Effective dynamic moduli and density of fiber-reinforced composites

A multiple scattering theory is developed to predict the effective dynamic material properties of elastic composites in two dimensions. The system consists of circular fibers distributed randomly in an elastic solid. The coherent wave propagation in the elastic composite is analyzed under the quasi-crystalline approximation. The effective medium equivalent to the original composite material is a medium with space and time dispersion, and hence, its parameters are functions of frequency of the incident field. Although the effective medium is homogeneous and isotropic, its effective dynamic moduli and density depend on the type of propagating wave, e.g., they are different for longitudinal and transverse incident waves. However, they coincide in the long-wave region as expected on physical grounds. Furthermore, the effective material properties are found to be complexvalued, in addition to their dynamic nature. For in-plane waves and in the long-wave limit the effective bulk modulus, mass density and shear modulus are independently determined by a set of monopolar, dipolar and quadrupolar scattering coefficients of the embedded fibers alone, respectively. Likewise, for anti-plane waves, the effective mass density and the shear modulus are specified, respectively, in terms of the monopolar and dipolar scattering coefficients of the corresponding fiberscattering problem. The emerging possibility of designing composite materials to form elastic metamaterials is discussed.