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Dynamic homogenization in periodic fibre reinforced media. Quasi-static limit for SH waves

Summary: Dynamic homogenization in periodic fibre reinforced
media. Quasi-static limit for SH waves
W.J. Parnell *, I.D. Abrahams
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Received 4 August 2005; received in revised form 22 February 2006; accepted 7 March 2006
Available online 4 May 2006
The effective response of a periodic fibre reinforced material to SH wave propagation is studied using the method of
asymptotic homogenization, complex variable theory and multipole expansions. The quasi-static limit of the effective prop-
erties is calculated when the wavelength is much longer than the defining lengthscale of the microstructure. The method
developed allows the determination of the elastic properties in the most general (monoclinic) fibre reinforced media and
the resulting expressions for the effective moduli are concise. The method is therefore both more general and provides neater
closed form solutions than extant methods. Results are shown to be excellent even for very high volume fractions of fibres.
2006 Elsevier B.V. All rights reserved.
Keywords: Asymptotic homogenization; SH waves; Quasi-statics; Monoclinic composites
1. Introduction
The subject of homogenization may be defined as the study of partial differential equations with rapidly
oscillating coefficients. As a result of these oscillations, direct numerical solution is extremely difficult and
it is therefore desirable to develop mathematical techniques to overcome these problems. Methods of homog-
enization date back to the end of the 19th century when various techniques were developed to find the effective


Source: Abrahams, I. David - Department of Mathematics, University of Manchester


Collections: Mathematics