Three-phase elliptical inclusions with internal uniform hydrostatic stresses in finite plane elastostatics

Abstract

We consider the internal stress field of a three-phase elliptical inclusion bonded to an infinite matrix through an interphase layer when the matrix is subjected to remote uniform stresses. The elastic materials comprising all the three phases belong to a particular class of harmonic materials, and the formed interfaces are two confocal ellipses. A condition leading to internal uniform hydrostatic stresses is derived. This condition relates the two remote principal stresses with the geometric parameters (the thickness of the interphase layer and the aspect ratio of the elliptical inclusion) of the three-phase elliptical inclusion. When this condition is met, the hoop stress in the interphase layer along the entire interphase/inclusion interface is also uniform. Five special situations of practical importance are discussed in considerable detail to demonstrate the unique phenomena inherent in harmonic materials. Our discussions indicate that when this condition is met, it is permissible for the two remote principal stresses to have opposite signs and that for given geometric and material parameters, the remote loading ratio is no longer constant and multiple external loading states exist leading to internal uniform hydrostatic stresses. It is found that this condition can be written into a hyperbola for the two remote principal stresses when the interphase layer is extremely compliant or relatively stiff or when the inclusion is almost rigid. When the magnitudes of the remote stresses are sufficiently large, this condition becomes a very simple one relating the remote loading ratio with the geometric parameters of the composite. Interestingly, it is clearly observed from the simple condition that for given geometric parameters of the three-phase elliptical inclusion, there exist two different values of the remote loading ratio, both of which lead to an internal uniform hydrostatic stress state.