Optimal bounds correlating electric, magnetic and thermal properties of two-phase, two-dimensional composites

Abstract

The effective conductivity function of a two-phase two-dimensional composite is known to be a tensor valued analytic function of the component conductivities, assumed here to be isotropic. Optimal bounds correlating the values this function can take are derived. These values may correspond to the measured effective magnetic permeabilities, thermal conductivities, or any other transport coefficient mathematically equivalent to the effective conductivity. The main tool in this derivation is a new fractional linear transformation which maps the appropriate class of conductivity functions passing through a given point to a similar class of functions which are not subject to the restriction of passing through a known point. Crude bounds on this class of functions give rise to sharp bounds on the original class of effective conductivity functions. These bounds are the best possible, being attained by sequentially layered laminate microgeometries.