On improving the Hashin-Shtrikman bounds for the effective properties of three-phase composite media

For a composite comprising an isotropic mixture of two isotropic dielectric materials, the Hashin-Shtrikman bounds for the overall dielectric constant tensor are attainable and hence are the best possible. Considering instead a three-phase composite, the Hashin-Shtrikman bounds are the best that are known in terms of volume fractions alone, and yet, in the limit of vanishing volume fraction of the material of greatest dielectric constant, the three-phase upper bound remains strictly greater than the two-phase bound. A similar comment applies to the lower bound, in relation to a small volume fraction of the material with the smallest dielectric constant. Although this phenomenon may reflect a limitation of the Hashin-Shtrikman methodology, it remains conceivable that some microgeometries exist for which all the «third» phase is positioned in regions of high field concentration, so that it always has a large effect. This paper resolves this problem to some extent, by generating a new upper bound that ranges continuously from the Hashin-Shtrikman two-phase bound to the HashinShtrikman three-phase bound as the volume fraction c 3 of the «third» material increases from zero. The Hashin-Shtrikman three-phase bound thus cannot be optimal, at least when c 3 is small. The method of derivation of the new bound relies on an application of the theory of functions of bounded mean oscillation, recently developed in the context of bounding the behaviour of nonlinear composites