On the optimal bounds for the effective conductivity of isotropic quasi-symmetric multiphase media

Abstract.

The bounds on the effective conductivity of isotropic multiphase media with certain symmetry among the phases' geometry are examined. The bounds of Phan-Thien and Milton and those of Pham partly coincide, partly differ. Wherever they differ, the bounds of Phan-Thien and Milton are more restrictive. Specifically, the bounds for the subclass of spherical cell materials are identical, while the bounds of Phan-Thien and Milton for the subclass of platelet cell materials, which coincide with those of Pham in the case of two-component materials, are tighter in the general multicomponent case. Phan-Thien-Milton upper bound on the conductivity of multicomponent platelet cell materials is attained by a geometric model of Pham, therefore is verified to be the optimal one. Subsequently, for the whole class of isotropic quasi-symmetric media, the respective bound is optimal over a range of parameters. An optimal lower bound is conjectured. The differential scheme is used to construct certain spherical and platelet cell models.