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Imperfections in microstructures such as voids, cracks and inhomogeneities cause
large changes in overall mechanical behavior of the structures. Determination of
stress fields inside/outside single and some inhomogeneities is of great
importance to understanding of fracture, fatigue strength and failure behavior of
the heterogeneous materials. Classical research was reported in the early 1960’s
by Eshelby [1-3], with many corresponding investigations [4-11]. Asaro and
Barnett [12] showed that when the eigenstrain inside an ellipsoidal inclusion is of
the form of a polynomial of an arbitrary order in Cartesian coordinates, an
induced strain field in the inclusion is also characterized by a polynomial of the
same order. The result for the polynomial eigenstrains is referred to as Eshellby’s
polynomial conservation theorem [13].