3 - The Eshelby approach to modelling composites

Summary

In the previous chapter a number of models were presented for estimating the partitioning of loads between the constituents of composites subjected to external loads. These models involve mathematical approximations ranging from the good to the very poor. Some are rather limited in terms of the properties which can be predicted, while others are computationally daunting. For an isolated inclusion (reinforcing constituent) having an ellipsoidal shape, the approach presented in this chapter is mathematically rigorous. Later we shall see that it is also a good model at higher inclusion volume fractions and for other inclusion shapes. This analysis, commonly named the Eshelby method, turns out to be useful for predicting a wide range of composite properties. On a practical level, the standard equations highlighted by boxes in the text can be used to predict many composite properties quickly and fairly accurately.

Internal stresses are commonplace in almost any material which is mechanically inhomogeneous. Typically, their magnitude varies according to the degree of inhomogeneity: for an externally loaded polycrystalline cubic metal, differently oriented crystallites will be stressed to different extents, but these differences are usually quite small. For a composite, consisting of two distinct constituents with different stiffnesses, these disparities in stress will commonly be much larger. Internal stresses arise as a result of some kind of misfit between the shapes of the constituents (matrix and reinforcement, i.e. fibre, whisker or particle). Such a misfit could arise from a temperature change, but a closely related situation is created during mechanical loading - when a stiff inclusion tends to deform less than the surrounding matrix.