Spectral theory of effective transport for continuous uniaxial polycrystalline materials

Abstract

Following seminal work in the early 1980s that established the existence and representations of the homogenized transport coefficients for two phase random media, we develop a mathematical framework that provides Stieltjes integral representations for the bulk transport coefficients for uniaxial polycrystalline materials, involving spectral measures of self-adjoint random operators, which are compositions of non-random and random projection operators. We demonstrate the same mathematical framework also describes two-component composites, with a simple substitution of the random projection operator, making the mathematical descriptions of these two distinct physical systems directly analogous to one another. A detailed analysis establishes the operators arising in each setting are indeed self-adjoint on an $L^2$-type Hilbert space, providing a rigorous foundation to the formal spectral theoretic framework established by Golden and Papanicolaou in 1983. An abstract extension of the Helmholtz theorem also leads to integral representations for the inverses of effective parameters, e.g., effective conductivity and resistivity. An alternate formulation of the effective parameter problem in terms of a Sobolev-type Hilbert space provides a rigorous foundation for an approach first established by Bergman and Milton. We show that the correspondence between the two formulations is a one-to-one isometry. Rigorous bounds that follow from such Stieltjes integrals and partial knowledge about the material geometry are reviewed and validated by numerical calculations of the effective parameters for polycrystalline media.