Limitation of average eshelby tensor and its application in analysis of ellipse approximation

By the aid of irreducible decomposition, the average Eshelby tensor can be expressed by two complex coefficients in 2D Eshelby problem. This paper proved the limitation of complex coefficients based on the span of elastic strain energy density. More discussions yielded the constraints on the sampling of module and phase difference of complex coefficients. Using this information, we obtained that the maximum relative error is 65.78% after an ellipse approximation. These results, as a supplement to our previous paper, further implied that Eshelby's solution for an ellipsoidal inclusion could not be applied to non-ellipsoidal inclusions without taking care.