Timeline for Two dimensional plasticity with radial return

Mar 3 '17 at 10:36 comment Thank you for the detailed answer and clarification. One last thing to clear up. Must the equation for $\sigma_{n+1}$ have to be solved iteratively with the consistency condition $f(\sigma_{n+1})=0$? I can't figure out how to solve for $\dot{\lambda}$ within the yield function for $n+1$ with $\sqrt{\frac{3\sigma_{n+a}^TT\sigma_{n+1}}{2}}=\sigma_y$.
Mar 2 '17 at 4:44 comment That should be $\bar{T}$ in the first relation and not $T$. I have corrected the answer. I'll try to write out the return algo when I get the time.
Mar 1 '17 at 10:12 comment Can you confirm that $T \sigma = [s_{11}\ s_{22} \ 2s_{12}]^T$ or have I made a mistake? Secondly, the yield function you have given is interchangeable with $f = \sqrt{\sigma_{11}^2-\sigma_{11}\sigma_{22}+\sigma_{22}^2+3\sigma_{12}^2} - \sigma_Y$ (in plane stress) which I have no problem with. My current issue is how to correctly compute the return in plane stress as in 3D as simple scaling factor (scalar value) can be used as show here. In the link given $\alpha = \sigma_Y / \sigma_{VM}$.