Representations for the conductivity functions of multicomponent composites

Abstract

The erective conductivity a * of a multicomponent composite material is considered. Integral representations for a* treated as a holomorphic function on a polydisk with values in a half-plane are analyzed. A representation for a* is introduced which is symmetric in the component conductivities and for which the moments of the positive measure in the integral are directly related to the coefficients in a perturbation expansion of a* around a homogeneous medium. This second feature, which is important for obtaining bounds on a*, was previously available only in the two-component case. In addition, a bound valid for any holomorphic function of the above type is proven.

Minimum variational principles for time-harmonic waves in a dissipative medium and associated variational principles of Hashin–Shtrikman type

“…Their variational principles were employed to give a simple derivation (Milton 1990) of existing bounds on the effective complex electrical permittivity of lossy multi-phase composites. (These bounds, which generalized the two-phase bounds of Milton (1981) and Bergman (1982), were first conjectured by Golden & Papanicolaou (1985) and Golden (1986) and subsequently proved by Bergman (1986), Milton (1987) and Milton & Golden (1990).) Their variational principle also provided entirely new bounds on the complex bulk and shear moduli of viscoelastic two-phase composites (Gibiansky & Lakes 1993Gibiansky & Milton 1993;Gibiansky et al 1999), which, in turn, led to bounds on the complex thermal expansion coefficient of viscoelastic two-phase composites (Berryman 2009).…”

Exact relations for Green’s functions in linear PDE and boundary field equalities: a generalization of conservation laws

“…Grabovsky's pioneering work, developed further with Sage in [19], provided essential clues that led to the breakthrough result [18] establishing conditions that guarantee an exact relation holds for all composites, and not just laminates. Using carefully devised perturbation expansions that had their basis in [32] Sect. 5, coupled with analytic continuation arguments, one sees [18] that finding exact relations which hold for all composite geometries is tied with identifying tensor subspaces K such that for all Fourier vectors k = 0 one has…”

Comparative study of total absorption of light by two-dimensional channel and hole array gratings

“…An interested reader can find a detailed work on homogenization theory in a recent review by G. W. Milton [19]. The specific form can be taken into account by the concept of polarisability, and the necessity to use the average values not only of the fields but of the modulus square of the electric field (see ch.16 of [19] and ref [20].). However, we are not able to directly apply this approach, because it is developed in the quasistatic limit as the frequency tends to zero, while in our case the frequency is finite, and the periods in the grating plane tend to zero.…”