GC 2009-267: TEACHING VON MISES STRESS: FROM PRINCIPAL AXES TO NONPRINCIPAL AXES
GC 2009-267: TEACHING VON MISES STRESS: FROM PRINCIPAL AXES TO NONPRINCIPAL AXES
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GC 2009-267：冯米塞斯应力教学：从主轴到非主轴
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
W. Springer
中科院分区：
文献类型：
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作者：
I. Jong;B. G. Rogers;W. Springer
冯米塞斯应力是一个等效或有效的应力，在该应力下，韧性材料预计会发生屈服。在大多数机械设计的教科书中，1-7这样的应力是根据主应力1 σ、2 σ和3 σ使用主轴导出的，如1/2 2 2 2 2 1 2 2 3 1 1（）2 σ σ ′ ε ε = − + − + − ε ε在其最新版本中，在这些机械设计的教科书中，有些开始表明，关于非主轴的冯米塞斯应力也可以表示为1/2 2 2 2 2 2 1（）6（）2 x y y z z x x y y z z x σ τ τ ′ = − ++然而，这些教科书没有提供关于前一个公式如何演变成后一个公式的解释。由于在教科书中或在课堂上教师对后一个公式缺乏很好的解释，学生们往往简单地相信这两个公式在某种程度上是等价的。本文旨在与机械设计的教育工作者和其他读者分享两种可供选择的途径，以达到后者的一般形式的冯米塞斯应力：（a）通过特征值的应力矩阵，（B）通过应力矩阵的应力不变量。当与教科书中现有的材料一起使用时，这两条路径中的任何一条都将使学生更好地理解冯米塞斯应力的一般形式。贡献的工作是为了加强教学和学习的一个重要的故障理论通常涵盖在高级设计课程，在大多数工程课程。
The von Mises stress is an equivalent or effective stress at which yielding is predicted to occur in ductile materials. In most textbooks for machine design, 1-7 such a stress is derived using principal axes in terms of the principal stresses 1 σ , 2 σ , and 3 σ as 1/2 2 2 2 1 2 2 3 3 1 1 ( ) ( ) ( ) 2 σ σ σ σ σ σ σ ′ = − + − + − In their latest editions, some of these textbooks for machine design began to show that the von Mises stress with respect to non-principal axes can also be expressed as 1/2 2 2 2 2 2 2 1 ( ) ( ) ( ) 6( ) 2 x y y z z x x y y z z x σ σ σ σ σ σ σ τ τ τ ′ = − + − + − + + + However, these textbooks do not provide an explanation regarding how the former formula is evolved into the latter formula. Lacking a good explanation for the latter formula in the textbooks or by the instructors in classrooms, students are often made to simply take it on faith that these two formulas are somehow equivalent to each other. This paper is written to share with educators of machine design and other readers two alternative paths that will arrive at the latter general form of the von Mises stress: (a) by way of eigenvalues of the stress matrix, (b) by way of stress invariants of the stress matrix. When used with the existing material presented in the textbooks, either of these two paths will provide students with a much better understanding of the general form of the von Mises stress. The contributed work is aimed at enhancing the teaching and learning of one of the important failure theories usually covered in a senior design course in most engineering curricula.