Composite materials display a wide range of conduction properties depending on the geometric configuration of the phases. A classical, but mostly unsolved problem, is to find the range of the effective conductivity, a constant but in general anisotropic matrix describing the overall electrical behavior of the composite (sometimes called G-closure).

Optimal bounds for isotropic mixtures are known in a few cases, beginning with the pioneering work of [Hashin and Shtrikman 1962]. However, the complete G-closure is known only in the special case of two isotropic phases [Murat & Tartar 1985, Lurie & Cherkaev 1986].

Building on previous works by  [Milton & Nesi 1990], we provide new anisotropic optimal microgeometries in the case of a three dimensional polycrystalline material, where the principal conductivities of the basic crystal are given, but the orientation of the crystal is allowed to change from point to point. 

Our results are obtained by exhibiting approximate solutions to the differential inclusion associated to the underlying system of pdes. 

(Joint work with Nathan Albin and Vincenzo Nesi.)