*4.2. Investigation of the Plastic Zone*

To analyze the influence of the partial heating strategy on springback and warping the location of the principal forming zone is important (Figure 11). The actual bending moment in the profile is evaluated using the bending force of the simulation data. The threshold bending moment is received by solving the bending moment equation (Equation (12)) for the minimum flow stress necessary to receive the set loaded radius. Temperature is assumed constant in the heated area. Plasticity starts at the first position the profile bending moment surpasses the threshold bending moment. For the room temperature case (Figure 11a) and 300 ◦C partial heating temperature (Figure 11b), the plastic deformation limit is surpassed at the position of the counter roll *x* = 0. The flow stress reduction for 300 ◦C is not sufficient for plasticity to start in the heated area. For the 600 ◦C case (Figure 11c), the bending moment reduction is sufficient for plasticity to start in the heated area at l<sup>p</sup> = 77 mm. Consequently, the bending moment is only decreased if the heating temperature is higher than 300 ◦C. The decreased bending moment results in lowered springback of the profile (Figure 9).

As plasticity for the 600 ◦C case does not start at the position of the counter roll, the moment arm also decreases which influences the set bending radius. The loaded radius is not adjusted for this effect in the analysis as it was not possible to determine the true radius before the analysis. The set radius *r<sup>i</sup>* for the 600 ◦C case would then be 447 mm, which would cause a deviation in the springback ratio of 11%. Consequently, the change of springback ratio between the 600 ◦C case and the room temperature case would be 44% (Figure 9b).

**Figure 11.** Bending moments and plastic threshold bending moments for room temperature (**a**), 300 ◦C partial heating temperature (**b**) and 600 ◦C partial heating temperature (**c**) case resulting from mechanical equilibrium and flow stress for the loaded radius.

#### *4.3. Analysis of the Bending Load and Profile Warping in the Kinematic Push Bending Phase*

 σ The longitudinal stresses σ<sup>x</sup> in the profile cross-section are displayed to examine the influence of the stress-free fiber position on the profile warping and to investigate the accuracy of the analytical model. (Figure 12). While the stress for the room temperature case and the 300 ◦C cases behave elastic-plastic, the stress for the 600 ◦C case is fully plastic. Compared to the room temperature case, the maximum stress in the tensile area is reduced by 4% for the 300 ◦C partial heating temperature and by 52% for 600 ◦C partial heating. The position of the stress-free fiber *y*<sup>m</sup> related to the profile width *b* is 0.013 for the room temperature case, 0.057 for 300 ◦C heating temperature, and 0.14 for 600 ◦C heating temperature. The warping reduction can be traced back to the change in stress free fiber position. The shift of stress-free fiber for 300 ◦C explains why warping is reduced (Figure 10b), but the bending moment is the same as for the room temperature case (Figure 11a,b). Through partial heating, stresses in the tensile area become lower. As force equilibrium between tensile and compressive area still needs to be fulfilled the stress-free fiber shifts in direction of the room temperature area. Through the flow stress reduction shear stresses also get reduced in the heated area. Consequently, with increasing stress-free fiber position, the torsion moment in the profile is reduced. As the increase in stress free fiber position is higher for 600 ◦C than for 300 ◦C, the warping reduction is higher. The mean deviation between experimental and analytical data is 7%. The model can therefore predict the stress curves in the profile cross-section and the shift of the stress-free fiber.

**Figure 12.** Comparison of numerical and analytical results for longitudinal stress over the profile y-axis at the onset of plastic deformation for the loaded radius of 600 mm and profile feed of 8 mm/s for different partial heating temperatures.

To investigate the accuracy at which the analytical model can predict the bending moment and the profile warping as well as to investigate the warping in the loaded state, bending moment and related warping angle are displayed (Figure 13). Both are evaluated at the start of the plastic zone (*x* = *lp*). For the analysis of the related warping angle, the point of zero warping for the 600 ◦C the curve is shifted to the origin of the diagram to allow an easier comparison with the room temperature and 300 ◦C cases. The bending moment (Figure 13a) decreases linearly until the position of the bending roll (*x* = 480 mm) is reached. The bending moment is the highest for room temperature, though the mean difference between room temperature case and 300 ◦C case is 1%. The bending moments for 300 ◦C and room temperature cases are the same because they both start plastic deformation at the same threshold bending moment (Figure 11a,b). Compared to the highest bending moment, the highest bending moment for the 600 ◦C case decreases by 40%. In all cases, the analytical moment overestimates the numerical bending moment with a mean deviation of 7%. The reason for the deviation could be the neglected influence of the normal stresses (assumption 5) or a cumulative effect of the error margins resulting from both numerical and analytical analysis compared to the experimental data. The related warping angles (Figure 13b) are only evaluated at the positions between counter roll and bending roll as the analytical model is only applicable for the loaded state. Thus, the data displayed here are not directly comparable to the unloaded warping angle (Figure 10b). The related warping angles increase approximately linearly as they result from the equation for torsion moment (Equation (10)). The torsion moment behaves linearly over the *x*-axis. The maximum related warping angle of the 300 ◦C case is decreased by 13% and the maximum related warping angle for the 600 ◦C case is decreased by 53% compared to the room temperature case. This is analogical to the unloaded state. The analytical model can predict the numerical data with a mean deviation of 10%. It is thus possible for the analytical model to predict the warping and bending moment. The analytical model can now be used for process development and to generalize the results of this study for other profile geometries and materials.

**Figure 13.** Analytical and numerical bending moment (**a**) and related waring angle (**b**) for the continuous push-bending phase at room temperature, 300 ◦C and 600 ◦C for a loaded radius of 600 mm and profile feed of 8 mm/s as a function of profile arc length.

## **5. Conclusions**

To reduce warping and springback in the bending of profiles with asymmetric geometry in the force application axis, partial heating of the cross-section can be used. It has been shown that partial heating of the cross section leads to a springback reduction of at least 44% and a warping reduction of 76% compared to the room temperature case. The warping and springback reduction can be attributed to a shift in the stress-free fiber position. Partial

heating reduces the flow stress in the heated area. Through the flow stress reduction, a shift in stress-free fiber to the compressive zone is noticeable, reducing the stresses in the cross section. To achieve reduced springback and warping, a threshold temperature of 300 ◦C must be reached.

Additionally, an analytical model has been developed which is able to predict warping with 90% and bending moment with 93% accuracy in a kinematic push-bending process. This model can be used for process design and control.

**Author Contributions:** Conceptualization, R.M. and A.E.T.; methodology, E.H.; formal analysis, E.H.; investigation, E.H.; resources, A.E.T.; writing-original draft preparation, E.H.; writing-review and editing, R.M. and A.E.T.; visualization E.H.; supervision, R.M. and A.E.T.; project administration, E.H. and R.M.; funding acquisition, A.E.T. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors would like to thank the German Research Foundation DFG for the support of the depicted research through project no. 408302329. The financial support is greatly acknowledged.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing is not applicable to this article.

**Acknowledgments:** The authors thank the German Research Foundation (DFG) for the financial support of project 408302329 "Kinematic profile bending with locally heated cross-section" (German: Kinematisches Profilbiegen mit partieller Erwärmung des Querschnitts). The authors would also like to thank Till Clausmeyer for his helpful comments on the manuscript. His help is greatly appreciated.

**Conflicts of Interest:** The authors declare no conflict of Interests.
