Abstract of the paper ``On the effective viscoelastic moduli of
two-phase media. II. Rigorous bounds on the complex shear modulus
in three dimensions'' with G. W. Milton
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 fluids and a suspension of solid particles
in a viscous fluid 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.
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