A ductile material under multiaxial stress cannot be assessed by comparing one arbitrary stress component with the uniaxial yield strength. A yield criterion reduces the complete stress state to a quantity that can be compared with material test data.

Tresca criterion

The maximum-shear-stress, or Tresca, criterion predicts yielding when the largest shear stress reaches the value associated with yielding in a uniaxial tensile test.

In terms of principal stresses, the governing difference is the largest value of

$$|\sigma_i-\sigma_j|.$$

von Mises criterion

The von Mises criterion uses distortion energy. For principal stresses $\sigma_1$, $\sigma_2$, $\sigma_3$, the equivalent von Mises stress is

$$\sigma_v=\sqrt{\frac{(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2}{2}}.$$

Yielding is predicted when $\sigma_v$ reaches the uniaxial yield strength.

Choosing a criterion

Tresca is somewhat more conservative in some stress states; von Mises often matches yielding of ductile metals well. Both describe static yielding, not fatigue or brittle fracture.