Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane
Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane
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DOI：
10.1007/s00707-002-0972-3
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发表时间：
2003-01-01
期刊：
影响因子：
2.7
通讯作者：
Ru, CQ
中科院分区：
文献类型：
--
作者：
Ru, CQ
各向异性非椭圆夹杂的Eshelby问题的解析解仍然是一个具有挑战性的问题。本文提出了一种简单的方法，求出了弹性常数相同的各向异性平面或半平面内任意形状夹杂的Eshelby问题的解析解。该方法是基于观察到的界面条件的任意夹杂形状可以写在一个解耦的形式，其中三个未知的斯特罗的功能是相互解耦。该解是由三个保角映射构造的三个辅助函数给出的，这三个保角映射将由三个斯特罗变量定义的包含边界的三条象曲线的外点映射到单位圆的外点上.借助于这些辅助函数，解析延拓技术可以应用于任何形状的包含。该解是在物理平面而不是在图像平面中给出的，并且是精确的，只要每个映射函数的展开式只包括有限数目的项。另一方面，如果至少一个映射函数包括无穷项，则应使用截断多项式映射，因此该方法给出了近似解。本方法的一个显著特点是给出了各向异性全平面内夹杂内部应力场的初等表达式。椭圆形和多边形包含被用来说明辅助函数的构造和方法的细节。
Analytical solution for Eshelby's problem of an anisotropic non-elliptical inclusion remains a challenging problem. In this paper, a simple method is presented to obtain an analytical solution for Eshelby's problem of an inclusion of arbitrary shape within an anisotropic plane or half-plane of the same elastic constants. The method is based on an observation that the interface conditions for arbitrary inclusion-shape can be written in a decoupled form in which three unknown Stroh's functions are decoupled from each other. The solution is given in terms of three auxiliary functions constructed by three conformal mappings which map the exteriors of three image curves of the inclusion boundary, defined by three Stroh's variables, onto the exterior of the unit circle. With aid of these auxiliary functions, techniques of analytical continuation can be applied to the inclusion of any shape. The solution is given in the physical plane rather than in the image plane, and is exact provided that the expansion of every mapping function includes only a finite number of terms. On the other hand, if at least one of the mapping functions includes infinite terms, a truncated polynomial mapping should be used, and thus the method gives an approximate solution. A remarkable feature of the present method is that it gives elementary expressions for the internal stress field within an inclusion in an anisotropic entire plane. Elliptical and polygonal inclusions are used to illustrate the construction of the auxiliary functions and the details of the method.