ON METHODS FOR BOUNDING THE OVERALL PROPERTIES OF NONLINEAR COMPOSITES - CORRECTION AND ADDITION

WILLIS (J. Mech. Phys. Solids 39, 73, 1991) concluded that a new bounding method for nonlinear composites, presented by PONTE CASTANEDA (J. Mech. Phys. Solids 39, 45, 1991) was equivalent to an earlier method which employed a nonlinear generalization of the Hashin-Shtrikman variational principle. This conclusion was reached by first showing that the nonlinear Hashin-Shtrikman bound is at least as good as the new bound and then that the new bound is at least as good as the older one. A fallacy in the latter part of this demonstration is exposed by considering a simple one-dimensional counter-example, corresponding to a nonlinear laminate. The conditions for coincidence identified by WILLIS (1991) are incomplete through failure to require explicitly that a stationary point defined by them yields a global minimum. Several cases have been studied previously, for which the two methods do yield the same bound; when they do, Ponte Castaneda's procedure has the potential to give an improvement by the use at an intermediate stage of an improved bound for a linear composite. When the methods yield different bounds, however, that produced by the nonlinear Hashin-Shtrikman procedure is the better.