Teaching von mises stress: From principal axes to non-principal axes

Abstract

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 σ1, σ2, and σ3 as σ1 = 1/√2[(σ1 - σ2)2 + (σ2 - σ3) 2 + (σ3 - σ1)2] 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 σ1 = 1/√2[(σ x - σy)2 + (σx - σz)2 + 6(τxy2 + τyz2 + τzx2]1/2 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 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. © American Society for Engineering Education, 2009.