## Abstract

U-bending tests are the most common method to predict springback and are influenced by the process and geometrical variables in addition to material behaviour. It needs a numerical study at a high level with many variables to reduce try-out time and loop. In this study, the U-bending test of DC01 steel has been researched numerically and experimentally to govern the influential parameters. The numerical analysis was conducted using AutoForm-Sigma code. The die radius has an excessive influence on the change of flange angle than the punch radius, but the punch radius has the greatest influence on the variation of the sidewall angle. The coefficient of friction played a great impact on both flange and sidewall angle deviation and its influence grows stronger as the blank holding force increases.

## 1 Introduction

Sheet metal bending is one of the most popular manufacturing techniques in which a homogenous material deforms around an axis that is orthogonal to the specimen's length direction and situated in the neutral plane [1]. Even though metal bending is a straightforward procedure, springback frequently interferes with it and is challenging to account for. The change in geometry of a stamped item must be compensated for, which raises the cost of manufacture [1]. After the tool is removed from the part, the component's dimensions change, and the stresses and strains in the material that underwent deformation change also [2]. The stamped portion is typically overbent to counteract this effect. One of the most crucial elements affecting contacting surface characteristics of the tool and sheet is the Blank Holding Force (BHF) [3]. Shape variation after sheet forming is mostly influenced by contact surface pressure and friction, and tool geometries [4–7].

By regulating a few process variables, Wang et al. [7] focused on analyzing springback that occurs during the stretch bending of sheet metal. It was found that a smaller die corner radius led to a smaller amount of sidewall springback due to the smaller required bending moment. Jiang et al. [1] studied the friction coefficient and blank holder force that affect the springback and have a reverse relationship. Angsuseranee et al. [8] explored springback and sidewall curl prediction in the U-bending process through the three-level tool geometry parameters and BHF at the constant coefficient of friction. The study confirmed that the springback decreases as increasing the BHF at a smaller punch radius. Tong et al. [9] numerically proposed a simplified method for obtaining material characteristics related to springback in addition to geometrical and process parameters. The investigation reached that the springback angle on the side wall is almost independent of radius of die. A detailed material model that accurately captures the Bauschinger effect is also necessary for accurate springback predictions. The springback was also impacted by the sheet thickness [10–12]. It is crucial to conduct a thorough optimization analysis of how changes to all examined parameters affect the springback. Optimization of design variables is very critical and it needs a systematic approach [13].

In the ongoing research, the impact of frictional coefficient, BHF, and punch and die profile radius on springback prediction of the U-bending test for cold-rolled DC01 steel was investigated both numerically and experimentally. A commercial code AutoForm-Sigma was used for numerical investigation. AutoForm-Sigma enables a systematic improvement of the forming process. This is accomplished by varying the process and design parameters within a range that enables the development of safe processes [14]. High-level with multiple variables of Design Of Experiment (DOE) matrix has been simulated using Systematic Process Improvement (SPI). SPI increases transparency in the forming process by demonstrating, which design parameters influence the forming process and to what extent and it also helps to reduce the invested time and tryout loops, which is common in a conventional trial-and-error iterative approach to get the most efficient result. It is particularly true to estimate the coefficient of friction in a U-bending test because of difficult to calculate or measure in the physical experiment. As a result, it is possible to carry out experimental research at a low level of the design of experiment matrix to confirm some results of numerical simulation.

## 2 Numerical simulation of U-bending test

U-bending of 250 × 20 × 0.45 mm rectangular cold rolled DC01 steel sheet was performed numerically using AutoForm commercial code. A uniaxial tensile test was used to determine the mechanical characteristic of the material as it is shown in Table 1.

Mechanical and formability behavior of DC01

Elastic Modulus, | Yield Stress, | Ultimate Tensile Strength (UTS), (MPa) | Total Elongation, |

206 | 192 | 322 | 24.5 |

*χ*is the saturation constant;

*γ*is the Young's reduction factor formulated as Eq. (2),

Kinematic hardening behavior of DC01

γ | χ | K |

0.24 | 40 | 0.003 |

Multiple U-bending tests were numerically carried out automatically using SPI in a single simulation. Table 3 lists the design factors and their levels that were taken into consideration for the numerical analysis. The maximum BHF was decided because of the maximum compression force generated by the spring in the real physical experiment apparatus. Figure 1 shows the initial and the final geometrical setup before the removal of the punch.

Design variables and their level for the simulation

Parameters | Level |

Radius of die profile, Rd, (mm) | 3–10 |

Radius of punch profile, Rp, (mm) | 3–10 |

BHF, (kN) | 3–9 |

Coefficient of friction, μ | 0.01–0.15 |

## 3 Experimental procedures

Some samples of the U-bending test experiment were conducted in dry and lubrication conditions at a constant 5 mm tool radius profiles with the intention of validating the influence of friction and BHF on springback parameters, which was achieved in the numerical investigation. Figure 2 shows the experimental setup of U-bending die equipment, which is installed on a 20 kN hydraulic press machine. Three-level spring-loaded BHF (at 3, 5, and 7 kN) was applied. The width of the punch was 40 mm and its half position matched the half position of the width of the sheet. The die and punch clearance on both sides was equal to double sheet thickness and the punch velocity was 5 mm s^{−1}.

## 4 Results and discussion

*μ*= 0.05 and

*BHF*= 3 kN are shown in Fig. 3a and all 3D profiles in the physical experiments are provided in Fig. 3b. The amount of springback of each experiment and simulation was measured using NUMISHEET ’93 benchmark standard [18] as it is shown in Fig. 4. The angle

*θ*1′ and

*θ*2′ were calculated using a simple relationship in Eq. (4),

Springback profiles, a) simulation, b) experimental

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Springback profiles, a) simulation, b) experimental

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Springback profiles, a) simulation, b) experimental

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Springback parameters in NUMISHEET ’93 benchmark [18]

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Springback parameters in NUMISHEET ’93 benchmark [18]

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Springback parameters in NUMISHEET ’93 benchmark [18]

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The 2D picture of each sample in numerical and experimental results is exported to AutoCAD commercial drafting software to measure the springback parameters. Figure 5 shows the 2D profiles of all physical experiment samples after removing the tool. Numerical simulation at *Rd* = 3 and 4 mm and at *Rp* = 10 mm revealed distortion and excessive thinning. But in all other causes of process parameters, there were free from any distortion and excessive thinning.

2D profiles of physical experiment samples after removal of the tool

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

2D profiles of physical experiment samples after removal of the tool

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

2D profiles of physical experiment samples after removal of the tool

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

### 4.1 Effect of die radius

In the numerical simulation, Fig. 6a and b show the effect of the *Rd* on the springback angle on the flange *θ*1′ and side wall *θ*2′ at a different constant value of punch radius respectively. BHF was 3 kN and the coefficient of friction was 0.15 as constant in all causes. It can be seen in Fig. 6a that in the numerical simulation the springback angle *θ*1′ decrease with the increase of *Rd* because when *Rd* increases by a specific amount, the elastic strain created is no longer sufficient to cause any springback. It is interesting that the behavior of the angle *θ*1′ becomes negative when *Rd* > 8 mm at a constant value of *Rp* > 6 mm. This might be brought on by a further reduction in elastic strain close to the die corner. On the other hand, *θ*2′ has no significant effect on the *Rd* in the numerical simulation as it is shown in Fig. 6b. But, the value of constant *Rp* has a great impact on both *θ*1′ and *θ*2′, *θ*1′ decreases as increasing constant *Rp* but *θ*2′ increases with increasing constant *Rp*.

Effect of the die radius in the numerical simulation, a) on the flange angle, b) on the sidewall angle

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Effect of the die radius in the numerical simulation, a) on the flange angle, b) on the sidewall angle

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Effect of the die radius in the numerical simulation, a) on the flange angle, b) on the sidewall angle

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

### 4.2 Effect of punch radius

Figures 7a and b show the effect of the *Rp* on the springback angle *θ*1′ and *θ*2′ respectively, in the numerical simulation. BHF was 3 kN and the coefficient of friction was 0.15 as constant in all causes. *θ*1′ is not affected by *Rp* but is highly affected by the constant value of *Rd* as it is shown in Fig. 7a. Angle *θ*1′ decreases as increasing of constant value of *Rd*. It is clearly shown in Fig. 7b that *θ*2′ is directly proportional to *Rp*. It increases as increasing *Rp* due to a reduction in the required contact pressure. The constant value of *Rd* has no discernible effect for *θ*2′.

Effect of the punch radius in the numerical simulation, a) on the flange angle, b) on the sidewall angle

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Effect of the punch radius in the numerical simulation, a) on the flange angle, b) on the sidewall angle

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Effect of the punch radius in the numerical simulation, a) on the flange angle, b) on the sidewall angle

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

*θ*1′ and Δ

*θ*2′ at different level constant variables of

*Rp*and

*Rd*is shown in Fig. 8 using Eq. (5),

Springback angle change, a) at constant *Rp* but variable *Rd*, b) at constant *Rd* but variable *Rp*

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Springback angle change, a) at constant *Rp* but variable *Rd*, b) at constant *Rd* but variable *Rp*

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Springback angle change, a) at constant *Rp* but variable *Rd*, b) at constant *Rd* but variable *Rp*

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In Fig. 7a, Δ*θ*1′ and Δ*θ*2′ was calculated from the maximum and minimum value found from variable *Rd* at a different constant level of *Rp*. But in Fig. 8b, Δ*θ*1′ and Δ*θ*2′ were calculated from variable *Rp* at a different constant level of *Rd*. The angle Δ*θ*1′ is higher at every constant *Rp* but smaller at constant *Rd*. However, there was an overall decreasing tendency with the increase of constant *Rp* and fluctuating in the cause of constant *Rd* but the opposite relationship was found for Δ*θ*2′. After all, *Rd* is the most influential parameter for *θ*1′ and *Rp* is the most influential for *θ*2′. To generalize, the result clearly shows the smaller value of *θ*1′ and *θ*2′ was found from *Rd* = 10 mm and *Rp* = 3 mm.

### 4.3 Effect of BHF and coefficient of friction

Controlling the BHF and coefficient of friction is one of the traditional methods to compensate springback. In order to study the relationship of BHF on springback, different BHF levels were selected from 3 to 9 kN. The advantage of finite element simulations is that easy to study the influences of coefficients of friction on the deformation process. In this study, the friction coefficient between tools and workpieces in the numerical analysis and it was varied from 0.01 to 0.15. The *Rd* and *Rp* were set at 5 mm as a constant value. Figure 9 shows the springback parameters as a function of BHF at different levels of constant coefficient of friction. The angle *θ*1′ and *θ*2′ have inverse relationships with BHF in all simulations. Springback decreases as increasing BHF because of a more uniform distribution of strains through the sheet thickness. It is clearly seen from Fig. 9 that springback parameters were almost closer amount in dry condition forming at *μ* = 0.15 in the numerical simulation, on the other hand, the experimental test with lubrication condition forming was closer at between *μ* = 0.05–0.10.

Effect of BHF and coefficient of friction: a) on the flange angle; b) on the sidewall angle

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Effect of BHF and coefficient of friction: a) on the flange angle; b) on the sidewall angle

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Effect of BHF and coefficient of friction: a) on the flange angle; b) on the sidewall angle

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In Fig. 10, Δ*θ*1′ and Δ*θ*2′ was also calculated from the maximum and minimum value found from variable BHF at a different constant level of coefficient of friction. Δ*θ*2′ is higher at every constant coefficient of friction and it increases as the increasing coefficient of friction but Δ*θ*1′ has the opposite relationship. Therefore, the coefficient of friction is most influential for *θ*2′ than *θ*1′ and the influence becomes higher as increasing BHF.

Springback angle change at different level of constant coefficient of friction but variable BHF

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Springback angle change at different level of constant coefficient of friction but variable BHF

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

Springback angle change at different level of constant coefficient of friction but variable BHF

Citation: Pollack Periodica 18, 2; 10.1556/606.2023.00799

## 5 Conclusion

The effects of process and geometrical parameters on springback phenomenon in the U-bending process have been explored numerically and experimentally. The numerical investigation with experimental verification is important and capable of predicting springback very accurately. A numerical investigation of the high-level with multiple variables design of the experiment matrix has been conducted using SPI and it is important to reduce the real physical design of the experimental study. Based on this study, the following remarks are drawn:

The results produced by the analysis of the springback phenomenon using the finite element approach can be regarded as being sufficiently precise and valid;

Die radius was the most influential geometrical parameter at the flange springback but punch radius is the most influential at the sidewall springback;

The friction coefficient is one of the process variables that have a considerable impact on springback prediction, especially on the sidewall deviations. The influence becomes higher as increasing BHF.

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