Non-uniform eigenstrain induced stress field in an elliptic inhomogeneity embedded in orthotropic media with complex roots

This paper deals with an elastic orthotropic inhomogeneity problem due to non-uniform eigenstrains. The specific form of the distribution of eigenstrains is assumed to be a linear function in Cartesian coordinates of the points of the inhomogeneity. Based on the polynomial conservation theorem, the induced stress field inside the inhomogeneity which is also linear, is determined by the evaluation of 10 unknown real coefficients. These coefficients are derived analytically based on the principle of minimum potential energy of the elastic inhomogeneity/matrix system together with the complex function method and conformal transformation. The resulting stress field in the inhomogeneity is verified using the continuity conditions for the normal and shear stresses on the boundary. In addition, the present analytic solution can be reduced to known results for the case of uniform eigenstrain.