The fast decompositions method in analytical form was used to solve the elastic problem with mixed boundary conditions for stresses in a sharp wedge-shaped cutting tool of finite dimensions in the form of a truncated sector. A load is applied to the faces of the wedge and its nose, which rapidly decreases with distance from the nose. At a larger radius, the wedge is rigidly fixed. The resulting solution is valid for any angle of the wedge spread. If we take a small enough inner radius and the angle of the spread, then the shape of the proposed cutter will be close enough to real. The influence of the angle of the spread and the inner radius of the wedge on the magnitude and location of the highest stress σ̃ was investigated. The critical dimensions of the wedge were found, but for any of its sizes the stresses are finite everywhere.

中文翻译：

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使用快速分解方法解决锋利的楔形四刃切削刀具中的应力问题以及边界条件匹配问题

使用解析形式的快速分解方法来解决带有有限边界的尖锐楔形切削工具（呈截头扇形）形式的应力混合边界条件下的弹性问题。负载会施加到楔形及其鼻子的表面，该载荷会随着距鼻子的距离而迅速减小。在较大的半径处，楔块被牢固地固定。所得的解对于楔形展开的任何角度均有效。如果我们采用足够小的内半径和展开角度，则建议的刀具形状将足够接近真实形状。扩展角和楔形内半径对最大应力σ̃的大小和位置的影响被调查了。发现了楔形的临界尺寸，但是对于任何尺寸的楔形来说，应力在任何地方都是有限的。