Effective properties of graded elliptical cylindrical composites

The effective property has been investigated theoretically in graded elliptical cylindrical composites consisting of inhomogeneous graded elliptical cylinders and an isotropic matrix under external uniform electric field. As a theoretical model, the dielectric gradient profile in the elliptical cylinder is modeled by a power-law function of short semi-axis variable parameter (ξ2-1) in the elliptical cylindrical coordinates, namely εi(ξ)=ck(ξ2-1)k, where ck and k are the parameters, and ξ is the long semi-axis space variable in an elliptical cylindrical inclusion region. In the dilute limit, the local analytical potentials in inclusion and matrix regions are derived exactly by means of the hyper-geometric function, and the formulas are given for estimating the effective dielectric responses under the external field along x^- and y^-directions, respectively. Furthermore, we have demonstrated that our effective response formulas can be reduced to the well-known results of homogeneous isotropic elliptical cylindrical composites if we take the limit k→0 in graded elliptical cylindrical composites.