Dielectric behavior of anisotropic inhomogeneities: interior and exterior point Eshelby tensors

In this work we analyze the problem of finding the electric behavior of an anisotropic ellipsoid (arbitrarily shaped) placed in a dielectric anisotropic environment. We suppose that the whole system is exposed to a uniform electric field remotely applied. In order to find the resulting electric quantities inside the particle and outside it we adopt a technique largely utilized for solving similar problems in elasticity theory. The inhomogeneity problems in elastostatics are solved within the framework of the Eshelby theory, which adopts, as crucial points, the concepts of eigenstrains and inclusions. The generalization and assessment of such an approach for the dielectric inhomogeneity problems is here addressed by means of the introduction of the concepts of eigenfields and inclusions in electrostatics. The advantages of this methodology are mainly two: firstly, we can consider completely arbitrary dielectric anisotropic behavior both for the particle and the host matrix. Secondly, we easily find explicit expressions for the electric quantities both inside and outside the inhomogeneity. The problem under consideration was solved in earlier literature by analyzing the singularity of the dyadic Green function, expressed as a two-dimensional integral. Here we propose a reformulation described by a one-dimensional integral obtained from explicitly electrostatic analysis, which can have both pedagogical and computational importance. We also introduce a method to generalize these results to the case of an arbitrary nonlinear anisotropic ellipsoid embedded in a linear anisotropic matrix. Finally, we show some applications to the dielectric characterization of anisotropic composite materials.