# Evolution of Residual Stress Based on Curvature Coupling in Multi-Roll Levelling
## 4 Evolution of the Residual Stress in Multi-Roll Levelling
### Evolution of Residual Stress in Multiple Bendings  
The evolution of residual stress in each bending after the first bending is similar and the second bending is analysed first. Figure 4 demonstrates the evolution process of residual stress for the second bending on the basis of the first bending, shown in Figure 3b, with the same initial residual stress. The second bending starts from the springback position \(OA_{c}^{N+1}\) after the first bending to the position \(OA_{N+2}\) and ends with a springback, which also includes the elastic region and plastic region. The elastic region is between 0 and \(H_{t}^{2}\), and the plastic region is between \(H_{t}^{2}\) and \(H/2\). Due to the gradual decrease of the intermeshing of the of the work rolls in multi-roll levelling, the bending curvature is gradually decreased, and the elastic region is gradually increased, that is, \(H_{t}^{1}<H_{t}^{2}\).  
In the second springback, the elastic region \(OB\) returns to the initial position \(OO_{1}O_{2}\) after the first springback and the plastic region \(BA_{N+2}\) returns to \(O_{2}A_{c}^{(N+2)^{\prime}}\) by a springback \(\varepsilon_{e}\). Thus, after the second springback, the initial line for free position is \(OO_{1}O_{2}A_{c}^{(N+2)^{\prime}}\), and the final springback position is \(OA_{c}^{N+2}\) due to the deformation coordination also. The residual stress at this time is three triangular regions surrounded by \(OO_{1}O_{2}A_{c}^{(N+2)^{\prime}}\) and \(OA_{c}^{N+2}\), wherein the triangles \(OO_{1}P_{1}\) and \(P_{2}A_{c}^{(N+2)^{\prime}}A_{c}^{N+2}\) represent tensile stress and \(P_{1}P_{2}O\) represents compressive stress.  
In Figure 4, the rolling residual stress after the first bending corresponds to the triangles \(OO_{1}C\) and \(CA_{c}^{N+1}A_{c}^{(N+1)^{\prime}}\), and the rolling residual stress after the second reverse bending corresponds to the triangles \(OO_{1}P_{1}\), \(P_{1}P_{2}O\), and \(P_{2}A_{c}^{(N+2)^{\prime}}A_{c}^{N+2}\). Since the absolute mean value \(\overline{|\sigma_{r}|}\) of the residual stress is proportional to the area of the shadow in the figure, the rolling residual stress after the second bending is found to be smaller than after the first bending by comparing the areas of the triangles. The main concern of the second bending process is the rolling residual stress caused by multi-roll levelling. As for the initial residual stress, the portion distributed in the plastic region has been eliminated in the first bending, and the portion distributed in the elastic region cannot be changed in the second and subsequent bendings. Therefore, the first bending is the only chance to eliminate the  
Figure 4: Evolution of residual stress in the second bending.  
initial residual stress and the subsequent bendings can only reduce the rolling residual stress caused by the first bending.  
The third bending is similar to the second one. The thickness of the elastic region increases again, from \(H_{t}^{2}\) to \(H_{t}^{3}\). At this time, the initial line is \(OO_{1}O_{2}O_{3}A_{c}^{(N+3)^{\prime}}\), and the corresponding distribution of residual stress becomes a region surrounded by \(OO_{1}O_{2}O_{3}A_{c}^{(N+3)^{\prime}}\) and \(OA_{c}^{N+3}\). For the \(i\)-th bending, the distribution of residual stress is the region surrounded by \(OO_{1}\cdots O_{i}A_{c}^{(N+i)^{\prime}}\) and \(OA_{c}^{N+i}\).  
Figure 5 demonstrates the evolution of residual stress in the multi-roll levelling of a sheet. After the first bending, the initial residual stress in the elastic region is maintained, the initial residual stress in the plastic region is eliminated, and a new rolling residual stress is caused. The second and subsequent bendings gradually reduce the rolling residual stress but have no effect on the remaining initial residual stress in the elastic region.