Exact relations between magnetoresistivity tensor components of conducting composites with a columnar microstructure

A recently discovered duality transformation for a three-dimensional columnar medium (i.e., a medium that is uniform along some fixed direction, called the columnar axis) is applied to a two-constituent composite medium. This leads to a generalization of Keller’s theorem: Exact relations are found between in-plane components (i.e., components that are perpendicular to the columnar axis) of the bulk effective resistivity or conductivity tensor of the columnar composite medium and similar quantities of a dual problem. These are used to find exact relations between in-plane bulk effective magnetoresistivity and magnetoconductivity components of a conductor/insulator mixture and those of a normal conductor/perfect conductor mixture with the same microstructure. The exact relations are applied to and tested on a number of available results on in-plane magnetoresistance in columnar composites, including an effective medium approximation, numerical computations on periodic microstructures, and closed form expressions for the asymptotic behavior at strong magnetic fields in periodic microstructures.