On the G-closure in the polycrystalline problem

This work is concerned with the G-closure problem for electrical conductivity in three space dimensions in the context of polycrystalline composites. Its specific goal is to explore the attainability of a certain lower bound. This bound was proved by Avellaneda et al. [J. Appl. Phys., 63 (1988), pp. 4989-5003]. In the case of uniaxial basic crystal and isotropic composites, the bound was conjectured by Schulgasser [J. Appl. Phys., 54 (1983), p. 1380]. The former work established the attainability of this bound only under very special circumstances, e.g., when the basic crystal is uniaxial. The main conclusion of the present work is that, in the general case, when the conductivity tensor of the basic crystal has distinct eigenvalues, the lower bound is always attained by a two-dimensional set of rank-one laminates.