Von Mises

2D (Plane) Stresses

The general equation for plane stress is

$$\sigma_{e} = \sqrt{\sigma_1^2-\sigma_1\sigma_2+\sigma_2^2+3\tau^2}$$

If the principle stresses are known then

$$\sigma_e=\sqrt{\sigma_1^2-\sigma_1\sigma_2+\sigma_2^2}$$

3D Stresses

The general equation is

$$\sigma_e=\sqrt{\frac{1}{2}\left((\sigma_{11}-\sigma_{22})^2+(\sigma_{22}-\sigma_{33})^2+(\sigma_{33}-\sigma_{11})^2\right)+3(\tau_{12}^2+\tau_{23}^2+\tau_{31}^2)}$$

If the principle stresses are known then

$$\sigma_e=\sqrt{\frac{1}{2}\left((\sigma_{1}-\sigma{2})^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2\right)}$$