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Research on Transient Wave Propagation Across Nonlinear Joints Filled with Granular Materials

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Abstract

A nonlinear thin layer interface model overcoming the long-wavelength assumption of displacement discontinuity model is developed to analyze the full-wavelength wave propagation across nonlinear parallel joints. In this study, the filling material is treated as a thin layer with reduced mechanical properties to reveal multiple reflections and time shifting within the filled joint. The nonlinearity of the filling material is considered, and a recursive matrix is derived in time domain. An experimental study on wave propagation across a filled joint was carried out by SHPB test. The quartz sand layer of different filling thicknesses sandwiched between Hopkinson bars was pressured by compressional waves to investigate the wave attenuation. Comparisons of the joint thickness, the wave frequency and the incident angle are carried out between the present model and the existing displacement discontinuity model. The results indicate that the thin layer interface model considering the thickness of the joint is capable of extending the long-wavelength assumption to full-wavelength research and it is more appropriate for filled joints with thick thickness that is comparable to wavelength. Then, this model is extended to parallel joints, and the properties of the filling material (i.e., initial elastic modulus and maximum closure), impact velocity, incident angle and wave frequency on wave attenuation are discussed for a joint set. The spacing dependency of the transmission coefficient for parallel joints is compared with displacement discontinuity model.

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Abbreviations

DDM:

Displacement discontinuity method

TLIM:

Thin layer interface model

ZTIM:

Zero thickness interface model

MPM:

Matrix propagation method

TRM:

Time recursive method

MC:

Method of characteristics

EMM:

Equivalent medium method

PPV:

Peak particle velocity

\( \alpha_{k} ,\beta_{k} \) :

Incident angles of P and S wave for intact rock (k = r)and filled joint (k = f)

\( z_{p,k} ,z_{s,k} \) :

Wave impendence of P and S wave for intact rock (k = r)and filled joint (k = f)

\( v_{rp} ,v_{rs} ,v_{lp} ,v_{{ls}} \) :

Velocities of the right-running P wave, right-running S wave, left-running P and left-running S wave

\( v_{\sigma } ,v_{\tau } \) :

Normal and tangential stresses on the interface

\( cp_{k} ,cs_{k} \) :

Velocities of P and S wave for intact rock (k = r) and filled joint (k = f)

\( \lambda_{{{\text{Lame}},k}} ,\mu_{{{\text{Lame,}}k}} \) :

Lamé constants for intact rock (k = r)and filled joint (k = f)

\( \rho_{k} \) :

Density intact rock (k = r)and filled joint (k = f)

\( \omega ,f \) :

Angular frequency and frequency of the incident wave

\( t \) :

Time

\( j \) :

Joint number

\( \sigma ,\tau \) :

Normal and tangential stresses on the interface

\( k_{\text{n}} ,k_{\text{s}} \) :

Normal and tangential stiffness of the joint

\( \varepsilon_{\sigma } ,\varepsilon_{\tau } \) :

Normal and tangential strains on the interface

\( d_{\sigma } ,d_{\tau } \) :

Normal and tangential closure of the joint

\( \dot{\varepsilon }_{\sigma } ,\dot{\varepsilon }_{\tau } \) :

Normal and tangential strain rates on the interface

\( h_{k} \) :

Thickness of the intact rock (k = r)and filled joint (k = f)

\( E_{k} \) :

Young’s modulus of the intact rock (k = r)and filled joint (k = f)

\( v_{\text{I}} \) :

Incident wave peak particle velocity

\( \xi ,\xi_{{{\text{cr}},1}} ,\xi_{{{\text{cr}},2}} \) :

Normalized spacing, first and second critical normalized spacing

\( \alpha_{\text{c}} ,\beta_{\text{c}} \) :

Critical angles of P and S wave

\( N \) :

Layer number

\( \lambda_{{p}} ,\lambda_{{s}} \) :

Wavelength of P and S wave

\( T_{{{{p}} - {{p}}}} ,T_{{{{s}} - {{s}}}} \) :

Transmission coefficients of P wave and S wave

\( d_{c} ,d_{ \hbox{max} } \) :

Current and maximum joint closure

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Acknowledgements

The authors are very grateful to the editor and the two referees for suggestions which have helped to improve the quality of this paper. In addition, we wish to acknowledge the financial support by National Natural Science Foundation of China (Grant Nos. 41572307 and 51439008).

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Authors and Affiliations

  1. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071, China

    X. F. Li, H. B. Li, J. C. Li & Z. W. Li

  2. University of Chinese Academy of Sciences, Beijing, 100049, China

    X. F. Li

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  1. X. F. Li
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  2. H. B. Li
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  3. J. C. Li
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  4. Z. W. Li
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Li, X.F., Li, H.B., Li, J.C. et al. Research on Transient Wave Propagation Across Nonlinear Joints Filled with Granular Materials. Rock Mech Rock Eng 51, 2373–2393 (2018). https://doi.org/10.1007/s00603-018-1471-8

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