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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.

The Hashin-Shtrikman methodology for nonlinear composite problems relies on the use of a comparison medium and in many of the examples studied so far the comparison medium has been taken to be homogeneous. A related approach originated by P. Ponte Castaneda employs a comparison medium which is itself a linear composite with the same microgeometry as the nonlinear composite. When the method is applicable, the bounds for the nonlinear problem then involve bounds for the energy of the linear comparison composite which could include, for example, three-point information about the microstructure of the composite. It is, however, only possible to obtain at most one bound using a linear comparison material. A recent approach involves the use of a nonlinear comparison medium and trial fields with the property of bounded mean oscillation. In this paper the approach is extended by using a nonlinear comparison composite so that both upper and lower bounds can be obtained which incorporate three-point information. The development is in the context of bounding the properties of nonlinear dielectric composites, although it has wider application.

The study of the effective properties of a nonlinear composite dielectric begun in the companion to this paper is continued in the context of a simple cubic array of inclusions embedded in a matrix. Use of a nonlinear comparison material in the nonlinear Hashin-Shtrikman variational principles permits the generation of both upper and lower bounds for any composite. It is straightforward to apply the new approach to the simple composite considered here as the trial electric field which is substituted into the Hashin-Shtrikman variational principle is finite everywhere. The results obtained indicate the likely improvement over the simple classical bounds that can be obtained using the Hashin-Shtrikman principles.

One method which is widely used for bounding the overall properties of linear composites makes use of the variational principles of Hashin and Shtrikman [2,3]. The bounds obtained depend only on the properties and volume fractions of the constituent phases, although they rely on the isotropy of the two-point geometrical correlations. Less is known when the constituent phases exhibit nonlinear behaviour although a generalization of the Hashin-Shtrikman principles, developed by Talbot and Willis [7], has been successfully used to bound the properties of some nonlinear composites. Initially, these bounds relied on the introduction of a linear comparison material which was required to be either more stiff or less stiff than the response of any of the phases at high field values. This limits the range of bounds that can be found as, for most nonlinear composites, it is only possible to find a linear material which satisfies at most one of these requirements.

A nonlinear fibre reinforced composite is considered whose local behaviour is described by a complementary energy function. The composite is regarded as nonlinearly elastic but subject to geometrically linear strains and attention is restricted to fields which are independent of the coordinate in the direction of the generators of the fibres. The overall complementary energy is defined and a generalization of the Hashin-Shtrikman variational structure is applied to the resulting problem. The results take the form of bounds for the complementary energy and results are presented for a random distribution of linearly elastic fibres embedded in a nonlinear matrix.

An extension of the Hashin-Shtrikman variational structure to nonlinear problems, initiated by J. R. Willis (J. appl. Mech. 50, 1202-1209 (1983)) and developed generally by D. R. S. Talbot & J. R. Willis (IMA J. appl. Math.35, 39-54 (1985)) is applied to the problem of homogenization of the semilinear steady-state diffusion equation ∇2c- (∂W/∂c) (c, x) +K(x)= 0, where the functionWis convex incbut varies randomly with positionx. A strict definition for an ‘overall’ functionŴwhich defines the homogenized medium is given and its properties are discussed. Bounds forŴare then developed from variational principles. Classical energy principles yield bounds analogous to the Voigt and Reuss bounds of elasticity theory while the new ‘Hashin-Shtrikman’ structure yields tighter bounds, which are sensitive to the two-point statistics of the medium. Explicit results are presented for a particular two-phase com­posite medium for which one phase is linear (so thatWis quadratic) whereas the second has a quadratic nonlinearity (so thatWis cubic).