Bounds for effective thermal, electrical, and magnetic properties of heterogeneous materials using high order statistical information

Publisher Website
Google Scholar
Abstract
Bounds have been developed for the effective thermal, electrical, and magnetic properties (K*) of a two phase statistically homogeneous and isotropic material. These bounds came in terms of the ratio of the properties of the two phases, volume fractions, and the constants G, G2, G3, M1, and M2. Bounds on the values of these constants are obtained. It is shown that the constants G, G2, and G3 are geometric parameters. They are calculated in the general case of spheroidal inclusions in terms of the axial ratio A. The constants M1 and M2 are packing parameters. General expressions of these constants in terms of packing information are obtained. It is found that certain combinations of the constants M1 and M2 for spheres and plates give exact solutions for the effective property K*. The exact solutions were shown in some cases to be equal to Miller's upper bound and in other cases to Miller's lower bound. A self‐consistent scheme for the case of spheres is carried out. The corresponding values of M1 and M2 were identified and used to plot the bounds. The bounds for plates are also calculated. It is found that these bounds introduce a great improvement over Miller's bounds. The small perturbation limit was considered. It is found that our bounds coincide to order η5 and all values of v (η = K1/K2− 1, Ki is the property of material i). Moreover, they include Miller's bounds to order η3 and Hashin bounds to order η2.