This study focuses on investigating the elastoplastic behavior and predicting plastic instabilities that occur during the large deformations in the metal and alloy forming process. The behavior was modelled using a phenomenological model with velocity-independent plasticity. This model takes into account the various physical phenomena encountered in the cold forming of metal sheets, in particular the initial anisotropy of the material and work hardening. The resulting constitutive equations were written in an incremental form, and the behavior law was introduced in a three-dimensional kinematic framework based on a particular writing of the gradient tensor of the deformations and gradient of the velocities in the form of an upper triangular matrix. The results demonstrate that the observed behavior is contingent upon the selection of a rotating reference frame. The impact of the selection of a rotating reference frame on the response of the behavior law is considerable. However, for loads with minimal rotation, this influence is negligible. When analysing the instabilities, we tested various prediction to understand the conditions that lead to deformations and to compare these different instability criteria. As a case study, the forming limits curves were analyzed.