An implicit formulation of a two-grain cluster type homogenization approach for polycrystals

The crystal-plasticity-based finite element method has received considerable attention in the field of sheet-metal forming. When the size of stamped sheets is hundreds to thousands of millimeters, a homogenization approach with a lower computational cost is required. A previous study proposed a two-grain cluster (TGC)-type homogenization method, which allows relaxation from the fully-constrained Taylor model (Yoshida, 2022. An alternative formulation of a two-grain cluster model for homogenization of elastoviscoplastic behavior of polycrystal, International Journal of Plasticity, 156, 103368). In the present study, a fully implicit numerical procedure was developed for the TGC model, and it was applied to a cup drawing simulation. The present implicit TGC model used the fully implicit stress integration method. In addition, the relaxation vector that satisfies the equilibrium and compatibility conditions at the end of increment was determined. Moreover, the algorithmic tangent moduli that are consistent with the Newton–Raphson method were derived analytically. These features facilitate the implementation of the TGC model in the FEM. The developed model was implemented in static implicit FEM software (MSC Marc), and cylindrical cup drawing was simulated. Consequently, the FE simulation in conjunction with the TGC model was found to more accurately predict the cup height and thickness than that in conjunction with the Taylor model. The better prediction accuracy of R-values obtained by the TGC model contributed to the accurate prediction of the drawn cup profile.