Why only the second invariant is considered? What about the third invariant?
The invariants affect the shape of the yield surface. The von Mises condition assumes that the yield surface remains cylindrical in principal stress space. If you want pressure-dependence (the circular cylinder becomes a circular cone), then you add the first invariant into the mix. If the yield surface varies depending on whether you are in pure triaxial tension or triaxial compression, then you need the third invariant to represent the shape. See, for example, the Willam-Warnke condition. https://en.wikipedia.org/wiki/Willam%E2%80%93Warnke_yield_criterion