Skip to main content
SpringerLink
Log in

Analytical Time-Domain Solution of Plane Wave Propagation Across a Viscoelastic Rock Joint

  • Original Paper
  • Published:
Rock Mechanics and Rock Engineering Aims and scope Submit manuscript

Abstract

The effects of viscoelastic filled rock joints on wave propagation are of great significance in rock engineering. The solutions in time domain for plane longitudinal (P-) and transverse (S-) waves propagation across a viscoelastic rock joint are derived based on Maxwell and Kelvin models which are, respectively, applied to describe the viscoelastic deformational behaviour of the rock joint and incorporated into the displacement discontinuity model (DDM). The proposed solutions are verified by comparing with the previous studies on harmonic waves, which are simulated by sinusoidal incident P- and S-waves. Comparison between the predicted transmitted waves and the experimental data for P-wave propagation across a joint filled with clay is conducted. The Maxwell is found to be more appropriate to describe the filled joint. The parametric studies show that wave propagation is affected by many factors, such as the stiffness and the viscosity of joints, the incident angle and the duration of incident waves. Furthermore, the dependences of the transmission and reflection coefficients on the specific joint stiffness and viscosity are different for the joints with Maxwell and Kelvin behaviours. The alternation of the reflected and transmitted waveforms is discussed, and the application scope of this study is demonstrated by an illustration of the effects of the joint thickness. The solutions are also extended for multiple parallel joints with the virtual wave source method and the time-domain recursive method. For an incident wave with arbitrary waveform, it is convenient to adopt the present approach to directly calculate wave propagation across a viscoelastic rock joint without additional mathematical methods such as the Fourier and inverse Fourier transforms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

(after Zhu et al. 2011, Fig. 5c, p. 1324)

Fig. 13

Similar content being viewed by others

References

  • Cai JG, Zhao J (2000) Effects of multiple parallel fractures on apparent attenuation of stress waves in rock masses. Int J Rock Mech Min Sci 37:661–682

    Article  Google Scholar 

  • Cook NGW (1992) Natural joint in rock: mechanical, hydraulic and seismic behaviour and properties under normal stress. Int J Rock Mech Min Sci Geomech Abstr 29:198–223

    Article  Google Scholar 

  • Deng XF, Chen SG, Zhu JB, Zhou YX, Zhao ZY, Zhao J (2015) UDEC–AUTODYN hybrid modeling of a large-scale underground explosion test. Rock Mech Rock Eng 48:737–747

    Article  Google Scholar 

  • Goodman RE (1976) Methods of geological engineering in discontinuous rock. West Publishing, St. Paul

    Google Scholar 

  • Gu BL, Suárez-Rivera R, Nihei KT, Myer LR (1996) Incidence of plane wave upon a fracture. J Geophys Res 101:25337–25346

    Article  Google Scholar 

  • Hao H, Wu C, Zhou YX (2002) Numerical analysis of blasting-induced stress wave in anisotropic rock mass with continuum damage models. Part I: equivalent material approach. Rock Mech Rock Eng 35:79–94

    Article  Google Scholar 

  • Jaeger JC (1971) Friction of rocks and stability of rock slopes. Géotechnique 21:97–137

    Article  Google Scholar 

  • Jaeger JC, Cook NGW, Zimmerman RW (2007) Fundamentals of rock mechanics, 4th edn. Blackwell Publishing, Malden

    Google Scholar 

  • King MS, Myer LR, Rezowalli JJ (1986) Experimental studies of elastic-wave propagation in a columnar-jointed rock mass. Geophys Prospect 34:1185–1199

    Article  Google Scholar 

  • Li JC (2013) Wave propagation across non-linear rock joints based on time-domain recursive method. Geophys J Int 193:970–985

    Article  Google Scholar 

  • Li JC, Ma GW (2009) Analysis of blast wave interaction with a rock joint. Rock Mech Rock Eng 43:777–787

    Article  Google Scholar 

  • Li N, Zhang P, Duan QW (2003) Dynamic damage model of the rock mass medium with microjoints. Int J Damage Mech 12:163–173

    Article  Google Scholar 

  • Li JC, Li HB, Ma GW, Zhao J (2012) A time-domain recursive method to analyse transient wave propagation across rock joints. Geophys J Int 188:631–644

    Article  Google Scholar 

  • Li JC, Li HB, Jiao YY, Liu YQ, Xia X, Yu C (2014) Analysis for oblique wave propagation across filled joints based on thin-layer interface model. J Appl Geophys 102:39–46

    Article  Google Scholar 

  • Myer LR (2000) Fractures as collections of cracks. Int J Rock Mech Min Sci 37:231–243

    Article  Google Scholar 

  • Perino A, Orta R, Barla G (2012) Wave propagation in discontinuous media by the scattering matrix method. Rock Mech Rock Eng 45:901–918

    Google Scholar 

  • Pyrak-Nolte LJ (1988) Seismic visibility of fractures. PhD Thesis. University of California, Berkeley

  • Pyrak-Nolte LJ, Myer LR, Cook NGW (1990) Transmission of seismic-waves across single natural fractures. J Geophys Res 95:8617–8638

    Article  Google Scholar 

  • Richer NH (1977) Transient waves in visco-elastic media. Elsevier, Amsterdam

    Google Scholar 

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

    Article  Google Scholar 

  • Stephansson O, Lande G, Bodare A (1979) A seismic study of shallow jointed rocks. Int J Rock Mech Min Sci Geomech Abstr 16:319–327

    Article  Google Scholar 

  • Suárez-Rivera R (1992) The influence of thin clay layers containing liquids on the propagation of shear waves. Ph.D. thesis, University of California, Berkeley

  • Wu W, Zhu JB, Zhao J (2013a) Dynamic response of a rock fracture filled with viscoelastic materials. Eng Geol 160:1–7

    Article  Google Scholar 

  • Wu W, Zhu JB, Zhao J (2013b) A further study on seismic response of a set of parallel rock fractures filled with viscoelastic materials. Geophys J Int 192:671–675

    Article  Google Scholar 

  • Wu W, Li JC, Zhao J (2014) Role of filling materials in a P-wave interaction with a rock fracture. Eng Geol 172:77–84

    Article  Google Scholar 

  • Yang JH, Lu WB, Hu YG, Chen M, Peng Yan (2015) Numerical simulation of rock mass damage evolution during deep-buried tunnel excavation by drill and blast. Rock Mech Rock Eng 48:2045–2059

    Article  Google Scholar 

  • Zhang QB, Zhao J (2014) A review of dynamic experimental techniques and mechanical behaviour of rock materials. Rock Mech Rock Eng 47:1411–1478

    Article  Google Scholar 

  • Zhao H, Gary G (1997) A new method for the separation of waves. Application to the SHPB technique for an unlimited duration of measurement. J Mech Phys Solids 45:1185–1202

    Article  Google Scholar 

  • Zhao J, Zhou YX, Hefny AM, Cai JG, Chen SG, Li HB, Liu JF, Jain M, Foo ST, Seah CC (1999) Rock dynamics research related to cavern development for ammunition storage Tunn Undergr Sp Tech 14:513–526

    Google Scholar 

  • Zhao XB, Zhao J, Cai JG (2006) P-wave transmission across fractures with nonlinear deformational behaviour. Int J Numer Anal Meth Geomech 30:1097–1112

    Article  Google Scholar 

  • Zhou YX, Zhao J (eds) (2011) Advances in rock dynamic and applications. CRC Press, Boca Raton

    Google Scholar 

  • Zhu JB, Zhao J (2013) Obliquely incident wave propagation across rock joints with virtual wave source method. J Appl Geophys 88:23–30

    Article  Google Scholar 

  • Zhu JB, Perino A, Zhao GF, Barla G, Li JC, Ma GW, Zhao J (2011) Seismic response of a single and a set of filled joints of viscoelastic deformational behaviour. Geophys J Int 186:1315–1330

    Article  Google Scholar 

Download references

Acknowledgements

This work was sponsored by the Swiss National Science Foundation (200020_140329). The financial supports from China Scholarship Council (CSC) and the National Nature Science Foundation of China (No. 41525009) are also appreciated.

Author information

Authors and Affiliations

  1. Laboratory of Soil Mechanics, School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne (EPFL), Station 18, 1015, Lausanne, Switzerland

    Yang Zou & Lyesse Laloui

  2. School of Civil Engineering, Southeast University, Nanjing, 210096, China

    Jianchun Li

  3. Department of Civil Engineering, Monash University, Melbourne, VIC, 3800, Australia

    Jian Zhao

Authors
  1. Yang Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianchun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lyesse Laloui
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jian Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianchun Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zou, Y., Li, J., Laloui, L. et al. Analytical Time-Domain Solution of Plane Wave Propagation Across a Viscoelastic Rock Joint. Rock Mech Rock Eng 50, 2731–2747 (2017). https://doi.org/10.1007/s00603-017-1246-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00603-017-1246-7

Keywords

Search

Navigation