Reciprocal relations for effective conductivities of anisotropic media

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Abstract
Any pair of two-dimensional anisotropic media with local conductivity tensors that are functions of position and that are related to one another in a certain reciprocal way are considered. It is proved that their effective conductivity tensors are related to each other in the same way for both spatially periodic media and statistically stationary random media. An inequality involving the effective conductivity tensors of two three-dimensional media that are reciprocally related is also proved. These results extend the corresponding results for locally isotropic media obtained by Keller, Mendelsohn, Hansen, Schulgasser, and Kohler and Papanicolau. They also yield a relation satisfied by the effective conductivity tensor of a medium reciprocal to a translated or rotated copy of itself.