Bounds on the effective conductivity of statistically isotropic multicomponent materials and random cell polycrystals

Upper and lower bounds on the effective conductivity of statistically isotropic multicomponent materials in d dimensions (d=2 or 3) are constructed from the minimum energy principles and appropriate trial fields. The trial fields, involving harmonic potentials and free parameters to be optimized, lead to the bounds containing up to three-point correlation information about the microgeometry of a composite. The bounds are applied to give estimates for the symmetric cell materials, which are optimal over some ranges of parameters, and asymmetric multicoated spheres, which yield the exact effective conductivity in certain cases. The results also agree with many known ones. New bounds for random cell polycrystals are obtained and illustrated on a number of polycrystalline aggregates.