A mathematical scaling approach to macroscopic heterogeneity of composite and porous media is introduced. It is based on weak limits of uniformly bounded measurable functions. The limiting local porosity distributions that were introduced in the work [Adv. Chem. Phys. XCII, 299–424 (1996)] are found to be related to Young measures of a weakly convergent sequence of local volume fractions. The Young measures determine frequency dependent complex dielectric functions of multiscale media within a generalized self-consistent effective medium approximation. The approach separates scales by scale factor functions of regular variation. It renders upscaled results independent of the shape of averaging windows upon reaching the scaling limit.
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Research Article| October 26 2018
Multiscale local porosity theory, weak limits, and dielectric response in composite and porous media
R. Hilfer; Multiscale local porosity theory, weak limits, and dielectric response in composite and porous media. J. Math. Phys. 1 October 2018; 59 (10): 103511. https://doi.org/10.1063/1.5063466
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