# Experimental and Numerical Investigation of the Influence of Process Parameters in Incremental Sheet Metal Forming on Residual Stresses

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Method

#### 2.1. Experimental Set-Up of SPIF

^{®}block. The sheet blanks are clamped between the blank holder and the backing plate. The hemispherical tool follows a sequence of square paths to manufacture the pyramidal frustum. The dimension of the manufactured pyramids is 150 × 150 × 35 mm (Figure 1). Pyramidal parts are manufactured by systematically varying process parameters. The varied parameters are tool diameter, tool step-down, and the wall angle. The design of experiments (DoE) is presented in Table 1. For three different values of the three process parameters, a total of 7 experiments are required to analyze the effect of each value of the process parameter.

^{®}by GOM GmbH, was used to digitize and analyze the respective principle strains.

#### 2.2. Hole-Drilling-Method

^{®}was used. A drill head of 0.8 mm rotating at 200,000 rpm was used to determine the residual stresses at the specified location (Figure 4a), containing an attached strain-rosette (Figure 4b). The high rotations of the drill head suppressed the creation of the supplementary stresses. The calculation steps for the residual stress values were automatically adjusted along the depth of the drilled hole. The stresses are calculated from the measured strain by using the relation [23]

#### 2.3. The Finite Element (FE) Model of the SPIF Process

#### 2.4. Solution Procedure

## 3. Results and Discussions

#### 3.1. Validation of the Numerical Simulation

^{®}. The final formed geometry was exported as a triangular mesh and can be further converted into common formats of CAD geometry i.e., iges or step data. As a basis of comparison, the major principle strains, thickness reduction, and the formed geometry of the part were considered. The results of the major principle strain and the thickness reduction from the experiment and simulation were compared along a section on the part, whereas the side walls of the pyramids were used to compare the experimental and numerical geometry. This is due to the fact that the geometric deviations along the corners and edges of the pyramids are negligible and are most pronounced in the side wall region.

#### 3.2. Influence of Process Parameters on Residual Stresses

#### 3.2.1. Tool Step-Down

- By increasing the tool step-down, the magnitude of the compressive residual stresses on the non-contact surface increased more compared to tensile stresses on the tool contact side as indicated by the ‘before unclamping’ state.
- For larger tool step-down values, the magnitude of the residual stresses changed more upon unclamping in the transverse direction of the tool motion.
- The change in the magnitude of the residual stresses was greater upon trimming compared to unclamping. There was also a significant change in the magnitude of the residual stresses for outer and inner surfaces. The final state of the residual stress after trimming for the inner surface was compression, similar to the initial unclamped state, i.e., before unclamping and vice-versa for the outer surface.

- The bending moment change was highest with the largest tool step-down and vice-versa. Therefore, the geometric deviations will be highest with the largest tool step-down.

#### 3.2.2. Tool Diameter

- In the initial clamped state, the magnitude of the residual stresses, by increasing the tool diameter, increased both in tension and compression.
- The ‘before unclamping’ state indicates that the residual stress, by increasing the tool diameter, increased more pronouncedly in the transverse direction of the tool motion.
- A notable change in the transverse direction in the magnitude and state of the residual stresses occurred upon unclamping, represented by ‘after unclamping’ in Figure 11, and this change was largest with the largest tool diameter.
- Upon trimming, the respective change in the magnitude of the residual stresses was slightly larger with the smallest tool diameter.

- The change in the bending moment was highest with the largest tool diameter upon unclamping in the transverse direction.
- The change in the bending moment was highest with the smallest tool diameter upon trimming in the transverse direction.

#### 3.2.3. Wall Angle

- In contrast to tool diameter and tool step-down, by increasing the wall angle, the through thickness distribution of the residual stresses changed to tension.
- The magnitude of the residual stresses was higher in the transverse direction.
- A significant change in the magnitude of the residual stress occurred upon trimming, and this was highest with the largest wall angle.

- The change in the bending moment was highest with the largest wall angle, and this change is significant as compared to the other process parameters, i.e., tool diameter and wall angle.

#### 3.3. Residual Stresses vs. Geometrical Accuracy

## 4. Conclusions

- (1)
- The magnitude of the residual stresses in the clamped state increased when the tool diameter, tool step-down, and the wall angle increased. Upon unclamping, the respective change in the magnitude of the residual stresses and the bending moments was highest with the greatest tool-diameter and tool step-down. Upon trimming, the change in the magnitude of the residual stresses and bending moments was greatest with the highest tool step-down value. However, the change in the magnitude of the residual stresses and bending moment was highest with the smallest tool diameter in the transverse direction of the tool motion. Moreover, for greater wall angles, the respective change in the magnitude of residual stresses and bending moment was highest and occurred significantly in the transverse direction of the tool motion.
- (2)
- The widely known fact from the literature that a smaller tool step-down and a smaller tool diameter have a positive effect on geometrical accuracy is explained in terms of residual stress. This is because, during unclamping, the greatest changes in the magnitude of the residual stresses occurred with the greatest tool step-down and tool diameter. During unclamping, these changes in the magnitude of the residual stresses were directly proportional to the elastic portion of the deformation. The elastic portion of the deformation, which was recovered upon unclamping, increased when the tool step-down and tool diameter increased. Hence, geometrical accuracy in the unclamped state increases when tool step-down and tool diameter decrease.
- (3)
- However, upon trimming, the largest changes in the magnitude of the residual stresses occurred with the smallest tool diameter was significant in the transverse direction of the tool motion. Hence, geometric deviations were largest with the smallest tool diameter in the trimmed state.
- (4)
- In contrast to tool diameter and tool step-down, the state of the residual stresses in the trimmed state with the greater wall angles was such that it caused the strips to curl toward the tool contact face, opposite to other process parameters.
- (5)
- The most significant parameter for the effect on the residual stresses and geometrical accuracy is the wall angle. The geometrical accuracy decreased significantly in the transverse direction when the wall angle was increased. However, this parameter is usually fixed, and its value depends on the geometry of the part to be formed. The other two parameters, i.e., tool diameter and tool step-down, also have a considerable effect on residual stress and geometrical accuracy.
- (6)
- The effect of changing process parameters on the residual stresses was more evident in the transverse direction of the tool motion.

## Author Contributions

## Funding

## Acknowledgement

## Conflicts of Interest

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**Figure 2.**(

**a**) Point pattern after incremental forming. (

**b**) Major principle strain on the outer surface obtained by ARGUS

^{®}. (

**c**) Geometry after forming obtained from ARGUS

^{®}.

**Figure 5.**(

**a**) Geometry of the tension-compression specimen. (

**b**) Comparison of the stress–strain curves for optimal parameters.

**Figure 7.**Comparison of the (

**a**) major principle strains, (

**b**) thickness reduction, and (

**c**) formed geometries from the experiment and simulation for Experiment 1.

**Figure 8.**Comparison of the residual stresses along the tool motion direction, i.e., ${\mathsf{\sigma}}_{\mathrm{x}}$ at selected location for Experiments 1 and 6.

**Figure 9.**Distribution of the residual stresses in the center of the strip along and in the transverse direction of the tool motion for different tool step-down values (

**a**) before unclamping, (

**b**) after unclamping, and (

**c**) after trimming.

**Figure 10.**Change in the bending moment for different tool step-down upon unclamping (Stage 1) and upon trimming (Stage 2) along a section.

**Figure 11.**Distribution of the residual stresses in the center of the strip along and in the transverse direction of the tool motion for different tool diameters: (

**a**) before unclamping, (

**b**) after unclamping, and (

**c**) after trimming.

**Figure 12.**Change in the bending moment for different tool diameters upon unclamping (Stage 1) and upon trimming (Stage 2) along a section.

**Figure 13.**Distribution of the residual stresses in the center of the strip along and in the transverse direction of the tool motion for different wall angles: (

**a**) before unclamping, (

**b**) after unclamping, and (

**c**) after trimming.

**Figure 14.**Change in the bending moment for different wall angles upon unclamping (Stage 1) and upon trimming (Stage 2) along a section.

**Figure 15.**Comparison of the geometric profile with the target geometry for (

**a**) different tool diameters and (

**b**) different tool step-down values.

**Figure 16.**Curvature of the strips cut along and in the transverse direction of tool motion for (

**a**) tool step-down Z, (

**b**) tool diameter D, and (

**c**) wall angle α.

**Figure 17.**Curvature of the strips toward the lower surface. The contact side undergoes compression, and the non-contact side undergoes tension.

**Figure 18.**(

**a**) Curvature of a strip for large wall angle in the transverse direction. (

**b**) Curling of a strip cut along the tool motion toward the transverse direction for large wall angle. (

**c**) Curvature of a strip for small wall angle in the transverse direction.

**Figure 19.**Curvature of the strip only due to the residual stresses upon trimming from numerical simulations for (

**a**) three tool step-down values along the tool motion, (

**b**) three tool diameters in the transverse direction, and (

**c**) three wall angles along the transverse direction.

Experiment No. | Tool Diameter (D/mm) | Wall Angle α | Tool Step-Down (Z/mm) |
---|---|---|---|

1 | 5 | 30° | 0.25 |

2 | 5 | 30° | 0.5 |

3 | 5 | 30° | 0.75 |

4 | 5 | 45° | 0.25 |

5 | 5 | 60° | 0.25 |

6 | 10 | 30° | 0.25 |

7 | 20 | 30° | 0.25 |

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**MDPI and ACS Style**

Maqbool, F.; Bambach, M.
Experimental and Numerical Investigation of the Influence of Process Parameters in Incremental Sheet Metal Forming on Residual Stresses. *J. Manuf. Mater. Process.* **2019**, *3*, 31.
https://doi.org/10.3390/jmmp3020031

**AMA Style**

Maqbool F, Bambach M.
Experimental and Numerical Investigation of the Influence of Process Parameters in Incremental Sheet Metal Forming on Residual Stresses. *Journal of Manufacturing and Materials Processing*. 2019; 3(2):31.
https://doi.org/10.3390/jmmp3020031

**Chicago/Turabian Style**

Maqbool, Fawad, and Markus Bambach.
2019. "Experimental and Numerical Investigation of the Influence of Process Parameters in Incremental Sheet Metal Forming on Residual Stresses" *Journal of Manufacturing and Materials Processing* 3, no. 2: 31.
https://doi.org/10.3390/jmmp3020031