Von Mises stress calculation

[SerafeimBakalakos]

1. Reminder of formulas: https://en.wikipedia.org/wiki/Von_Mises_yield_criterion#Summary

2. For 3D problems and 2D plane stress problems, we can compute the equivalent von Mises stress just by knowing the Cauchy stress tensor. For 2D plane strain problems we need components that are not stored in the 2D stress tensor (σzz != 0, σyz = σzx = 0). For linear materials we can use Hook's law to get:
   σzz = Ev / ((1+v)(1-2v) * (εxx + εyy).

3. What about non linear materials?

4. When should the Von Mises stress be calculated, if we want to plot it using its values at element nodes?
   a. Calculate the Cauchy stress tensor at Gauss points, then the von Mises stress at Gauss points, then extrapolate von Mises stress to nodes, then average its values over all elements that meet at each node. Only this approach has E,v available at the same points where von Mises is calculated (E,v are only needed for plane strain problems).
   b. Extrapolating the Cauchy stress tensor from Gauss points to the nodes, then compute von Mises stress to nodes, then calculate von Mises at nodes, then average its values over all elements that meet at each node.
   c. Calculate the Cauchy stress tensor at Gauss points, extrapolate it to nodes, average its components over all elements that meet at each node and finally calculate von Mises stress.