Dilute limits and effective medium theory

One of the few exact structure-property results in 3-D is for a dilute concentration of spheroidal inclusions with bulk and shear moduli K_iand G_idispersed in a solid matrix with moduli K_s and G_s. In this case, where unsubscripted variables stand for effective quantities (Hashin, 1983),

K = K_s + c_i P^si (K_i-K_s) (2.1)

G = G_s + c_i Q^si (G_i-G_s) (2.2)

Here c_i = 1 - p denotes the concentration (volume fraction) of inclusions, and for the case of spherical inclusions,

$$\frac{3K_s+4G_s}{3K_i+4G_s},\;\; Q^{si}=\frac{G_s+F_s}{G_i+F_s}, F_s=\frac{G_s}6 \frac{9K_s+8G_s}{K_s+2G_s}.$$ P^si = (2.3)

The form of P^si and Q^si for spheroidal inclusions (Wu, 1966) is given by (Berryman 1980). Young's modulus and Poisson's ratio are obtained via the relations E = 9KG / (3K + G) and = (3K - 2G) / (6K + 2G). When the inclusions are pores, $\nu$ exhibits the interesting property that $\nu=0.2$ = 0.2 when $\nu _s=0.2$_s = 0.2 for any value of E. This has been shown before (Garboczi & Day, 1995). In 2-D, this value is 1/3.

To adapt the dilute formulas to the case of a finite concentration of inclusions, a number of proposals have been made. The approximate equations that result are usually called effective medium theories. The most common approximation is the so-called self consistent method (SCM) of ( Hill 1965) and (Budiansky 1965). In this model the equations of elasticity are solved for a spherical inclusion embedded in a medium of unknown effective moduli. The effective moduli K and G are then derived. In the dilute case the embedding medium is just the matrix. The Hill-Budiansky result can be stated as (Berryman, 1980)

c_i P ^{* i} (K_i - K_* ) + c_s P^{* s}(K_s - K_* ) = 0 (2.4)

c_i Q ^{* i}(G_i - G_* ) + c_s Q^{* s}(G_s - G_* ) = 0 (2.5)

where K_* and G_* denote the effective moduli and P^{* m} and Q^{* m}are given in Eq. (2.3). Here c_s = 1 - c_i = p. Numerical methods are usually used to solve for K_* and G_* (see (Hill 1965) and (Berryman 1980) for details). (Garboczi and Day 1995) showed that, for spherical zero-moduli inclusions, a value of the matrix Poisson's ratio $\nu _s=0.2$_s = 0.2 gave $\nu=0.2$ = 0.2 for all inclusion concentrations that gave non-zero effective moduli. In d-dimensions, the critical value of $\nu$ was found to be 1/(2d - 1).

Two other forms of the SCM are relevant to our numerical results. When the inclusions are voids the SCM predicts a vanishing modulus for c_i $c_i\geq 0.5$ 0.5, although many materials remain rigid above this threshold. To address this problem (Christensen 1990) derived an alternate SCM based on concentric spheres embedded in a matrix of unknown moduli. The result is complicated and not reproduced here. (Wu 1966) also proposed an SCM for spheroidal inclusions that we will employ.

The differential effective medium theory (DEM), reviewed by (McLaughlin 1977), provides an alternative to the SCM using a similar philosophy. Suppose that the effective moduli of a composite medium are known to be K_* and G_*. Now, if a small additional concentration of inclusions are added, the change in K_* and G_* is approximated to be that which would arise if a dilute concentration of inclusions were added to a uniform, homogeneous matrix with moduli K_* and G_*. This leads to a pair of coupled non-linear differential equations which must be solved to find K^*(c_i) and G^*(c_i),

$$\frac{dK_*}{d c_i} P^{*i} \frac {K_i-K_*}{1-c_i};\; K_*(c_i=0)=K_s$$ = (2.6)

$$\frac{dG_*}{d c_i} Q^{*i} \frac {G_i-G_*}{1-c_i};\; G_*(c_i=0)=G_s.$$ = (2.7)

Zimmerman (Zimmerman, 1994) has shown that $\nu _s=0.2$_s = 0.2 is also a fixed point for the differential effective medium theory, for any value of inclusion concentration.

(Milton 1984) and (Norris 1985) have shown that the predictions of the DEM and SCM correspond to the properties of materials with spheroidal inclusions at widely different length scales. These types of structures are not commonly observed (Christensen, 1990). Therefore, except in the dilute limit, neither method can be accurately used to interpret experimental results, or guide the improvement of materials because of the unrealistic microstructural assumptions underlying each kind of theory. Nevertheless, the results are widely used, and until recently (Torquato, 1998) were the only predictive theories used for moderate and high porosity random interpenetrating porous materials.