Rigorous bounds

Some of the rigorous bounds that are expressible in terms of the canonical functions for J = 2 are listed in TABLE 1. Functions and averages required as definitions for some of the more complex terms in TABLE 1 are:  

$$\Theta(K,\mu) = \frac{\mu}{6}\left(\frac{9K + 8\mu}{K + 2\mu}\right),$$ (4)

and the expressions needed for the McCoy-Silnutzer (MS) bounds (McCoy, 1970; Silnutzer, 1972), which are  

$$\begin{array} {rl} X & = \left[10\mu_V^2\left<K\right\gt _\z... ...^2\left<\mu\right\gt _\eta\right]/(K_V+2\mu_V)^2, \end{array}$$ (5)

 

$$\begin{array} {rl} \Xi & = \left[10K_V^2\left<K^{-1}\right\g... ...t<\mu^{-1}\right\gt _\eta\right]/(9K_V+8\mu_V)^2. \end{array}$$ (6)

The averages $\left<M\right\gt = v_1M_1+v_2M_2$,$\left<M\right\gt _\eta = \eta_1M_1+\eta_2M_2$, and $\left<M\right\gt _\zeta = \zeta_1M_1+\zeta_2M_2$ are defined for any modulus M. The volume fractions are v₁,v₂, while $\zeta_1,\zeta_2$ and $\eta_1,\eta_2$ are the microgeometry parameters or Milton numbers (Milton, 1981; 1982), related to spatial correlation functions of the composite microstructure. The Voigt averages of the moduli are $K_V = \left<K\right\gt$ and $\mu_V = \left<\mu\right\gt$.For symmetric cell materials: $\zeta_1 = \eta_1 = v_1$ for spherical cells, $\zeta_1 = \eta_1 = v_2$ for disks, while $\zeta_1 = (v_2+3v_1)/4$and $\eta_1 = (v_2+5v_1)/6$ for needles.

Alternative bounds that are at least as tight as the McCoy-Silnutzer (MS) bounds for any choice of microstructure were given by Milton and Phan-Thien (1982) as  

$$\hat{X} = \frac{<3\mu\gt _\eta<6K+7\mu\gt _\zeta-5<\mu\gt^2_\zeta} {<2K-\mu\gt _\zeta + <5\mu\gt _\eta}$$ (7)

and  

$$\hat{\Xi} = \frac{N}{<128/K+99/\mu\gt _\zeta + <45/\mu\gt _\eta},$$ (8)

where  

$$\begin{array} {r} N = <5/\mu\gt _\zeta<6/K-1/\mu\gt _\zeta + \ <1/\mu\gt _\eta<2/K+21/\mu\gt _\zeta.\end{array}$$ (9)

It has been shown numerically that the two sets of bounds (MS and MPT) using the transform parameters X,$\Xi$ and $\hat{X}$,$\hat{\Xi}$ are nearly indistiguishable for the penetrable sphere model (Berryman, 1985).

Note that ``improved bounds'' are not necessarily improved for every choice of volume fraction, constituent moduli, and microgeometry. It is possible in some cases that ``improved bounds'' will actually be less restrictive, than say the Hashin-Shtrikman bounds, for some range of the parameters. In such cases we obviously prefer to use the more restrictive bounds when our parameters happen to fall in this range.

Milton (1987; 2002) has shown that, for the commonly discussed case of two-component composites, the canonical functionals can be viewed as fractional linear transforms with the arguments $\beta$ and $\theta$ of the canonical functionals as the transform variables. In light of the monotonicity properties of the functionals, this point of view is very useful because the problem of determining estimates of the moduli can then be reduced to that of finding estimates of the parameters $\beta$ and $\theta$. Furthermore, properties of the canonical functions also imply that excellent estimates of the moduli can be obtained from fairly crude estimates of the transformation parameters $\beta$ and $\theta$. (Recall, for example, that estimates of zero and infinity for these parameters result in Reuss and Voigt bounds on the moduli.) Milton calls this transformation procedure the Y-transform, where Y stands for one of these transform parameters (i.e., $\beta$ and $\theta$ in elasticity, or another combination when electrical conductivity and/or other mathematically analogous properties are being considered).