Prediction of limit strain in sheet metal-forming processes by 3D analysis of localized necking

Theoretical prediction of forming limit strain of sheet metal is developed in the framework of the three-dimensional general bifurcation theory. The onset of the three-dimensional discontinuous velocity field in the biaxially stretched uniform sheet is predicted. Three fundamental mode vectors, i.e. shear horizontal, shear vertical and normal modes are introduced and it is demonstrated that any bifurcation mode is represented by the linear combination of them. The onset of the bifurcation is numerically analyzed in terms of the modes by the use of the linear comparison solid originally introduced by Hill in 1959. In this study, a linear constitutive relation is adopted for the linear comparison solid, which is developed based on the constitutive theory proposed by Goya and Ito and is capable of incorporating the directional dependence of the plastic strain rate on the stress rate. The numerical results show that forming limit strains predicted by the three-dimensional mode theory is much higher in general than that given by Stören and Rice in 1975. Then, it is revealed from the three-dimensional mode analysis that the bifurcation mode that arises can be changed from one type to another according to the sign of stress ratio. It is also shown that the strain limit predicted by the three-dimensional mode analysis gives upper limit lines for the bifurcation lines proposed in the past for any linear strain-path directions.