Experimental and Visco-Plastic Self-Consistent evaluation of forming limit diagrams for anisotropic sheet metals: An efficient and robust implementation of the M-K model

Abstract

In the present work, an efficient formulation for the prediction of forming-limit diagrams (FLDs) based on the well-known Marciniak and Kuczynski (MK) theory using a Visco- Plastic Self-Consistent (VPSC) crystal-plasticity model has been detailed. The present model extends the previous MK-VPSC implementation (Signorelli et al., Predictions of forming limit diagrams using a rate-dependent polycrystal self-consistent plasticity model, International Journal of Plasticity 25 (2009) 1e25) based on the NewtoneRaphson (N-R) method, which gives no guarantee of a robust iterative procedure. In order to avoid convergence problems and to reduce the computational cost of the coupled MK-VPSC scheme, a direct approach (DA) is proposed. The DA eliminates the need of the Jacobian evaluation associated with the N-R method as well as the iterative procedure tied to other possible minimization techniques. Moreover, the mechanical states outside and inside the groove are solved in the sample reference frame, avoiding the need to rotate the crystallographic orientations and the internal variables to the current band reference frame at each increment. In this way, only two calls to the material law are required per MK increment, obtaining a more robust numerical procedure with a significant computational cost reduction. Interestingly, the requirement of more complex boundary conditions does not substantially increase the number of internal VPSC iterations to achieve a given tolerance. Simulation results show that the direct MK-VPSC approach is consistent with that based on the N-R method. The generalized boundary conditions in the polycrystal model allowed us to calculate either strain-rate ratio or stress ratio based FLDs. The effect of using either strain-rate ratio or stress ratio paths on the FLDs has been investigated by imposing three types of pre-straining on the sheet metals. Formability predictions for a randomly-textured FCC material and for textured FCC, BCC and HCP polycrystals are presented and discussed. Finally, by considering dissimilar metals e extra deep-drawing quality steel (EDDQ), dual-phase steel (DP-780) and pure zinc (Zn20) e we evaluated the MK-VPSC model´s ability to predict forming-limit strains irrespective of microstructure and crystallography. The predicted results have been compared with experimental data and good agreement was found.