Conductivity of a periodic particle composite with transversely isotropic phases

The effective conductivity tensor of a particle composite with transversely isotropic phases is evaluated accurately. The geometric model of a composite material is a continuous matrix containing aligned spheroidal inclusions arranged in a periodic array. The rigorous solution is obtained for the model periodic boundary-value problem, the exact analytical expressions for all the components of the effective conductivity tensor are derived. The essence of the method used is the series representation of the solution in each phase of the composite and the reduction of the boundary-value problem to an infinite set of linear algebraic equations. This last step is obtained by full satisfaction of the interface contact conditions with the use of addition theorems developed for partial solutions of Laplace's equation in a spheroidal basis. The numerical results are presented demonstrating how the shape of inclusions and the anisotropy of phase materials influences the overall properties of the composite.