Canonical functions

Another topic of broad and continuing interest in the field of composite materials is the study of heterogeneous conductors, dielectrics, and -- for porous media -- fluid permeability (Beran, 1968; Milton, 2002; Torquato, 2002). Because of the wide range of applications, including both thermal and electrical conduction, and the theoretical interest in analysis of critical phenomena such as percolation thresholds in resistor networks and localization (Kirkpatrick, 1971; 1973), this topic has surely been studied as much or more than any other in the field of heterogeneous media.

Many results in this field of research can also be expressed in terms of canonical functions. First define  

$$\Sigma(\sigma) \equiv \left[\sum_{i=1}^J \frac{v_i}{\sigma_i + 2\sigma}\right]^{-1} - 2\sigma,$$ (28)

where $\sigma_i$ is the conductivity in the ith component, and v_i is the corresponding volume fraction, again having the space filling constraint that $\sum_{i=1}^J v_i = 1$.Hashin-Shtrikman bounds (Hashin and Shtirkman, 1962) on conductivity for a multicomponent composite material can then be expressed as  

$$\sigma_{HS}^\pm = \Sigma(\sigma_\pm),$$ (29)

where $\sigma_\pm$ are the largest and smallest values of the J isotropic conductivities present. These bounds are generally improvements on the mean and harmonic mean bounds:  

$$\sigma_M = \sum_{i=1}^J v_i\sigma_i \qquad \hbox{and} \qquad \sigma_H = \left[\sum_{i=1}^J \frac{v_i}{\sigma_i}\right]^{-1}.$$ (30)

Beran (1965; 1968) used variational methods to arrive at improved bounds on conductivity for two-component media, again based on information in spatial correlation functions. His results are also expressible in terms of the canonical functions as  

$$\sigma_{B}^+ = \Sigma(\left<\sigma\right\gt _\zeta)$$ (31)

and  

$$\sigma_{B}^- = \Sigma(\left<1/\sigma\right\gt _\zeta^{-1}),$$ (32)

where $\sigma_{B}^+$ ($\sigma_{B}^-$) is the upper (lower) bound and the $\zeta$ averages are the same ones we introduced here previously [following Eq.(6)]. Since some of the same measures of microstructure (in this case the $\zeta_i$'s) can be used to bound both conductivity and elastic constants, it has been noticed before that this fact and similar relations for other systems can be used to produce various cross-property bounds (Berryman and Milton, 1988; Gibiansky and Torquato, 1995), thereby measuring one physical property in order to bound another.