Greens's functions for an anisotropic elliptic inclusion under generalized plane strain deformations

Abstract

Green's functions for an infinite anisotropic elastic medium with an elliptic inclusion of dissimilar material are obtained for a line force and a line dislocation that may be located outside, inside, or on the interface of the elliptic inclusion. From the expressions of the Green's functions it is clear that they are identical in the limit when the line force and the line dislocation are on the elliptic boundary. This is achieved by carefully including all image singularities in the solutions. While the solutions contain an infinite series, it is shown that closed-form solutions are possible when (i) the impedances of the two materials are identical, or (ii) the Stroh eigenvalues p α (α = 1, 2, 3) for the material inside the inclusion are triple roots and are given by p 1 = p 2 = p 3 = ia/b, where a and b are the major and minor axes of the ellipse. A special case of (ii) is when the ellipse is a circle, and the material inside the circle is isotropic. The material outside the ellipse can be arbitrary.