平面任意形状等剪切模量异质夹杂问题的解析解

Analytical Solution for the Eshelby Problem of an Arbitrarily Shaped Inhomogeneous Inclusion with the Same Shear Modulus as the Matrix in an Infinite Plane

复合材料内部的异质严重影响着材料整体的力学性能,研究基体材料由于不规则形状的异质而引发的弹性扰动场对于开发和设计复合材料有着非常重要的作用.论文就任意形状异质夹杂在受到远端均匀载荷和均匀本征应变作用下的弹性扰动场问题展开研究,其中夹杂和基体的材料不同但具有相同的剪切模量.首先利用等效理论将远端均匀荷载的作用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;然后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,利用柯西积分公式和Faber多项式求解了等剪切模量下夹杂和基体的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.将所得解析解退化到椭圆情况,与现有的椭圆的异质解和同质解是完全一致的;同时还将所得解析解退化到同质内旋轮线情况,与现有文献的结果也完全一致.最后,通过编程计算了均匀剪切应变作用下的特定材料的三角形和正方形夹杂的应力场,展示了其分布特征.论文结果具有一定的应用价值,也是获得更一般的异质问题的解析解的前奏.

Composites usually contain a variety of inhomogeneous inclusions, which can severely affect the global mechanical properties of materials. So, exploring the elastic disturbance field due to the irregular shape of the inhomogeneous inclusion is very important for the development and design of composites. In this paper, the elastic fields of the matrix in two-dimensional space including an inhomogeneous inclusion undergoing a uniform eigenstrain and/or a uniform remote load are studied, where the inclusion shaped by the Laurent polynomial has different properties from the matrix but the same shear modulus as it. By means of the equivalent method, the problem of disturbance field due to remote uniform loading is converted into that of an equivalent uniform eigenstrain in the inhomogeneous inclusion, and the planar elastic fields and the continuity conditions at the interface are expressed by the famous Kolosov-Muskhelishvili(K-M) potentials. Based on the Riemann mapping theorem, the exterior of the inclusion is mapped onto the exterior of the unit disk with the center at the origin by the Laurent polynomial. Using the Cauchy integral formula and the Faber polynomial, the explicit analytical solutions of the K-M potentials are carried out for points inside and outside the inclusion, where the relative rigid-body displacement of the inclusion to the matrix is considered and worked out. The analytical results reduced to the elliptical case are completely consistent with the existed solutions for inhomogeneous and homogeneous elliptical inclusions. At the same time, the analytical solution degenerated to the case of homogeneous inclusion shaped by the hypocycloidal curve is also in complete agreement with the existed results in literature. The stress fields of triangular and square inclusions of specific materials under uniform shear strain are calculated, and their distribution properties are illustrated. The solution of this study can be a prelude for obtaining the analytical solutions of the problems of more general heterogeneous inclusions.