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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.

The problem of bounding the effective conductivity of a two–phase composite with an imperfect interface is considered. The interface can be either highly conducting or resistive, and both the material properties and geometric arrangement of the phases can be anisotropic. The problem is formulated variationally and by choosing appropriate trial fields, new bounds are obtained in terms of upper and lower bounds on the effective conductivity of a composite with the same microgeometry in which the phases are perfectly bonded. The methodology also applies to composites with a nonlinear interface, and a particular example is described.

The construction of generalized Hashin‐Shtrikman bounds for nonlinear composite problems relies on the introduction of a comparison material. In recent work, a nonlinear comparison medium has been used; however, this requires detailed knowledge of the properties of the trial fields that are employed. The fields used have the property of ‘bounded mean oscillation’ and this enables the size of the penalty exacted by using a nonlinear comparison material to be controlled. Some recent results concerning Riesz transforms and the Beurling operator are used here to reduce the effect of the penalty and hence to generate improved bounds. In addition, an exact solution is established for a particular class of composites.

Lower bounds for the overall energy functions of a class of nonlinear composite conductors are established by two different methods: the translation method, and the use of a comparison linear composite. When the properties of the comparison linear composite are themselves estimated by the translation method, the resulting comparison bound is demonstrated never to be superior to the bound obtained by direct application of the translation method to the nonlinear composite. Examples show, however, that the bounds obtained by these two methods in fact often coincide; when they do, the use of the comparison medium permits the development of more refined bounds, by exploiting better bounds for the linear composite. Particular emphasis is placed on a limiting case of material behaviour for which explicit results can be obtained: a composite dielectric that displays breakdown when the local electric field exceeds a critical value. The formalism delivers an upper bound for the value of the mean field in the composite which induces breakdown in the composite. Bounds are also established for a composite with a power-law relation between current and electric field.

The nonlinear Hashin–Shtrikman variational structure relies on the introduction of a comparison material. In recent work a nonlinear comparison composite has been employed. This has enabled both upper and lower bounds to be found that depend on the three–point statistics of the microstructure through a parameter introduced by G. W. Milton. At least one of the bounds requires an estimate of the size of the trial field to be known and even if the best estimate available is used and the Milton parameter is known for a particular microstructure, the bounds can still be widely separated. O. P. Bruno has obtained bounds for a linear composite consisting of a matrix containing spherical inclusions where the inclusions are coated with the matrix phase. The bounds remain non–trivial even when the properties of the inclusions tend to either zero or infinity. These bounds are used to improve the bounds for a nonlinear composite comprising either a nonlinear matrix containing linear inclusions or a linear matrix containing nonlinear inclusions. For the microgeometry considered here, it is possible to construct a bounded trial field and a simple example involving this approach is also described.