1 Three-Phase Elliptical Inclusions with an Internal Stress Field of Linear Form Xu Wang1,*, Weiqiu Chen2 1 School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China 2 Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China * Corresponding author: xuwang@ecust.edu.cn Abstract This paper studies the internal stress field of a three-phase elliptical inclusion which 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 prescribed non-constant field. Keywords Elliptical inclusion, Interphase layer, Non-uniform loading, Harmonic inclusion, Inverse problem 1. Introduction 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, [15] 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. Elementary-form solution is derived when these two conditions are met. Some special examples are presented to demonstrate the solution. In particular, 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. 2. The Internal Stress Field of Linear Form We study the stress-field of a three-phase elliptical inclusion with two confocal interfaces when the matrix is subjected to a linearly distributed stress field at infinity. Let S1, S2 and S3 denote the inclusion, the interphase layer and the matrix, respectively, which are perfectly bonded across two confocal elliptical interfaces L1 and L2, as shown in Fig. 1. Throughout the paper, the subscripts 1, 2 and 3 are used to identify the respective quantities in S1, S2 and S3.
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