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.