Differential replacement procedure and bounds on conductivities of heterogeneous media with graded anisotropic cylindrical fibres

The effective conductivities of heterogeneous media containing either discretely suspended cylindrical fibres with a graded anisotropic interphase or graded anisotropic fibres, whose conductivity in the radial direction is different from that in the tangential direction, are obtained by two methods in this Letter. First, a differential replacement procedure (DRP) based on an energy equivalency condition is presented to replace the graded anisotropic constituents by equivalent homogeneous cylindrical fibres. This allows many approximate schemes to be used to predict the effective conductivities of the heterogeneous media containing graded anisotropic constituents from the conductivity of the equivalent homogeneous cylindrical fibres. Second, the optimized upper and lower bounds on the effective conductivities of these heterogeneous media are presented by introducing comparison materials. It is shown that the DRP predictions are within these bounds for the considered media.