Variational second-order estimates for nonlinear composites

It is shown here that the approximate 'second-order' method of Ponte Castaneda for estimating the effective behaviour of nonlinear composite materials can be given a variational status. This is accomplished by means of a slight generalization of the variational structure of Talbot & Willis, consisting of the replacement of the infimum over the polarization field by the more general corresponding stationary operation. Since the relevant functions are not convex in general, this replacement provides additional flexibility that can be exploited advantageously. The net result is a stationary principle, from which stationary variational estimates can be generated. Under certain hypotheses on the choice of the linear comparison composite, it is shown that the resulting estimates are identical to the 'second-order' estimates, thus showing that these second-order estimates are, indeed, variational in character, although only of the stationary variety. The connection of these second-order estimates to earlier bounds, obtained from related extremum variational principles, is also explored.