Multiphase composites with extremal bulk modulus

                                   by

                           Leonid V. Gibiansky
                               Globomax LLC

                        in collaboration with

                              Ole Sigmund
                    Department of Solid Mechanics,  
                   Technical University of Denmark

               INSCC 110, 3:30pm  Monday, January 25, 1999


                                Abstract

This talk is devoted to the analytical and numerical study of
isotropic elastic composites made of three or more isotropic phases. The
ranges of their effective bulk and shear moduli are restricted by the  
Hashin-Shtrikman-Walpole (HSW) bounds. For two-phase composites, 
these bounds are attainable, that is, there
exist composites with  extreme bulk and shear moduli.  For
multiphase composite, they may or may not be attainable depending on phase  
moduli and volume fractions. Sufficient conditions of attainability of the
bounds and various previously known and new types of optimal composites
are described. A numerical topology optimization  procedure that 
solves the inverse homogenization problem
is adopted and used to look for two-dimensional three-phase
composites with maximal effective bulk modulus. For the
combination of parameters where the HSW bound is known to be attainable, new
microstructures are found numerically that possess bulk moduli close to 
the bound. Moreover, new types of microstructures with the bulk moduli close
to the bound are found  numerically for the situations where the
aforementioned attainability conditions  are not met.  
Based on the numerical results, several new types of structures
that possess extremal bulk modulus are suggested and studied analytically. 
The bulk moduli of the new structures are either equal to the HSW bound or
higher than the bulk modulus of any other known composite with
the same phase moduli and volume fractions. It is proved that the HSW
bound is attainable in a much wider range than it was previously
believed. Results are readily applied to
two-dimensional three-phase isotropic conducting composites with
extremal conductivity. They can also be used to study transversely isotropic
three-dimensional three-phase composites with cylindrical inclusions 
of arbitrary cross-sections.