The problem of the prediction of the effective electric or thermal conductivity of a polycrystal in terms of the conductivities of the constituting single crystals is treated by a variational method. It is assumed that the crystals are all of the same kind and randomly oriented. Consequently, the polycrystal is assumed to be homogeneous and isotropic in the large. However, no assumptions about the shape of the crystals have to be introduced.

Lower and upper bounds for the effective conductivity are derived on the basis of a new variational formulation of the conductivity problem in anisotropic and nonhomogeneous media. For reasons of mathematical analogy the results are also valid for the effective dielectric constant and magnetic permeability of polycrystals. The bounds obtained are close when the anisotropy of the single crystals is not too large.

• Received 17 September 1962

DOI:https://doi.org/10.1103/PhysRev.130.129