Bounds on the complex permittivity of polycrystalline materials by analytic continuation

Abstract

An analytic continuation method for obtaining rigorous bounds on the effective complex permittivity * of polycrystalline composite materials is developed. It is assumed that the composite consists of many identical anisotropic crystals, each with a unique orientation. The key step in obtaining the bounds involves deriving an integral representation for * , which separates parameter information from geometrical information. Forward bounds are then found using knowledge of the single crystal permittivity tensor and mean crystal orientation. Inverse bounds are also developed, which recover information about the mean crystal orientation from * . We apply the polycrystalline bounds to sea ice, a

Spectral measure computations for composite materials

“…The self-consistent mathematical framework developed here helps lay the groundwork for studies in the effective transport properties of a broad range of important composites, such as electrorheological fluids [57], multiscale sea ice structures, and bone [32]. Remarkably, the ACM has also been adapted to provide Stieltjes integral representations for effective transport coefficients underlying a wide variety of transport processes, such as: the effective diffusivity for steady [50,2] and time-dependent [3] fluid velocity fields, the effective complex permittivity for uniaxial polycrystalline media [5,37], and the effective elastic moduli of two-phase elastic composites [60,61]. The Golden-Papanicolaou formulation of the ACM has been pivotal in the development of these mathematical frameworks, and in the understanding of these important transport processes.…”

Modeling Sea Ice

“…More recently, again motivated by sea ice processes, there have been several advances in extending the ACM to larger-scale problems. These include homogenization for polycrystalline materials [GLCG15], advection diffusion processes involving incompressible velocity fields [S4,S66], such as thermal transport through sea ice enhanced by brine convection [S51], and ocean surface wave propagation through the sea ice pack treated as a two-phase composite of ice floes and sea water [S84].…”

“…In extending the analytic continuation approach beyond two-phase composites, a Stieltjes integral representation and bounds were obtained for * of polycrystalline composites in general, and sea ice in particular, considered as a three-dimensional, transversely isotropic or uniaxial polycrystalline composite material [GLCG15]. The forward bounds on the components of * use information about the complex permittivity tensor of the individual crystals and the mean crystal orientation.…”

“…In Figure 4 (left) they are in good agreement with measurements [S2], and (right) they are compared with the classic two-component bounds. The inverse bounds [GLCG15,S16] for the mean orientation are obtained from measurements of * , and lay the groundwork for determining ice type using remote sensing techniques.…”

Approximating the Effective Tensor as a Function of the Component Tensors in Two-Dimensional Composites of Two Anisotropic Phases

“…Note that Π 2 = I so the eigenvalues of Π are either +1, corresponding to eigenfunctions h s (x 1 , x 2 ) that are symmetric vector fields satisfying 23) or −1, corresponding to eigenfunctions h a (x 1 , x 2 ) that are antisymmetric vector fields satisfying and then (I+Π)/2 is the projection onto H s , while (I−Π)/2 is the projection onto H a . Now the actual Hilbert space of interest, thus defined, is infinite-dimensional.…”