Three-phase elliptical inclusions with an internal stress field of linear form

Abstract

This paper studies the internal stress field of a three-phase elliptical inclusion that is bonded to an infinite matrix through an interphase layer when the matrix is subjected to a linearly distributed in-plane stress field at infinity. Two conditions are found that ensure that the internal non-uniform stress field is simply a linear function of the two coordinates. For given material and geometric parameters of the composite, these conditions can be considered as two restrictions on the applied non-uniform loadings. When these two conditions are met, elementary-form expressions of the stresses in all the three phases are derived. In particular, it is found that the mean stress within the interphase layer is also a linear function of the coordinates. If the interphase layer and the matrix have the same elastic constants, the satisfaction of the two conditions will result in a harmonic inclusion under a prescribed non-constant field.