---

In the past decades, for prediction of necking phenomenon, several models such as the vertex theory have been proposed. Here, a vertex model considering strain and stress rate discontinuities in the necking band is extended. This model is based on the J_2 deformation theory of classical plasticity to predict the evolution of a bounded deformation in necking modes. Although consideration of the strain rate hardening effect plays an important role to obtain accurate results, but, usually imposing it leads to emerging, relatively the main complex constitutive equations. Therefore, a delicate connective bridge between the diffuse and localized models is made using the maximum force assumption to overcome on the complexity of the problem. Also, by investigating the strain rate behavior on the plastic instability, the forming limit diagrams are obtained by illustrating more accurate results as compared to existing models. Effect of stress triaxiality is investigated on the localization in bifurcation analysis. Also, a modified maximum force criterion is applied to predict diffuse necking considering loading conditions. The anisotropy effects is studied by application of a quadratic Hill's criteria. The necking band angle will be investigated per different conditions through extending the vertex model coupled with the angle-dependent yield criterion.

---

In this paper, an analytical path-dependent plastic instability model is proposed for the thin metallic sheets through considering a linear pre-straining path for the both diffuse and localized necking. The model is introduced by extending a modified maximum force (MMFC) that the MMFC considers the strain hardening on the diffuse necking as well as the loading conditions. Also, the vertex criterion will be used to prediction of localized necking. The vertex criterion presented by Sto ̈ren and Rice are usually based on the J_2 deformation theory of classical plasticity, which explores the localized necking through the rate discontinuity assumption at the necking band. Both models will be combined with the strain path effect through a linear adoption of an equivalent strain. It will be investigated by applying a pre-strain in the major and minor directions for prediction of the formability in the non-proportional loading. Moreover, a dependent to yield criterion (DYC) - angle is used for prediction of the necking band angle in the vertex theory. Finally, the quadratic Hill criterion is used to investigate the anisotropy effect. The model is verified by experimental results presented by other authors.