Viscoelastic Composites:
     Variational Principles and Bounds on the Effective
                         Properties

                             by

         Leonid V. Gibiansky, Princeton University


           JTB 120,  3:20pm Monday, March 3, 1997

                          Abstract


This talk is a review of the results obtained during last
five years in collaboration with A. Cherkaev, R. Lakes, G.
Milton, and S. Torquato.

We consider the dynamic response of isotropic composites of
two viscoelastic isotropic phases in the frequency range
where the acoustic wavelength is much larger than the
inhomogeneities. Harmonic oscillation of the viscoelastic
media in such a limit can be described by the elasticity
equations with complex moduli and complex stress and strain
fields, and the properties of the isotropic composite can be
described by complex bulk and shear moduli.

We discovered new minimum variational principles that
describe the behavior of the media with complex moduli. Then
we applied these new principles, in conjunction with the
Hashin-Shtrikman and the Translation method, to extend the
effective bulk and shear moduli bounds of Hashin, Shtrikman,
and Walpole to viscoelasticity. A specific structure of the
new bounds allowed us to get bounds on the effective complex
moduli of composite with arbitrary phase volume fraction by
using the fixed-volume-fraction bounds. We also propose a
new method that combines the advantages of the translation
and the Beran procedures and allows to obtain tight
three-point bounds.  Such bounds take into account
three-point microstructural information in the form of the
so-called geometrical parameters. The new method is applied
to obtain three-point bounds on the effective complex bulk
modulus of viscoelastic composites. An intriguing feature of
all the results is their simplicity: the effective bulk and
shear moduli were shown to be constrained to a lens-shaped
regions of the complex bulk or shear moduli planes bounded
by the outermost pair of several circular arcs.
Microstructures were identified which have bulk or shear
moduli that correspond to various points on each of the
circular arcs. Thus these microstructures have extremal
viscoelastic behavior when the associated arc forms one of
the outermost pair.


Requests for preprints and reprints to: gibian@cherrypit.princeton.edu
WWW sources: http://cherrypit.princeton.edu/~gibian
This source can be found at http://www.math.utah.edu/research/