Effective wave propagation in a prestressed nonlinear elastic composite bar

The problem of determining the effective incremental response of nonlinearly elastic composite materials given some initial prestress is of interest in numerous application areas. In particular, the case when small-amplitude elastic waves pass through a prestressed inhomogeneous structure is of great importance. Of specific interest is how the initial finite deformation affects the microstructure and thus the subsequent response of the composite. Modelling this effect is in general extremely difficult. 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, given the constitutive behaviour (strain energy function) of the phases in question. 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 length scale of the microstructure. We close by considering the arbitrary frequency case and illustrate how the initial deformation affects the location of stop bands and pass bands of the material. Work is under way to confirm these results experimentally.