Analytical solution for the displacement of a polygonal inclusion with a special application to the case of linear eigenstrain

Abstract

Previous studies on a polygonal inclusion have been concerned primarily with the stress/strain solutions of uniform eigenstrains, while the analyses of the displacements are less elaborated in literature. By employing the method of Green's function, the displacement solution may be formulated in an area integral, which is then converted to a contour integral around the boundary of the inclusion. This work presents a straightforward and closed-form evaluation of the displacements when the boundary of the inclusion is composed of line elements. The proposed method of solution can not only deal with the widely investigated problem of uniform eigenstrains, but is also effective for handling linearly distributed eigenstrains. The strain Eshelby tensor may be derived in a straightforward manner from the displacement solution. Unlike the classical Eshelby ellipsoidal inclusion, where the solutions for the exterior field are complicated, the displacements for both interior and exterior points of a polygonal inclusion can be written in a unified algebraic form of elementary functions. It is well-known that the stress and strain fields have logarithmic singularities at the corners of a polygonal inclusion. In contrast, the present work shows that the displacement is continuous and finite at the vertices of the polygon.