Bounds for the effective constitutive relation of a nonlinear composite

For a nonlinear composite, a bound on its effective energy density does not induce a corresponding bound on its constitutive relation, because differentiating a bound on a function does not automatically bound its derivative. In this work, a method introduced by Milton and Serkov for bounding directly the constitutive relation is refined by employing a linear comparison material, in a way similar to that employed by the present authors to obtain bounds of ‘Hashin–Shtrikma’ type for the effective energy of a nonlinear composite. The original Milton–Serkov approach produces bounds with a close relationship to the classical energy bounds of Voigt and Reuss type. The bounds produced in the present implementation are closely related to bounds of Hashin–Shtrikman type for the composite. Indeed, in the case of a linear composite, the present method delivers the Hashin–Shtrikman bounds in their generalized form, valid for any two–point statistics. It is demonstrated for nonlinear examples that the approximate constitutive relation that is obtained by differentiating the energy bound can be on the boundary of the bounding set for the exact constitutive relation, but a simple counterexample is presented to show that this is not always the case.