||The paper is mainly devoted to a comparative analysis of two plasticity models with a nonlinear hardening law, namely the Armstrong and Frederick model, and a recent modification proposed in literature in the framework of Generalized Standard Materials (GSM). We first provide a detailed mathematical analysis of the two models by appropriately resorting to the bipotential theory. This delivers for the GSM model a closed form expression of a bipotential. Moreover, it is demonstrated for the first time that the Armstrong and Frederick model does not admit a convex potential; this result confirms the necessary requirement of a non associated framework for this model. Then, for the modified model, making use of the above bipotential-based tools, we carry out a shakedown analysis of a thin walled tube under constant tension and alternating cyclic torsion. The accuracy of the obtained results is checked by comparing them to data obtained by numerical solving the corresponding shakedown bounds problems. Finally, the predictions of the two models are compared and illustrated their significant differences.