# Evolution of Residual Stress Based on Curvature Coupling in Multi-Roll Levelling
## 2 Residual Stress in Multi-Roll Levelling  
During the multi-roll straightening process, the sheet metal is bent by the upper and lower work rolls, resulting in multiple alternating elastoplastic bending deformations, as shown in Figure 1. Considering the bending radius is much larger than the thickness of the sheet, the modelling of the levelling process is based on simple bending theory [14], thus, some basic assumptions are applied: (1) The cross sections remain on a plane and only rotate along the intersection lines with neutral layer during bending; (2) the strain neutral layer and stress neutral layer always coincide with the geometric centre layer of the sheet section during deformation; and (3) the stress state of any mass point on the cross section is uniaxial stretching or compression. Based on the above assumptions, the bending process gives rise to a symmetric strain distribution along the sheet thickness [25]. If the bending radius is \(\rho\) and the curvature is \(A\), the strain of the fibre with a distance \(z\) from the neural layer can be expressed as,  
\[\varepsilon_{z}=(\rho+z)A-\rho A=zA \tag{1}\]  
If the strain of the surface fibre reaches elastic limit \(\varepsilon_{e}\), the current curvature \(A_{t}\) is called the elastic limit curvature:  
\[A_{t}=\frac{2\varepsilon_{e}}{H} \tag{2}\]  
The elastic limit stress \(\sigma_{e}\) and yield limit stress \(\sigma_{s}\) of typical elastoplastic materials are very close, thus \(\varepsilon_{e}=\sigma_{e}/E=\sigma_{s}/E\), \(E\) is the Young's elastic modulus. When the bending curvature exceeds \(A_{t}\), the outer material will be plastically deformed. \(H_{t}\) denotes the distance from the elastic-plastic demarcation point to the neutral layer at which the strain reaches \(\varepsilon_{e}\). Then, the elastic region is between 0 and \(H_{t}\), and the plastic region is between \(H_{t}\) and \(H\)/2. The elastoplastic deformations occur only in the area before the sheet metal contacts the roller. \(A_{i}\) represents the curvature of each bending at the contact point between the metal sheet and roller. Applying a certain degree of plastic deformation is considered an effective method to reduce residual stress [40]. The mechanism of this phenomenon is that the grain sliding during the plastic deformation can alleviate the misfits between different regions or between different phases within the material, resulting in stress relaxation [41; 42]. Experimental studies have also shown that when the material strain reaches the yield limit, the residual stress is significantly reduced [43; 44]. Based on these results, in our model, the residual stress in the plastic deformation region before each bending is considered to be eliminated by the following bending. Correspondingly, the residual stress in the elastic region remains unchanged. For each bending, the next bending is a reverse one, therefore, from the contact point with roller \(i\) to the contact point with roller \(i+1\) the sheet curvature will have a continuous change from \(A_{i}\) to zero and from zero to \(A_{i+1}\), during which both the plastic and elastic regions will have a springback process. If the material at the different layers springback freely, the material in elastic region returns to the initial position, and the material in plastic region returns to the position by a certain springback \(\varepsilon_{e}\). However, the deformation of the material must meet the deformation coordination condition to keep the cross sections on a plane, which means the two regions interact with each other in the springback. The permanent plastic deformation will hold the elastic region, prevent it from full springback, and the rebounding stress of elastic region will also force the plastic region to cause excessive springback [45]. Such interaction leads to new internal stresses, that is, rolling residual stress. The final residual stress after levelling is the superposition of initial residual stress and the rolling residual stress.