This work describes a homogenization scheme that gives rise to explicit, analytical forms for the effective antiplane shear modulus of a periodic fibre reinforced medium. The expressions take the form of rational functions in the volume fraction . Previous work developed by my supervisors (Professors Will Parnell & David Abrahams) invoked a (non-dilute) approximation in order to determine simple leading order expressions. Here the theory is extended in order to determine higher order terms in the rational function expansions.
The methodology is attractive in that the expressions can be derived for a large class of fibres with non-circular cross section. Furthermore, terms are clearly identified as being associated with the lattice geometry of the periodic structure, fibre cross-sectional shape, and host/fibre material properties.
In addition, through investigating these non-circular cross sections, a scheme was developed to efficiently produce polynomial expansions of famous integral tensors associated with inclusion shape, the generalized Hill and Eshelby tensors.
The expressions for the effective properties are derived in the context of antiplane elasticity but are broadly applicable 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 the associated cell problem is usually solved by some computational scheme, e.g. finite element methods.