## 3 Case Studies of Pure Bending and Springback
### The Modified Analytical Method for Springback Prediction of SUS304 under V-Bending  
In this section, a modified analytical method is developed for springback prediction of V-bending accounting for friction effects, noting that the friction has to be considered in the simulations, but it was always ignored in analytic modelling. Figure 14 shows the variation of the springback obtained from numerical, analytical and modified analytical models versus the friction coefficient for 3 bending radii of 3, 5 and 8 mm. As it can be seen, for R = 5 mm, the numerical springback values increase by increasing the friction coefficient from 0 to around 0.1. For greater values of the friction coefficient, the springback is approximately constant. It is interesting that as the friction coefficient increases, the numerical values get closer to the analytical values of springback. For example, for the bending radius of 5 mm and the friction coefficients of higher than 0.1, the numerical and analytical springback results are almost the same. This indicates that the springback behaviours in V-bending are sensitive to friction between the sheet and the die for lower friction coefficients. This indicates that the underlying parameter, which causes the difference between the numerical and analytical springback results, is the friction. Therefore, in this section, the analytical model is modified to take the frictional effects into account. Based on simulation results, it was observed that in the frictionless case, during the bending process, the sheet slipped towards the die and reached the bottom of the die before the bending  
Figure 14: Springback relationship with the friction coefficient for different bending in the (**a**) rolling direction and (**b**) transverse direction. Figure 13: Comparison of experimental and numerical springback in V-bending. process ended. This caused the sheet to bend extra at a lower degree (almost 45\({}^{\circ}\)) in the middle of the sheet and, consequently, forced it to unbend to the die radius at the end of the bending process. This behaviour causes non-uniform bending in the sheet which consequently reduces the springback. By increasing the friction coefficient, the sheet slip reduces and almost stops for higher friction. Therefore, the radius reduction due to the slip was measured from the simulation results for different friction coefficients. As it can be seen in Figure 15, the radius reduction increases with increasing the coefficient friction and almost reaches zero for higher friction coefficients. This is in line with the variation of springback angle with the friction coefficient which indicates the relationship of radius reduction and springback. The variation of the radius reduction is different for TD and RD bending which shows that the radius reduction is related to material properties. In addition, the radius reduction is related to the bending radius, as it is different for different bending radii. To include the effects of friction in analytical solution, the springback caused by the radius reduction, which is called slip springback in this study, was deducted from the analytical springback results. Therefore, to find the analytical values of springback in the V-bending, the following relationship was defined:  
\[\Delta\theta_{VB}=\Delta\theta_{PB}-\Delta\theta_{SS} \tag{34}\]  
where \(\Delta\theta_{VB}\) is the V-bending springback, \(\Delta\theta_{PB}\) is the pure bending springback and \(\Delta\theta_{SS}\) is the slip springback, respectively. Based on Equation (34), springback values of V-bending in RD and TD for the bending radius of 5 mm were calculated. According to Figure 15, for the bending radius of 5 mm and for friction coefficients higher than 0.1, the radius reduction is zero. Therefore, in this case, the slip springback was zero, and the pure bending and V-bending springbacks were the same. Figure 16 shows the experimental and calculated springback values of V-bending in RD and TD for 16 types of SUS304 with different strengths.