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Effective Medium Theory

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 [3], a 3-D EMT developed for spherical inclusions [16], and a 3-D EMT developed for inclusions shaped like ellipsoids of revolution [17]. 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 [16] 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 γ_i 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 c_i 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

where

depends only on the (unknown) Poisson ratio of the composite.

As has been pointed out [16], when = β the bulk and shear moduli have the same functional form and so the Poisson ratio remains constant for any values of the c_i. 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 [2]. The formulas for ellipsoidal inclusions are much more complicated, and are given in detail elsewhere [17].

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 [18]. 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 [18]:

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 [18]. 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: