Investigation of formability limit of metallic sheets subjected to non-proportional biaxial loading based on anisotropic diffuse and vertex localized plastic instability models

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.