A compressibility-based derivation of simple expressions for the transverse Poisson's ratio and shear modulus of an aligned long fibre composite

Abstract

There is continuing interest in the elastic behaviour of long fibre composites. This concerns not only the axial and transverse Young's moduli, but also the shear moduli and Poisson's ratios. As an example of the significance of such parameters, interlaminar coupling stresses in a composite laminate may be influenced by the transverse Poisson's ratios of the individual plies: in a cross-ply laminate the through-thickness Poisson contraction of the transverse ply will tend to be much greater than that of the axial ply (i.e. 1'23 >~> v21 , where 1 is the fibre axis direction), which can be relevant for calculation of the coupling stresses. Unfortunately, analysis becomes complex when transverse stresses, which are very inhomogeneously distributed, are significant. Advanced laminate stress analysis is routinely carried out, requiring as input data the elastic constants of an individual lamina. These are obtained by use of: (i) equations derived from the convenient but very crude "slab" or "sandwich" model [1-3], (ii) equations derived from some attempt to analyse the transverse stress distribution [4], or at least to represent its effect by a semi-empirical approach [5], or (iii) accurate but complex models which do not lead to simple analytical equations. This last group may be subdivided into numerical methods [6, 7] and complex analytical approaches, notably the Eshelby equivalent homogeneous inclusion analysis [8, 9], in which the effect of a reinforcement with different elastic constants from those of the matrix can, for the case of an ellipsoidal shape, be predicted after replacing it with matrix which has first been subjected to an appropriate "stress-free" or ' transformation" strain. The analysis is mathematically exact only for a single inclusion in an infinite matrix [10], but it can also be used to predict elastic constants and other thermophysical properties [11, 12] of a real composite (with short or long fibres), providing a suitable assumption is made about how the matrix and the reinforcement sample the mean stress. The mean field model [13] is now commonly employed and has been used to produce the predictions reported below. Calculations are normally made on the basis of both constituents being themselves elastically isotropic. The slab model is quite widely used, where f is the volume fraction of fibre although it is accurate only for the axial Young's modulus, El, and, to a good approximation, for the axial Poisson's Ratio, vii, of a long fibre composite.