Comparison between the Marciniak and Kuczyński imperfection approach and bifurcation theory in the prediction of localized necking for porous ductile materials

To prevent the occurrence of localized necking, the concept of forming limit diagram is often used, thus playing an important role in sheet metal forming processes. The aim of the present study is to develop a numerical tool for the theoretical prediction of forming limit diagrams, which would be a cost-effective procedure as compared to experimental measurements. The proposed numerical tool is based on the Marciniak and Kuczyński imperfection approach combined with the Gurson–Tvergaard–Needleman damage model, which is implemented into the MATLAB program within the framework of plane-stress conditions. Forming limit diagrams have been predicted by assuming both geometric (thickness) as well as material initial imperfections in the Marciniak and Kuczyński imperfection approach. These forming limit diagrams, for different sizes of geometric or material imperfections, are also compared with the forming limit diagram obtained by using the bifurcation theory. It is shown that the bifurcation-based forming limit diagram provides an upper bound as compared to the Marciniak and Kuczyński imperfection approach predictions. The results also reveal that irrespective of the imperfection type considered in the Marciniak and Kuczyński imperfection approach, the corresponding forming limit diagram tends to that predicted by bifurcation theory when the size of initial imperfection tends to zero. Additionally, the predicted ductility limits are lowered as the magnitude of initial imperfection increases; however, the decrease in the ductility limits at balanced biaxial tension is more significant than for the other strain-path ratios. The results for the forming limit diagrams indicate that the predicted ductility limits are more sensitive to the initial imperfection in the thickness and the isotropic hardening coefficient as compared to the other types of material imperfections. Moreover, the initial imperfection in the critical porosity is the most influential one among the Gurson–Tvergaard–Needleman damage parameters.