USING COHERENT WAVES TO EVALUATE DYNAMIC MATERIAL PROPERTIES

A method of estimating effective macroscopic material properties (stiffness and density) for heterogeneous elastic materials containing distributions of identical infinite cylindrical objects, randomly and uniformly distributed, is formulated based on the theory of elastic-wave multiple scattering. We show that the equivalent homogeneous medium possesses effective dynamic complex-valued density and stiffness, which are frequency-dependent. Dynamic explicit expressions, requiring minimal computational effort, are established in terms of both the wavenumber in the object-free host medium and the single-particle forward and backward scattering amplitudes in the far field. Predictions for the dynamic properties are presented in the cases of a SH wave propagating in a solid containing empty cavities and of an incident longitudinal wave interacting with a set of water-immersed solid bars. In both cases, the qualitative behavior of the dynamic properties is in agreement with physical expectations. Introduction Modeling of linear wave propagation in heterogeneous media, such as bubbly liquid [1] or even fiber-reinforced material composite [2, 3], is still a challenging problem in the investigation of internal microstructure. Derivations have been performed to predict both propagation of plane wave [1, 2, 4-7] and dynamic (frequency-dependent) effective material properties [3, 8]. The term effective is associated with the long wavelength assumption used to denote the macroscopic behavior of the continuous phase (host medium) combined with the discontinuous one (objects). The aim of this work is to describe macroscopically the dynamic behavior of a two-phase medium in which the energy dissipation mechanisms are only due to the multiple scattering between the elastic objects contained in the lossless host medium. Independent explicit analytical expressions are derived for the effective frequency-dependent material properties (density and stiffness) of a two-phase medium, in terms of the scattering amplitudes for a single object in the host medium. Unlike previous works [1], no assumption about the expression of the effective density has been made. When the energy losses are induced only by multiple scattering, we show that the effective medium can be seen as a linearly dissipative material in which inertial coupling between the two phases cannot be neglected. In other words the effective stiffness and density are both frequencydependent complex-valued functions. After stating the problem, formulae for the dynamic material properties are established from the responses in transmission and reflection of a two-phase slab viewed by an incident plane wave as either heterogeneous or homogeneous. In this paper we refer to the notion of heterogeneity when we take into account the multiple scattering phenomena between the objects. Problem statement We consider a two-phase medium in which both phases are purely linearly elastic and isotropic. The two-dimensional objects are assumed to be identical, parallel, infinite and distributed randomly and uniformly. The incident wave travels normal to the symmetry axes of the objects. Material properties and acoustic field quantities of the host medium are indicated by the superscript 1 and those of the object by the superscript 0. While the propagation of a plane monochromatic wave in the lossless object-free host medium is governed by the real-valued wavenumber , it is well known [4, 6] that the coherent plane wave propagation in the two-phase medium can be conveniently described by an effective wavenumber, given by 1 k