## 3 Results and Discussion
### Tensile Test  
The change of length with elongation during the tensile test enables the calculation of the generally used engineering stress-strain curve (Figure 4). Alternatively, the true stress-strain relation, which better reflects the real nature of deformation compared to the engineering counterparts, can be expressed in the form of the Ramberg-Osgood (R-O) model (see Figure 5). The model parameters of the R-O equation were evaluated from the pre-neck formation stage. As Figure 5 shows, the measured strain-stress diagram can be successfully reproduced by implementing Equation (2) with the fitting parameters listed in Table 2. The peak stress for both the simulated and measured curves is observed at the true strain of 0.49 and this result is consistent with the one reported elsewhere [19], while the yield stress was estimated to be 15.3 MPa. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Parameter** & **Asym.** & **Value** \\ \hline Roll radii & \(R\) (mm) & 75 & 75 \\ Initial thickness of the sheet & \(h_{i}\) (mm) & 1.92 & 5.02 \\ Final thickness of the sheet & \(h_{f}\) (mm) & 1.3 & 3.61 \\ Initial sheet length & \(L\) (mm) & 20 & 20 \\ Material & - & Al-1050 & Al-1050 \\ Minimal friction coefficient necessary & \(\mu_{min}\) & 0.0672 & 0.118 \\ for rolling & \(\omega_{t}\) (rad/s) & 1.1023 & 1.1023 \\ Angular velocity of the bottom roll & \(\omega_{b}\) (rad/s) & 0.748 & 1.1023 \\ Velocity rate & \(\omega_{t}/\omega_{b}\) & 1.474 & 1.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Technological parameters used in FEM simulation of cold rolling. Figure 4: Engineering strain–stress curve measured by tensile test for 1050 Al alloy. The engineering stress-strain relation (Figure 4) clearly shows that, when the strain value exceeds \(\sim\)0.37 (at the max. stress value of 69.11 MPa), the stress shows a declining tendency due to the reduced cross-section. This phenomenon can be foreseen by using Equation (4), which predicts the formation of necking. The d\(\sigma\)/d\(\varepsilon\) relation plotted together with the true stress-strain curve (see Figure 5) reveals that the formation of necking occurs at the straining level of 0.3703. Table 2 shows that the Ramberg-Osgood model parameters (Equation (2)) derived from the experimental data are comparable with the fitting parameters determined for the Hollomon model (Equations (5) and (6)). The difference between the model constants is negligibly small. As Figure 6 shows, the FEM simulation (blue dash-dotted line) is capable of reproducing the experimental strain-stress curve (continuous line) with great accuracy (correlation coefficient = 0.9996) under the condition that the proper material model is employed. In the present case, the R-O material model (dashed line in Figure 5) was used in the FEM calculations. It is important to underline that the FEM captures the beginning of the neck formation quite accurately (Figure 6). However, the advanced stages of the contraction unfolded by the simulated results reveal some differences with respect to the experimental evidence; i.e., the failure of the investigated alloy occurs earlier compared to the results of the simulation. This discrepancy can be attributed to the fact that the heat-treated 1050 Al alloy is textured and therefore the anisotropy of the plastic strain ratio will affect the contraction. In the case of the isotropic material, the so-called average Lankford value (width-to-thickness reduction ratio or \(r\)-value) is equal to one, implying that the deformation is equally distributed in all directions. However, Al alloys are strongly textured with the prevailing {100}\(<\)001\(>\) component [42] (which is highly anisotropic in terms of the plastic strain ratio), and therefore the resulting \(r\)-value in this material will be below one. In turn, this leads to the thinning of a sheet during deformation and, consequently, the textured material fails faster compared to the isotropic one.