The transformation problem of an elliptical homoeioid inclusion with a uniform eigenstrain embedded in an unbounded homogeneous isotropic medium is studied in the context of plane elasticity. The term homoeoid is used to name a region of a plane medium bounded by two concentric, similar and similarly-oriented elliptic contours. The solution to the problem is achieved by solving first an auxiliary problem corresponding to the case in which the region of the medium ( core ) surrounded by the inclusion is replaced by a hole. A particular feature of the elastic field of the auxiliary problem is the unmoving of the hole boundary. This result suggests that the solution to the auxiliary problem is, also, the solution to the problem under consideration; additionally, it is the solution whatever is the mechanical property of the core and its bonding conditions with the inclusion. The solution to the problem is obtained in closed form, in terms of the complex potentials of the inclusion and its surrounding ( matrix ). Based on the complex potentials obtained, a simple expression for the total elastic energy stored in the unbounded medium is derived. It is shown that the total area change of the unbounded medium is that of the inclusion, which is determined in a simple form.

中文翻译：

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平面弹性中的椭圆同型体包裹体

在平面弹性的背景下研究了嵌入在无界均匀各向同性介质中的具有均匀本征应变的椭圆同质包裹体的转换问题。术语homoeoid 用于命名由两个同心、相似且方向相似的椭圆轮廓包围的平面介质区域。该问题的解决是通过首先解决对应于其中被夹杂物包围的介质（核心）区域被孔代替的情况的辅助问题来实现的。辅助问题的弹性场的一个特殊特征是孔边界的不动。这个结果表明辅助问题的解决方案也是正在考虑的问题的解决方案；此外，无论芯的机械性能如何以及它与夹杂物的结合条件如何，它都是解决方案。该问题的解决方案是在封闭形式中获得的，就包含物及其周围环境（矩阵）的复势而言。基于获得的复势，推导出存储在无界介质中的总弹性能量的简单表达式。结果表明，无界介质的总面积变化是夹杂物的总面积变化，用简单的形式确定。