| The concept of sheet metal forming limits diagram (FLD) dates back to the 1960s. It was during that time that S. P. Keeler, W. A. Backhofen and G. M. Goodwin, discovered through experimentation the extraordinary phenomenon of sheet metal failure under in-plane stretching following a pattern in major and minor strain space. In their work, different widths of sheet metals were pulled over a spherical punch with large curvature. This generated bi-axial stretching which covered a range of strain ratios. Localized necking in the sheets were identified and plotted in major/minor strain space, thus generating the FLD.Although discovery of the FLD was based mostly on experimental evidence, it has generated tremendous interests in the prediction theoretically based FLD's. One of the most important theoretical models is the Marciniak-Kuczynski (M-K) model. In this model an initial imperfection in the form of a groove, with slightly decreased thickness, is assumed in an otherwise homogenous flat piece of sheet metal subjected to an orthogonal loading. In this method, forming limits are reached based on a strain increment criterion. The M-K method has been used by many researchers in theoretical prediction of localized necking in flat sheet metal under in-plane loading.Recently, many studies have shown that the presence of out-of-plane loading enhances the forming limits. In this study, the M-K method is modified to include the bending component so that it is no longer restricted to a flat geometry and in-plane loading. Sheet forming limits for the right hand side of the FLD were calculated numerically from an integral equation as a result of force equilibrium. The force equilibrium was derived using the associated plastic flow rule, a power law material hardening relation, the deformation theory, and two different yield criteria. The proposed theoretical predictions compared well with numerical results from other researchers using a different theoretical model. These predictions are also in good agreement with the limited set of experimental data. The model introduced through this research represents a valuable amendment to the current FLD models in literature intended to capture bending effects. |
