Modelling of Stoneley wave transference at the frictional interface between ice and rock medium

Abstract

The present article attempts to study the propagation of surface seismic waves (Stoneley waves) in the layered structure composed of rock and ice medium. The interface between the two media (ice and rock) is considered to be frictional. Mathematical model of the present problem is formulated by adapting the Coulomb frictional boundary conditions. The frequency relation is obtained in the determinant form. The non-dispersive nature of the Stoneley wave is observed through the frequency relation. The non-dimensional phase velocity and damping parameter curves have been plotted against the non-dimensional angular frequency. Effect of different parameters (viscoelastic coefficient of the rock medium, frictional interface parameter, anisotropy parameter and initial stress of both the media) on the phase velocity and damping has been distinctly marked and shown graphically.

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References

  1. 1.

    Kumar, A., Kumar, N., Mukhopadhyay, S., Baidya, P.R.: Crustal and uppermost mantle structures in the frontal Himalaya and Indo-Gangetic basin using surface wave: tectonic implications. Quatern. Int. 462, 34–49 (2017)

    Article  Google Scholar 

  2. 2.

    Chen, L., Capitanio, F.A., Liu, L., Gerya, T.V.: Crustal rheology controls on the Tibetan plateau formation during India-Asia convergence. Nat. Commun. 8, 15992 (2017)

    Article  Google Scholar 

  3. 3.

    Gupta, I.D., Trifunac, M.D.: Attenuation of strong earthquake ground motion-I: dependence on geology along the wave path from the Hindu Kush subduction to Western Himalaya. Soil Dyn. Earthq. Eng. 114, 127–146 (2018)

    Article  Google Scholar 

  4. 4.

    Miller, R.K.: An approximate method of analysis of the transmission of elastic waves through a frictional boundary. J. Appl. Mech. 44(4), 652–656 (1977)

    Article  Google Scholar 

  5. 5.

    Miller, R.K.: The effects of boundary friction on the propagation of elastic waves. B Seismol. Soc. Am. 68(4), 987–998 (1978)

    Google Scholar 

  6. 6.

    Walsh, J.B.: Seismic wave attenuation in rock due to friction. J. Geophys. Res. 71(10), 2591–2599 (1966)

    Article  Google Scholar 

  7. 7.

    Ostoja-Starzewski, M.: Propagation of Rayleigh, Scholte and Stoneley waves along random boundaries. Probab. Eng. Mech. 2(2), 64–73 (1987)

    Article  Google Scholar 

  8. 8.

    Stoneley, R.: Elastic waves at the surface of separation of two solids. Proc. R. Soc. Lond. Ser. A Contain. Pap. Math. Phys. Char. 106(738), 416–428 (1924)

    MATH  Google Scholar 

  9. 9.

    Murty, G.S.: A theoretical model for the attenuation and dispersion of Stoneley waves at the loosely bonded interface of elastic half spaces. Phys. Earth Planet. Int. 11(1), 65–79 (1975)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Murty, G.S.: Reflection, transmission and attenuation of elastic waves at a loosely-bonded interface of two half spaces. Geophys. J. Int. 44(2), 389–404 (1976)

    Article  Google Scholar 

  11. 11.

    Dasgupta, A.: Effect of high initial stress on the propagation of Stoneley waves at the interface of two isotropic elastic incompressible media. Indian J. Pure Appl. Math. 12(7), 919–926 (1981)

    MATH  Google Scholar 

  12. 12.

    Goda, M.A.: The effect of inhomogeneity and anisotropy on Stoneley waves. Acta Mech. 93(1), 89–98 (1992)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Tajuddin, M., Reddy, G.N.: Effect of curvature on Stoneley waves in poroelastic solids. Indian J. Pure Appl. Mater. 33(3), 391–402 (2002)

    MATH  Google Scholar 

  14. 14.

    Vinh, P.C., Giang, P.T.H.: Uniqueness of Stoneley waves in pre-stressed incompressible elastic media. Int. J. Non linear Mech. 47(2), 128–134 (2012)

    Article  Google Scholar 

  15. 15.

    Biot, M.A.: The influence of initial stress on elastic waves. J. Appl. Phys. 11(8), 522–530 (1940)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Abd-Alla, A.M., Abo-Dahab, S.M., Hammad, H.A.H.: Propagation of Rayleigh waves in generalized magneto-thermoelastic orthotropic material under initial stress and gravity field. Appl. Math. Model. 35(6), 2981–3000 (2011)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Nam, N.T., Merodio, J., Ogden, R.W., Vinh, P.C.: The effect of initial stress on the propagation of surface waves in a layered half-space. Int. J. Solids Struct. 88, 88–100 (2016)

    Article  Google Scholar 

  18. 18.

    Shams, M.: Effect of initial stress on Love wave propagation at the boundary between a layer and a half-space. Wave Motion 65, 92–104 (2016)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Singh, A.K., Negi, A., Verma, A.K., Kumar, S.: Analysis of stresses induced due to a moving load on irregular initially stressed heterogeneous viscoelastic rock medium. J. Eng. Mech. 143(9), 04017096 (2017)

    Article  Google Scholar 

  20. 20.

    Kuznetsov, S. V.: Observance of Stoneley waves at the Lamb wave dispersion in stratified media. Arch. Appl. Mech. 90(5), 957–965 (2020)

  21. 21.

    Hearmon, R.F.S., Maradudin, A.A.: An introduction to applied anisotropic elasticity. Phys. Today 14, 48 (1961)

    Article  Google Scholar 

  22. 22.

    Anderson, D.L.: Preliminary results and review of sea ice elasticity and related studies. Trans. Eng. Inst. Can. 2(3), 2–8 (1958)

    Google Scholar 

  23. 23.

    Chandrasekharaiah, D.S., Keshavan, H.R.: Thermoelastic plane waves in a transversely isotropic body. Acta Mech. 87(1–2), 11–22 (1991)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Honarvar, F., Enjilela, E., Sinclair, A.N., Mirnezami, S.A.: Wave propagation in transversely isotropic cylinders. Int. J. Solids Struct. 44(16), 5236–5246 (2007)

    Article  Google Scholar 

  25. 25.

    Lay, T., Wallace, T.C.: Modern Global Seismology, vol. 58. Elsevier, Amsterdam (1995)

    Google Scholar 

  26. 26.

    Singh, A.K., Negi, A., Chattopadhyay, A., Verma, A.K.: Analysis of different types of heterogeneity and induced stresses in an initially stressed irregular transversely isotropic rock medium subjected to dynamic load. Int. J. Geomech. 17(8), 04017022 (2017)

    Article  Google Scholar 

  27. 27.

    Carcione, J.M.: Wave propagation in anisotropic linear viscoelastic media: theory and simulated wavefields. Geophys. J. Int. 101(3), 739–750 (1990)

    Article  Google Scholar 

  28. 28.

    Chadwick, P., Borejko, P.: Existence and uniqueness of Stoneley waves. Geophys. J. Int. 118(2), 279–284 (1994)

    Article  Google Scholar 

  29. 29.

    Biot, M.A.: Mechanics of Incremental Deformations. Wiley, New York (1964)

    Google Scholar 

  30. 30.

    Iwan, W.D.: A generalization of the concept of equivalent linearization. Int. J. Non linear Mech. 8(3), 279–287 (1973)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Kanai, K.: A new problem concerning surface waves. B Earthq. Res. Inst. 39, 359–366 (1961)

    MathSciNet  Google Scholar 

  32. 32.

    Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland Publishing, New York (1975)

    Google Scholar 

  33. 33.

    Ewing, M.W.: Elastic Waves in Layered Media. McGraw-Hill, New York (1957)

    Google Scholar 

  34. 34.

    Hearmon, R.F.S.: An Introduction to Applied Anisotropic Elasticity. Oxford University Press, Oxford (1961)

    Google Scholar 

  35. 35.

    Zhang, L.: Drilled Shafts in Rock: Analysis and Design. Taylor and Francis Group, Hoboken, NJ (2004)

    Google Scholar 

  36. 36.

    Tsvankin, I.: Anisotropic parameters and P-wave velocity for orthorhombic media. Geophysics 62(4), 1292–1309 (1997)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are thankful to Indian Institute of Technology (ISM), Dhanbad, for providing the necessary research facilities to the authors. On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Correspondence to Shreeta Kumari.

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Appendix

Appendix

$$\begin{aligned} {X_1}= & {} \left( {\frac{{2{c_{44}} - {P_1}}}{2}} \right) {D^2} - {k^2}{c_{11}} + {p^2}{\rho _1}, \end{aligned}$$
(51)
$$\begin{aligned} {X_2}= & {} \frac{{ - ik}}{2}\left( {2{c_{13}} + 2{c_{44}} + {P_1}} \right) D, \end{aligned}$$
(52)
$$\begin{aligned} {X_3}= & {} {c_{33}}{D^2} - {k^2}\left( {{c_{44}} + \frac{{{P_1}}}{2}} \right) + {p^2}{\rho _1}, \end{aligned}$$
(53)
$$\begin{aligned} {X_4}= & {} \frac{{ - ik}}{2}\left( {2{c_{13}} + 2{c_{44}} - {P_1}} \right) D, \end{aligned}$$
(54)
$$\begin{aligned} {X_5}= & {} \left( {\mu + \mu 'ip - \frac{{{P_2}}}{2}} \right) {D^2} - {k^2}\left\{ {\lambda + \lambda 'ip + 2\left( {\mu + \mu 'ip} \right) } \right\} + {p^2}{\rho _2}, \end{aligned}$$
(55)
$$\begin{aligned} {X_6}= & {} \left\{ { - ik\left( {\lambda + \mu + \frac{{{P_2}}}{2}} \right) + kp\left( {\lambda ' + \mu '} \right) } \right\} D, \end{aligned}$$
(56)
$$\begin{aligned} {X_7}= & {} \left\{ {\lambda + \lambda 'ip + 2\left( {\mu + \mu 'ip} \right) } \right\} {D^2} - {k^2}\left( {\mu + \mu 'ip + \frac{{{P_2}}}{2}} \right) + {p^2}{\rho _2}, \end{aligned}$$
(57)
$$\begin{aligned} {X_8}= & {} \left\{ { - ik\left( {\lambda + \mu - \frac{{{P_2}}}{2}} \right) + kp\left( {\lambda ' + \mu '} \right) } \right\} D,~~D =\frac{\partial }{\partial z}, \end{aligned}$$
(58)
$$\begin{aligned} {D_1}= & {} \left\{ \begin{array}{l} \left( {\mu + ip\mu ' - \frac{{{P_2}}}{2}} \right) \left( { - {k^2}m_j^2} \right) + {p^2}{\rho _2} - {k^2}\left( {\lambda + \lambda 'ip + 2\mu + 2\mu 'ip} \right) \end{array} \right\} , \end{aligned}$$
(59)
$$\begin{aligned} {D_2}= & {} \left\{ \begin{array}{l} {k^2}{m_j}\left( {\lambda + \mu + \frac{{{P_2}}}{2}} \right) + i{k^2}{m_j}p\left( {\lambda ' + \mu '} \right) \end{array} \right\} , \end{aligned}$$
(60)
$$\begin{aligned} {D_3}= & {} \left\{ \begin{array}{l} \left\{ {\lambda + 2\mu + ip\left( {\lambda ' + 2\mu '} \right) } \right\} \left( { - {k^2}m_j^2} \right) - {k^2}\left( {\mu + \frac{{{P_2}}}{2}} \right) + {p^2}{\rho _2} - i{k^2}p\mu ' \end{array} \right\} , \end{aligned}$$
(61)
$$\begin{aligned} {D_4}= & {} \left\{ {{k^2}{m_j}\left( {\lambda + \mu - \frac{{{P_2}}}{2}} \right) + ip\left( {\lambda ' + \mu '} \right) } \right\} , \end{aligned}$$
(62)
$$\begin{aligned} {D_5}= & {} {K_3}{m_3} + \overline{{K_3}}, \end{aligned}$$
(63)
$$\begin{aligned} {D_6}= & {} {K_4}{m_4} + \overline{{K_4}}, \end{aligned}$$
(64)
$$\begin{aligned} {D_7}= & {} \left( {\lambda + 2\mu + ip\left( {\lambda ' + 2\mu '} \right) } \right) \overline{{K_3}} {m_3}+ \left( {\lambda + ip\lambda '} \right) {K_3}, \end{aligned}$$
(65)
$$\begin{aligned} {D_8}= & {} \left( {\lambda + 2\mu + ip\left( {\lambda ' + 2\mu '} \right) } \right) \overline{{K_4}} {m_4} + \left( {\lambda + ip\lambda '} \right) {K_4}, \end{aligned}$$
(66)
$$\begin{aligned} {\delta _j}= & {} \frac{{ - \left( {\frac{{2{c_{44}} - {P_1}}}{{2{c_{44}}}}} \right) \left( {m_j^2 - {m_j}} \right) - \frac{{{c_{11}}}}{{{c_{44}}}} + \frac{{{c^2}}}{{\beta _1^2}}{{\left( {1 + i\kappa } \right) }^2} + \frac{{{c_{13}}}}{{{c_{44}}}}{m_j}}}{{ - \frac{{{c_{33}}}}{{{c_{44}}}}m_j^2 - \left( {\frac{{2{c_{44}} + {P_1}}}{{2{c_{44}}}}} \right) + \frac{{{c^2}}}{{\beta _1^2}}{{\left( {1 + i\kappa } \right) }^2} - {m_j}\left( {\frac{{2{c_{13}} + 2{c_{44}} + {P_1}}}{{2{c_{44}}}}} \right) }};j = 1,2 \end{aligned}$$
(67)
$$\begin{aligned} {\delta _j}= & {} \frac{{\left[ {\left\{ {\left( {1 - \frac{{{P_2}}}{{2\mu }}} \right) + \frac{{\mu '\omega i\left( {1 + i\kappa } \right) }}{\mu }} \right\} \left( { - m_j^2} \right) - \left( {\frac{\lambda }{\mu } + 2} \right) + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}\frac{{{{\left( {1 + i\kappa } \right) }^2}}}{{{\beta ^2}}} - {D_9} + {D_{10}} + {D_{11}}} \right] }}{{\left[ {\left\{ {\frac{\lambda }{\mu } + 2 + {D_9}} \right\} \left( { - m_j^2} \right) - \left( {1 + \frac{{{P_2}}}{{2\mu }}} \right) + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}\frac{{{{\left( {1 + i\kappa } \right) }^2}}}{{{\beta ^2}}} - i\left( {1 + i\kappa } \right) \frac{{\mu '\omega }}{\mu } + {D_{10}} + {D_{11}}} \right] }};j = 3,4 \end{aligned}$$
(68)
$$\begin{aligned} {D_9}= & {} i\left( {1 + i\kappa } \right) \left( {\frac{{\lambda '\omega }}{\mu } + \frac{{2\mu '\omega }}{\mu }} \right) ,~~{D_{10}} = {m_j}i\left( {1 + i\kappa } \right) \left( {\frac{{\lambda '\omega }}{\mu } + \frac{{\mu '\omega }}{\mu }} \right) ,~~{D_{11}} = {m_j}\left( {\frac{\lambda }{\mu } + 1 - \frac{{{P_2}}}{{2\mu }}} \right) , \end{aligned}$$
(69)
$$\begin{aligned} {a_1}= & {} \frac{{{c_{33}}}}{{{c_{44}}}}\left( {1 - \frac{{{P_1}}}{{2{c_{44}}}}} \right) , \end{aligned}$$
(70)
$$\begin{aligned} {a_2}= & {} \left( {1 - \frac{{{P_1}}}{{2{c_{44}}}}} \right) \left\{ {1 - \frac{{{P_1}}}{{2{c_{44}}}} - {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}{{\left( {1 + i\kappa } \right) }^2}} \right\} \nonumber \\&+ \frac{{{c_{33}}}}{{{c_{44}}}}\left( {\frac{{{c_{11}}}}{{{c_{44}}}} - {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}{{\left( {1 + i\kappa } \right) }^2}} \right) - \left\{ {1 - {{\left( {\frac{{{c_{13}}}}{{{c_{44}}}} + \frac{{{P_1}}}{{2{c_{44}}}}} \right) }^2}} \right\} , \end{aligned}$$
(71)
$$\begin{aligned} {a_3}= & {} \left\{ {\frac{{ - {c_{11}}}}{{{c_{44}}}} + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}{{\left( {1 + i\kappa } \right) }^2}} \right\} \left\{ { - 1 - \frac{{{P_1}}}{{2{c_{44}}}} + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}{{\left( {1 + i\kappa } \right) }^2}} \right\} , \end{aligned}$$
(72)
$$\begin{aligned} {b_1}= & {} {D_{12}} \times {D_{13}}, \end{aligned}$$
(73)
$$\begin{aligned} {b_2}= & {} \left\{ {{D_{12}} - {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}\frac{{{{\left( {1 + i\kappa } \right) }^2}}}{{{\beta ^2}}}} \right\} {D_{12}} + {D_{13}}\left\{ {{D_{13}} - {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}\frac{{{{\left( {1 + i\kappa } \right) }^2}}}{{{\beta ^2}}}} \right\} \,\nonumber \\&+ {\left\{ {1 + \frac{\lambda }{\mu } + i\left( {1 + i\kappa } \right) \left( {\frac{{\omega \lambda ' + \omega \mu '}}{\mu }} \right) } \right\} ^2}\nonumber \\&- {\left( {\frac{{{P_2}}}{{2\mu }}} \right) ^2}, \end{aligned}$$
(74)
$$\begin{aligned} {b_3}= & {} \left\{ { - {D_{12}} + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}\frac{{{{\left( {1 + i\kappa } \right) }^2}}}{{{\beta ^2}}}} \right\} \left\{ { - \left( {1 + \frac{{\mu '\omega i\left( {1 + i\kappa } \right) }}{\mu } + \frac{{{P_2}}}{{2\mu }}} \right) + {{\left( {\frac{c}{{{\beta _1}}}} \right) }^2}\frac{{{{\left( {1 + i\kappa } \right) }^2}}}{{{\beta ^2}}}} \right\} , \end{aligned}$$
(75)
$$\begin{aligned} {D_{12}}= & {} 2 + \frac{\lambda }{\mu } + i\left( {1 + i\kappa } \right) \left( {\frac{{\omega \lambda ' + 2\omega \mu '}}{\mu }} \right) , \end{aligned}$$
(76)
$$\begin{aligned} {D_{13}}= & {} \left( {1 - \frac{{{P_2}}}{{2\mu }} + \frac{{\mu '\omega i\left( {1 + i\kappa } \right) }}{\mu }} \right) . \end{aligned}$$
(77)

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Sahu, S.A., Kumari, S. & Pankaj, K.K. Modelling of Stoneley wave transference at the frictional interface between ice and rock medium. Arch Appl Mech (2021). https://doi.org/10.1007/s00419-021-01894-5

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Keywords

  • Stoneley wave
  • Frictional interface
  • Ice medium
  • Rock medium
  • Phase velocity
  • Damping parameter