A Unified Bifurcation Analysis of Sheet Metal Forming Limits

Abstract

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.

Damage evaluation in tube spinnability test with ductile fracture criteria

“…For example, the coupling of DFCs with finite element (FE) simulation was used to predict crack initiation, propagation, and final rupture during metal forming. Compared with the forming limit diagram (FLD), which is based on necking models [10][11][12][13][14][15], damage models based on DFC/FE have distinct advantages in predicting rupture during the spinning process. On the one hand, the three-dimensional body element has been widely used in modeling spinning process, while the FLD method can only be applied to finite elements with plane stress states.…”

Prediction of Formability of Bi-axial Pre-strained Dual Phase Steel Sheets Using Stress-Based Forming Limit Diagram

“…They found that the discontinuity of shear stress inside and outside of the localized band is actually zero. The bifurcation analysis with vertex theory, proposed by Stören and Rice, was later revised by Zhu et al (2001). The modified vertex theory, which involves a generalized material constitutive relation, is achieved by applying Hosford's higher-order yield criterion (Hosford 1979) and the concepts of plasticity theory.…”

In-plane anisotropy in tensile deformation and its influence on the drawability of Nimonic c–263 alloy sheets

“…Since this method of determining limit strain values used in the present study is proposed for the first time, the method is to be validated for its correctness. The validation is carried out using the vertex theory developed by Storen & Rice (1975) and modified by Zhu and co-workers (Zhu et al 2001) and the forming limits are derived by Jie and co-workers (Jie et al 2007) for the material with power law plasticity model with Hosford's yield criterion. The same is presented in the following equations.…”