We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

中文翻译：

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弹性半条带的不均匀问题：精确的解决方案

在Papkovich-Fadle本征函数的级数形式为自由长边的半带弹性理论中，我们构造了两个不均匀边值问题的精确解：（a）半带端是自由的，（b）半条带末端已牢固夹紧。最初，我们构造了无限条带不均匀问题的解决方案。随后，将用于半带的相应解添加到该解中，从而满足最后的边界条件。Papkovich正交关系用于解决带钢中的不均匀问题。