Question

Consider the two states of stress as shown in configurations I and II in the figure below. From the standpoint of distortion energy (von-Mises) criterion, which one of the following statements is true?

• A. I yields after II
• B. Both yield simultaneously
• C. Nothing can be said about their relative yielding
• D. II yields after I

Answer 1

The correct option is B Both yield simultaneously

Simplified Von Mises equation for general load scenario is given by
${\sigma }_v=\sqrt{\frac{1}{2}[{({\sigma }_{11}-{\sigma }_{22})}^2+({\sigma }_{22}-{\sigma }_{33}{)}^2+({\sigma }_{33}-{\sigma }_{11}^2)+6({\sigma }_{12}^2+{\sigma }_{23}^2+{\sigma }_{31}^2)]}$

Case - I
${\sigma }_{11}=0;{\sigma }_{22}=\sigma ;{\sigma }_{33}=0;{\sigma }_{12}=0{\sigma }_{23}=\tau ;{\sigma }_{31}=0$

$\therefore {\sigma }_v=\sqrt{\frac{1}{2}(2{\sigma }^2+6{\tau }^2)}=\sqrt{{\sigma }^2+3{\tau }^2}$

Case - II
${\sigma }_{11}=0;{\sigma }_{22}=\sigma ;{\sigma }_{33}=0;{\sigma }_{12}=0{\sigma }_{23}=0;{\sigma }_{31}=\tau$

$\therefore {\sigma }_v=\sqrt{\frac{1}{2}(2{\sigma }^2+6{\tau }^2)}=\sqrt{{\sigma }^2+3{\tau }^2}$

In both the cases, ${\sigma }_v$ is same, so both yield simultaneously.