On the elastic moduli of unidirectional fibre reinforced composites

Abstract

A simple analytical technique is presented for obtaining lower bounds to the effective moduli of unidirectional fibre-reinforced composites with any repetitive array of fibres. The analysis depends upon a direct method involving stress-equilibrium and strain-continuity considerations, and thus it differs from the type of analysis due to Hashin and Hill (and others) involving variational methods. The composite is envisaged as being sliced along all 1,2 planes so that conditions of plane stress exist throughout and the stiffness properties of each vanishingly thin slice are readily determined in terms of the constituent properties of the fibre and matrix. The effective elastic moduli are obtained by suitable integration of such slice properties over a repeating fibre/matrix pattern. Comparison with known accurate values of the longitudinal shear modulus shows that the technique underestimates the modulus by factors of about 1.1 for the square array and 1.2 for the hexagonal array, depending on the fibre volume fraction and fibre/matrix stiffness ratio. By judicious use of such correction factors it should be possible to estimate the longitudinal shear modulus over a wide range of design parameters to within about 5 per cent; for the square and hexagonal arrays Symm's accurate but limited values have been augmented by new results which have an accuracy of 0.3 per cent. New results are also presented for the transverse modulus.