Multiphase laminates of extremal effective conductivity in two dimensions

This paper deals with two-dimensional composites made of several isotropic linearly conducting phases in prescribed volume fractions. The primary focus is on the three-phase case; the generalization to a larger number of phases is straightforward. A class of high- but finite-rank laminates is introduced. The laminates saturate the known inequality bounds—due to the work of Hashin and Shtrikman, Lurie and Cherkaev, Tartar, and Murat and Tartar—on the effective conductivity tensor of any composite. These bounds depend only on the constituent material properties and volume fractions and not on the placement of these materials in the composite. The bounds are known not to be optimal for all admissible choices of the conductivities and volume fractions. However, they are now known to be realizable in a much larger range of these parameters than was previously known. The range of effective properties of our multiphase laminates strictly includes those corresponding to the composites found earlier by Milton and Kohn, Lurie and Cherkaev, and Gibiansky and Sigmund. The new optimal laminates are found in a systematic fashion by satisfying sufficient conditions on the fields in each layer. This leads to a simple algorithm for generating optimal laminates. In addition a new supplementary bound for multiphase structures is also proven which must be satisfied by composites with smooth interfaces.