The construction of dynamic effective medium theories (EMT) for inhomogeneous fluid or elastic continua

Abstract

A deterministic technique is described that predicts the dynamic (i.e., frequency‐dependent) effective properties of inhomogeneous media. Situations are considered in which a host medium sustaining a distribution of spherical inhomogeneities is either fluid or solid. In the Rayleigh region (viz., a/λ < 1) in which most effective medium theories (EMTs) hold, inhomogeneities of any shape can be assumed spherical without much error. However, assuming them to be penetrable implies that they realistically admit interior fields, which are coupled to the exterior ones by stress and displacement boundary conditions. The outcome of dynamic EMTs is a set of expressions to predict the frequency dependence of the effective (sound) wave speed c̄eff, and the effective (sound) wave attenuation ᾱeff in the mixture. These quantities emerge from the real and imaginary parts of the effective wavenumber k̄eff, which are connected by Kramers‐Kronig type relations, as has been discussed by Dr. Beltzer. An important consideration for the practical usefulness of an EMT is its simplicity. Often, the resulting predictive expressions are so cumbersome that they can only be evaluated by means of quite formidable computer codes. The principal EMT to the discussed here, yields closed‐form simple predictive expressions for c̄eff and ᾱeff, which, in many instances, are valid over significantly broad frequency bands. These expressions reduce to (and contain) almost all existing earlier results for static (f→0) cases. The (present form of this) EMT is constructed by exploiting the presence of the monopole resonance of the inhomogeneities, and thus it works best in the cases in which this resonance is dominant. This EMT has to be further refined to account for cases where the dipole resonance is dominant. In this regard, Dr. Martinez, in a later lecture, will discuss an approach to more accurately deal with heavy/rigid inclusions in a rubberlike matrix. [Work supported by NSWC and ONR.]