On the Eeective Viscoelastic Moduli of Two-phase Media. Ii. Rigorous Bounds on the Complex Shear Modulus in Three Dimensions

Cherkaev-Gibiansky and Hashin-Shtrikman variational principles are utilized in order to obtain rigorous bounds on the shear modulus of two-phase viscoelastic composites in three dimensions. The simplest class of bounding 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 on the complex shear modulus. A compact algorithm for computing this region of intersection is formulated and tested. Several examples of bounding sets computed using the method are presented. When the phases have equal and real Poisson's ratio, the bounding set reduces to a simple lens-shaped region in the complex shear modulus plane. A mixture of two viscous uids and a suspension of solid particles in a viscous uid provide physical motivations for two other examples of bounds that have been computed. In the important limiting case when all the constituent moduli are real, the new shear modulus bounds are shown to reduce precisely to the well-known Hashin-Shtrikman-Walpole bounds.