Effective Conductivity in Effective Media and Mixtures
Three methods are available to compute the averaged electrical conductivity of the mixture.
Volume Average, Conductivity
If the electric conductivities of the two materials are not so different from each other, a simple form of averaging can be used, such as a volume average:
where σi is the conductivity of the material i. This is equivalent to a “parallel” system of resistivities.
If the conductivities are defined by second order tensors (such as for anisotropic materials), the volume average is applied element by element.
Volume Average, Resistivity
A similar expression for the effective conductivity can be used, which mimics a “series” connection of resistivities. Equivalently, the effective conductivity is obtained from
If the conductivities are defined by second order tensors, the inverse of the tensors are used.
Power Law
A power law gives the following expression for the equivalent conductivity:
The effective conductivity calculated by Volume Average, Conductivity is the upper bound, the effective conductivity calculated by Volume Average, Resistivity is the lower bound, and the Power Law average is somewhere between these two.