Elliptical inhomogeneity with polynomial eigenstrains embedded in orthotropic materials

A closed-form solution for elastic field of an elliptical inhomogeneity with polynomial eigenstrains in orthotropic media having complex roots is presented. The distribution of eigenstrains is assumed to be in the form of quadratic functions in Cartesian coordinates of the points of the inhomogeneity. Elastic energy of inhomogeneity–matrix system is expressed in terms of 18 real unknown coefficients that are analytically evaluated by means of the principle of minimum potential energy and the corresponding elastic field in the inhomogeneity is obtained. Results indicate that quadratic terms in the eigenstrains induce zeroth-order elastic strain components, which reflect the coupling effect of the zeroth- and second-order terms in the polynomial expressions on the elastic field. In contrast, the first-order terms in the eigenstrains only produce corresponding elastic fields in the form of the first-order terms. Numerical examples are given to demonstrate the normal and shear stresses at the interface between the inhomogeneity and the matrix. Furthermore, the solution reduces to known results for the special cases.