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Existence of longitudinal waves in anisotropic poroelastic solids

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Abstract

In anisotropic fluid-saturated porous solids, four waves can propagate along a general phase direction. However, solid particles in different waves may not vibrate in mutually orthogonal directions. In the propagation of each of these waves, the displacement of pore–fluid particles may not be parallel to that of solid particles. The polarization for a wave is the direction of aggregate displacement of the particles of the two constituents of a porous aggregate. These polarizations, for different waves, are not mutually orthogonal. Out of the four waves in anisotropic poroelastic medium, two are termed as quasi-longitudinal waves. The prefix ‘quasi’ refers to their polarization being nearly, but not exactly, parallel to the direction of propagation. The existence of purely longitudinal waves in an anisotropic poroelastic medium is ensured by the stationary characters of two expressions. These expressions involve the elastic (stiffness and coupling) coefficients of a porous aggregate and the components of phase direction. Necessary and sufficient conditions for the existence of longitudinal waves are discussed for different anisotropic symmetries. Conditions are also discussed for the existence of the apparent longitudinal waves, i.e., the propagation of wave motion with the particle displacement parallel to the ray direction instead of the phase direction. A graphical solution of a numerical example is shown to check the existence of these apparent longitudinal waves for general directions of phase propagation.

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References

  1. Borgnis F.E.: Specific directions of longitudinal wave propagation in anisotropic media. Phys. Rev. 98, 1000–1005 (1955)

    Article  MATH  Google Scholar 

  2. Brugger K.: Pure modes for elastic waves in crystals. J. Appl. Phys. 36, 759–768 (1965)

    Article  MathSciNet  Google Scholar 

  3. Truesdell C.: Existence of longitudinal waves. J. Acoust. Soc. Am. 40, 729–730 (1996)

    Article  Google Scholar 

  4. Cazzani A., Rovati M.: Extrema of Young’s modulus for elastic solids with tetragonal symmetry. Int. J. Solids Struct. 42, 5057–5096 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ting T.C.T.: Longitudinal and transverse waves in anisotropic elastic materials. Acta Mech. 185, 147–164 (2006)

    Article  MATH  Google Scholar 

  6. Biot M.A.: The theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low-frequency range, II. Higher frequency range. J. Acoust. Soc. Am. 28, 168–191 (1956)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Biot M.A.: Generalized theory of acoustic propagation in porous dissipative media. J. Acoust. Soc. Am. 34, 1254–1264 (1962)

    Article  MathSciNet  Google Scholar 

  9. Mengi Y., McNiven H.D.: Propagation and decay of waves in porous media. J. Acoust. Soc. Am. 64, 1125–1131 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  10. Schmitt D.P.: Acoustical multipole logging in transversely isotropic poroelastic formations. J. Acoust. Soc. Am. 86, 2397–2421 (1989)

    Article  Google Scholar 

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

    Article  Google Scholar 

  12. Thomsen L.: Elastic anisotropy due to aligned cracks in a porous rock. Geophys. Prosp. 43, 805–829 (1995)

    Article  Google Scholar 

  13. Rathore J.S., Fjaer E., Holt R.M., Renlie L.: P- and S-wave anisotropy of a synthetic sandstone with controlled crack geometry. Geophys. Prosp. 43, 711–728 (1995)

    Article  Google Scholar 

  14. Sharma M.D.: Surface wave propagation in a cracked poroelastic half-space lying under a uniform layer of fluid. Geophys. J. Int. 127, 31–39 (1996)

    Article  Google Scholar 

  15. Hudson J.A., Liu E., Crampin S.: The mechanical properties of materials with interconnected cracks and pores. Geophys. J. Int. 124, 105–112 (1996)

    Article  Google Scholar 

  16. Carcione J.M.: Wave Fields in Real Media: Wave propagation in anisotropic, anelastic and porous media. Pergamon Press, New York (2001)

    Google Scholar 

  17. Tod S.R.: The effects of stress and fluid pressure on the anisotropy of interconnected cracks. Geophys. J. Int. 149, 149–156 (2002)

    Article  Google Scholar 

  18. Sharma M.D.: Three-dimensional wave propagation in a general anisotropic poroelastic medium: phase velocity, group velocity and polarization. Geophys. J. Int. 156, 329–344 (2004)

    Article  Google Scholar 

  19. Sharma M.D.: Wave propagation in a general anisotropic poroelastic medium with anisotropic permeability: phase velocity and attenuation. Int. J. Solids Struct. 41, 4587–4597 (2004)

    Article  MATH  Google Scholar 

  20. Johnson D.L., Koplik J., Dashen R.: Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J. Fluid Mech. 176, 379–402 (1987)

    Article  MATH  Google Scholar 

  21. Albert D.G.: A comparison between wave propagation in water-saturated and air-saturated porous materials. J. Appl. Phys. 73, 28–36 (1993)

    Article  Google Scholar 

  22. Sharma M.D.: Group velocity along general direction in a general anisotropic medium. Int. J. Solids Struct. 39, 3277–3288 (2002)

    Article  MATH  Google Scholar 

  23. Rasolofosaon P.N.J., Zinszner B.E.: Comparison between permeability anisotropy and elasticity anisotropy of reservoir rocks. Geophysics 67, 230–240 (2002)

    Article  Google Scholar 

  24. Helbig K.: Foundations of Anisotropy for Exploration Seismics. Pergamon Press, New York (1994)

    Google Scholar 

  25. Synge J.L.: Flux of energy for elastic waves in anisotropic media. Proc. R.I.A. 58, 13–21 (1956)

    MATH  MathSciNet  Google Scholar 

  26. Chadwick P., Whitworth A.M., Borejko P.: Basic theory of small-amplitude waves in a constrained elastic body. Arch. Rat. Mech. Anal. 87, 339–354 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  27. Borejko P.: Inhomogeneous plane waves in a constrained elastic body. Q. J. Mech. Appl. Math. 40, 71–87 (1987)

    Article  MATH  Google Scholar 

  28. Ben-Menahem A., Sena A.G.: Seismic source theory in stratified anisotropic media. J. Geophys. Res. 95, 15395–15427 (1990)

    Article  Google Scholar 

  29. Crampin S., McGonigle R.: The variations of delays in stress induced polarisation anomalies. Geophys. J. R. Astro. Soc. 64, 115–131 (1981)

    Google Scholar 

  30. Aki K., Richards P.G.: Quantitative Seismology, Theory and Methods. Freeman, New York (1980)

    Google Scholar 

  31. Ben-Menahem A., Singh S.J.: Seismic Waves and Sources. Springer, New York (1981)

    MATH  Google Scholar 

  32. Ben-Menahem A., Gibson R.L. Jr., Sena A.G.: Green’s tensor and radiation patterns of point sources in general anisotropic inhomogeneous elastic media. Geophys. J. Int. 107, 297–308 (1991)

    Article  Google Scholar 

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  1. Department of Mathematics, Kurukshetra University, Kurukshetra, 136 119, India

    Mohan D. Sharma

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Sharma, M.D. Existence of longitudinal waves in anisotropic poroelastic solids. Acta Mech 208, 269–280 (2009). https://doi.org/10.1007/s00707-009-0153-8

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  • DOI: https://doi.org/10.1007/s00707-009-0153-8

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