Effect of the kinematic hardening on the plastic anisotropy parameters for metallic sheets

The initial plastic anisotropy parameters are conventionally determined from the Lankford strain ratios defined by $r^{\psi }=\frac{{\epsilon }_{22}^{\mathrm{p}\psi }}{{\epsilon }_{33}^{\mathrm{p}\psi }}$ (ψ being the direction of the loading path). They are usually considered as constant parameters that are determined at a given value of the plastic strain far from the early stage of the plastic flow (i.e. equivalent plastic strain of ${\epsilon }_{\mathrm{eq}}^{\mathrm{p}}=0.2\text{\%}$) and typically at an equivalent plastic strain in between 20% and 50% of plastic strain failure (or material ductility). What prompts to question about the relevance of this determination, considering that this ratio does not remain constant, but changes with plastic strain. Accordingly, when the nonlinear evolution of the kinematic hardening is accounted for, the Lankford strain ratios are expected to evolve significantly during the plastic flow.In this work, a parametric study is performed to investigate the effect of the nonlinear kinematic hardening evolution of the Lankford strain ratios for different values of the kinematic hardening parameters. For the sake of clarity, this nonlinear kinematic hardening is formulated together with nonlinear isotropic hardening in the framework of anisotropic Hill-type (1948) yield criterion. Extension to other quadratic or non-quadratic yield criteria can be made without any difficulty. This parametric study is completed by studying the effect of these parameters on simulations of sheet metal forming by large plastic strains.