Yield surface

Remarks

If the material is isotropic, $f$ is a symmetric function of the principal stresses $\sigma_1, \sigma_2, \sigma_3$. The yield surface is then drawn in the space with coordinates $(\sigma_1, \sigma_2, \sigma_3)$. If $f$ also does not depend on the mean stress $\tfrac13 \tr \sigma$, the surface is a cylinder with axis the line $\sigma_1 = \sigma_2 = \sigma_3$. Example: $f(\sigma) = \sqrt{\tfrac32 s : s} - \sigma_y$, where $s = \sigma - \tfrac13 (\tr \sigma) I$ and $\sigma_y > 0$, gives the von Mises surface, a circular cylinder of radius $\sqrt{2/3}\,\sigma_y$ about the hydrostatic axis. Non-example: $f(\sigma) = \tr \sigma + 1$ has $f(0) = 1 > 0$, so it does not have the zero stress in its elastic domain and is not a yield function. When the material hardens, $f$ also depends on internal variables, and the yield surface moves or grows with plastic deformation.