Linear theory of disordered fibre-reinforced composites

Abstract

A theory is presented for obtaining the effective elastic Green function G(k) of well-bonded fibre-reinforced materials at low volume-fractions of fibre in the presence of disorder. The centre of mass positions of the fibres are taken to be random. The theory is applied to the case where the alignment is also random, and comment is made on its suitability for less totally disordered geometries. The shear stress at the fibre-matrix interface is also calculated, and effective elastic moduli obtained by taking the low k limit. Finite frequency effects are considered; the damping of long wavelength acoustic modes by disorder is calculated. Results are given for general length distribution, Young modulus and number density of fibres in a matrix of arbitrary bulk and shear moduli. The fibre radius is taken to be small compared with the fibre length. In a limiting case, that of high overlap of fibres and high values of Young modulus for the fibres in an incompressible matrix, the results for the composite Young modulus and interface shear stress obtained by a full Green function treatment take forms similar to those first derived heuristically by shear-lag analysis (H. L. Cox, Br. J. appl. Phys. 3, 72 (1952)). The increment in shear modulus of an incompressible elastic medium when inflexible, inextensible random-flight wires or fibres are embedded in it is also calculated.