Repository logo

An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems

Published version



Change log


Joyce, D 
Parnell, WJ 
Assier, RC 
Abrahams, ID 


In Parnell & Abrahams (2008 Proc. R. Soc. A 464, 1461–1482. (doi:10.1098/rspa.2007.0254)), a homogenization scheme was developed that gave rise to explicit forms for the effective antiplane shear moduli of a periodic unidirectional fibre-reinforced medium where fibres have non-circular cross section. The explicit expressions are rational functions in the volume fraction. In that scheme, a (non-dilute) approximation was invoked to determine leading-order expressions. Agreement with existing methods was shown to be good except at very high volume fractions. Here, the theory is extended in order to determine higher-order terms in the expansion. Explicit expressions for effective properties can be derived for fibres with non-circular cross section, without recourse to numerical methods. Terms appearing in the expressions are identified as being associated with the lattice geometry of the periodic fibre distribution, fibre cross-sectional shape and host/fibre material properties. Results are derived in the context of antiplane elasticity but the analogy with the potential problem illustrates the broad applicability of the method to, e.g. thermal, electrostatic and magnetostatic problems. The efficacy of the scheme is illustrated by comparison with the well-established method of asymptotic homogenization where for fibres of general cross section, the associated cell problem must be solved by some computational scheme.



homogenization, potential problem, antiplane elasticity, fibre-reinforced composite

Journal Title

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Conference Name

Journal ISSN


Volume Title



The Royal Society Publishing
Engineering and Physical Sciences Research Council (EP/K032208/1)
D.J. is grateful to Thales UK and the Engineering and Physical Sciences Research Council (EPSRC) for funding via a CASE PhD studentship. W.J.P. acknowledges the EPSRC for funding his research fellowship (EP/L018039/1). I.D.A. undertook part of this work whilst in receipt of a Royal Society Wolfson Research Merit Award, and part was supported by the Isaac Newton Institute under EPSRC Grant Number EP/K032208/1. R.C.A. would like to acknowledge support from the EPSRC (EP/N013719/1).