OPTIMAL BOUNDS CORRELATING ELECTRIC, MAGNETIC AND THERMAL-PROPERTIES OF 2-PHASE, 2-DIMENSIONAL COMPOSITES

The effective conductivity function of a two-phase two-dimensional com posite is known to be a tensor valued analytic function of the compone nt conductivities, assumed here to be isotropic. Optimal bounds correl ating 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 equi valent 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 sim ilar class of functions which are not subject to the restriction of pa ssing through a known point. Crude bounds on this class of functions g ive rise to sharp bounds on the original class of effective conductivi ty functions. These bounds are the best possible, being attained by se quentially layered laminate microgeometries.