Von Mises yield criterion

Remarks

The criterion is a constitutive assumption confirmed by experiments on metals, not a theorem. The quantity $\sigma_{vM} = \sqrt{3J_2}$ is the von Mises equivalent stress; the yield surface $\sigma_{vM} = \sigma_y$ is a circular cylinder about the line $\sigma_1 = \sigma_2 = \sigma_3$. Hydrostatic pressure has no effect, since $J_2$ does not change when a multiple of $I$ is added to $\sigma$. Check in uniaxial tension, $\sigma_1 = \sigma$, $\sigma_2 = \sigma_3 = 0$: the left side is $\abs{\sigma}$, so yielding starts at $\sigma = \sigma_y$, as it should. In pure shear, $\sigma_1 = \tau$, $\sigma_2 = 0$, $\sigma_3 = -\tau$: the left side is $\sqrt{3}\abs{\tau}$, so the shear yield stress is $\sigma_y / \sqrt{3}$. $J_2$ is proportional to the elastic energy of distortion, the part of the strain energy that changes shape but not volume. So the criterion is also called the distortion energy criterion.

Going deeper

For a student

The criterion is the standard yield condition for metals in engineering design and in finite element codes. Combined with the associated flow rule it gives the Prandtl–Reuss equations, in which the plastic strain rate is parallel to the deviatoric stress and plastic flow keeps the volume constant. It is smooth everywhere, which makes it easier to use than the Tresca criterion. When both are matched in uniaxial tension, the Tresca hexagon lies inside the von Mises circle, and the two differ by at most about $15$ percent.