## 5 Discussion
### Formability Properties  
As shown in Section 4.3, a large number of simulations under different loading conditions allows performing a detailed characterization of the formability behavior of the material. In Figure 6, the r-values or Lankford coefficient and stress levels for different directions and plastic strain values are presented. In order to evaluate the formability of the material, it is convenient to also calculate the aggregate properties \(\bar{r}\) and \(\Delta r\), as explained in Section 3.7. The evolution of \(\bar{r}\) and \(\Delta r\) with respect to (uniaxial) plastic strain is presented in Figure 11. The figure also includes curves for the analogous magnitudes \(\bar{r}\) and \(\Delta r\), obtained from the stress values in Figure 6.  
The graphs show that better formability is obtained for the LT steel. The \(\bar{r}\) value for LT starts at an upper level and a narrow \(\pm\Delta r\) region is maintained until the end. On the other hand, the HT steel initially has a lower \(\bar{r}\) value, but it increases as the material hardens. The stress \(\bar{r}\) values for the LT steel are also higher than for the HT one. Although the total amount of martensite is almost the same in both materials, and the amount of the MA phase is larger in the HT steel, as Figure 9 shows, the individual phases of LT are stronger. This is the result of smaller grain size, higher carbides content and, depending on loading direction, also of grain shape and crystallographic texture. Yield surfaces at different strain rates and strain levels are shown in Figure 7. The simulations used to generate these graphs were performed under the same conditions as experimental tests employed for the fitting of phenomenological plasticity models used in  
Figure 11: Predicted evolution of \(r\pm\Delta r\) and \(\bar{r}\pm\Delta r\) with respect to uniaxial plastic strain for the LT and HT steels calculated from the values in Figure 6.  
finite element simulations [33; 34]. In these works, a phenomenological expression of the yield surface is found using biaxial tension experiments on cruciform specimens, and then correlating the measured angles of stress with the strain rate angle. This information can be easily calculated from the simulation results. For instance, Figure 12 shows the curves obtained for the LT and HT steels at a strain rate of 0.01 s\({}^{-1}\). The same experimental data used to calculate yield surfaces can be employed for the calculation of forming limit diagrams, such as the one presented in Figure 8. These forming limits are comparable to previous results in the literature. However, it is important to remark that, most often, finite element simulations with phenomenological models are employed [70; 71; 72; 73]. Instead, the approach used here is based on crystal plasticity, allowing additional analysis that is not possible with a phenomenological model. As an example of possible techniques that could be applied, it is studied how strain inhomogeneities between different grains might be used to investigate the formability of the material. In [74; 75], the crystallographic texture around voids in damaged material was identified and clearly differentiated from the texture of grains far from these voids. In [43], a correlation was found between the orientations of grains surrounding the voids and the orientations of the pairs of grains for which a maximum difference was observed in the shear stress component on the plane perpendicular to the loading direction. This maximum difference is calculated by VPSC90 for every simulation step, so it can be easily used to calculate limit curves to be represented in the FLD axis, as shown in Figure 13.  
Figure 12: Angle of strain rate with respect to loading direction for the LT and HT steels at a strain rate of 0.01 s\({}^{-1}\), calculated from the yield surfaces in Figure 7.  
Figure 13: Forming limit diagrams superimposed with isolevel lines indicating the maximum difference on the shear strain component perpendicular to the loading direction between two grains.