ON THE EFFECTIVE VISCOELASTIC MODULI OF 2-PHASE MEDIA .2. RIGOROUS BOUNDS ON THE COMPLEX SHEAR MODULUS IN 3 DIMENSIONS

Cherkaev-Gibiansky and Hashin-Shtrikman variational principles are use d in order to obtain rigorous bounds on the shear modulus of two-phase viscoelastic composites in three dimensions. The simplest class of bo unding regions is composed of circles in the complex plane containing four points related to the viscoelastic moduli of the constituents. By taking the intersection of all such circles, we obtain tight bounds o n the complex shear modulus. A compact algorithm for computing this re gion of intersection is formulated and tested. Several examples of bou nding sets computed using the method are presented. When the phases ha ve equal and real Poisson's ratio, the bounding set reduces to a simpl e lens-shaped region in the complex shear modulus plane. A mixture of two viscous fluids and a suspension of solid particles in a viscous fl uid provide physical motivations for two other examples of bounds that have been computed. In the important limiting case when all the const ituent moduli are real, the new shear modulus bounds are shown to redu ce precisely to the well-known Hashin-Shtrikman-Walpole bounds.