June 14, 2009
June 14, 2009
June 17, 2009
14.1159.1 - 14.1159.9
Teaching von Mises Stress: From Principal Axes To Non-Principal Axes
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 , 2 , and 3 as 1 2 2 2 1/2 ( 1 2) ( 2 3) ( 3 1) 2 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 1/2 ( x y) ( y z) ( z x) 6 ( x2y y2z z2x ) 2 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 text- books 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.
Poncelet and Saint-Venant8 proposed using the strain-energy in a ductile material to determine when failure of the material would occur. Maxwell8 expanded the idea by showing that the total strain energy could be divided into distortional and volumetric terms, but he never extended the idea further. A distortion-energy theory was prompted from the observation that ductile materials stressed hydrostatically exhibited yield strengths greatly in excess of the values given by simple tension tests. It was then postulated that yielding was related somehow to the angular distortion of the stressed element, rather than that yielding was a simple tensile or compressive phenome- non. Nowadays, the distortion-energy theory for ductile materials states that yielding occurs when the distortion strain energy per unit volume reaches or exceeds the distortion strain energy per unit volume for yield in simple tension or compression of the same material.
Using principal axes, we can first decompose the state of stress at a point that is given in terms of the principal stresses 1 , 2 , and 3 into the sum of two states: (a) a state of hydrostatic stress
Jong, I., & Springer, W. (2009, June), Teaching Von Mises Stress: From Principal Axes To Nonprincipal Axes Paper presented at 2009 Annual Conference & Exposition, Austin, Texas. https://peer.asee.org/5353
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