Neutrality in the case of N-phase elliptical inclusions with internal uniform hydrostatic stresses

A novel method is proposed to design neutral N-phase (N ⩾ 3) elliptical inclusions with internal uniform hydrostatic stresses. We focus on the study of the internal and external stress states of an N-phase elliptical inclusion which is bonded to an infinite matrix through (N − 2) interphase layers. The interfaces of the N-phase elliptical inclusion are (N − 1) confocal ellipses. The design of the resulting overall composite material consists of four stages: (i) an inner perfectly bonded interphase/inclusion interface which is necessary to make the internal uniform stress state hydrostatic; (ii) outer imperfect interphase layers properly designed to make the coated inclusion harmonic (i.e., the uniform mean stress of the original field within the matrix is unperturbed); (iii) the aspect ratio of the elliptic inclusion uniquely chosen for a given material and thickness parameters to make the resulting coated inclusion neutral (i.e., the prescribed uniform stress field in the matrix remains undisturbed); and finally (iv) the derivation of a simple condition relating the remote uniform stresses and the thickness parameters of the (N − 2) interphase layers for given material parameters which lead to internal uniform hydrostatic stresses. We note that another interesting feature of the present results is that the mean stress is found to be constant within each interphase layer, and the hoop stress in the innermost interphase layer is uniform along the entire interphase/inclusion interface.