Sphere assemblage model for polycrystals and symmetric materials

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.