Evolutionary vertex forming limit model in development of diffuse to localized viscoplastic instability of sheets

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.