth rock;
- ρm
- density of the mth rock;
- τ
- dynamic shear stress across bedding plane;
- τc
- frictional shear strength of bedding plane;
- τ0
- in situ shear stress acting on bedding plane;
- φ
- friction angle of bedding plane;
- σ
- dynamic normal stress across bedding plane; and
- σ0
- in situ normal stress acting on bedding plane.
References
Bandis, S. C., A. C. Lumsden, and N. R. Barton. 1983. “Fundamentals of rock joint deformation.” Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 20 (6): 249–268. https://doi.org/10.1016/0148-9062(83)90595-8.
Cai, J. G., and J. Zhao. 2000. “Effects of multiple parallel fractures on apparent attenuation of stress waves in rock masses.” Int. J. Rock Mech. Min. Sci. 37 (4): 661–682. https://doi.org/10.1016/S1365-1609(00)00013-7.
Chandran, D., and P. Anbazhagan. 2020. “2D nonlinear site response analysis of typical stiff and soft soil sites at shallow bedrock region with low to medium seismicity.” J. Appl. Geophys. 179 (2): 104087. https://doi.org/10.1016/j.jappgeo.2020.104087.
Cui, Z., Q. Sheng, and X. Leng. 2017. “Analysis of S wave propagation through a nonlinear joint with the continuously yielding model.” Rock Mech. Rock Eng. 50 (1): 113–123. https://doi.org/10.1007/s00603-016-1108-8.
Du, H., H. Wang, P. Feng, R. Tian, and Y. Wang. 2022. “Mechanical responses of underground unparallel-fissured rocks subjected to coupled static-dynamic loading.” Lithosphere 2022: 1790417. https://doi.org/10.2113/2022/1790417.
Du, H.-b., F. Dai, Y. Xu, Z. Yan, and M.-d. Wei. 2020. “Mechanical responses and failure mechanism of hydrostatically pressurized rocks under combined compression-shear impacting.” Int. J. Mech. Sci. 165: 105219. https://doi.org/10.1016/j.ijmecsci.2019.105219.
Fan, L. F., and H. Y. Sun. 2015. “Seismic wave propagation through an in-situ stressed rock mass.” J. Appl. Geophys. 121: 13–20. https://doi.org/10.1016/j.jappgeo.2015.07.002.
Huang, X., S. Qi, Y. Liu, and Z. Zhan. 2014. “Stress wave propagation through viscous-elastic jointed rock masses using propagator matrix method (PMM).” Geophys. J. Int. 200 (1): 452–470. https://doi.org/10.1093/gji/ggu407.
Huang, X., S. Qi, K. Xia, and X. Shi. 2018. “Particle crushing of a filled fracture during compression and its effect on stress wave propagation.” J. Geophys. Res.: Solid Earth 123 (7): 5559–5587. https://doi.org/10.1029/2018JB016001.
Hudson, J. A., F. H. Cornet, and R. Christiansson. 2003. “ISRM suggested methods for rock stress estimation—Part 1: Strategy for rock stress estimation.” Int. J. Rock Mech. Min. Sci. 40 (7–8): 991–998. https://doi.org/10.1016/j.ijrmms.2003.07.011.
Kayestha, P., E. R. Ferreira, and A. C. Wijeyewickrema. 2013. “Finite-amplitude shear horizontal waves propagating in a pre-stressed layer between two half-spaces.” Int. J. Solids Struct. 50 (22–23): 3586–3596. https://doi.org/10.1016/j.ijsolstr.2013.07.002.
Li, J., G. Ma, and J. Zhao. 2011. “Stress wave interaction with a nonlinear and slippery rock joint.” Int. J. Rock Mech. Min. Sci. 48 (3): 493–500. https://doi.org/10.1016/j.ijrmms.2010.11.013.
Li, J. C., H. B. Li, G. W. Ma, and J. Zhao. 2012. “A time-domain recursive method to analyse transient wave propagation across rock joints.” Geophys. J. Int. 188 (2): 631–644. https://doi.org/10.1111/j.1365-246X.2011.05286.x.
Li, T., M. F. Cai, and M. Cai. 2007. “A review of mining-induced seismicity in China.” Int. J. Rock Mech. Min. Sci. 44 (8): 1149–1171. https://doi.org/10.1016/j.ijrmms.2007.06.002.
Liu, T., X. Li, Y. Zheng, F. Meng, and D. Song. 2020. “Analysis of seismic waves propagating through an in situ stressed rock mass using a nonlinear model.” Int. J. Geomech. 20 (3): 04020002. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001621.
Perino, A., J. B. Zhu, J. C. Li, G. Barla, and J. Zhao. 2010. “Theoretical methods for wave propagation across jointed rock masses.” Rock Mech. Rock Eng. 43 (6): 799–809. https://doi.org/10.1007/s00603-010-0114-5.
Pyrak-Nolte, L. J., L. R. Myer, and N. G. W. Cook. 1990. “Anisotropy in seismic velocities and amplitudes from multiple parallel fractures.” J. Geophys. Res. 95 (B7): 11345–11358. https://doi.org/10.1029/JB095iB07p11345.
Resende, R., L. N. Lamas, J. V. Lemos, and R. Calcada. 2010. “Micromechanical modelling of stress waves in rock and rock fractures.” Rock Mech. Rock Eng. 43 (6): 741–761. https://doi.org/10.1007/s00603-010-0098-1.
Sari, M., A. Seren, and S. Alemdag. 2020. “Determination of geological structures by geophysical and geotechnical techniques in Kırklartepe Dam site (Turkey).” J. Appl. Geophys. 182 (9): 104174. https://doi.org/10.1016/j.jappgeo.2020.104174.
Schoenberg, M. 1980. “Elastic wave behavior across linear slip interfaces.” J. Acoust. Soc. Am. 68: 1516–1521. https://doi.org/10.1121/1.385077.
Sharma, M. D., and N. Garg. 2006. “Wave velocities in a pre-stressed anisotropic elastic medium.” J. Earth Syst. Sci. 115 (2): 257–265. https://doi.org/10.1007/BF02702040.
Singh, A. K., A. Das, A. Chattopadhyay, and S. Dhua. 2015. “Dispersion of shear wave propagating in vertically heterogeneous double layers overlying an initially stressed isotropic half-space.” Soil Dyn. Earthquake Eng. 69: 16–27. https://doi.org/10.1016/j.soildyn.2014.10.021.
Sitharam, T. G., J. Sridevi, and N. Shimizu. 2001. “Practical equivalent continuum characterization of jointed rock masses.” Int. J. Rock Mech. Min. Sci. 38 (3): 437–448. https://doi.org/10.1016/S1365-1609(01)00010-7.
Tian, H. M., W. Z. Chen, D. S. Yang, and J. P. Yang. 2015. “Experimental and numerical analysis of the shear behaviour of cemented concrete–rock joints.” Rock Mech. Rock Eng. 48 (1): 213–222. https://doi.org/10.1007/s00603-014-0560-6.
Wang, L. J., C. Q. Wu, L. F. Fan, and M. Wang. 2023. “Investigation of wave reflection at the joint with different wave impedances on two sides.” Waves Random Complex Medium 33: 237–253. https://doi.org/10.1080/17455030.2021.1876960.
Zhang, N., Y. Zhang, Y. Gao, R. Y. S. Pak, Y. Wu, and F. Zhang. 2019. “An exact solution for SH-wave scattering by a radially multilayered inhomogeneous semicylindrical canyon.” Geophys. J. Int. 217 (2): 1232–1260. https://doi.org/10.1093/gji/ggz083.
Zhang, Q., F. Dai, and Y. Liu. 2022. “Experimental assessment on the dynamic mechanical response of rocks under cyclic coupled compression-shear loading.” Int. J. Mech. Sci. 216 (15): 106970. https://doi.org/10.1016/j.ijmecsci.2021.106970.
Zhao, J., and J. G. Cai. 2001. “Transmission of elastic P-waves across single fractures with a nonlinear normal deformational behavior.” Rock Mech. Rock Eng. 34 (1): 3–22. https://doi.org/10.1007/s006030170023.
Zhao, J., J. G. Cai, X. B. Zhao, and H. B. Li. 2008. “Dynamic model of fracture normal behaviour and application to prediction of stress wave attenuation across fractures.” Rock Mech. Rock Eng. 41 (5): 671–693. https://doi.org/10.1007/s00603-006-0127-2.
Zhao, W., W. Chen, D. Yang, X. Tan, H. Gao, and C. Li. 2019. “Earthquake input mechanism for time-domain analysis of tunnels in layered ground subjected to obliquely incident P- and SV-waves.” Eng. Struct. 181: 374–386. https://doi.org/10.1016/j.engstruct.2018.12.050.
Zhao, W., W. Chen, and K. Zhao. 2018. “Laboratory test on foamed concrete–rock joints in direct shear.” Constr. Build. Mater. 173: 69–80. https://doi.org/10.1016/j.conbuildmat.2018.04.006.
Zhao, X. B., J. Zhao, A. M. Hefny, and J. G. Cai. 2006. “Normal transmission of S-wave across parallel fractures with Coulomb slip behavior.” J. Eng. Mech. 132 (6): 641–650. https://doi.org/10.1061/(ASCE)0733-9399(2006)132:6(641).
Zhao, X. B., J. B. Zhu, J. Zhao, and J. G. Cai. 2012. “Study of wave attenuation across parallel fractures using propagator matrix method.” Int. J. Numer. Anal. Methods Geomech. 36 (10): 1264–1279. https://doi.org/10.1002/nag.1050.
Zhu, J. B., A. Perino, G. F. Zhao, G. Barla, J. C. Li, G. W. Ma, and J. Zhao. 2011. “Seismic response of a single and a set of filled joints of viscoelastic deformational behaviour.” Geophys. J. Int. 186 (3): 1315–1330. https://doi.org/10.1111/j.1365-246X.2011.05110.x.
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Published In
International Journal of Geomechanics
Volume 23 • Issue 12 • December 2023
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Sep 27, 2022
Accepted: Jun 10, 2023
Published online: Sep 26, 2023
Published in print: Dec 1, 2023
Discussion open until: Feb 26, 2024
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