## 3 Results and Discussion
### Rolling Processes: Simulation and Experiment  
In order to test the material model as well as the model parameters derived from both the tensile test and FEM calculations, the process of cold rolling was simulated with Deform 2D by employing the Ramberg-Osgood approximation. The roll-gap geometry shown in Figure 3 was used for the simulation of the symmetric and asymmetric rolling trials. The geometry and technological parameters for both cases are listed in Table 1. In addition to the processing parameters, the coefficient of friction \(\mu\) should be determined. In the present case, the minimal value of \(\mu\) necessary for cold rolling was estimated by Equation (19), while the FEM simulations were conducted for a range of \(\mu\), exceeding \(\mu_{min}\). The flow of the investigated material during the rolling trials was examined experimentally by analyzing the distortion of the initially straight lines made by hardness indents on the transverse direction plane of the sheets (see Figures 9 and 10). It is obvious that the symmetric and asymmetric rolling processes accounted for different deformation patterns. The diversity in the degree of distortion can be attributed to the differences in the shear strain distribution across the thickness. When the material is deformed symmetrically (both  
Figure 8: FEM-computed and analytically fitted curves (Equations (20) and (21)) of surface pressure over indentation per indenter size ratio \(d/D\). rolling cylinders rotate with an identical angular velocity), the symmetry plane is located in the mid-thickness part of the rolled sheet and the parabolic-type distortion is attributed to the shear strain localization within the subsurface region. In the case of asymmetric rolling, the symmetry is broken and the shear distribution is strongly affected by the roll-gap geometry as well as by the ratio of the angular velocities of the upper and lower rolls [45]. Figures 11 and 12 show the simulated displacement patterns obtained by the FEM for various friction conditions and experimentally observed counterparts. The results of the calculation reveal that the displacement fields across the thickness can be successfully reproduced under the condition that the friction coefficient is known. In both cases, the successful simulation is carried out with the \(\mu\approx 1.1\mu_{min}\) (for symmetric rolling, the best fit is observed for \(\mu\approx 1.1\mu_{min}\), while for the asymmetric trial, \(\mu\approx 1.13\mu_{min}\)). This result is consistent with the recently reported simulations [36]. Knowing the displacement profile, one can compute the distribution of the equivalent strain across the thickness by Equations (15)-(18). Since asymmetric rolling introduces shear across the whole thickness [28; 45], the strain distribution is also expected to be quite homogeneous. An investigation of the substructure and texture evolution in the asymmetrically rolled sheets [28] showed that the asymmetric ratio of \(\sim\)1.5 can ensure extensive shear even in the middle of the rolled sheets. This is justified by the current investigation of the displacement field. Furthermore, an analysis of the strain distribution  
Figure 10: Hardness indents before (**a**) and after (**b**) symmetric cold rolling. Figure 9: Hardness indents before (**a**) and after (**b**) asymmetric cold rolling. suggests that a nearly uniform distribution can be obtained by asymmetric rolling when the optimum rate of angular velocity is [45; 46]:  
\[\frac{\omega_{t}}{\omega_{b}}=\frac{h_{i}}{h_{f}}(\approx 1.475), \tag{22}\]  
where \(\omega_{t}\) is the angular velocity of the upper roll, and \(\omega_{b}\) is the angular velocity of the lower roll. In the present investigation, the ratio of \(\omega_{t}/\omega_{b}=1.474\) and \(h_{i}/h_{f}=1.476\), and therefore the homogeneous strain distribution can be ensured, as shown in Figure 13. Comparable values are predicted by the model of [PERSON] et al. [46] (\(\omega_{t}/\omega_{b}=1.52\) and \(h_{i}/h_{f}=1.52\)).