Mean field homogenization schemes for composites with prolate and oblate spheroids: use of the orientation tensors and computation of the strain second-moments

Abstract

In this document we provide the homogenized stiffnesses from Mori-Tanaka scheme and Ponte-Casta{ñ}eda and Willis scheme applied to composites with spheroidal inclusions. The inclusions can be prolate spheroids ('fibers') or oblate spheroids ('penny-shapes'). We show how to compute the homogenized stiffnesses for particular distribution of spheroid orientations, using the second-order and fourth-order orientation tensors. We also provide formulas to compute the derivatives of these quantities w.r.t. the material parameters, which is of particular interest for computing the second-moments of the strains.