Abstract This paper presents the analytical solution of a class of plane elasticity problems for circular inhomogeneous inclusions with non-uniform axisymmetric eigenstrain distribution, which includes both radial and hoop eigenstrains. The complex variable potential solution for a point-wise eigenstrain in an infinite plane solid is presented first, in which two principal strains and their directions are taken into account. Directly employing it as influence function, the complex variable potential for the circular homogeneous inclusion problem is formulated with Green's function method. The novelty of this approach is that it is able to take intrinsic advantage of complex variable approach and effectively tackle the mathematical difficulties encountered during formulation. Next, by using the principle of equivalent eigenstrain, the main challenge in solving inhomogeneous inclusion problems is overcome, allowing the general explicit analytical solution to be derived. Based on these results, three illustrative examples of practical significance are given: (i) dissimilar cylinder interference-fits within an infinite body, (ii) a pure dilatational eigenstrain problem within a circular inclusion, and (iii) a circular inclusion problem with a wedge disclination eigenstrain distribution. The fundamental formulation introduced here will find application in other aspects in the mechanics of fiber composites, thermoelasticity, and nano-mechanics of defects in solids.

中文翻译：

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具有非均匀轴对称特征应变分布的圆形非均匀夹杂问题的解析解

摘要 本文提出了一类具有非均匀轴对称特征应变分布的圆形非均匀夹杂物的平​​面弹性问题的解析解，包括径向和环向特征应变。首先给出无限平面实体中点状本征应变的复变势解，其中考虑了两个主应变及其方向。直接将其作为影响函数，用格林函数法对圆形均匀包含问题的复变势进行了公式化。这种方法的新颖之处在于它能够利用复变量方法的内在优势，有效地解决公式化过程中遇到的数学难题。接下来，利用等效本征应变原理，解决非均匀包含问题的主要挑战已被克服，从而可以推导出一般的显式解析解。基于这些结果，给出了三个具有实际意义的示例：(i) 无限体内不同圆柱干涉配合，(ii) 圆形夹杂物内的纯膨胀本征应变问题，以及 (iii) 圆形夹杂物问题楔形向错特征应变分布。这里介绍的基本公式将应用于纤维复合材料力学、热弹性和固体缺陷纳米力学的其他方面。(i) 无限体内的不同圆柱干涉配合，(ii) 圆形夹杂物内的纯膨胀特征应变问题，以及 (iii) 具有楔形向错特征应变分布的圆形夹杂物问题。这里介绍的基本公式将应用于纤维复合材料力学、热弹性和固体缺陷纳米力学的其他方面。(i) 无限体内的不同圆柱干涉配合，(ii) 圆形夹杂物内的纯膨胀特征应变问题，以及 (iii) 具有楔形旋错特征应变分布的圆形夹杂物问题。这里介绍的基本公式将应用于纤维复合材料力学、热弹性和固体缺陷纳米力学的其他方面。