Eshelby's problem of inclusion with arbitrary shape in an isotropic elastic half-plane

Eshelby's inclusion problem in an isotropic elastic half-plane is revisited in this paper. Following Zou et al. (2012, Int. J. Solids Struct. 49(13), 1627-1636), the general solution for Eshelby's inclusion problem in a half-plane is decomposed into the fundamental solution of Eshelby's problem with the same inclusion but in a full-plane, and the solution of an auxiliary boundary value problem of the half-plane. By virtue of the Kolosov–Muskhelishvili potentials of 2D isotropic elasticity, the auxiliary solution of the half-plane problem is derived in terms of the boundary integrals in the inclusion domain. For some inclusions of particular geometries, i.e., an arbitrary N-sided polygon, an arbitrary oriented ellipse and a circle (as a special case of the ellipse), the integrals involved are carried out explicitly, and eventually the potential functions are given with elementary functions. Some specified displacement, stress and strain energy density fields are illustrated and discussed. The obtained solutions are verified in more than one ways. Compared with the known methods, the one presented in this paper is more compact and more versatile, and will no doubt be helpful in achieving more explicit solutions besides those listed in this paper. Another feature of the present work is that both the Dirichlet and Neumann boundary conditions are accounted for in a unified form.