Effective properties of a composite half-space: exploring the relationship between homogenization and multiple-scattering theories. (English) Zbl 1273.74415

Parnell, W. J.; Abrahams, I. D.; Brazier-Smith, P. R.

Q. J. Mech. Appl. Math. 63, No. 2, 145-175 (2010).

Summary: A classical problem in applied mathematics is the determination of the effective wavenumber of a composite material consisting of inclusions distributed throughout an otherwise homogeneous host phase. This problem is discussed here in the context of a composite half-space and a new integral equation method is developed. As a means of obtaining the effective material properties (density and elastic moduli) associated with the material, we consider low-frequency elastic waves incident from a homogeneous half-space onto the inhomogeneous material. We restrict attention to dilute dispersions of inclusions and therefore results are obtained under the assumption of small volume fractions \({\varphi}\). We consider how this theory relates to associated predictions derived from multiple-scattering theories (MSTs) in the low-frequency limit. In particular, we show that predictions of the effective elastic properties are exactly the same as those derived via either the non-isotropic Foldy or the Waterman-Truell MSTs.

MSC:

74Q15	Effective constitutive equations in solid mechanics
74Q10	Homogenization and oscillations in dynamical problems of solid mechanics
74J20	Wave scattering in solid mechanics
74E30	Composite and mixture properties

Keywords:

dilute dispersions; low-frequency limit