ON IMPROVING THE HASHIN-SHTRIKMAN BOUNDS FOR THE EFFECTIVE PROPERTIESOF 3-PHASE COMPOSITE MEDIA

For a composite comprising an isotropic mixture of two isotropic diele ctric materials, the Hashin-Shtrikman bounds for the overall dielectri c constant tenser are attainable and hence are the best possible. Cons idering instead a three-phase composite, the Hashin-Shtrikman bounds a re the best that are known in terms of volume fractions alone, and yet , in the limit of vanishing volume fraction of the material of greates t dielectric constant, the three-phase upper bound remains strictly gr eater than the two-phase bound. A similar comment applies to the lower bound, in relation to a small volume fraction of the material with th e smallest dielectric constant. Although this phenomenon may reflect a limitation of the Hashin-Shtrikman methodology, it remains conceivabl e that some micro-geometries exist for which all the 'third' phase is positioned in regions of high field concentration, so that it always h as 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 Hashin-Shtrikman three-phase bound a s the volume fraction c(3) of the 'third' material increases from zero . The Hashin-Shtrikman three-phase bound thus cannot be optimal, at le ast when c(3) is small. The method of derivation of the new bound reli es on an application of the theory of functions of bounded mean oscill ation, recently developed in the context of bounding the behaviour of nonlinear composites.