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.
REFERENCES
1.
Some of the results of the present study were briefly reported previously. See
Z.
Hashin
and S.
Shtrikman
, J. Franklin Institute
271
, 423
(1961
).2.
C. J. F. Böttcher, Theory of Electric Polarization (Elsevier Publishing Company, Houston, Texas, 1952), pp. 415–420.
3.
G. P. DeLoor, “Dielectric Properties of Heterogenous Mixtures,” Thesis, University of Leiden, Leiden (1956).
4.
5.
6.
7.
8.
These principles are an extension of a variational theorem formulated by W. F. Brown, Jr., in a private communication.
9.
Note that because of (2.15) the definition (3.2) is also equivalent to defining the effective permeability as the ratio of the average induction to the average field intensity.
10.
11.
W. F. Brown (private communication).
12.
A similar conclusion was reached in a study of the effective elastic moduli of multiphase materials. Z. Hashin and S. Shtrikman (to be published).
13.
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© 1962 The American Institute of Physics.
1962
The American Institute of Physics
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