Elasticity/VonMises stress

Von Mises is a criteria used in predicting the onset of yield in ductile materials. The von Mises stress/equivalent stress/distortional strain energy is given by the relation

${\displaystyle \sigma _{\text{eq}}={\sqrt {I_{1}^{2}-3I_{2}}}={\sqrt {\sigma _{11}^{2}+\sigma _{22}^{2}+\sigma _{33}^{2}-\sigma _{11}\sigma _{22}-\sigma _{22}\sigma _{33}-\sigma _{33}\sigma _{11}+3\sigma _{12}^{2}+3\sigma _{23}^{2}+3\sigma _{31}^{2}}}}$

Introduction to Elasticity