Analytic Solution for Eshelby's Problem of an Inclusion of Arbitrary Shape in a Plane or Half-Plane
Analytic Solution for Eshelby's Problem of an Inclusion of Arbitrary Shape in a Plane or Half-Plane
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DOI：
10.1115/1.2791051
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发表时间：
1999-06
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影响因子：
--
通讯作者：
C. Ru
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文献类型：
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作者：
C. Ru
尽管对简单形状包裹体的Eshelby问题进行了广泛的研究，但对任意形状包裹体的研究却很少。本文利用解析延拓和保角映射技术，给出了平面或半平面上任意形状包含的Eshelby问题解析解的一种新方法。包体的边界用保角映射来表征，保角映射将包体的外部映射到单位圆的外部。然而，边值问题是在物理平面上而不是在图像平面上研究的。利用保角映射构造一个辅助函数，利用该辅助函数可将解析延拓技术应用于任意形状的包含。当映射函数的展开式只包含有限个数的项时，该方法得到的解是精确的。另一方面，如果精确映射函数包含无限项，则应使用截断的多项式映射函数，然后该方法给出近似解。特别地，这种方法得到了整个平面内夹杂物内部应力的简单初等表达式。讨论了几个有实际意义的例子来说明该方法及其有效性。与已有的二维Eshelby问题的方法相比，该方法具有基本的性质和对平面或半平面上任意形状的内含物的适用性。
Despite extensive study of the Eshelby's problem for inclusions of simple shape, little effort has been made to inclusions of arbitrary shape. In this paper, with aid of the techniques of analytical continuation and conformal mapping, a novel method is presented to obtain analytic solution for the Eshelby's problem of an inclusion of arbitrary shape in a plane or a half-plane. The boundary of the inclusion is characterized by a conformal mapping which maps the exterior of the inclusion onto the exterior of the unit circle. However, the boundary value problem is studied in the physical plane rather than in the image plane. The conformal mapping is used to construct an auxiliary function with which the technique of analytic continuation can be applied to the inclusion of arbitrary shape. The solution obtained by the present method is exact, provided that the expansion of the mapping function includes only a finite number of terms. On the other hand, if the exact mapping function includes infinite terms, a truncated polynomial mapping function should be used and then the method gives an approximate solution. In particular, this method leads to simple elementary expressions for the internal stresses within the inclusion in an entire plane. Several examples of practical interest are discussed to illustrate the method and its efficiency. Compared to other existing approaches for the two-dimensional Eshelby's problem, the present method is remarked by its elementary characters and applicability to inclusions of arbitrary shape in a plane or a half-plane.