Coated non-elliptical harmonic inclusions with internal uniform hydrostatic stresses

This paper is concerned with the question of whether a single-coated, non-elliptical inclusion can be designed in such a way as to satisfy the harmonic condition of Bjorkman and Richards (the so-called ‘harmonic inclusion’). The minimization of stress concentration arising from both the internal and external stress fields of the inclusion is accomplished in the sense that the internal stress field is uniform and hydrostatic while the first invariant of the external stress field in the matrix remains constant. The inclusion is taken to be perfectly bonded to the coating at the inclusion-coating interface while the coating–matrix interface is assumed to be inhomogeneously imperfect of the spring-layer type. It is found that only a coated hypotrochoidal inclusion can be made harmonic and only in the case when the coating and the matrix have different Poisson’s ratios and the matrix is subjected to a uniform remote isotropic field. Both exact and approximate solutions are derived.