Acoustic propagation in a random saturated medium: the biphasic case

Abstract

Homogenized acoustic properties of a non-periodic porous linear elastic solid filled with a viscous Newtonian fluid are derived. A small parameter of the problem is the ratio between typical size of the microstructural inhomogeneity and the macroscopic length scale. We consider the ratio of elasticity coefficients to viscosity coefficients to be of order which leads to the biphasic macroscopic behaviour. To pass to the limit in the governing equations, we employ stochastic two-scale convergence in the mean. Periodic approximation is then used and elimination of corrector terms is carried out in detail for the periodic geometry. The effective equations for the solid displacement, fluid displacement and fluid pressure are obtained and compared to the classical Biot system.

Keywords:

• homogenization
• composite media

AMS Subject Classifications:

• 35A35
• 74Q99

Acknowledgments

Work of R.P. Gilbert was partially supported by the NSF Research Grant DMS-0920850. Work of A. Panchenko was partially supported by DOE Grant DE-FG02-05ER25709, and by NSF Grant OISE-0438765. Work of A. Vasilic was partially supported by UAE University research Grant FOS/IRG-20/11 G00000874.

Notes

Dedicated to Alexander Pankov on the occasion of his 65th birthday.