Uniform hydrostatic stresses inside a coated non-parabolic inhomogeneity in the vicinity of a concentrated couple

Abstract

We establish the existence of an internal uniform hydrostatic state of stress inside a coated non-parabolic open elastic inhomogeneity when the surrounding matrix is subjected to a concentrated couple and uniform remote in-plane stresses. The proof of existence is accomplished through analytic continuation and the introduction of a specialized form of the conformal mapping function containing an infinite number of first-order poles. The complex constants appearing in the mapping function can be uniquely determined from a resulting set of non-linear recurrence relations for given material, geometric and loading parameters. The convergence criterion for the series in the mapping function is rigorously established. A simple condition on the remote loading in the matrix is found for given elastic constants of the inhomogeneity and the matrix in order to ensure that the internal stresses inside the non-parabolic inhomogeneity are uniform and hydrostatic. The analysis indicates that the internal uniform hydrostatic stress field, the constant mean stress in the coating and the constant hoop stress along the inhomogeneity-coating interface on the coating side are all independent of the specific open shapes of the two interfaces and are also independent of the existence of the nearby concentrated couple. However, the non-parabolic open shapes of the two interfaces are significantly influenced by the concentrated couple. We also accomplish the design of an uncoated non-parabolic harmonic elastic inhomogeneity in the presence of an arbitrary number of concentrated couples applied in the matrix.