# Evolution of Residual Stress Based on Curvature Coupling in Multi-Roll Levelling
## 4 Evolution of the Residual Stress in Multi-Roll Levelling
### Evolution of the Residual Stress in the First Bending  
There are two cases of the residual stress evolution in the first bending depending on whether the sheet contains initial residual stress. The case of no initial residual stress is shown in Figure 3a, where a half-thickness sheet is used as an analysis object, the abscissa indicates strain, and the ordinate indicates thickness. \(OA_{c}^{0}\) represents the strain distribution corresponding to the initial residual curvature of the sheet and \(OA_{N+1}\) represents the strain distribution corresponding to the curvature of the first valid bending. The first valid bending refers to the bending that causes the maximum strain of the sheet  
Figure 2: Distribution of the residual stresses before and after levelling. with the initial curvature. Only the first and subsequent bendings are significant for the evolution of the residual stress because the effect of the previous bending on the sheet will be eliminated as an initial residual stress. The first bending in this article refers to the first valid bending and \(N\) is the number of instances of bending before the first valid bending. \(H_{t}^{1}\) is defined as the thickness of the elastic region of the sheet corresponding to the first bending then, the elastic region is between 0 and \(H_{t}^{1}\), and the plastic region is between \(H_{t}^{1}\) and \(H\)/2. If the sheet is bent to the position \(OA_{N+1}\) and then the external force is removed, a free springback will be presented. If the interaction between the two regions in the springback is not taken into account, the elastic region \(OB\) returns to the initial position \(OO_{1}\) by a full springback, and the plastic region \(BA_{N+1}\) returns to the position \(O_{1}A_{c}^{(N+1)^{\prime}}\) by a certain springback. \(\varepsilon_{e}\). \(\varepsilon_{e}\) is the elastic strain limit, that is, the strain when the sheet is bent from the free state to the critical position of the elastic region and the plastic region. The lines that indicate the free position of the sheet under stress-free conditions similar to \(OO_{1}A_{c}^{(N+1)^{\prime}}\) are defined herein as the initial line. However, because the two regions interact with each other in the springback, the actual springback position of the sheet that satisfies the deformation coordination condition is located at line \(OA_{c}^{N+1}\) instead of \(OO_{1}A_{c}^{(N+1)^{\prime}}\). The inconsistency between the unstressed free position \(OO_{1}A_{c}^{(N+1)^{\prime}}\) and the actual position \(OA_{c}^{N+1}\) forms an internal stress region, including a compressive stress region \(OO_{1}P_{1}\) and a tensile stress region \(P_{1}A_{c}^{N+1}A_{c}^{(N+1)^{\prime}}\). The numbers above the lines \(OA_{c}^{0}\), \(OA_{N+1}\), and \(OA_{c}^{N+1}\) indicate the order of bending and springback of the sheet. According to the above-mentioned analysis, a sheet without initial residual stress exhibits internal stress caused after the first bending. Therefore, the internal stress of the sheet due to the roll levelling is defined as the rolling residual stress to distinguish it from the initial residual stress. The springback of the sheet with initial residual stress is similar to that without residual stress, as shown in Figure 3b. \(Y_{1}\) represents the initial residual stress of the sheet, including two regions, \(OCD\) and \(DEA_{c}^{0}\), which are tensile stress and compressive stress, respectively. If the sheet is bent from the initial position \(OA_{c}^{0}\) to position \(OA_{N+1}\) and then the external force is removed, the elastic region \(OG\) returns to the initial position \(OO_{1}\) with a free springback, the plastic section \(GA_{N+1}\) returns to the position \(O_{1}A_{c}^{(N+1)^{\prime}}\) with a springback \(\varepsilon_{e}\), and finally the sheet returns to the position \(OA_{c}^{N+1}\) due to the deformation coordination, accompanied by a certain residual stress.