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IMA Journal of Applied Mathematics Advance Access originally published online on January 30, 2007
IMA Journal of Applied Mathematics 2007 72(2):223-244; doi:10.1093/imamat/hxl033
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Effective wave propagation in a prestressed nonlinear elastic composite bar

W. J. Parnell**

School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK

** Email: William.Parnell{at}manchester.ac.uk


  Abstract

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.

Keywords: finite deformation; composites; homogenization; effective wave propagation; incremental moduli; pass bands and stop bands.

Received on 26 September 2006. accepted on 16 November 2006.


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