Bounds and approximations for the 
       effective properties of nonlinear composites.

                             by

          Leonid V. Gibiansky, Princeton University

          JTB120 3:20pm   Friday, February 27, 1998


                          ABSTRACT

At present, two approaches are developed to obtain bounds on
the effective properties of nonlinear composites. One is the 
non trivial generalization of the Hashin-Shtrikman procedure 
due to Talbot and Willis. The other is the Ponte-Castaneda 
variational method that allows one to use bounds (or approxi-
mations) on the effective moduli of linear composites to 
obtain corresponding results for  the nonlinear materials. 
In our work we apply the translation method to derive bounds 
for nonlinear composites. The translation method is a very 
powerful tool for obtaining bounds on the effective properties 
of linear composites. It is natural to expect that it should 
yield good results in the nonlinear case as well. We show that 
the Ponte-Castaneda bounds immediately follow when the energy 
of the corresponding linear comparison composite is used as a 
quasiaffine quadratic form in the translation method procedure. 
We demonstrate that the same bounds can be obtained by direct
implementation of the translation method by using the same 
quasiconvex quadratic forms that are used to obtain 
corresponding linear bounds. There are evidences that all
these methods give similar or identical results although they 
are quite different in implementation. 

To illustrate the results we use the Ponte-Castaneda and the 
translation methods to obtain bounds on the effective energy 
of the two-phase nonlinear conductor, and the two-phase plane 
elastic composite with the power-law phase energy functions.

Request for reprints to gibian@cherrypit.princeton.edu