Summary

This paper investigates the validity of two different analytical homogenization methods: the Mori–Tanaka
mean-field theory and Milton’s correlation function-dependent bounds. We focus on biphase linearly elastic
transversely isotropic composites. The composites consist of a matrix reinforced with long fibers of either
circular or irregular cross section shapes formed by overlapping circles, with different degrees of radius
polydispersity. The Mori–Tanaka effective stiffness depends on the phase moduli, volume fractions, and on a
few geometric descriptors of the fibers that can be readily evaluated. In contrast, the computation of Milton’s
bounds requires finer knowledge of the microstructure, in terms of two and three-point spatial correlation
functions, which are not always analytically tractable. We thus consider very specific random microstructure
geometries with known correlation functions. The effective moduli estimates of the two methods are validated
against the results of numerical homogenization using the finite element method. It is shown that the Mori–
Tanaka predictions of the effective transverse bulk modulus are significantly more accurate than those of
the transverse and axial shear moduli. In addition, the predictions of the scheme generally deteriorate with
an increasing fiber volume fraction. By contrast, the average of Milton’s upper and lower bounds provides
a highly accurate estimate for all three independent effective moduli, without any limitation on the fiber
concentration. This study highlights the indisputable effect of the spatial correlation functions on the effective
properties of composites, and aspires to pave the way towards the development of more predictive, correlation
function-dependent homogenization methods.