11.1 Eshelby Method

In this chapter, we present a brief overview of the homogenization of composite materials. Homogenization refers to the process of considering a statistically homogeneous representation of the composite material called a representative volume element (RVE). This homogenized element is considered for purposes of calculating the stresses and strains in the matrix and fibers. We will emphasize mainly the Eshelby method in the homogenization process. For more details, the reader is referred to the book An Introduction to Metal Matrix Composites by Clyne and Withers.

Since the composite system is composed of two different materials (matrix and fibers) with two different stiffnesses, internal stresses will arise in both the two constituents. Eshelby in the 1950s demonstrated that an analytical solution may be obtained for the special case when the fibers have the shape of an ellipsoid. Furthermore, the stress is assumed to be uniform within the ellipsoid. Eshelby's method is summarized by representing the actual inclusion (i.e fibers) by one made of the matrix material (called the equivalent homogeneous inclusion). This equivalent inclusion is assumed to have an appropriate strain (called the equivalent transformation strain) such that the stress field is the same as for the actual inclusion. This is the essence of the homogenization process.

The following is a summary of the steps followed in the homogenization procedure according to the Eshelby method (see Fig. 11.1):

1. Consider an initially unstressed elastic homogeneous material (see Fig. 11.1a). Imagine cutting an ellipsoidal region (i.e.