Elastic Wave Scattering by Two Spherical Inclusions in a Poroelastic Medium
Publication: Journal of Engineering Mechanics
Volume 131, Issue 9
Abstract
This study considers the most fundamental problem of multiple scattering in a poroelastic medium. It treats the interaction of a plane compressional elastic wave with a cluster of two of spherical inhomogeneities in a boundless fluid-saturated porous elastic formation. The novel features of Biot classic model for dynamic description of poroelastic material behavior along with the appropriate wave field expansions, the pertinent boundary conditions, and the translational addition theorems for spherical wave functions are employed to develop a closed-form solution in the form of infinite series. The analytical results are illustrated with numerical examples in which a pair of spherical inclusions is insonified by a plane (fast) compressional wave at end-on incidence. The effects of incident wave frequency, proximity of the two inclusions, and inclusion type are examined. Particular attention has been focused on multiple scattering interactions in addition to the slow wave coupling effects which is known to be the primary distinction of the scattering phenomenon in poroelasticity from the classical elastic case. The limiting case involving two elastic spheres submerged in an ideal unbounded fluid medium is considered and fair agreement with a well-known solution is established.
Get full access to this article
View all available purchase options and get full access to this article.
References
Abramowitz, M., and Stegun, I. A. (1964). Handbook of mathematical functions, National Bureau of Standards, Washington, D.C.
Allard, J. F. (1993). Propagation of sound in porous media, modeling sound absorbing materials, Elsevier Applied Science, London.
Atalla, N., and Sgard, F. (2000). “Transmission loss of heterogeneous porous materials.” J. Acoust. Soc. Am., 108, 2625–2644.
Berryman, J. G. (1985). “Scattering by a spherical inhomogeneity in a fluid-saturated porous medium.” J. Math. Phys., 26, 1408–1419.
Biwa, S., Idekoba, S., and Ohno, N. (2002). “Wave attenuation in particulate polymer composites: independent scattering/absorption analysis and comparison to measurement.” Mech. Mater., 34, 671–682.
Bourbie, T., Coussy, O., and Zinszner, B. E. (1987). Acoustics of porous media, Gulf Publishing, Houston.
Datta, S. K., Ledbetter, H. M., Shindo, Y., and Shah, A. H. (1988). “Phase velocity and attenuation of plane elastic waves in a particulate-reinforced composite medium.” Wave Motion, 10, 171–182.
Deresiewicz, H., and Skalak, R. (1963). “On uniqueness in dynamic poroelasticity.” Bull. Seismol. Soc. Am., 53, 783–788.
Domany, E., Entin-wholman, O., and Mizrachi, L. (1984). “Multiple scattering formalism: Application to scattering by two spheres.” J. Appl. Phys., 56, 132–136.
Dutta, N. C., and Ode, H. (1979). “Attenuation and dispersion of compresional waves in fluid-filled porous rocks with partial gas saturation (white model) EM dash 1. Biot theory.” Geophysics, 44, 1777–1788.
Eringen, A. C., and Suhubi, E. S. (1975). Elastodynamics, Vol. II, Academic, New York.
Fu, L. S., and Sheu, Y. C. (1984). “Ultrasonic wave propagation in two-phase media:Spherical inclusions.” Compos. Struct., 2, 289–303.
Gabrielli, P., and Mercier-Finidori, M. (2001). “Acoustic scattering by two spheres: Multiple scattering and symmetry considerations.” J. Sound Vib., 241, 423–439.
Gaunaurd, G. C., Huang, H., and Strifors, H. C. (1995). “Acoustic scattering by a pair of spheres.” J. Acoust. Soc. Am., 98, 495–507.
Gurevich, B., Kelder, O., and Smeulders, D. M. J. (1999). “Validation of the slow compressional wave in porous media: comparison of experiments and numerical simulations.” Transp. Porous Media, 36, 149–160.
Gurevich, B., Sadovnichaja, A. P., Lopatnikov, S. L., and Shapiro, S. A. (1992). “Born approximation in the problem of elastic wave scattering by a spherical inhomogeneity in a fluid-saturated porous medium.” Appl. Phys. Lett., 61, 1275–1286.
Haberman, M. R., Berthelot, Y. H., and Jarzynski, J. (2002). “Micromechanical modeling of viscoelastic voided composites in the low-frequency approximation.” J. Acoust. Soc. Am., 112, 1937–1943.
Hasheminejad, S. M., and Badsar, S. A. (2004). “Acoustic scattering by a pair of poroelastic spheres.” Q. J. Mech. Appl. Math., 57, 95–113.
Hinders, M. K., Rhodes, B. A., and Fang, T. M. (1995). “Particle-loaded composites for acoustic anechoic coatings.” J. Sound Vib., 185, 219–246.
Ivanov, Y. A. (1970). Diffraction of electromagnetic waves on two bodies, National Aeronautics and Space Administration, Washington, D.C.
Johnson, D. L., Koplik, J., and Dashen, R. (1987). “Theory of dynamic permeability and tortuosity in fluid-saturated porous media.” J. Fluid Mech., 76, 379–402.
Johnson, D. L., Plona, T. J., and Kojima, H. (1994). “Probing porous media with first and second sound, II. Acoustic properties of water-saturated porous media.” J. Appl. Phys., 76, 115–125.
Kargl, S. G., and Lim, R. A. (1993). “Transition-matrix formulation of scattering in homogeneous, saturated, porous media.” J. Acoust. Soc. Am., 94, 1527–1550.
Kargl, S. G., Williams, K. L., and Lim, R. (1998). “Double monopole resonance of a gas-filled, spherical cavity in sediment.” J. Acoust. Soc. Am., 103, 265–274.
Kim, J. (2003). “Extinction and propagation of elastic waves in inhomogeneous materials.” Mech. Mater., 35, 877–884.
Krutin, V. N., Markov, M. G., and Yumatov, A. Yu. (1984). “Scattering of a longitudinal wave by a spherical cavity with a fluid in an elastic porous saturated medium.” Appl. Math. Mech., 48, 238–241.
Lim, R. (1996). “Acoustic scattering by targets in porous ocean sediments.” J. Acoust. Soc. Am., 100, 2765–2765.
Lim, R., and Hackman, R. H. (1992). “Formulation of multiple scattering by many bounded obstacles using a multicentered T supermatrix.” J. Acoust. Soc. Am., 91, 613–638.
Maslov, K., Kinra, V. K., and Henderson, B. K. (2000). “Elastodynamic response of a coplanar periodic layer of elastic spherical inclusions.” Mech. Mater., 32, 785–795.
Norris, A. N. (1985). “Scattering of elastic waves by spherical inclusions with applications to low frequency wave propagation in composites.” J. Acoust. Soc. Am., 77, 2012–2022.
O’Neill, T. J., Tebbutt, J. S., and Challis, R. E. (2001). “Convergence criteria for scattering models of ultrasonic wave propagation in suspensions of particles.” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 48, 419–424.
Sato, H., and Shindo, Y. (2003). “Multiple scattering of plane elastic waves in a particle-reinforced-composite medium with graded interfacial layers.” Mech. Mater., 35, 83–106.
Sayers, C. M., and Smith, R. L. (1983). “Ultrasonic velocity and attenuation in an epoxy matrix containing lead inclusion.” J. Phys. D, 16, 1189–1194.
Schafbuch, P. J., Rizzo, F. J., and Thompson, R. B. (1991). “Elastic wave scattering by multiple inclusions.” Proc., ASME Symp. on Enhancing Analysis Techniques Composite Materials, Atlanta, 10, ASME, New York, 103–111.
Shindo, Y., Nozaki, H., and Datta, S. K. (1995). “Effect of interface layers on elastic wave propagation in a metal matrix composite reinforced by particles.” J. Appl. Mech., 62, 178–185.
Simpson, H. J., Houston, B. H., and Lin, R. (2003). “Laboratory measurements of sound scattering from a buried sphere above and below the critical angle.” J. Acoust. Soc. Am., 113, 39–42.
Tourin, A., Fink, M., and Derode, A. (2000). “Multiple scattering of sound.” Waves Random Media, 10, 31–60.
Velea, D., Hickey, C., and Sabatier, J. M. (2001). “Acoustic scattering by buried objects in a rigid porous material.” Proc. SPIE, 4394, 595–606.
Velea, D., Waxier, R., and Sabatier, J. M. (2004). “An effective fluid model for landmine detection using acoustic to seismic coupling.” J. Acoust. Soc. Am., 115, 1993–2003.
Wang, T., and Wang, Y. (2003). “Acoustic wave scattering by a poroelastic sphere embedded in solid matrix.” Proc., World Congress on Ultrasonics, Paris, 1351–1354.
Xiang, N., and Sabatier, J. M. (2003). “An experimental study on antipersonnel landmine detection using acoustic-to-seismic coupling.” J. Acoust. Soc. Am., 113, 1333–1341.
Yamakawa, N. (1962). “Scattering and attenuation of elastic waves.” Geophys. Mag., 31, 63–103.
Yamamoto, K., and Kitahara, M. (2002). “Elastic wave scattering analysis of cavities in poroelastic media using three-dimensional boundary element formulation.” Proc., 2nd Biot Conf. on Poromechanics, Grenoble, France, 857–863.
Yang, R. B. (2003). “A dynamic generalized self-consistent model for wave propagation in particulate composites.” J. Appl. Mech., 70, 575–582.
Yin, Y., Takemasa, F., Hu, N., and Shi, H. (2002). “Rigid-plastic meso-damage constitutive theory for porous composites reinforced by particles.” Compos. Sci. Technol., 62, 697–708.
Ying, C., and Truell, R. (1956). “Scattering of plane longitudinal wave by a spherical obstacle in an isotropically elastic solid.” J. Appl. Phys., 27, 1086–1097.
Zeng, Y. Q., and Liu, Q. H. (2001). “Acoustic detection of buried objects in 3-D fluid saturated porous media: Numerical modeling.” IEEE Trans. Geosci. Remote Sens., GE-39(6), 1165–1173.
Zimmerman, C. (1993). “Scattering of plane compressional waves by a spherical inclusion in a poroelastic medium.” J. Acoust. Soc. Am., 94, 527–536.
Information & Authors
Information
Published In
Copyright
© 2005 ASCE.
History
Received: Jul 28, 2003
Accepted: Jun 2, 2004
Published online: Sep 1, 2005
Published in print: Sep 2005
Notes
Note. Associate Editor: Alexander H.-D. Cheng
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.