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.
Highlights
Rigorous analysis of a composite system consisting of an internal inclusion, an intermediate interphase layer and an outer matrix is challenging, especially when the inclusion is non-circular
It has been found that the internal stress field within a three-phase confocal elliptical inclusion can be uniform and hydrostatic when the remotely applied uniform in-plane stresses satisfy a condition [2]
When the interphase layer and the matrix have identical elastic constants, the satisfaction of the two conditions will result in a harmonic elliptical inclusion under non-uniform loadings
Summary
Rigorous analysis of a composite system consisting of an internal inclusion, an intermediate interphase layer (or coating) and an outer matrix is challenging, especially when the inclusion is non-circular (see for example, [1 5] and the references cited therein). It has been found that the internal stress field within a three-phase confocal elliptical inclusion can be uniform and hydrostatic when the remotely applied uniform in-plane stresses satisfy a condition [2]. How about the stress field within a three-phase elliptical inclusion when the matrix is subjected to non-uniform in-plane stresses at infinity? Is there any elementary solution for the case of non-uniform loading?. In this paper two conditions are found that ensure that the internal stress field within a three-phase confocal elliptical inclusion is a linear function of the coordinates when the matrix is subjected to a linearly distributed (non-uniform) in-plane stress field at infinity. When the interphase layer and the matrix have identical elastic constants, the satisfaction of the two conditions will result in a harmonic elliptical inclusion under non-uniform loadings
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