Three-phase elliptical inclusions with internaluniform hydrostatic stresses

Abstract

For an elastic inclusion embedded within an elastic matrix, it is the interfacialstresses that control mechanical integrity of the inclusion/matrix system. To eliminate the stresspeaks at the interface, the uniform hydrostatic stress state within the inclusion is of particularinterest because it achieves both uniform normal stress and vanishing tangential stress along theentire interface. Motivated by practical significance of interphase layer, the present paper studiesthe internal stress state of a three-phase elliptic inclusion which is bonded to an infinite matrixthrough an intermediate elastic layer. What is essential is that the interfaces of the three-phaseelliptic inclusion considered are two confocal ellipses. A simple condition is found that ensuresthat the internal stress state within the elliptic inclusion is uniform and hydrostatic. For givenremote stresses and material parameters, this condition gives a simple relationship between thethickness of the interphase layer and the aspect ratio of the elliptic inclusion. The exact stressfield is obtained in elementary form when this condition is met. In particular, the hoop stress inthe interphase layer is found to be uniform along the entire interphase/inclusion interface. It isbelieved that the availability of this condition relies on the confocal character of the ellipticinterfaces.