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Modeling Wave Propagation in Rock Masses Using the Contact Potential-Based Three-Dimensional Discontinuous Deformation Analysis Method

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Abstract

The most recently proposed three-dimensional (3D) contact potential-based discontinuous deformation analysis (3D-CPDDA) method is further applied for wave propagation problems in rock masses. A viscous non-reflecting boundary is incorporated into the 3D-CPDDA method to minimize the wave reflections. Furthermore, a force input method is incorporated to eliminate the contamination of scattered waves in the numerical solution and to accurately input the incident wave. Several benchmark problems about P-wave/S-wave propagation in homogeneous rock masses and jointed rock masses are solved to validate the modified 3D-CPDDA method. The numerical results assessed by the modified 3D-CPDDA method agree well with those obtained from analytical methods, which means that the modified 3D-CPDDA method can reliably and correctly simulate wave propagation in rock masses. The modified 3D-CPDDA method warrants further investigation.

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Abbreviations

\({\varvec{u}}\left( {\varvec{x}} \right)\) :

The displacement vector for an arbitrary point \({\varvec{x}}\)

\({\varvec{N}}\) :

The interpolation matrix corresponding to \({\varvec{u}}\left( {\varvec{x}} \right)\)

\({\varvec{d}}\) :

Degree of freedom for a block

\(\left( {x_{0} , { }y_{0} , { }z_{0} } \right)\) :

The centroid coordinates of concerned block

\(\left( {u_{0} , { }v_{0} , w_{0} } \right)\) :

The translation displacement of the rigid body at the centroid coordinates

\(\left( {\alpha_{0} , { }\beta_{0} , \gamma_{0} } \right)\) :

The rigid body rotation components of the concerned block

\(\left( {\varepsilon_{x} , { }\varepsilon_{y} , \varepsilon_{z} } \right)\) :

The normal strains of the concerned block

\(\left( {\gamma_{yz} , { }\gamma_{zx} , \gamma_{xy} } \right)\) :

The shear strains of the concerned block

\({\varvec{\sigma}}\) :

Cauchy stress tensor

\(\rho\) :

Mass density

\(t\) :

Time

D :

The elastic Hooke matrix

\({\Gamma }_{{\text{t}}}\) :

Traction boundary

\({\Gamma }_{{\text{u}}}\) :

Displacement boundary

\(\delta\) and \(\alpha\) :

The Newmark parameters

\(\sigma_{{\text{n}}}\) :

Normal viscous traction

\(\tau_{{{\text{s1}}}}\) and \(\tau_{{{\text{s2}}}}\) :

Two shear tractions

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

Normal component of velocity vector

\(v_{{{\text{s1}}}}\) and \(v_{{{\text{s2}}}}\) :

Two shear components of velocity vector

\(c_{{\text{p}}}\) :

P-wave velocity

\(c_{{\text{s}}}\) :

S-wave velocity

T :

The coordinate transformation matrix

W :

The wave velocity matrix

\(C_{{\text{A}}}\) :

The wave impedance

\(\Delta t\) :

Time step size

\(\left| {T_{1} } \right|\) :

Transmission coefficient

\(K\) :

The normalized value of the joint stiffness

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Acknowledgements

This study is supported by the Youth Innovation Promotion Association CAS, under Grant no. 2020327; and the National Natural Science Foundation of China, under Grant no. 12072357.

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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

    Yongtao Yang & Hong Zheng

  2. Key Laboratory of Geotechnical Mechanics and Engineering of Ministry of Water Resources, Changjiang River Scientific Research Institute, Wuhan, 430010, China

    Dongdong Xu

  3. Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology, Beijing, 100124, China

    Hong Zheng

  4. School of Civil Engineering, Wuhan University, Wuhan, 430072, China

    Zhijun Wu

  5. School of Civil and Transportation Engineering, Hebei University of Technology, Tianjin, 300401, China

    Da Huang

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  1. Yongtao Yang
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Correspondence to Yongtao Yang.

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Yang, Y., Xu, D., Zheng, H. et al. Modeling Wave Propagation in Rock Masses Using the Contact Potential-Based Three-Dimensional Discontinuous Deformation Analysis Method. Rock Mech Rock Eng 54, 2465–2490 (2021). https://doi.org/10.1007/s00603-020-02359-x

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