Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T13:28:33.341Z Has data issue: false hasContentIssue false
  • English
  • Français

Elastic waves in layered media: Two-scale homogenization approach

Published online by Cambridge University Press:  01 August 2012

V. V. SHELUKHIN  and
A. E. ISAKOV
V. V. SHELUKHIN
Affiliation:
Lavrentyev Institute of Hydrodynamics, Novosibirsk 630090, Russia emails: shelukhin@list.ru, foru86@mail.ru
A. E. ISAKOV
Affiliation:
Lavrentyev Institute of Hydrodynamics, Novosibirsk 630090, Russia emails: shelukhin@list.ru, foru86@mail.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

Using the two-scale convergence approach, we derive equations which govern transversal time-harmonic waves through a layered medium taking the form of a poroelastic composite saturated with a viscous fluid. To improve convergence, we construct a corrector. We study how wave speed and attenuation time depend on porosity and frequency. We prove that the Darcy permeability and the acoustic permeability in the Biot equations do not coincide.

Keywords

74Q1074Q1576S0535B27
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 6 , December 2012 , pp. 691 - 707
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aki, K. & Richards, P. (1980) Quantitative Seismology: Theory and Methods, W.H. Freean and Co., San Francisco.Google Scholar
[2]Allaire, G. (1992) Homogenization and two-scale convergence. SIAM J. Math. Anal. 32, 1482–1518.CrossRefGoogle Scholar
[3]Amirat, Y. & Shelukhin, V. (2008) Electroosmosis law via homogenization of electrolyte flow equations in porous media. J. Math. Anal. Appl. 342 (1), 1227–1245.CrossRefGoogle Scholar
[4]Auriault, J.-L. (1997) In: Homogenization and Porous Media, Interdisciplinary Applied Mathematics, Chapter 8. Poroelastic media. Hornung, U. (Ed.). Springer, Berlin, pp. 163–182.Google Scholar
[5]Biot, M. A. (1955) Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range. II. High frequency range. J. Acoust. Soc. Am. 28, 168–191.Google Scholar
[6]Biot, M. A. (1962) Mechanics of deformation and acoustic propagation in porous media. J. Appl. Phys. 33, 1482–1498.CrossRefGoogle Scholar
[7]Berryman, J. G. (1980) Confirmation of Biot's theory. Appl. Phys. Lett. 37, 382–384.Google Scholar
[8]Brekhovskikh, L. M. (1960) Waves in Layered Media, Academic Press, New York.Google Scholar
[9]Burridge, R. & Keller, J. B. (1981) Poroelasticity equations derived from microstructure. J. Acoust. Soc. Am. 70, 1140–1146.CrossRefGoogle Scholar
[10]Conca, C. (1985) Numerical results on the homogenization of Stokes and Navier–Stokes equations modelling a class of problems from fluid mechanics. Comput. Methods Appl. Mech. Eng. 53 (3), 223–258.Google Scholar
[11]Gilbert, R. P. & Mikelic, A. (2000) Homogenizing the acoustic properties of the seabed: Part I. Nonlinear Anal. 40, 185–212.Google Scholar
[12]Jyothi, N. V. N., Prasanna, M., Prabha, S., Seetha Ramaiah, P., Srawan, G. & Sakarkar, S. N. (2009) Microencapsulation techniques, factors influencing encapsulation efficiency: A review. Internet J. Nanotechnol. 3 (1).Google Scholar
[13]Kennett, B. L. N. (1983) Seismic Wave Propagation in Stratified Media, Cambridge University Press, Cambridge.Google Scholar
[14]Levy, T. (1979) Propagation of waves in a fluid-saturated porous elastic solid. Int. J. Eng. Sci. 17, 1005–1014.Google Scholar
[15]Lukkassen, D., Nguetseng, G. & Wall, P. (2002) Two-scale convergence. Int. J. Pure Appl. Math. 2 (1), 35–86.Google Scholar
[16]Nguetseng, G. (1989) A general convergence result for a functional related to the theory of homogenization. Math. Anal. 20, 608–623.Google Scholar
[17]Phona, T. J. (1980) Observation of a second bulk compressional wave in a porous medium at ultrasonic frequencies. Appl. Phys. Lett. 36, 259–261.Google Scholar
[18]Sanchez-Palencia, E. (1980) Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics, Springer, New York.Google Scholar
[19]Shelukhin, V., Yeltsov, I. & Paranichev, I. (2011) The electrokinetic cross-coupling coefficient: Two-scale homogenization approach. World J. Mech. 1 (1), 127–136.CrossRefGoogle Scholar
[20]Tsvakin, I. (1995) Seismic Wave Fields in Layered Isotropic Media, Samizdat Press, USA.Google Scholar