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The simplest exact composite relations that are sometimes useful are the well-known series and parallel results for two different phases with volume fractions of x for phase 1 and 1 − x for phase 2. If σ1 and σ2 are the isotropic conductivities of the two phases, then if the phases are arranged in parallel, the effective conductivity will be given by σ = xσ1 + (1 − x) σ2. The field in both phases will be uniform and equal to the applied field. If the two phases are arranged in series, then the effective conductivity will be
In this case, the field in each phase will be uniform, but different. The
current, however, must be
the same in both phases, as the direction of current flow is normal to the
interface, so that
it is simple to
work out these fields,
which can then be used to check the phase average of the field result for the program.
It is important that any numerical scheme be checked on more that just the series and
parallel problems, as these do not give realistic phase boundaries.