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Variational theorems are established and applied to the derivation of bounds for the effective magnetic permeability of macroscopically homogeneous and isotropic multiphase materials. For reasons of mathematical analogy the results are also valid for the dielectric constant, electric conductivity, heat conductivity, and diffusivity of such materials. For the case of two-phase materials, the bounds derived are the most restrictive ones that can be given in terms of the phase permeabilities and volume fractions. Comparison of present theoretical results with existing experimental data shows good agreement.
Hashin et al. (1962) studied this question.