 
Summary: A new integral equation approach to
elastodynamic homogenization
BY WILLIAM J. PARNELL* AND I. DAVID ABRAHAMS
School of Mathematics, University of Manchester, Oxford Road,
Manchester M13 9PL, UK
A new theory of elastodynamic homogenization is proposed, which exploits the integral
equation form of Navier's equations and relationships between length scales within
composite media. The scheme is introduced by focusing on its leadingorder
approximation for orthotropic, periodic fibrereinforced media where fibres have
arbitrary crosssectional shape. The methodology is general but here it is shown for
horizontally polarized shear (SH) wave propagation for ease of exposition. The resulting
effective properties are shown to possess rich structure in that four terms account
separately for the physical detail of the composite (associated with fibre crosssectional
shape, elastic properties, lattice geometry and volume fraction). In particular, the
appropriate component of Eshelby's tensor arises naturally in order to deal with
the shape of the fibre cross section. Results are plotted for circular fibres and compared
with extant methods, including the method of asymptotic homogenization. The leading
order scheme is shown to be in excellent agreement even for relatively high
volume fractions.
Keywords: homogenization; integral equations; effective moduli
