The Newtonian potential inhomogeneity problem: non-uniform eigenstrains in cylinders of non-elliptical cross section

Understanding the fields that are set up in and around inhomogeneities is of great importance in order to predict the manner in which heterogeneous media behave when subjected to applied loads or other fields, e.g., magnetic, electric, thermal, etc. The classical inhomogeneity problem of an ellipsoid embedded in an unbounded host or matrix medium has long been studied but is perhaps most associated with the name of Eshelby due to his seminal work in 1957, where in the context of the linear elasticity problem, he showed that for imposed far fields that correspond to uniform strains, the strain field induced inside the ellipsoid is also uniform. In Eshelby's language, this corresponds to requiring a uniform eigenstrain in order to account for the presence of the ellipsoidal inhomogeneity, and the so-called Eshelby tensor arises, which is also uniform for ellipsoids. Since then, the Eshelby tensor has been determined by many authors for inhomogeneities of various shapes, but almost always for the case of uniform eigenstrains. In many application areas in fact, the case of non-uniform eigenstrains is of more physical significance, particularly when the inhomogeneity is non-ellipsoidal. In this article, a method is introduced, which approximates the Eshelby tensor for a variety of shaped inhomogeneities in the case of more complex eigenstrains by employing local polynomial expansions of both the eigenstrain and the resulting Eshelby tensor, in the case of the potential problem in two dimensions.