Next: Tests of elastic
There are many ways to develop effective medium theories (EMT's) that attempt to predict the effective elastic properties of a composite [8,9]. The usual approach is to exactly solve a one-inclusion problem, in the dilute limit, and then use some sort of averaging process to generate a formula that predicts effective properties at general volume fractions. We use a 2-D EMT for elliptical inclusions , a 3-D EMT developed for spherical inclusions , and a 3-D EMT developed for inclusions shaped like ellipsoids of revolution . We will also discuss the equivalent EMT for hyperspherical inclusions in d-dimensions.
The 2-D EMT has been previously discussed [1,2]. The equations for the 3-D EMT in the case of spherical inclusions can be derived  by requiring that for a composite subjected to a uniform external shear stress o, the average shear strain γ for the composite is just
where γi is the average strain in the i'th phase, G is a shear modulus, and where
which depends only on the (as yet unknown) effective Poisson ratio of the composite. In eqs. (4)- (6) the subscripts refer to the phase label, unsubscripted variables are the unknown effective quantities, and ci is the volume fraction of the i'th phase. A similar calculation for the bulk modulus leads to the coupled equations for the effective moduli
depends only on the (unknown) Poisson ratio of the composite.
As has been pointed out , when = β the bulk and shear moduli have the same functional form and so the Poisson ratio remains constant for any values of the ci. This occurs in 3-D for 1 = 2 = * = 1/5. The value * = 1/5 is also the EMT prediction for the fixed point to which the Poisson's ratio is drawn, for any starting value of Poisson's ratio, when one phase has zero moduli and a zero-modulus percolation threshold is approached . The formulas for ellipsoidal inclusions are much more complicated, and are given in detail elsewhere .
The d-dimensional EMT that is based on hyperspherical inclusions has previously been presented for the case where one phase has zero moduli, though without any details . The general form of the equations is exactly eqs. (7) and (8) but and β will be different, depending on the dimension d. If we make the assumption that in general and β will only depend on the effective Poisson's ratio of the composite, as was the case both in 2-D and in 3-D, the general forms can be extracted from the zero moduli result :
Once again, eqs. (10) and (11) give the same functional form for G and K when = β, or when = 1/(2d-1), which is then a fixed point of the effective medium theory, for any stiffness ratio E1/E2. The value * = 1/(2d-1) is the EMT prediction for the fixed point to which the Poisson's ratio is drawn, for any starting value of Poisson's ratio, when one phase has zero moduli and a zero-modulus percolation threshold is approached . Kantor and Bergman have presented this result in terms of the ratio (K/G)* = 4/d. This is seen to be equivalent by substitution into the expression for the d-dimensional Poisson's ratio: