As mentioned, the stress field inside and outside of an elliptic inhomogeneity can be obtained using the equivalent inclusion method. To give a flavour of such a solution, here I show the xx, yy, and xy components of the stress field for an elliptical inhomogeneity along the x and y axes. The major and minor axes of the inhomongenety are, respectively, 50 and 25 units long and the major axis of the ellipse was aligned along the x axis. The eigenstrain was assumed to be dilatational (and 1%). Both the matrix and the precipitate were assumed to have the same Poisson's ratio of 1/3. The Lame's constants of the matrix were 100 and 150 (mu and lambda respectively, in reduced units), and that of the precipitate were 50 and 75.
The equivalent eigenstrain for this particular problem turned out to be 6.534527e-03 for the 11 component of eigenstrain and 9.219949e-03 for the 22 component of eigenstrain. Thus, in general the eigenstrain in the inclusion will not be dilatational even if the eigenstrain in the inhomogeneity was dilatational. For the special case of a circular inhomogeneity, dilatational eigenstrains in the inhomogeneity translate to dilatational eigenstrain in the equivalent inclusion, albeit with a different numerical value.
Note that stress tensor has a non-zero trace in the matrix. The cases where the stress tensor in the matrix will be traceless are the following: In 2D, circular inhomogeneity and elliptic inclusion with dilatational eigenstrains and in 3D, a sphere with dilatational eigenstrain.