There were a lot of observations performed to evaluate the behavior within and in the setting of inhomogeneities placed within a homogeneous material. Poisson conducted a study that shows the generated magnetic field within an ellipsoid is constant for a constant electric polarization. This is with respect to a disturbed field resulting from an isolated ellipsoid which is part of the subject of Newtonian potential problem. An expression was formulated by Maxwell for this type of field. It was also proved by Dive and Nickliborc that in a far field the polarization of electricity is the same as what will be generated in an ellipsoid. Eshelby proposed that ellipsoid is the only shape that can generate a uniform strain with respect to an arbitrary strain in a far-field. This provides a strong claim for a specific far-field. However, this is a weak proposition for an arbitrary far-field strain. Both are considered accurate for two dimensions only. Beyond that this two propositions will become untrue. Eshelby tensor and Hill tensor considered equivalent, are both efficient for ellipsoidal inhomogeneity problems. Their goal is to provide functional attributes of media with inhomogeneity. For non-ellipsoidal inhomogeneities the classical Eshelby tensors are not applicable. Instead, a generalised Eshelby tensor is preferred.
Researchers Duncan Joyce and William J. Parnell submitted this article, " The Newtonian potential inhomogeneity problem: non-uniform eigenstrains in cylinders of non-elliptical cross section ", published in CrossMark ; volume 107, pages: 283–303. This article was published to present how Eshelby tensor were approximated for different shapes with inhomogeneities for complicated eigenstrains both uniform and non-uniform using polynomial expansion. Also, how Eshelby tensor were developed from potential problem in two dimensions.
There are two main reasons why the generalized Eshelby tensor was explored. One is the presence of inhomogeneity placed within a homogeneous material. Secondly, a region that is included behaves as is with a homogeneous material but within an eigenstrain field.
For inhomogeneity problems, a cylindrical inhomogeneity was placed within an extensive channel that was modeled absolutely two-dimensional. The possible complication found is with respect to the steady state thermal conductivity.
For the eigenstrain, the tensor for the included region is the same as the outside. This will generate a disturbed activity of its field. Purely two-dimensional as well but a specific field within infinity is not forced.
Obtaining a direct solution for tensors in inhomogenetities was extremely strenuous unless a particular inhomogeneity is given. Otherwise, a particular solution that is based on polynomial expansion will be used. The method started by letting Eshelby tensor equal to zero and by approximating an elliptical inhomogeneity. Then moved into a more complex domain such as superellipse still equating Eshelby tensor to zero. As the polynomial approximations increases, the error decreases.
Generalised Eshelby tensor and classical Eshelby tensor for non-uniform and uniform eigentrains; that is a strain due to temperature or phase change has been developed in this study to provide a polynomial-approximation to a given integral over a non-eleptical two-dimensional domains. A promising result was obtained from the performances of the two methods. After determining the integral by evaluating it at every point of its domain of interest, its polynomial-approximation can then be evaluated at any point. A study about two-dimensional in-plane plasticity and full three-dimensional scalar and tensor scenarios is already on its way as part of the progress of this current study.