Effective wave propagation in a prestressed nonlinear elastic composite bar

The problem of determining the effective incremental responseof nonlinearly elastic composite materials given some initialprestress is of interest in numerous application areas. In particular,the case when small-amplitude elastic waves pass through a prestressedinhomogeneous structure is of great importance. Of specificinterest is how the initial finite deformation affects the microstructureand thus the subsequent response of the composite. Modellingthis effect is in general extremely difficult. In this article,we consider the simplest problem of this type where the materialis a one dimensional composite bar consisting of two distinctphases periodically distributed. Neglecting lateral contractions,the initial deformation is thus piecewise homogeneous and wecan therefore determine the incremental behaviour semi-analytically,given the constitutive behaviour (strain energy function) ofthe phases in question. We apply asymptotic homogenization theoryin the deformed configuration in order to find the effectiveresponse of the deformed material in the low-frequency limitwhere the wavelength of the propagating waves is much longerthan the characteristic length scale of the microstructure.We close by considering the arbitrary frequency case and illustratehow the initial deformation affects the location of stop bandsand pass bands of the material. Work is under way to confirmthese results experimentally.