Uncertainty ranges for the macroscopic resistivities and permeabilities of random polycrystalline aggregates

Abstract

The completely random polycrystals considered here are aggregates of the anisotropic grains, the crystalline and shape orientations of which are uncorrelated. Because of irregular shapes of the grains ͑uncertainty of the microgeometry͒, one expects some uncertainty in observed macroscopic behavior of the aggregates. Our recently derived bounds on the possible ranges for polycrystal transport properties, which are proportional coefficients relating solenoidal and irrotational vector fields, are applied to evaluate electrical resistivities, permeabilities, and thermal conductivities of a number of practical aggregates, and considered for possible improvements by an interpolation method. We argue that the statistical homogeneity and isotropy hypotheses for the random aggregates are valid only within some uncertainty, and our bounds can still provide measures of this uncertainty asymptotically.