Theory

The basic information required for the evaluation of the effective moduli are the volume fractions, and elastic moduli, of each phase; p (fraction of phase 1), q=1-p, $\kappa_i$ _i and µ _i (i=1,2). A common approximation for the effective moduli is the self-consistent method (SCM) of Hill [22] and Budiansky [11] which involves solving the equations of elasticity for a spherical particle of phase 1 surrounded by a medium of unknown effective moduli $\kappa_e$ _e µ _e. The results are obtained by solving

$$\frac{p}{\mu_e-\mu_2}+\frac{q}{\ mu_e-\mu_1}$ $\displaystyle \frac{6(\kappa_e+2\mu_e)}{5\mu_e(3\kappa_e+4\mu_e)}$$ (11)

for $\kappa_e$_e, µ _e. In the case where one of the phases is perfectly soft or rigid the results exhibit a percolation threshold of p = ½. The formula is also symmetric to phase interchange [ $\kappa_e(k_1,g_1,k_2,g_2,p)=\kappa_e(k_2,g_2,k_1,g_1,1-p)$_e (k₁, g ₁, k₂, g₂ , p) =  _e(k₂, g₂, k ₁, g₁, 1 - p) etc.]. These facts limit the applicability of the SCM, since most composites have lower percolation thresholds and many are not symmetric [19]. A more realistic formula is obtained using a generalized SCM (GSCM) [13] for the case of a particle of phase 1 surrounded by a spherical shell of phase (embedded in a medium of the effective moduli). The result is complicated [12] and not reproduced here. The GSCM has zero percolation threshold, and is not symmetric under phase interchange. For non-particulate media it is not clear which phase should be associated with the inclusions and which with the matrix. Below we consider both cases. Christensen [12] found that the GSCM provided a better prediction of composite properties than other common methods, so we shall not consider these here. It should also be noted that the volume fraction is the only microstructural information included in the SCM and GSCM results. This means that these formulae are insensitive to the distribution of each phase, or rather that each formula has a ``built-in" microstructure, which may or may not match the experimental one.

The difficulty in deriving general theoretical results for predicting the elastic properties of random composites has provided the impetus for the development of rigorous bounds [41]. For orientationly isotropic materials the bounds take the general form [31],

$$\left( \langle \kappa^{-1} \rangle -\frac {4pq(\kappa_2^{-1}... ...\kappa_2-\kappa_1)^2} {3\langle\tilde{\kappa}\rangle+4\Lambda} \end{displaymath} img84.gif$$ (12)

$$\left( \langle \mu^{-1} \rangle -\frac {pq(\mu_2^{-1}-\mu_1^... ...-\frac{6pq(\mu_2-\mu_1)^2} {6\langle\tilde{\mu}\rangle+\Theta} \end{displaymath} img85.gif$$ (13)

where for a variable b, $\langle b\rangle\equiv pb_1+qb_2$ and $\langle \tilde{b} \rangle\equiv qb_1+pb_2$. The additional parameters $\Gamma$, $\Lambda$, $\Xi$ and $\Theta$ depend on the level of microstructural information available. If any of the moduli ($\kappa_i$ _i or µ _i) are zero then the lower bounds vanishes. Similarly if any of the moduli are infinite the upper bound diverges.

If only the volume fractions of the composite are known

$$\Gamma=\mu_1^{-1},\;\;\; \Lambda=\mu_2,\;\;\; \Xi= \frac{\kap... ...\;\;\; \Theta=\frac{\mu_2(9\kappa_2+8\mu_2)}{\kappa_2+2\mu_2}, \end{displaymath} img92.gif$$ (14)

and Eqns. (12) & (13) are the bounds of Hashin and Shtrikman [20] for the case µ ₂ $\mu_2\geq\mu_1$ µ ₁ and $\kappa_2\geq\kappa_1$_{2 1}. The bounds only apply to well-ordered materials [($[(\kappa_2-\kappa_1)(\mu_2-\mu_1) \geq 0]$₂ - ₁)(µ ₂ - µ ₁) 0] and the inequality signs in the bounds must be reversed if µ ₂ < µ ₁ and $\kappa_2<\kappa_1$₂ < ₁. If further information is available in the form of three-point statistical correlations it is possible to derive more restrictive bounds in terms of the microstructure parameters [27],

where t²=r²+s²-2rs u and P₂(u) = ½(3u ² - 1) and P₄(u) = 1/8(35u ⁴ - 30u² + 3) are Legendre polynomials. In this case we have

$$\Xi=\frac{5\langle \mu^{-1}\rangle_\zeta\langle 6\kappa^{-1}... ...^{-1}+99\mu^{-1}\rangle_\zeta+45\langle \mu^{-1}\rangle_\eta}. \end{displaymath} img105.gif$$ (17)

Here we have used the standard notation $\langle b\rangle_\zeta\equiv \zeta_1b_1+\zeta_2b_2$ and $\langle b\rangle_\eta\equiv \eta_1b_1+\eta_2b_2$ where b_i is any function of µ_i and and $\kappa_i$ _i, ₂ = 1 - ₁ and ₂ = 1 - ₁. and $\eta_2=1-\eta_1$ ₂ = 1 - ₁. With the parameters given in Eqn. (17), the bounds on $\kappa$ are due to Beran and Molyneux [5] while the bounds on µ are those of Milton and Phan-Thien [28]. The development of these bounds have been recently reviewed [41,31]. Below we consider the Young's modulus [ E = 9$E=9\kappa\mu/(3\kappa+\mu)$ µ /(3 + µ) ] and Poisson's ratio
[ $\nu=(3\kappa-2\mu)/(6\kappa+2\mu)$ = (3 - 2 µ) / ( 6 + 2 µ) ] of a composite. The bounds on E [19] and $\nu$ [50] are

$$\frac{9\kappa_l\mu_l}{3\kappa_l+\mu_l}\leq E_e \leq \frac{9\... ...u_u}\leq \nu_e \leq \frac{3\kappa_u-2\mu_l}{6\kappa_u+2\mu_l}, \end{displaymath} img114.gif$$ (18)

where the subscripts refer to the upper and lower bound of Eqns. (12) & (13). Depending on the level of microstructural information employed to find the bounds on $\kappa$ and µ we refer to Eqn. (18) as the Hashin and Shtrikman (HS) or Beran, Molyneux, Milton and Phan-Thien (BMMP) bounds respectively. The microstructure parameters $\zeta$ and $\eta$ have been evaluated for hard and overlapping spheres [41], level cut Gaussian random-field models [37] and can be evaluated for other Boolean models [23].