A Unified Bifurcation Analysis of Sheet Metal Forming Limits

The purpose of this study is to determine analytically the orientations of localized necks occurring in sheet metal forming processes, and obtain the corresponding forming limit diagrams (FLDs). In addition to the force equilibrium condition as adopted by other researchers, we include the moment equilibrium in this study. The shear terms due to the perturbation are found to vanish inside the localized neck of a region of deformation. This simplifies the two-dimensional problem to a one-dimensional problem. Furthermore, it is found that there are only three possible orientations for the initiation of a localized neck, i.e., two principal directions and one zero extension direction (which applies only to negative strain ratio deformations). A special case study using the von Mises yield criterion is also presented in this paper. Predictions from our unified analysis matches with the results of Hill, R., 1952, “On Discontinuous Plastic States, With Special Reference to Localized Necking in Thin Sheets,” J. Mech. Phys. Solids, 1, pp. 19–30. For the negative strain ratio regime (left-hand side of the FLDs), and with the results of Storen, S., and Rice, R., 1975, “Localized Necking in Thin Sheets,” J. Mech. Phys. Solids, 23, pp. 421–441. For the positive strain ratio regime (right hand-side of the FLD). When the localized neck is assumed to be in the zero extension direction for the negative strain ratio deformation, deformation theory and flow theory of plasticity give the same limit strains, and a unified solution to the limit strain is obtained. This solution is independent of the specific yield criterion used.