力学与实践 ›› 2019, Vol. 41 ›› Issue (4): 445-448.DOI: 10.6052/1000-0879-18-486

• 教育研究 • 上一篇    下一篇

斜面应力公式的适用条件 1)

胡恒山2), 宋永佳   

  1. 哈尔滨工业大学航天科学与力学系,哈尔滨 150001
  • 收稿日期:2018-12-10 出版日期:2019-08-27 发布日期:2019-08-26
  • 通讯作者: 胡恒山
  • 作者简介:2) 胡恒山,教授,从事材料力学与波动力学教学与研究。E-mail: hhs@hit.edu.cn
  • 基金资助:
    1) 国家自然科学基金资助项目(11734017,11802074)

CONDITIONS FOR THE APPLICATION OF THE STRESS TRANSFORMATION FORMULA 1)

HU Hengshan2), SONG Yongjia   

  1. Department of Astronautics and Mechanics, Harbin Institute of Technology, Harbin 150001,China
  • Received:2018-12-10 Online:2019-08-27 Published:2019-08-26
  • Contact: HU Hengshan

摘要:

斜面应力公式,即一点应力的坐标旋转变换公式,是材料力学和弹性力学里最常用的公式之一,并广泛用于固体力学和工程设计中。一个让学生感觉例外的典型例子是含V 形切口的薄板在切口尖端的应力情况,如果利用斜面应力公式和切口面的自由面条件,就会得出切口尖点处于零应力状态的结果,而这与线弹性断裂力学给出切口处应力趋于无限大的结果不符。为消除这一疑虑,考察了尖端应力的特性,指出:只有在过一点的各个斜面上的应力是单值连续的情况下,斜面应力公式才能适用,此时,该点的全部应力分量组成应力张量。在V形切口的尖端、裂纹尖端,自由面与顺其延伸至介质内侧的面上的应力不同,应力在该面上就不是单值连续的,该点的应力状态就不能用张量表示,斜面应力公式在切口或裂纹尖端就不适用了。

关键词: 应力变换, 单元体, V形切口, 应力张量, 单值

Abstract:

The stress transformation formula, which is also called the Cauchy's formula for stresses on slanted surfaces, is widely used in solid mechanics. No explicit statement has been found on its limitations in application. The stresses at the sharp point of a V-shaped free-surface notch on a bar under axial loading at both ends are examined as an example. With the two intersecting surfaces being free, it is deduced from the transformation equations that the sharp point is stress free. This deduction is however contradictory to the fact that the stress concentrates near the sharp point. Clearly the stress transformation formula leads to incorrect results and thus is not valid at that sharp point. It is known that if the stress transformation formula is valid, the stress components constitute a stress tensor, but the stress at the sharp concave corner is shown to be discontinuous and double-valued. It is further pointed out that stress transformation formula is not applicable to any point where a stress component has discontinuity or lacks of uniqueness at a surface passing through that point. The tip of any crack is the case.

Key words: stress transformation, stress element, V shaped cut, stress tensor, single-valued

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