# Evolution of Residual Stress Based on Curvature Coupling in Multi-Roll Levelling
## 5 Modelling and Calculation of Residual Stress
When point \(O_{i}\) and point \(O_{i+1}\) are on opposite sides of line \(OP_{m+1}\), \(P_{i}\) is the intersection of line \(O_{i}O_{i+1}\) and line \(OP_{m+1}\), and its coordinate \(\left(x_{i}^{p},y_{i}^{p}\right)\) is:  
\[\left\{ \begin{array}{l}x_{i}^{p}=\left(-1\right)^{N+i+1}\frac{1}{k_{n}-k_{N+ i}}\\ y_{i}^{p}=\left(-1\right)^{N+i+1}\frac{k_{n}}{k_{n}-k_{N+i}}\end{array} \right. \tag{12}\]  
Equation (9) to (12) determine the calculation methods for points \(O\) through \(O_{m+1}\), \(Q_{1}\) through \(Q_{m+1}\), and \(P_{1}\) through \(P_{m+1}\). All of the slopes \(k_{0}\), \(k_{1}\), \(\ldots\), \(k_{n}\) are solved by calculating the ratio of the curvature at the contact point of each roll to the elastic curvature limit via the coupled curvature integral model. After the coordinates of these points are determined, the area \(f_{i}\) of each part can be solved. When \(O_{i}\) and \(O_{i+1}\) are on the same side and opposite sides of the line \(OP_{m+1}\), \(f_{i}\) is a trapezoidal region and two triangular regions, respectively. The method for determining the position of point \(O_{i}\) and point \(O_{i+1}\) is that the coordinates of the two points are substituted into the equation of line \(OP_{m+1}\) and their product is calculated by Equation (13):  
\[g=\left(y_{i+1}-k_{n}x_{i+1}\right)\left(y_{i}-k_{n}x_{i}\right) \tag{13}\]  
They are on the same side if \(g>0\) and on opposite sides if \(g<0\). According to Figure 6, \(f_{i}\) is calculated by Equation (14):  
\[\left\{\begin{array}{l}f_{0}=\frac{1}{2}|O_{0}|_{1}\cdot|O_{1}Q_{1}|\\ \left\{\begin{array}{l}f_{i}=\frac{1}{2}\Big{(}|O_{i}\Omega_{i}|+\Big{|}O_{i +1}Q_{i+1}\Big{|}\Big{)}\cdot\Big{|}Q_{i}Q_{i+1}\Big{|}\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad  
the residual stress as small as possible. Additionally, the intermesh of the work rolls at the exit is adjusted to make the residual curvature close to zero. The plastic deformation is typically the largest caused by the second or third work roll at the entry and then gradually decreases. The influence of the intermesh of the work rolls at the entry on the maximum plastic deformation ratio (the ratio of the thickness of the plastic deformation zone to the total thickness of the sheet metal) based on the coupled curvature integral model is shown in Figure 7. As the intermesh of the work rolls at the entry increases, the maximum plastic deformation ratio of the sheet increases, which makes the elimination of the initial residual stress more sufficient, but the rolling residual stress increases accordingly. The variation of the residual stress calculated by the model described in Section 4 with the intermesh of the work rolls at the entry in the levelling is shown in Figure 8, where the total residual stress is the sum of the rolling residual stress and the remaining initial residual stress. Figure 8 shows that, as the intermesh of the work rolls at the entry increases, the rolling residual stress increases due to the large plastic deformation, but the remaining initial residual stress decreases. The result is a reduction in total residual stress after levelling, with the absolute mean value decreasing from 40 MPa before levelling to 18.5 MPa. Figure 8: Variation of the residual stress with the intermesh of the work rolls at the entry. Figure 7: Influence of the intermesh of the work rolls at the entry on the maximum plastic deformation ratio.