The problem of the prediction of the effective conductivity of a polycrystal given the conductivity of the single crystal is considered in the light of what can be learned from a constructible polycrystal model for which the effective conductivity can be exactly calculated. It is shown that if the only information known about the internal geometry of the polycrystal is that the aggregate is statistically homogeneous and isotropic it is not possible to narrow appreciably the well-known ‘‘average conductivity-average resistivity’’ bounds on the effective conductivity. The model also casts some light on the analogous problem for two phase symmetric materials.
The problem of the prediction of the effective electrical conductivity of a polycrystal from the electrical conductivity of a single crystal is considered. It is shown that the familiar Voigt-Reuss bounds on the behavior of a polycrystal are the very best generally valid bounds that have thus far been proposed and that the various methods that are claimed to predict exact effective conductivity (or narrow bounds) all include implicit restrictions on the internal geometry of the polycrystal. This is accomplished by constructing a series of statistically homogeneous and isotropic polycrystal models for which the effective conductivity can be exactly calculated. It is hence to be expected that no universal relationship between single-crystal and polycrystal conductivity exists. Experimental evidence is adduced to support this conclusion. The results are also applicable to the analogous problems of thermal conductivity and electrical permittivity.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.