平面任意形状等剪切模量异质夹杂问题的解析解

本文研究任意形状夹杂域在受到远端均匀荷载和均匀本征应变作用下的弹性场问题，其中基体和夹杂的材料不同但具有相同的剪切模量。利用等效理论将远端均匀荷载引起的扰动转化为等效均匀本征应变的作用，再利用K-M势函数表达扰动场问题的界面连续条件；借助于黎曼映射定理，用洛朗多项式将平面光滑闭合曲线外部区域映射到单位圆外部区域，借助柯西积分公式和Faber多项式求解了等剪切本征应变下夹杂和基体的K-M势函数的显式解析解，其中考虑了夹杂相对于基体的刚体位移。将得到的结果与相关文献的结果进行对比，表明了本论文的方法和结果是有效的和正确的。

Analytical solution for the Eshelby problem of an inhomogeneous inclusion with the same shear module as the matrix and arbitrary shape in an infinite plane

The elastic fields of the matrix in two-dimensional space with an inhomogeneous inclusion undergoing a uniform eigenstrain and/or due to a uniform remote load are studied, where the inclusion shaped by the Laurent polynomial has distinct properties to the matrix but shares the same shear modulus. The equivalent method is used to convert the problem of the perturbance field due to the remote uniform loadings into that of an equivalent uniform eigenstrain, and the interface continuity conditions are expressed by the K-M potentials. Then, by virtue of the Riemann mapping theorem, the exterior of the inclusion is mapped on to the exterior of the unit disk by the Laurent polynomial and making use of the Cauchy integral formula and the Faber polynomial, the explicit analytical solutions of the K-M potentials are carried out in the inclusion and the matrix, where the relative rigid-body displacement of the inclusion to the matrix is considered. The obtained results are compared with those of previous literatures, to show that the method and results of this paper are effective and correct.