# Evolution of Residual Stress Based on Curvature Coupling in Multi-Roll Levelling
## 1 Introduction  
Rolling is a primary processing method for sheet metal, in which the inevitable internal stresses typically cause some explicit and implicit flatness defects [1]. An explicit defect means that the shape of the sheet is not an ideal plane because the longitudinal internal stress caused by the uneven distribution of the longitudinal fibre length of the sheet in the width direction exceeds the stress limit of the sheet. An implicit defect means that the sheet is flat in appearance, but when it is cut along the longitudinal direction, the explicit flatness defects reappear due to the release of internal stress [2; 3; 4; 5]. The internal stress that causes the flatness defects of the sheet in rolling is called residual stress, which is a main factor affecting the quality of the sheet [6; 7]. Therefore, reducing or eliminating residual stress as much as possible to improve the flatness of the sheet is the basis for ensuring that the quality of the sheet is good in subsequent processing [8]. The primary method of reducing and eliminating residual stress is multi-roll levelling. The basic principle of multi-roll levelling is that a series of alternating and progressively decreasing loads are applied to the sheet by a set of work rolls of the leveller to cause multiple elastoplastic deformations and the residual stress in the sheet is gradually reduced until it is eliminated [9; 10]. The current research on analysis of the evolution of residual stress in multi-roll levelling mainly includes analytical methods, numerical methods, and experimental methods. For analytical methods, [PERSON] presented an iteration method with a relaxation factor to calculate the entry and exit stress variation of a cold rolling strip, considering the relationship between stressvariation and velocity variation and the incoming flatness propagation efficiency [11]. [PERSON] et al. established a mechanical model of the strip to evaluate the distributions of contact pressure and residual stress of the strip, especially along the width direction, and calculated the longitudinal residual warpage accordingly by the virtual layers method [6]. [PERSON] et al. presented a 3D axisymmetric elastic-plastic bending mathematical model of plates based on the Prandle-Reuss increment theory according to the relevant initial conditions and boundary conditions of the plate and leveller to predict the residual stress and shape of the plate [12]. [PERSON] et al. developed an analytical model with a computational procedure for describing the continuous bending, reversed bending process and the unloading process, evaluating the levelling process and residual stress distributions for typical H-beams [13]. [PERSON] et al. developed an analytic forming model that analysed the roll levelling process with sufficient precision in a shorter time than is possible with the finite element method, investigating the states of multiple forming under bending conditions, and calculating the residual stresses and residual bend of the sheet metal [14]. For numerical methods, [PERSON] et al. calculated the shape and amplitude of the flatness defects of strips generated by residual stresses in strip rolling, with and without tension, by using a shell finite element model based on the asymptotic numerical method [7]. [PERSON] et al. analysed the rolling process using finite element analysis, determined the distribution of residual stresses, and obtained the theoretical moment curvature characteristics using the output of this analysis as the input to the modelling for pure bending [15]. [PERSON] et al. used a coupled finite element method to compute stresses and strains in-bite and out-of-bite in cold rolling of thin strips and predicted the manifested flatness defect, taking into account the stress field in the pre- and post-bite areas reorganized by the buckling [16]. [PERSON] et al.