Identifying simple shear in plane stress states

Modern phenomenological damage models use Lode parameter L and triaxiality \(\eta \) to describe the stress state of an isotropic material. Value pairs in the region between \(L, \eta = (0, 0)\) and \(L, \eta = (0, \frac{1}{\sqrt{3}})\) in plane stress condition can lead to ambiguous descriptions of the deformation. The case of simple shear is not defined separately. By using the difference in angles between the principal strain and principal stress axes, cases of coaxial stretch superposed with simple shear can be distinguished from cases of coaxial stretch without simple shear. In the case of anisotropic material or large elements, the distinction between these ambiguous cases can be utilized to optimize failure models. This study proposes a method to recover the deformation gradient and shear direction for proportional and non-proportional loading with an elastoplastic von Mises material. The deformation gradient is suitable for distinguishing stress states with simple shear from stress states without simple shear in plane stress condition.