# Evolution of Residual Stress Based on Curvature Coupling in Multi-Roll Levelling
## 1 Introduction
developed a model of residual stresses in hot-rolled sheets based on the elastic-plastic material model, taking into account the nonuniform distribution of elastic-plastic deformations in the volume and unloading of the sheet material and phase transformation during cooling, and performed experimental verification of the model under industrial conditions [29]. [PERSON] presented an inverse routine to predict residual stress in sheet material and collected data in a free bending test using the measurement of residual stress by X-ray diffraction to determine residual stress distribution in sheet material [30]. For combinations of numerical methods and experimental methods, [PERSON] et al. carried out three-point cold bending experiments and tension tests on a thick steel plate, performing a numerical simulation with the finite element solver to replicate the experimentally determined residual stresses and strains [31]. [PERSON] and [PERSON] investigated the residual stress evolution of cold-rolled and hot-rolled steel sheets respectively by means of finite-element analysis simulations and X-ray diffraction [32; 33]. [PERSON] et al. studied the numerical simulation of the tension levelling process of thin strip steel with the aid of finite element software, analysed the influence of the roll intermeshes in the anti-cambering on the distribution and magnitude of residual stresses in levelled strip steel, and carried out several experiments with the tension leveller based on the results from the simulation [34]. [PERSON] et al. investigated the experimental and numerical modelling of flatness defects in strip cold rolling, proposed an experimental setup to analyse the interaction between residual stress and buckling for wavy edge flatness defects, and used finite element simulation for test validation [35; 36]. [PERSON] et al. used a numerical model of a seven-roll leveller to determine roll positions resulting in a flat sheet and a defined residual stress distribution, measured the residual stresses across the sheet thickness after levelling using diffraction analysis [9; 10]. In combinations of analytical methods, numerical methods, and experimental methods, the curvature integration method is a particularly noteworthy method. [PERSON] et al. developed a mechanics model for predicting the levelling process of plates based on the curvature integration method and evaluated the residual stress using a global nonlinear unconstrained optimization method [8]. [PERSON] et al. introduced the curvature integration by elastic-plastic differences to analyse the effect of bending on the levelling results and studied the plastic deformation ratio, residual stress, residual curvature, and levelling force for different levelling strategies to find an effectual strategy [37; 38; 39]. In multi-roll levelling, each elastoplastic bending deformation of the sheet will cause a change in the residual stress distribution, which in turn changes the curvature of the sheet and then affects the distribution of residual stress again. Therefore, the evolution of the residual stress of the sheet in multi-roll levelling is a multi-factor coupled process, which is particularly closely related to the bending curvature. However, there are few studies on the curvature coupling between bendings. In addition, improper parameter settings in multi-roll levelling of the sheet not only do not reduce the residual stress but also cause new residual stress. The previous research is mainly focused on eliminating the initial residual stress of the sheet and the new residual stress caused by the plastic deformation during the levelling process has not received enough attention. Some studies analyse the initial residual stress evaluation of the sheet in multi-roll levelling, but the systematic calculation method of the residual stress after levelling is not proposed, and the key factors affecting the residual stress are not discussed.