On the effective conductivity of polycrystals and a three-dimensional phase-interchange inequality

We derive optimal bounds on the effective conductivity tensor of polycrystalline aggregates by introducing an appropriate null-Lagrangian that is rotationally invariant. For isotropic aggregates of uniaxial crystals an outstanding conjecture of Schulgasser is proven, namely that the lowest possible effective conductivity of isotropic aggregates of uniaxial crystals is attained by a composite sphere assemblage, in which the crystal axis is directed radially outwards in each sphere. By laminating this sphere assemblage with the original crystal we obtain anisotropic composites that are extremal, i.e., attaining our bounds. These, together with other results established here, give a partial characterization of the set of all possible effective tensors of polycrystalline aggregates. The same general method is used to prove a conjectured phase interchange inequality for isotropic composites of two isotropic phases. This inequality correlates the effective conductivity of the composite with the effective tensor when the phases are interchanged. It leads to optimal bounds on the effective conductivity when another effective constant, such as the effective diffusion coefficient, has been measured, or when one has information about ζ1 which is a parameter characteristic of the microgeometry, or when one knows the material is symmetric, i.e., invariant under phase interchange like a three-dimensional checkerboard.