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Surface waves in nonlocal transversely isotropic liquid-saturated porous solid

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Abstract

In this paper, propagation of surface waves in nonlocal transversely isotropic liquid-saturated porous solid half-space has been investigated. The model is proposed for two different situations of which one is for excluding fluid nonlocal effect, and the other is for including fluid nonlocal effect. The existence of a new wave that arises due to the presence of nonlocality parameter in the medium as well as some critical circular frequencies is derived. Dispersion equation for the propagation of Rayleigh-type surface waves and their conditions of existence at the free surface of transversely isotropic liquid-saturated porous solids has been obtained. The boundary may be opened surface pores or sealed surface pores. Dependence of the velocities of surface waves, Rayleigh wave type on the direction of propagation and frequency has been illustrated. These numerical results report the phenomena such as anomalous negative dispersion of wave velocities with frequency.

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References

  1. Baferani, A.H., Ohadi, A.R.: Analytical investigation of the acoustic behavior of nanocomposite porous media by using modified nonlocal Biot’s equations. J. Vib. Control (2017). https://doi.org/10.1177/1077546317693184

    Article  Google Scholar 

  2. Barak, M.S., Kaliraman, V.: Reflection and transmission of elastic waves from an imperfect boundary between micropolar elastic solid half space and fluid saturated porous solid half space. Mech. Adv. Mater. Struct. 26(14), 1226–1233 (2019)

    Article  Google Scholar 

  3. Barak, M.S., et al.: Inhomogeneous wave propagation in partially saturated soils. Wave Motion 93, (2020). https://doi.org/10.1016/j.wavemoti.2019.102470

  4. Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am. 28(2), 179–191 (1956)

    Article  MathSciNet  Google Scholar 

  5. Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. J. Acoust. Soc. Am. 28(2), 168–178 (1956)

    Article  MathSciNet  Google Scholar 

  6. Biot, M.A.: Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33(4), 1482–1498 (1962)

    Article  MathSciNet  Google Scholar 

  7. Chakraborty, A.: Wave propagation in anisotropic media with nonlocal elasticity. Solids Struct. 44, 5723–5741 (2007)

    Article  Google Scholar 

  8. Chakraborty, A.: Prediction of negative dispersion by a nonlocal poroelastic theory. Acoust. Soc. Am. 123(1), 56–67 (2008)

    Article  Google Scholar 

  9. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer-Verlag New York, Inc, New York (2001)

    MATH  Google Scholar 

  10. Liu, K., Liu, Y.: Propagation characteristic of Rayleigh waves in orthotropic fluid-saturated porous media. J. Sound Vib. 271, 1–13 (2004)

    Article  Google Scholar 

  11. Marutyan, K.G., Holland, M.R., Miller, J.G.: Anomalous negative dispersion in bone can result from the interference of fast and slow waves. J. Acoust. Soc. Am. 120(5) (2006). https://doi.org/10.1121/1.2357187

  12. Sharma, M.D.: Surface waves in a general anisotropic poroelastic solid half-space. Geophys. J. Int. 159, 703–710 (2004)

    Article  Google Scholar 

  13. Sharma, M.D.: Rayleigh wave at the surface of a general anisotropic poroelastic medium: derivation of real secular equation. Proc. R. Soc. 474 (2018). https://doi.org/10.1098/rspa.2017.0589

  14. Sharma, M.D., Gogna, M.L.: Wave propagation in anisotropic liquid-saturated porous solids. J. Acoust. Soc. Am. 90(2), 1068–1073 (1991)

    Article  Google Scholar 

  15. Tomar, S., Khurana, A.: Rayleigh type waves in nonlocal micropolar solids half-space. Ultrasonics 73, 162–168 (2016)

    Google Scholar 

  16. Tong, L.H., Yu, Y., Hu, W., Shi, Y., Xu, C.: On wave propagation characteristics in fluid saturated porous materials by a nonlocal Biot theory. J. Sound Vib. 379, 106–118 (2016)

    Article  Google Scholar 

  17. Tong, L.H., Lai, S.K., Zeng, L.L., Xu, C.J., Yang, J.: Nonlocal scale effect on Rayleigh wave propagation in porous fluid-saturated materials. Int. J. Mech. Sci. 148, 459–466 (2018)

    Article  Google Scholar 

  18. Tung, D.X.: Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space. Vietnam J. Mech. 41(4), 363–371 (2019)

    Article  Google Scholar 

  19. Vinh, P.C., Aoudia, A., Giang, P.T.H.: Rayleigh waves in orthotropic fluid-saturated porous media. Wave Motion 61, 73–82 (2016)

    Article  MathSciNet  Google Scholar 

  20. Vinh, P.C., Aoudia, A., Anh, V.T.N.: Rayleigh waves in anisotropic porous media and the polarization vector method. Wave Motion 83, 202–213 (2018)

    Article  MathSciNet  Google Scholar 

  21. Wear, K.A.: Group velocity, phase velocity, and dispersion in human calcaneus in vivo. J. Acoust. Soc. Am. 121(4), 2431–2437 (2007)

    Article  Google Scholar 

Download references

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  1. Faculty of Civil Engineering, Hanoi Architectural University, Km 10 Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

    Do Xuan Tung

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  1. Do Xuan Tung
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Appendix A

Appendix A

1.1 The elements \(a_{i}\) and \(a_{i}^{'}\)

$$\begin{aligned} a_{1}= & {} L-\rho _{11}\epsilon ^2k^2c^2, \quad a_{2}=(A+2N)-\rho _{11}c^2-\rho _{11}\epsilon ^2k^2c^2, \quad a_{3}=L+F\\ a_{4}= & {} -\rho _{12}\epsilon ^2k^2c^2,\quad a_{5}=M-\rho _{12}c^2-\rho _{12}\epsilon ^2k^2c^2,\quad a_{6}=M\\ a_{7}= & {} C-\rho _{11}\epsilon ^2k^2c^2,\quad a_{8}=L-\rho _{11}c^2-\rho _{11}\epsilon ^2k^2c^2, \quad a_{9}=Q, \quad a_{10}=Q-\rho _{12}\epsilon ^2k^2c^2\\ a_{11}= & {} -\rho _{12}c^2-\rho _{12}\epsilon ^2k^2c^2, \quad a_{12}=-\rho _{22}\epsilon ^2k^2c^2,\quad a_{13}=R-\rho _{22}c^2-\rho _{22}\epsilon ^2k^2c^2\\ a_{14}= & {} R,\quad a_{15}=R-\rho _{22}\epsilon ^2k^2c^2,\quad a_{16}=-\rho _{22}c^2-\rho _{22}\epsilon ^2k^2c^2\\ a_{5}^{'}= & {} M-\rho _{12}c^2,\quad a_{10}^{'}=Q,\quad a_{11}^{'}=-\rho _{12}c^2, \quad a_{13}^{'}=R-\rho _{22}c^2\\ a_{15}^{'}= & {} R, \quad a_{16}^{'}=-\rho _{22}c^2 \end{aligned}$$

1.2 The coefficients of characteristic equation—including fluid nonlocal effect case

$$\begin{aligned} t_{8}&=\left( a_{1}a_{12}-a_{4}^2\right) \left( a_{7}a_{15}-a_{10}^2\right) \\ t_{6}&=2a_{4}a_{5}a_{10}^2 - a_{1}a_{7}a_{14}^2 - a_{1}a_{10}^2a_{13} - a_{2}a_{10}^2a_{12} - a_{1}a_{9}^2a_{15} + 2a_{4}^2a_{10}a_{11} - a_{6}^2a_{7}a_{12} - a_{4}^2a_{7}a_{16}\\&\quad - a_{4}^2a_{8}a_{15} - a_{3}^2a_{12}a_{15} - 2a_{4}a_{6}a_{9}a_{10} + 2a_{3}a_{4}a_{9}a_{15} - 2a_{3}a_{4}a_{10}a_{14} + 2a_{3}a_{6}a_{10}a_{12} - 2a_{4}a_{5}a_{7}a_{15}\\&\quad + 2a_{4}a_{6}a_{7}a_{14} + 2a_{1}a_{9}a_{10}a_{14} - 2a_{1}a_{10}a_{11}a_{12} + a_{1}a_{7}a_{12}a_{16} + a_{1}a_{7}a_{13}a_{15} + a_{1}a_{8}a_{12}a_{15} + a_{2}a_{7}a_{12}a_{15}\\ t_{4}&=a_{4}^2a_{11}^2 + a_{5}^2a_{10}^2 + a_{6}^2a_{9}^2 + a_{3}^2a_{14}^2 - a_{1}a_{8}a_{14}^2 - a_{2}a_{7}a_{14}^2 - a_{1}a_{11}^2a_{12} - a_{2}a_{10}^2a_{13} - a_{1}a_{9}^2a_{16} - a_{2}a_{9}^2a_{15}\\&\quad - a_{6}^2a_{7}a_{13} - a_{6}^2a_{8}a_{12} - a_{5}^2a_{7}a_{15} - a_{4}^2a_{8}a_{16} - a_{3}^2a_{12}a_{16} - a_{3}^2a_{13}a_{15} + 4a_{4}a_{5}a_{10}a_{11} - 2a_{4}a_{6}a_{9}a_{11}\\&\quad - 2a_{5}a_{6}a_{9}a_{10} + 2a_{3}a_{4}a_{9}a_{16} - 2a_{3}a_{4}a_{11}a_{14} + 2a_{3}a_{5}a_{9}a_{15} - 2a_{3}a_{5}a_{10}a_{14} - 2a_{3}a_{6}a_{9}a_{14}\\&\quad + 2a_{3}a_{6}a_{10}a_{13} + 2a_{3}a_{6}a_{11}a_{12} - 2a_{4}a_{5}a_{7}a_{16} - 2a_{4}a_{5}a_{8}a_{15} + 2a_{4}a_{6}a_{8}a_{14} + 2a_{5}a_{6}a_{7}a_{14}\\&\quad + 2a_{1}a_{9}a_{11}a_{14} - 2a_{1}a_{10}a_{11}a_{13} + 2a_{2}a_{9}a_{10}a_{14} - 2a_{2}a_{10}a_{11}a_{12} + a_{1}a_{7}a_{13}a_{16} + a_{1}a_{8}a_{12}a_{16}\\&\quad + a_{1}a_{8}a_{13}a_{15} + a_{2}a_{7}a_{12}a_{16} + a_{2}a_{7}a_{13}a_{15} + a_{2}a_{8}a_{12}a_{15}\\ t_{2}&=2a_{4}a_{5}a_{11}^2 - a_{2}a_{8}a_{14}^2 - a_{1}a_{11}^2a_{13} - a_{2}a_{11}^2a_{12} + 2a_{5}^2a_{10}a_{11} - a_{2}a_{9}^2a_{16} - a_{6}^2a_{8}a_{13} - a_{5}^2a_{7}a_{16}\\&\quad - a_{5}^2a_{8}a_{15} - a_{3}^2a_{13}a_{16} - 2a_{5}a_{6}a_{9}a_{11} + 2a_{3}a_{5}a_{9}a_{16} - 2a_{3}a_{5}a_{11}a_{14} + 2a_{3}a_{6}a_{11}a_{13} - 2a_{4}a_{5}a_{8}a_{16}\\&\quad + 2a_{5}a_{6}a_{8}a_{14} + 2a_{2}a_{9}a_{11}a_{14} - 2a_{2}a_{10}a_{11}a_{13} + a_{1}a_{8}a_{13}a_{16} + a_{2}a_{7}a_{13}a_{16} + a_{2}a_{8}a_{12}a_{16} + a_{2}a_{8}a_{13}a_{15}\\ t_{0}&=\left( a_{2}a_{13}-a_{5}^2\right) \left( a_{8}a_{16}-a_{11}^2\right) \end{aligned}$$

1.3 The coefficients of characteristic equation—excluding fluid nonlocal effect case

$$\begin{aligned} t_{6}^{'}&=a_{4}a_{6}a_{7}a_{14} - a_{1}a_{9}^2a_{15}^{'} - a_{4}a_{6}a_{9}a_{10} - a_{1}a_{7}a_{14}^2 + a_{1}a_{9}a_{10}a_{14} - a_{3}a_{4}a_{14}a_{10}^{'} + a_{1}a_{9}a_{14}a_{10}^{'}\\&\quad + a_{4}a_{10}a_{5}^{'}a_{10}^{'} + a_{3}a_{4}a_{9}a_{15}^{'} - a_{4}a_{7}a_{5}^{'}a_{15}^{'} - a_{1}a_{10}a_{10}^{'}a_{13}^{'} + a_{1}a_{7}a_{13}^{'}a_{15}^{'}\\ t_{4}^{'}&=a_{6}^2a_{9}^2 + a_{3}^2a_{14}^2 - a_{1}a_{8}a_{14}^2 - a_{2}a_{7}a_{14}^2 - a_{6}^2a_{7}a_{13}^{'} - a_{2}a_{9}^2a_{15}^{'} - a_{1}a_{9}^2a_{16}^{'} - a_{3}^2a_{13}^{'}a_{15}^{'} - a_{4}a_{6}a_{9}a_{11}\\&\quad - a_{5}a_{6}a_{9}a_{10} - 2a_{3}a_{6}a_{9}a_{14} + a_{4}a_{6}a_{8}a_{14} + a_{5}a_{6}a_{7}a_{14} + a_{1}a_{9}a_{11}a_{14} + a_{2}a_{9}a_{10}a_{14} - a_{3}a_{10}a_{14}a_{5}^{'}\\&\quad + a_{6}a_{7}a_{14}a_{5}^{'} - a_{3}a_{5}a_{14}a_{10}^{'} + a_{2}a_{9}a_{14}a_{10}^{'} - a_{3}a_{4}a_{14}a_{11}^{'} + a_{1}a_{9}a_{14}a_{11}^{'} + a_{3}a_{6}a_{10}a_{13}^{'} + a_{4}a_{11}a_{5}^{'}a_{10}^{'} \\&\quad + a_{5}a_{10}a_{5}^{'}a_{10}^{'} - a_{6}a_{9}a_{5}^{'}a_{10}^{'} + a_{3}a_{5}a_{9}a_{15}^{'} + a_{4}a_{10}a_{5}^{'}a_{11}^{'} + a_{3}a_{4}a_{9}a_{16}^{'} + a_{3}a_{9}a_{5}^{'}a_{15}^{'} - a_{4}a_{8}a_{5}^{'}a_{15}^{'} \\&\quad - a_{5}a_{7}a_{5}^{'}a_{15}^{'} - a_{4}a_{7}a_{5}^{'}a_{16}^{'} + a_{3}a_{6}a_{10}^{'}a_{13}^{'} - a_{1}a_{11}a_{10}^{'}a_{13}^{'} - a_{2}a_{10}a_{10}^{'}a_{13}^{'} - a_{1}a_{10}a_{11}^{'}a_{13}^{'} + a_{1}a_{8}a_{13}^{'}a_{15}^{'} \\&\quad + a_{2}a_{7}a_{13}^{'}a_{15}^{'} + a_{1}a_{7}a_{13}^{'}a_{16}^{'}\\ t_{2}^{'}&=a_{5}a_{6}a_{8}a_{14} - a_{6}^2a_{8}a_{13}^{'} - a_{2}a_{9}^2a_{16}^{'} - a_{3}^2a_{13}^{'}a_{16}^{'} - a_{5}a_{6}a_{9}a_{11} - a_{2}a_{8}a_{14}^2 + a_{2}a_{9}a_{11}a_{14}\\&\quad - a_{3}a_{11}a_{14}a_{5}^{'} + a_{6}a_{8}a_{14}a_{5}^{'} - a_{3}a_{5}a_{14}a_{11}^{'} + a_{2}a_{9}a_{14}a_{11}^{'} + a_{3}a_{6}a_{11}a_{13}^{'} + a_{5}a_{11}a_{5}^{'}a_{10}^{'} \\&\quad + a_{4}a_{11}a_{5}^{'}a_{11}^{'} + a_{5}a_{10}a_{5}^{'}a_{11}^{'} - a_{6}a_{9}a_{5}^{'}a_{11}^{'} + a_{3}a_{5}a_{9}a_{16}^{'} - a_{5}a_{8}a_{5}^{'}a_{15}^{'} + a_{3}a_{9}a_{5}^{'}a_{16}^{'} - a_{4}a_{8}a_{5}^{'}a_{16}^{'}\\&\quad - a_{5}a_{7}a_{5}^{'}a_{16}^{'} - a_{2}a_{11}a_{10}^{'}a_{13}^{'} + a_{3}a_{6}a_{11}^{'}a_{13}^{'} - a_{1}a_{11}a_{11}^{'}a_{13}^{'} - a_{2}a_{10}a_{11}^{'}a_{13}^{'} + a_{2}a_{8}a_{13}^{'}a_{15}^{'} \\&\quad + a_{1}a_{8}a_{13}^{'}a_{16}^{'} + a_{2}a_{7}a_{13}^{'}a_{16}^{'}\\ t_{0}^{'}&=\left( a_{8}a_{16}^{'}-a_{11}a_{11}^{'}\right) \left( a_{2}a_{13}^{'}-a_{5}a_{5}^{'}\right) \end{aligned}$$

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Tung, D.X. Surface waves in nonlocal transversely isotropic liquid-saturated porous solid. Arch Appl Mech 91, 2881–2892 (2021). https://doi.org/10.1007/s00419-021-01940-2

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