We first study the plane strain problem associated with an incompressible liquid inclusion having an n-fold axis of symmetry which is embedded in an infinite isotropic elastic matrix subjected to uniform remote hydrostatic stresses. A closed-form solution is derived using Muskhelishvili’s complex variable formulation, a four-term conformal mapping function and the application of analytic continuation. The pair of analytic functions characterizing the elastic field in the matrix is completely determined in elementary closed-form. Explicit expressions are obtained and graphically illustrated for the internal uniform hydrostatic stresses within the liquid inclusion and the hoop stress along the liquid–solid interface on the matrix side. The closed-form solution for a linearly compressible liquid inclusion having an n-fold axis of symmetry is also obtained.

我们首先研究与具有n重对称轴的不可压缩液体夹杂物相关的平面应变问题，该液体夹杂物嵌入无限各向同性弹性基体中，承受均匀的远程静水应力。使用 Muskhelishvili 的复变量公式、四项共形映射函数以及解析延拓的应用导出了封闭式解。表征矩阵中弹性场的解析函数对完全以初等封闭形式确定。获得了明确的表达式并以图形方式说明了液体夹杂物内的内部均匀静水应力以及基体侧沿液固界面的环向应力。还获得了具有n重对称轴的线性可压缩液体包裹体的封闭形式解。