New robust algorithms for Marciniak–Kuczynski model to calculate the forming limit diagrams

Abstract

To solve the non-convergence of Newton–Raphson (N–R) method in the framework of Marciniak–Kuczynski (M–K) model for predicting sheet metal's forming limit, the relationship between the internal stress variables in the groove of sheet metal is investigated. The number of unknown variables in N–R method reduces to two from three. And the N–R method is changed as modified N–R method by considering this relationship. Since the convergences of N–R method and modified N–R method are not promised, an increment method for M–K model is developed, which can unconditionally guarantee its convergence. To improve the computational efficiency, a modified increment method is established by combining modified N–R method and increment method. For N–R method and modified N–R method, the accuracy of solutions is depended on the artificial defined errors for those equilibrium equations, while for increment method and modified increment method, the accuracy of solutions can be easily promised. To verify the validity of these methods, the material AA6111-T3 is adopted. It's found that increment method is convergent, whether combining the method with Hill’48 or Yld2000-2d yield criterion. The convergences of N–R method and modified N–R method are affected by strain path, necking angle and so on. The modified increment method is unconditionally convergent, and the computation efficiency is almost equal to N–R method. The modified increment method is recognized as the best method by considering from computation efficiency and convergence. The M–K model is also applied to Ti–6Al–4V, which shows that no matter for Hill’48 or Yld2000-2d yield criterion, the M–K model can predict the forming limit well under negative strain path, while overestimates the forming limit strains under positive strain path.