Modelling non-quadratic anisotropic yield criteria and mixed isotropic-nonlinear kinematic hardening: analysis of forward- and backward-Euler formulations

This paper focus on the simulation of plastic anisotropic effects, regarding both initial yielding and hardening, in sheet metal forming processes. A constitutive model for a general stress state, suitable for any yield criteria based on an anisotropic stress potential, is presented. In order to predict anisotropic hardening effects, such as the Bauschinger’s effect, the model also accounts for a mixed isotropic-nonlinear kinematic hardening rule. Aiming to the application of this model to the Finite Element Method, two generalized stress integration algorithms are derived. The first is formulated using the forward-Euler approach and uses the sub-incrementation technique. This technique allows larger time steps without losing convergence and improving the quality of the results. The second one is based on the backward-Euler approach and includes a multi-stage methodology in order to help the convergence of the return mapping procedure. The model was implemented in the commercial finite element code Abaqus by means of a “user-defined” material subroutine for Yld91 and Yld2004-18p yield criteria. The model and the algorithms were assessed in different sheet metal forming simulations: U-draw bending test and cylindrical cup drawing test. The obtained results were in good agreement with the experimental data available in the literature, and the comparison of springback results and earing profiles obtained with both algorithms are presented, along with the corresponding computational costs.