## 4 Discussion  
Overall, the indexed active slip system was limited to a fraction of 50% or less, even for grains with \(\{111\}//\)ND orientation. This limitation arises because the calculations in this study are based on a single slip system family; thus, if another slip system is active alongside the primary one being considered, the rotation axis from the experiment may not align with the calculated axis. Additionally, when deformation exceeds the \"early deformation\" stage, other phenomena, such as cross-slip, curved slip, diffuse slip, and/or intersecting slip, are likely to occur, significantly obstructing the rotation axis resulting from slip system activity [25, 26]. Nevertheless, Figure 8d reveals an intriguing trend where the \(\{111\}<1\overline{2}1>\) grain orientation exhibits the highest susceptibility to accommodate \(\{112\}<111>\) single slip system activity based on its relatively high fraction of solo active slip systems compared to other \(\{111\}//\)ND orientations, which exhibit nearly similar fractions. This indicates that the current method is sensitive to orientation, influencing slip system activity.  
In this study, the proposed method proved most effective for \(\{111\}//\)ND grains and primarily during the initial stage of plastic deformation. For other grain orientations, such as \(\{001\}//\)ND, the method can still be applied, but calculations of the rotation axis should incorporate additional active slip systems such as \(\{110\}<111>\) and \(\{123\}<111>\). Grains located at the center consistently showed higher statistical values compared to those at the top, likely due to the presence of additional shear strain at the top surface [27]. This shear strain induces a complex mixture of deformation modes, resulting in the activation of multiple slip systems. Consequently, the current method, which indexes single slip system activation, may not fully match the observed data in these cases. In such scenarios, slip systems may interact to collectively rotate the grain. [PERSON] et al. conducted simulations considering both single (\(\{110\}<111>\)) and dual (\(\{110\}<111>\) + \(\{112\}<111>\)) slip systems working in tandem [21].  
Despite its limitations, the current method reveals several key insights. Figure 8c indicates that DSR without lubrication leads to greater uniformity in the activity of the \(\{112\}\)\(<\)\(111>\) slip system across the surface and center under combined plane and shear strain or, in this case, termed as deep penetration of shear deformation. This aligns with findings that suggest DSR can enhance through-thickness equivalent plastic strain homogeneity [3]. Including shear mode in the calculations also corresponds well with the regions of the sample undergoing shear deformation. It is generally understood that shear promotes the activation of a broader range of slip systems compared to symmetrical rolling [28]. However, in this study, it was observed that shear primarily supports the activation of a single \(\{112\}<111>\) slip system, especially in grains with the \(\{111\}<1\overline{1}0>\) orientation.  
The application of viscoplastic self-consistent formulation for assessing rotation axis presents considerable potential. Beyond its capability to identify which deformation modes drive texture changes and to evaluate the relative activities of multiple slip and twinning modes on texture evolution, it also allows for the inclusion of additional factors influencing orientation changes. In rotation axis analysis, incorporating grain incompatibilities could serve as input to permit each grain to adopt a unique strain value, potentially impacting neighboring grains [21]. Classic studies have highlighted the importance of grain shape effects on lattice rotation, though this has been traditionally limited to elastic interactions [29]. In this context, incorporating rate-controlling dislocation mechanisms may also be bene ficial [30]. Therefore, refining the current method to account for these factors is essential before broader application.