# Evolution of Residual Stress Based on Curvature Coupling in Multi-Roll Levelling
## 1 Introduction
introduced a numerical modelling of strip conveying through an industrial leveller using FEM software and predicted the final strip shape from plastic strains and residual stresses via width and thickness [17]. [PERSON] et al. used numerical simulation to determine the influence of residual stress and the effect of plastic deformation on the material behaviour in roll forming, predicting the shape defects in the roll forming process if a residual stress profile exists in the material [18]. For experimental methods, the nanoindentation technique is widely used to examine the residual stresses distribution in the deformation machining process [19; 20]. [PERSON] et al. developed a slope cutting method for determining the residual stresses in roll formed products, which utilizes a modification of the current sectioning methods to obtain the membrane and bending residual stress components, and they employed a tapered removal approach to determine the layering stresses component by measuring the curvature variations of the remaining sample via its thickness direction [21]. The analytical methods predict the output variables by resolving the equations of the mathematical model that is almost instantaneous, however a complicated manufacturing process has to be simplified to a certain degree for mathematical modelling, which reduces the accuracy of the results. With the help of modern analysis software and hardware, finite element model may produce more accurate results but much slower in time. The experimental methods reveal the actual situation of the process, but lack of efficiency and have very high cost in capital and manpower. To take advantage of the benefits of each method, combinations of two or three of the methods may provide more information about the manufacture process. In combinations of analytical methods and numerical methods, [PERSON] et al. proposed an integrated analytical model and a closed-form solution for these residual stresses arising from the coiling and uncoiling process experienced by steel sheets before they become cold formed into sections and demonstrated the accuracy of the solution by comparing its predictions with those from a finite element simulation [22; 23]. [PERSON] et al. presented a semi-analytical model based on a discrete bending theory and classical beam theory for the process optimization of roll levelling, which is able to calculate the critical levelling parameters, i.e., force, plastification rate, and residual stresses, in a few seconds [24; 25]. [PERSON] et al. studied a multi-scale method for modelling thin sheet buckling under residual stresses in the context of strip rolling and proposed a two-scale model based on the generalized continuum approach for developing the unknowns in the Fourier series and solving von Karman equations of thin strip buckling in an energetic formulation [26]. [PERSON] et al. dealt with analytical and numerical considerations of buckling phenomena in thin plates under in-plane loads, which typically appear during the rolling and levelling of sheet metal, and investigated the buckling due to self-equilibrating residual stresses caused by the rolling process [2]. For combinations of analytical methods and experimental methods, [PERSON] et al. expressed springback and residual stresses as a function of geometric parameters and material properties of sheet metals, determined the residual stresses using a layer-removing method, and carried out an analytical measurement of the residual stress by simulating the layer-removing process [27]. [PERSON] et al. developed a mathematical model to calculate the residual stress in stretched aluminium alloy thick plates based on the balance of stresses, two-dimensional plasticity, and a conception of free size, verifying the model against experiments performed on 7075 aluminium plates [28]. [PERSON] et al.