Effective Properties of a Composite Half-Space: Exploring the Relationship Between Homogenization and Multiple-Scattering Theories

• Published in: Quarterly journal of mechanics and applied mathematics 2010-05, Vol.63 (2), p.145-175
• Main Authors: Parnell, W. J., Abrahams, I. D., Brazier-Smith, P. R.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Physics; Solid mechanics; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)
• ISSN: 0033-5614; 1464-3855
• DOI: 10.1093/qjmam/hbq002

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 φ. 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.