The paper is concerned with the classical Flamant problem of the theory of elasticity. The classical solution of this problem obtained by A. Flamant in the nineteenth century specifies the stresses and the displacements induced in a half-plane by a concentrated force applied to the half-plane boundary in the normal direction. Being presented in numerous textbooks in the theory of elasticity, this solution provides results that can be qualified as paradoxical and which traditionally are left without proper comments in the literature. Particularly, the half-plane displacements are singular at the point of the force application and are infinitely increasing with a distance from this point. The displacement of the boundary in the boundary direction is not continuous and experiences a jump at the point of the force application. A boundary element under any force, irrespective of how low it is, rotates at the point of the force application by 90 $$^{\circ }$$ ∘ which does not correspond to basic assumptions of the linear theory of elasticity; hence, the classical solution cannot be qualified as consistent. The consistent solution of the problem is constructed in the paper on the basis of the generalized theory of elasticity the equations of which are obtained for the solid element that has small but not infinitesimal dimensions. As a result, these equations provide regular solutions of the problems that are singular in the classical theory of elasticity. The obtained generalized solution of the Flamant problem demonstrates regular behavior of the displacements which are not singular at the point of the force application. The solution is supported by experimental results obtained for the plate of silicon rubber simulating the half-plane.

中文翻译：

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受力集中的半平面的弗拉芒问题

该论文涉及弹性理论的经典弗拉芒问题。由 A. Flamant 在 19 世纪获得的这个问题的经典解决方案指定了在法线方向上施加到半平面边界的集中力在半平面中引起的应力和位移。在弹性理论的众多教科书中，该解决方案提供的结果可以被认为是矛盾的，并且传统上在文献中没有适当的评论。特别是，半平面位移在受力点处是奇异的，并且随着与该点的距离而无限增加。边界在边界方向上的位移不是连续的，并且在施加力的点处会发生跳跃。任何力下的边界元，无论它有多低，在力施加点旋转 90 $$^{\circ }$$ ∘ 这不符合线性弹性理论的基本假设；因此，经典解决方案不能被认为是一致的。该问题的一致解是在广义弹性理论的基础上构建的，该理论的方程是针对具有小但不是无穷小的维度的实体单元获得的。因此，这些方程提供了经典弹性理论中奇异问题的正则解。获得的 Flamant 问题的广义解证明了位移的规则行为，这些行为在力施加点不是奇异的。