Abstract
The analytic continuation method for representing transport in composites provides rigorous Stieltjes integral representations for the effective transport coefficients of two-phase random media. Here we
develop a mathematical framework which extends this method to the case of random uniaxial polycrystalline media. The integral representations of the bulk transport coefficients involve spectral measures
of self-adjoint random operators which depend only on the crystallite micro-geometry of the medium.
When the crystallites have finite lattice arrangements, these random operators are represented by random
matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors,
which enables statistical analysis of the spectral data. We also provide the mathematical foundation for
rigorous computation of spectral measures for such polycrystalline media, and develop a numerically efficient projection method to facilitate such computations. We employ this projection method to directly
compute the spectral measures associated with various crystallite orientation statistics. The computed
spectral measures are in excellent agreement with known theoretical results.
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