Inclusion problem in a two-dimensional nonlocal elastic solid

We have examined the problem of incorporating an inclusion in a two-dimensional linear elastic solid that includes the nonlocal interactions via strain-gradient contributions, and determined an analytical solution for the strain field that minimizes the elastic energy. We have effected this analysis by representing the presence of the inclusion by an external stress field which the elastic solid relaxes around in order to accommodate the inclusion. This formulation of an old problem is extremely general, and is to be contrasted with the solution determined by Eshelby which requires the knowledge of the solid's elastic Green's function, an impractical requirement for either systems including nonlocal interactions, or anisotropic, or nonlinear elastic materials. One result that follows from our analytical solution is that we can say with certainty that the effect of the nonlocal interactions is restricted to at most a few angstroms near the parent-phase/inclusion interface in almost all materials. An analytic solution is presented for graphite, which is of special interest because it forms an exception to this rule.