Macroscopic conductivity tensor of a three-dimensional composite with a one- or two-dimensional microstructure

摘要

Exact linear relations are found among different elements of the macroscopic conductivity tensor of a three-dimensional, two-constituent composite medium with a columnar microstructure, without any further assumptions about the forms of the constituent conductivities: Those can be arbitrary nonscalar, nonsymmetric, and nonreal (i.e., complex valued) tensors. These relations enable all the elements of the macroscopic conductivity tensor of such a system to be obtained, from a knowledge of the macroscopic conductivity tensor components only in the plane perpendicular to the columnar axis. Exact linear relations are also found among different elements of the macroscopic resistivity tensor of such systems. Again, these relations enable all the elements of the macroscopic resistivity tensor of such a system to be obtained, from a knowledge of the macroscopic resistivity tensor components only in the plane perpendicular to the columnar axis. We also present simple exact expressions for all elements of the macroscopic conductivity tensor of a three-dimensional composite medium with a parallel slabs or laminar microstructure and an arbitrary number of constituents, again without making any assumptions about the forms of the constituent conductivities, which can be arbitrary nonscalar, nonsymmetric, and nonreal tensors. The latter results were obtained previously, but their great generality and extreme simplicity were not realized by most physicists.