A brief history of bounds derived using the analytic method

Bergman (1978) recognized that the analytic properties discussed in chapter 18 on page 369 provide a powerful tool for deriving bounds. He rederived the Hashin-Shtrikman bounds and obtained new bounds correlating different properties of composites. A major success of the approach was that it lead to tight bounds on the complex dielectric constant of a two-phase composite (Milton 1979, 1980, 1981a; Bergman 1980, 1982). These bounds are illustrated in figure 27.1 on the next page. [The first available bounds on complex dielectric constants were those of Schulgasser and Hashin (1976), but they were limited to materials with low-loss constituents, that is, with permittivities having small imaginary parts.] These complex dielectric constant bounds have been directly compared with experimental measurements: Niklasson and Granqvist (1984) applied them to bounding the optical properties of composite films; Korringa and LaTorraca (1986) applied them to bounding the complex electrical permittivity of rocks; Golden (1995) applied them to bounding the complex permittivity of sea ice; and Mantese, Micheli, Dungan, Geyer, Baker-Jarvis, and Grosvenor (1996) applied them to bounding the complex dielectric constant and magnetic permeability of composites of Barium Titanate and ferrite. In most cases the experimental measurements were consistent with the bounds. However, it is important to recognize that these bounds apply only in the quasistatic limit where the wavelength of the radiation is much larger than the inhomogeneities of the microstructure; see Aspnes (1982). McPhedran, McKenzie, and Milton (1982); McPhedran and Milton (1990); and Cherkaeva and Golden (1998) applied the bounds in an inverse fashion to obtain quite tight bounds on the volume fraction from measurements of the complex dielectric constant.

Variational principles have long been known in the context of both conductivity type problems and elasticity problems. Their application to composites was initiated by Hill (1952), who used them to show that the Voigt (1889, 1910) and Reuss (1929) estimates of the elastic moduli of polycrystals were in fact bounds. A new type of variational principle was discovered by Hashin and Shtrikman (1962a, 1962b, 1963), which become famous because it lead to optimal bounds on the conductivity, bulk, and shear moduli of isotropic composites of two isotropic phases. Hill (1963b) gave rigorous proof of their variational principles, showing how they could be derived from the classical variational principles. Subsequently, Hashin (1967) generalized the variational principles to inhomogeneous elastic bodies (not just composites) subject to body forces and mixed boundary conditions. Cherkaev and Gibiansky (1994) extended all of these variational principles to media with complex moduli. Talbot and Willis (1985) extended the Hashin and Shtrikman variational principles to nonlinear media. Other variational inequalities for nonlinear media, based on comparisons with linear inhomogeneous media, were obtained by Ponte Castañeda (1991). In a development that falls outside the range of this book, Smyshlyaev and Fleck (1994, 1996) extended the Hashin and Shtrikman variational principles to elastic composites, where the elastic energy depends not only on the strain, but also on the strain gradient.

Classical variational principles and inequalities

We have seen how to manipulate equations into a form where the tensor entering the constitutive law is self-adjoint and positive-definite.

One of the most powerful methods for bounding the effective tensors of composites is what has become known as the compensated compactness method or translation method. The method was introduced by Murat and Tartar (Tartar 1979b; Murat and Tartar 1985; Tartar 1985); see in particular theorem 8 of Tartar (1979b), and independently by Lurie and Cherkaev (1982, 1984a). While embodying many of the same ideas, there is a difference between their approaches. For nonlinear media the two approaches give different types of bounds, as discussed in section 25.1 on page 529: the compensated compactness method of Murat and Tartar gives bounds on the average fields, while the approach of Lurie and Cherkaev gives bounds on the energy. Since both approaches yield identical results for linear media, the term translation method [introduced in Milton (1990b)] will be used to encompass both. The name arises because the bounds can be obtained by shifting, that is, translating, the tensor field by a constant tensor and applying the classical bounds. [This approach has the advantage that by applying the same translation, but replacing the classical bounds by tighter correlation function dependent bounds, one generates improved bounds that include more detailed information about the composite microgeometry; see section 26.5 on page 560. It has the disadvantage that one does not see why it is natural to consider bounds on the translated medium in the first place; see section 25.1 on page 529.]

This book is intended to be a self-contained introduction to the theory of composite materials, encompassing the electrical, thermal, magnetic, thermoelectric, mechanical, piezoelectric, poroelastic, and electromagnetic properties. It is intended not only for mathematicians, but also for physicists, geophysicists, material scientists, and electrical and mechanical engineers. Consequently, the results are not stated in the format of lemmas, propositions, and theorems. Instead, the focus is on explaining the central ideas and providing proofs that avoid unnecessary technicalities. The book is suitable as a textbook in an advanced-level graduate course, and also as a reference book for researchers working on composites or in related areas.

The field of composite materials is enormous. That's good, because it means that there are many avenues of research to explore. The drawback is that a single book cannot adequately cover the whole field. The main focus of this book is on the relation between the microstructure of composites and the effective moduli that govern their behavior. This choice reflects my research interests, and is also the starting point for many other avenues of research on composites.

The method of variation of poles and zeros discussed in the previous chapter is a powerful tool that can be applied to bound any rational function of fixed degree provided that we have some information on the location of the poles and zeros of that function. For those special classes of analytic functions appropriate to composites there is another approach based on the use of fractional linear transformations. This approach has the advantage that it is easily generalized to matrix-valued analytic functions, and in particular to the matrix-valued conductivity tensor of anisotropic composites. Fractional linear transformations were used by Bergman (1978) as a tool in deriving some of the elementary bounds. In 1980, following remarks of Jim Berryman and John Wilkins (private communication), I became aware of the large body of literature on bounding Stieltjes functions and realized that the fractional linear transformations of Baker, Jr. (1969), among others, provided an alternative proof of many of the bounds discussed in the previous chapter. Independently, Golden and Papanicolaou (1983), Kantor and Bergman (1984), and Bergman (1986, 1993) recognized that one could use fractional linear transformations to generate most of the hierarchies of bounds discussed in the last chapter. Their fractional linear transformations are similar to the ones used by Baker, Jr. (1969); see the appendix in Milton (1986).

The hierarchical structure of the bounds on σe, as a nested sequence of intervals on the real line, or as a nested sequence of lens-shaped regions in the complex plane, suggests that there may be some recursive method for deriving the bounds.

Countless approximations for estimating effective moduli have been introduced; some are semi-empirical, some are based on ad-hoc assumptions, and some have a reasonable theoretical basis. Here we will only review those well-known approximations that have a reasonable theoretical foundation, and which have withstood the test of time; see also the reviews of Van Beek (1967), Landauer (1978), Willis (1981), Markov (2000), and Buryachenko (2001). In addition, we discuss various asymptotic formulas that are applicable in high-contrast media.

Polarizability of a dielectric inclusion

Many approximations for the effective moduli of composites are based on the solution for dilute suspensions. For simplicity, let us suppose that we are interested in approximating the effective dielectric constant and require the solution for a dilute suspension of inclusions embedded in an isotropic matrix of dielectric constant ε0I. Since the grains are well-separated from each other, the field acting on each inclusion will be approximately uniform. To a good approximation we can solve for the field in the neighborhood of any such inclusion by treating it as if it was embedded in an infinite homogeneous medium of dielectric constant ε0 and subject to a uniform applied field at infinity. The analysis of this problem is the focus of the this section.

Consider an isolated, possibly inhomogeneous, inclusion that is embedded in an isotropic matrix of dielectric constant ε0 and subject to a uniform applied electric field a at infinity.

Why are bounds useful?

Very efficient numerical algorithms are currently available for calculating the effective tensors of quite complicated two-dimensional microgeometries. The numerical evaluation of effective tensors for three-dimensional microgeometries is also progressing rapidly. In light of these advances one might ask: Why is there a need for developing bounds on effective tensors? One reason is that they often provide quick and simple estimates for the effective tensors.

Another reason for favoring bounds is that in most experimental situations we do not have a complete knowledge of the composite geometry. Even when an accurate determination of the three-dimensional composite microgeometry is possible, obtaining this information and numerically parameterizing it (which may involve the triangulation of boundaries between phases) can be a very time-consuming process. Cross-sectional photographs give only limited information. For example, in a two-phase microgeometry it can be difficult to judge whether a phase is connected if a cross-sectional photograph shows only islands of that phase surrounded by the second phase. In the three-dimensional microgeometry, does the first phase consist of connected wire-like filaments, or does it consist of isolated elongated inclusions? The answer could have a large influence on one's estimates for, say, the effective conductivity when both phases have widely different conductivities. The problem of reconstructing the three-dimensional microstructure from a cross-sectional photograph is the subject of active research; see Yeong and Torquato (1998) and references therein.

Analyticity of the effective dielectric constant of two-phase media

Consider an isotropic composite of two isotropic phases. When the microgeometry is fixed it has a complex effective dielectric constant ε⋆(ε1, ε2), which is a function of the complex dielectric constants ε1 and ε2 of the phases that depend on the frequency ω of the applied field. As a prelude to the proof given in the next section, we will now present a strong argument that shows why ε⋆(ε1, ε2) should have some rather special analytic properties. The argument is based on the premise that analyticity properties of the dielectric constant as a function of the frequency ω should extend to composite materials.

The properties of the function ε1(ω) [or ε2(ω)] are well-known and are discussed, for example, by Jackson (1975); see also section 11.1 on page 222. The function ε1(ω) is analytic in the upper half ω-plane, Im(ω) > 0. When Re(ω) = 0 the function takes real values of ε1(ω) ≥ 1, which decrease and approach 1 as │ω│ → ∞. Positive imaginary values of ε1(ω) occur when ω has a positive real part and negative imaginary values of ε1(ω) when ω has a negative real part. As ω ranges over the upper half-plane ε1(ω) can in principle range anywhere in the cut complex plane, where the cut extends along the real axis from −∞ to 1.

This section presents a sampling of general techniques for using solutions of the conductivity or elasticity equations in a given microgeometry to generate associated solutions in related microgeometries. In addition to the methods discussed here, it should be mentioned that conformal transformations are useful for finding exact solutions for the fields, in particular, two-dimensional geometries: See Berdichevskii (1985), who found the exact solution for the fields in a regular checkerboard geometry; Obnosov (1996, 1999), who obtained the explicit solution for rectangular and triangular checkerboards, as well as for certain rectangular arrays of rectangles at a volume fraction of 1/4; the many papers of Vigdergauz referred to in section 23.9 on page 481, who found various periodic microstructures where the field was constant within one phase; and Reuben, Smith, and Radchik (1995).

Modifying the material moduli so the field is not disturbed

Consider the formula (7.13) for the effective bulk modulus k* of the coated sphere assemblage. It does not depend on the shear modulus µ1 of the core material. The physical reason for this is quite clear: Under an externally applied hydrostatic loading the displacement field (7.11) in the core of each coated sphere is a pure dilation with no shear component, and so the effective bulk modulus is not influenced by the shear modulus of phase 1. This is an example of a more general principle, which says that if we modify the material moduli in any way that leaves the field undisturbed, then the response of the composite to that average field will remain unchanged.

The main difficulty in applying the translation method is in choosing the translation T. This is intimately tied with the question of what microgeometries attain a given bound, since one typically wants to choose the translations so that the resulting bounds are attained by at least some microgeometries. This chapter addresses these two issues.

Other derivations of the translation bounds and their extension to nonlinear problems

The argument leading to the translation bounds (24.17) is appealing because it gives an interpretation of the bounds in terms of the harmonic mean bounds applied to the translated medium. Also, as will be discussed later in section 26.5 on page 560, one immediately sees that tighter correlation function dependent bounds could be used instead of the harmonic mean bounds. However, it is somewhat mysterious as it does not explain why it is natural to consider bounds on the translated medium in the first place. Both the compensated compactness approach (Tartar 1979a, 1979b, 1985; Murat and Tartar 1985; Murat 1987) and the variational approach (Lurie and Cherkaev 1982, 1984 1986; Gibiansky and Cherkaev 1984; Firoozye 1991) provide this insight.

In the compensated compactness approach, which is outlined in theorem 8 in the paper of Tartar (1979b), one seeks to find the possible values of the average field pair (〈E〉, 〈J〉) given the differential constraints on the fields and given that the fields satisfy the constitutive relation.

The fractional linear transformations provide mappings between analytic functions satisfying the homogeneity and Herglotz properties. The field equation recursion method provides mappings at a deeper level, namely, at the level of the underlying Hilbert space. The field equation recursion method has two advantages over the analytic method: First, it has a natural generalization to multiphase composites, and, second, the method provides matrix representations for the relevant operators. Also, with the introduction of additional fields and operators, the method allows one to incorporate the differential constraints on the fields in a direct fashion.

The original papers (Milton 1987a, 1987b) describing the field equation recursion method are difficult to read, but fortunately many of the arguments have since been simplified. It is hoped that the presentation given here will convey the main ideas; see also Milton (1991), where a brief summary is given. We will focus on two-phase composites, since the analysis is simpler and one can immediately see the connection with the analytic method. Another simple case that we could have discussed is the conductivity of a two-dimensional polycrystal, where Clark (1997) has shown that the field equation recursion method leads to an elegant continued fraction expansion of the effective conductivity tensor as a function of the crystal conductivity.