Harmonic Cartesian polynomial and rational functions are shown to provide a simple way of obtaining a semi-analytical solution to the uncoupled, two-dimensional problem of thermomagnetoelasticity for a long, transversely isotropic, elastic cylinder carrying an axial, steady electric current. The proposed method involves the solution of a difficult inhomogeneous biharmonic equation for the stress function, and may be invariably used for general geometries of the normal cross-section of the cylinder, for various thermal and mechanical boundary conditions, and also to solve the problem of a long non-conducting cylinder in a transverse magnetic field. At all stages of the solution, there is no need to smoothen the cross-sectional contour in the presence of singular points, as for other methods. Results are provided for the elliptic and rectangular normal cross-sections under the Dirichlet thermal condition and prescribed uniform normal boundary displacement. Quantities of practical interest are represented graphically and discussed. The results may be of interest in determining the deformations occurring in current-carrying metallic parts of structures and machines, and in the central regions of busbars in electric power plants.

中文翻译：

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笛卡尔谐波多项式对长载流弹性圆柱导体变形问题的半解析解

谐波笛卡尔多项式和有理函数提供了一种简单的方法，可以为长的、横向各向同性的、带有轴向稳定电流的弹性圆柱体的热磁弹性的非耦合二维问题获得半解析解。所提出的方法涉及求解应力函数的一个困难的非齐次双调和方程，并且可以始终用于圆柱体法向横截面的一般几何形状，各种热和机械边界条件，也可以解决在横向磁场中的长不导电圆柱体。在求解的所有阶段，都不需要像其他方法那样在存在奇异点的情况下平滑横截面轮廓。提供了在狄利克雷热条件和规定的均匀法向边界位移下的椭圆和矩形法向截面的结果。实际感兴趣的数量以图形方式表示和讨论。结果可能对确定结构和机器的载流金属部件以及发电厂母线中心区域发生的变形感兴趣。