Effective wave propagation in a pre-stressed nonlinear elastic composite bar

The problem of determining the effective incremental response of nonlinearly elastic composite materials given some initial pre-stress, is of interest in numerous application areas. In particular the case when small amplitude elastic waves pass through a pre-stressed inhomogeneous structure (often known as acoustoelasticity) is of great importance. Of specific interest is how the initial finite deformation affects the microstructure and thus the subsequent response of the material. In this article we consider the simplest problem of this type where the material is a one dimensional composite bar consisting of two distinct phases, periodically distributed. Neglecting lateral contractions, the initial deformation is thus piecewise homogeneous and we can therefore determine the incremental behaviour semi-analytically. We apply asymptotic homogenization theory in the deformed configuration in order to find the effective response of the deformed material in the low frequency limit where the wavelength of the propagating waves is much longer than the characteristic lengthscale of the microstructure.