Bounds for Effective Electrical, Thermal, and Magnetic Properties of Heterogeneous Materials

Determining the effective dielectric constant is typical of a broad class of problems that includes effective magnetic permeability, electrical and thermal conductivity, and diffusion. Bounds for these effective properties for statistically isotropic and homogeneous materials have been developed in terms of statistical information, i.e., one‐point and three‐point correlation functions, from variational principles. Aside from the one‐point correlation function, i.e., the volume fraction, this statistical information is difficult or impossible to obtain for real materials. For a broad class of heterogeneous materials (which we shall call cell materials) the functions of the three‐point correlation function that appear in the bounds of effective dielectric constant are simply a number for each phase. Furthermore, this number has a range of values $\frac{1}{9}$ to ⅓ and a simple geometric significance. The number $\frac{1}{9}$ implies a spherical shape, the number ⅓ a cell of platelike shape, and all other cell shapes, no matter how irregular, have a corresponding number between. Each value of this number determines a new set of bounds which are substantially narrower and always within the best bounds in terms of volume fraction alone (i.e., Hashin‐Shtrikman bounds). For dilute suspensions the new bounds are so narrow in most cases as to be essentially an exact solution. There is a substantial improvement over previous bounds for a finite suspension and yet greater improvement for multiphase material where the geometric characteristics of each phase are known.