Abstract
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.
Highlights
A classical problem in the mechanics of inhomogeneous media is to attempt to replace the two-dimensional potential problem ∇ · (μ(x)∇w(x)) = 0, where x = (x1, x2) and μ(x) and w(x) are periodic functions that vary rapidly with x, by an equivalent problem of the form∇ · (μ∗ · ∇w∗(x)) = 0, (1.1)where w∗ is the leading-order displacement field and μ∗ is the second-order tensor of effective shear moduli [1]
To fix ideas and since general cross sections are of principal interest here, attention is restricted to the case of square lattices
Results derived using the methodology shall be compared with those obtained using the method of asymptotic homogenization (MAH) [1]. It transpires that the effective antiplane shear moduli as determined by the MAH can be written in the form (see eqns (3.25) and (3.26) of [1])
Summary
The method to be discussed extends the work in [18] (referred to as PA below), where a new homogenization scheme was devised based on the integral equation form of the governing equation, considering antiplane wave propagation in the low-frequency limit, so that an equation of the form (1.2) was derived, and the leading-order result was determined. Returning to the two-dimensional periodic medium considered here, the method itself may be applied to a material of any macroscopic elastic symmetry, attention shall be restricted to microstructure that gives rise to orthotropic effective properties for clarity of exposition In this case, μ∗ij = δ1iδ1jμ∗1 + δ2iδ2jμ∗2, (i, j = 1, 2) when written with respect to the principal axes of anisotropy. It is reiterated that the method is described here in the context of antiplane elasticity and expressions for the effective antiplane shear moduli μ∗1 and μ∗2 are derived, the method is applicable to any of the applications summarized in table 1
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