# Optimization of the heating parameters of a robotized hot incremental forming of high impact polystyrene

DUARTE ROCHA Valentin<fnref n="1" />,ª*, GIRAUD MOREAU Laurence<fnref n="1" />,ᵇ and CHEROUAT Abel<fnref n="2" />,ᶜ

**Keywords:** Robotized Incremental Forming, High Impact Polystyrene, FEA, Heating System, Optimization

**Abstract.** Single point incremental forming (SPIF) is a modern rapid manufacturing technology able to manufacturing complex sheet parts in small quantities. In comparison to conventional deep-drawing process, complex tools can be dispensed in order to reduce tool costs and the time required to achieve the first finished part. This new technology consists of locally and iteratively deforming plastically the sheet material by a punch that is generally hemispherical and of small dimensions, whose trajectory is programmed on a numerically controlled machine or a robot arm. The parts formed are mainly made from metallic materials (steel, aluminum, titanium or copper alloys). Very few studies focus on the incremental forming of thermoplastic parts. Because of the poor formability of thermoplastic parts and the large spring-back at room temperature, the use of heating systems of the work-piece during the forming process is required. In this work, a novel incremental sheet forming assisted by heat transfer was developed to improve the formability of thermoplastic sheet. By means of a series of experimental forming of High Impact Polystyrene pyramid frustum with constant wall angle at different heat temperatures, geometric accuracy was recorded and analyzed. This study aims to evaluate the behavior of polymer material in the hot incremental sheet forming with different pre-heating system using the finite element method.

**Introduction**

Incremental sheet forming is a promising process, well-suited for small batch production or prototyping, with many advantages compared to traditional forming processes such as low cost and higher flexibility. Most studies concern metallic parts or composite parts manufactured using classical three-axis CNC milling machines [1]. Nevertheless, these machines have a limited workspace and can be used to form only simple shape parts. Larger dimension parts and more complex shapes can be produced by industrial robots. Regarding the materials, the use of composite materials is now increasing in many areas (aeronautic, automotive and medical) due to their mechanical properties and environment concerns [2]. There is little to no literature on the incremental forming of high impact polystyrene, although this polymer is widely used in industry. However, this polymeric material having a poor formability at room temperature, a heating system must be considered. The aim of this research is to study experimentally and numerically the robotized hot incremental forming of polymer cone parts. With the help of design of experiment (DOE), the effects of heating system (tool or sheet or combination) were studied to optimize the process.

**Experimental process**

The High Impact Polystyrene part considered in this study is a truncated cone with a wall angle of $α = 30°$, a depth of $h = 35mm$ and a base radius of $R = 81 mm$. The High Impact Polystyrene sheet

<figure><img src="image_1434.png" /></figure>

<footnote n="1">¹UR-Lasmis, UTT, 12 rue Marie Curie CS 42060, 10004 Troyes cedex, France</footnote>

<footnote n="2">²UR-Gamma3, UTT, 12 rue Marie Curie CS 42060, 10004 Troyes cedex, France</footnote>

<footnote>ᵃvalentin.duarte_rocha@utt.fr, ᵇlaurence.moreau@utt.fr, ᶜabel.cherouat@utt.fr</footnote>
has a size of (200 x 200 mm²) and initial thickness of 0.8 mm. Three heating systems are considered:

1. Local heating approach where only the tool is heated by electrical resistor;

2. Global heating approach based on the use of hot air blower to heat the sheet;

3. Combination approach of the two heating methods.

The effects of the heating system (local, global, mixed) and the heating temperature (80, 100, 120°C) on the geometric accuracy and thickness were analyzed using the experimental robotized hot incremental forming set up (see Fig. 1). The forming tool is a copper cylinder with a semi-hemispheric end and a radius of 5 mm. It can be electrically heated to the performed temperature. For the global heating approach, the polymer material sheet is globally heated by hot air convection to a desired temperature with respect to the thermal properties of the polymer sheet.

The clamping system consists of two wooden planks with circular shaped holes of 190 mm diameter pinching evenly the material with enough clearance on each side to allow the deformation of the sheet part. It needs to be closed enough to the tool path to minimize the spring back effect. The aim was to study the influence of the heating systems on the accuracy of the formed part (especially to minimize the shape errors occurring during and after the incremental forming process).

<figure><img src="image_1431.png" /><figcaption>Fig. 1 – Experimental tool and clamping system for the hot robotized incremental forming</figcaption></figure>

The toolpath used during the incremental forming process is generally realized by computer-aided manufacturing (CAM) software. The tool trajectory has a significant effect on different aspects like dimensional accuracy, thickness distribution, processing time, and surface roughness. It is also necessary to develop specific toolpath generation to improve its efficiency. In this work, a procedure based on the mesh of the part has been developed to generate the incremental toolpath and to take into account the radius of the hemispherical tool (see Fig 2).
has a size of (200 x 200 mm²) and initial thickness of 0.8 mm. Three heating systems are considered:

1. Local heating approach where only the tool is heated by electrical resistor;

2. Global heating approach based on the use of hot air blower to heat the sheet;

3. Combination approach of the two heating methods.

The effects of the heating system (local, global, mixed) and the heating temperature (80, 100, 120°C) on the geometric accuracy and thickness were analyzed using the experimental robotized hot incremental forming set up (see Fig. 1). The forming tool is a copper cylinder with a semi-hemispheric end and a radius of 5 mm. It can be electrically heated to the performed temperature. For the global heating approach, the polymer material sheet is globally heated by hot air convection to a desired temperature with respect to the thermal properties of the polymer sheet.

The clamping system consists of two wooden planks with circular shaped holes of 190 mm diameter pinching evenly the material with enough clearance on each side to allow the deformation of the sheet part. It needs to be closed enough to the tool path to minimize the spring back effect. The aim was to study the influence of the heating systems on the accuracy of the formed part (especially to minimize the shape errors occurring during and after the incremental forming process).

<figure><img src="image_1435.png" /><figcaption>Fig. 1 – Experimental tool and clamping system for the hot robotized incremental forming</figcaption></figure>

The toolpath used during the incremental forming process is generally realized by computer-aided manufacturing (CAM) software. The tool trajectory has a significant effect on different aspects like dimensional accuracy, thickness distribution, processing time, and surface roughness. It is also necessary to develop specific toolpath generation to improve its efficiency. In this work, a procedure based on the mesh of the part has been developed to generate the incremental toolpath and to take into account the radius of the hemispherical tool (see Fig 2).
<figure><img src="image_1432.png" /><figcaption>Fig. 2 – Generated toolpath for the cone part manufacturing and the simulation</figcaption></figure>

In order to characterize the melting temperature $T_m$, glass transition temperature $T_g$, specific heat capacity and other related kinetic properties of High Impact Polystyrene, a Differential scanning calorimetry (DSC) is used. By analyzing the obtained curve (Fig. 3), there are two crystals (or structures molecular) inside the material, one of which it begins to melt at about $T_m$ = 125°C and the other at about 148°C. Therefore, the temperature of the used material should not exceed 125°C throughout the forming process. In order to minimize the friction between the thin sheet and the rigid heated tool, the forming speed was set at 10 mm/s and the depth increment of $\Delta z$ = 1 mm was used.

<figure><img src="image_1433.png" /><figcaption>Fig. 3 – DSC test on High Impact Polystyrene material.</figcaption></figure>

## Effect of heating system

The effect of the heating system (Local, Global and Mix) and the temperature value (80, 100, 120°C) on the final geometry was studied (see Table 1). In order to study the contribution of each factor and optimize the forming parameters, DOE method design was used to conduct the experiments. Fig. 4 presents the final forming part obtained with local heating at 120°C (test 3). Parts realized with the global heating system presented cracks and the forming process couldn't be performed completely. Fig. 5 presents the final obtained profile of all the experimental tests except those carried out with the global heating system (tests 4, 5 and 6). Experimental results
<figure><img src="image_1436.png" /><figcaption>Fig. 2 – Generated toolpath for the cone part manufacturing and the simulation</figcaption></figure>

In order to characterize the melting temperature $T_m$, glass transition temperature $T_g$, specific heat capacity and other related kinetic properties of High Impact Polystyrene, a Differential scanning calorimetry (DSC) is used. By analyzing the obtained curve (Fig. 3), there are two crystals (or structures molecular) inside the material, one of which it begins to melt at about $T_m$ = 125°C and the other at about 148°C. Therefore, the temperature of the used material should not exceed 125°C throughout the forming process. In order to minimize the friction between the thin sheet and the rigid heated tool, the forming speed was set at 10 mm/s and the depth increment of $\Delta z$ = 1 mm was used.

<figure><img src="image_1437.png" /><figcaption>Fig. 3 – DSC test on High Impact Polystyrene material.</figcaption></figure>

## Effect of heating system

The effect of the heating system (Local, Global and Mix) and the temperature value (80, 100, 120°C) on the final geometry was studied (see Table 1). In order to study the contribution of each factor and optimize the forming parameters, DOE method design was used to conduct the experiments. Fig. 4 presents the final forming part obtained with local heating at 120°C (test 3). Parts realized with the global heating system presented cracks and the forming process couldn't be performed completely. Fig. 5 presents the final obtained profile of all the experimental tests except those carried out with the global heating system (tests 4, 5 and 6). Experimental results
show that the parameter having the greatest effect on the final profile is the heating system followed by the local temperature value. The curves show that a reduction in global temperature allows to reduce the spring back and obtain a result closer to the desired profile angle. Test 3 is the best performing heat parameter with low spring-back effect and overall good deformation. However, the combination of the two heating systems did not bring interesting results (tests 7, 8 and 9).

<table><caption>Table 1 – Experimental plan of temperature.</caption><thead><tr><td>Experiment Number</td><td>Test</td><td>Input Temperature (°C)</td></tr></thead><tbody><tr><td>1</td><td>Local</td><td>T<sub>tool</sub> = 100</td></tr><tr><td>2</td><td>Local</td><td>T<sub>tool</sub> = 110</td></tr><tr><td>3</td><td>Local</td><td>T<sub>tool</sub> = 120</td></tr><tr><td>4</td><td>Global</td><td>T<sub>sheet</sub> = 100</td></tr><tr><td>5</td><td>Global</td><td>T<sub>sheet</sub> = 110</td></tr><tr><td>6</td><td>Global</td><td>T<sub>sheet</sub> = 120</td></tr><tr><td>7</td><td>Mix</td><td>T<sub>tool</sub> = 80 – T<sub>sheet</sub> = 60</td></tr><tr><td>8</td><td>Mix</td><td>T<sub>tool</sub> = 100 – T<sub>sheet</sub> = 60</td></tr><tr><td>9</td><td>Mix</td><td>T<sub>tool</sub> = 110 – T<sub>sheet</sub> = 60</td></tr></tbody></table>

<figure><img src="image_1434.png" /><figcaption>Fig. 4 – Final part for test 3 (local heating, 120°C)</figcaption></figure>

<figure><img src="image_1435.png" /><figcaption>Fig. 5 – Comparison between experimental and theoretical profile for the different cases</figcaption></figure>
show that the parameter having the greatest effect on the final profile is the heating system followed by the local temperature value. The curves show that a reduction in global temperature allows to reduce the spring back and obtain a result closer to the desired profile angle. Test 3 is the best performing heat parameter with low spring-back effect and overall good deformation. However, the combination of the two heating systems did not bring interesting results (tests 7, 8 and 9).

<table><caption>Table 1 – Experimental plan of temperature.</caption><thead><tr><td>Experiment Number</td><td>Test</td><td>Input Temperature (°C)</td></tr></thead><tbody><tr><td>1</td><td>Local</td><td>T<sub>tool</sub> = 100</td></tr><tr><td>2</td><td>Local</td><td>T<sub>tool</sub> = 110</td></tr><tr><td>3</td><td>Local</td><td>T<sub>tool</sub> = 120</td></tr><tr><td>4</td><td>Global</td><td>T<sub>sheet</sub> = 100</td></tr><tr><td>5</td><td>Global</td><td>T<sub>sheet</sub> = 110</td></tr><tr><td>6</td><td>Global</td><td>T<sub>sheet</sub> = 120</td></tr><tr><td>7</td><td>Mix</td><td>T<sub>tool</sub> = 80 – T<sub>sheet</sub> = 60</td></tr><tr><td>8</td><td>Mix</td><td>T<sub>tool</sub> = 100 – T<sub>sheet</sub> = 60</td></tr><tr><td>9</td><td>Mix</td><td>T<sub>tool</sub> = 110 – T<sub>sheet</sub> = 60</td></tr></tbody></table>

<figure><img src="image_1438.png" /><figcaption>Fig. 4 – Final part for test 3 (local heating, 120°C)</figcaption></figure>

<figure><img src="image_1439.png" /><figcaption>Fig. 5 – Comparison between experimental and theoretical profile for the different cases</figcaption></figure>
It can be observed that an error exists between the experimental and theoretical profile that neither the heating method nor the heating parameter can correct showing the necessity of optimizing the tool path.

Finite element simulation

The FEM simulation of single point incremental sheet forming of polymer sheet was performed with ABAQUS code using dynamic-temperature explicit solver. In this simulation multi-linear isotropic elasticity and bilinear isotropic hardening strain material model is used to model High Impact Polystyrene sheet blank. The blank is modeled by coupled displacement-temperature 8-node linear hexahedra finite element with Reduced Integration. Elastic and plastic material properties of the High Impact Polystyrene at each temperature were obtained by uniaxial tensile test given in Fig. 6. Other mechanical and thermal properties were obtained using literature review [3].

<figure><img src="image_1436.png" /><figcaption>Fig. 6 – Uniaxial tensile properties of high impact polystyrene at different temperatures</figcaption></figure>

The experimental tool path was used for the numerical simulation. Fig. 7 presents the final numerical profile obtained with a local heating of 100°C (with the same experimental condition as the test 1). This profile obtained numerically (Fig. 8) seems closed of the experimental and theoretical profiles. However, since the damage was not taken into account in the numerical model, the numerical simulation of the process with a global heating system was not able to predict the appearance of cracks and this result cannot be exploited at present. Further improvements of the numerical simulation are necessary to model thermal and mechanical effects as shearing due to the contact.

<figure><img src="image_1437.png" /><figcaption>Fig. 7 – Predicted profile compared with the desired shape</figcaption></figure>
It can be observed that an error exists between the experimental and theoretical profile that neither the heating method nor the heating parameter can correct showing the necessity of optimizing the tool path.

Finite element simulation

The FEM simulation of single point incremental sheet forming of polymer sheet was performed with ABAQUS code using dynamic-temperature explicit solver. In this simulation multi-linear isotropic elasticity and bilinear isotropic hardening strain material model is used to model High Impact Polystyrene sheet blank. The blank is modeled by coupled displacement-temperature 8-node linear hexahedra finite element with Reduced Integration. Elastic and plastic material properties of the High Impact Polystyrene at each temperature were obtained by uniaxial tensile test given in Fig. 6. Other mechanical and thermal properties were obtained using literature review [3].

<figure><img src="image_1440.png" /><figcaption>Fig. 6 – Uniaxial tensile properties of high impact polystyrene at different temperatures</figcaption></figure>

The experimental tool path was used for the numerical simulation. Fig. 7 presents the final numerical profile obtained with a local heating of 100°C (with the same experimental condition as the test 1). This profile obtained numerically (Fig. 8) seems closed of the experimental and theoretical profiles. However, since the damage was not taken into account in the numerical model, the numerical simulation of the process with a global heating system was not able to predict the appearance of cracks and this result cannot be exploited at present. Further improvements of the numerical simulation are necessary to model thermal and mechanical effects as shearing due to the contact.

<figure><img src="image_1441.png" /><figcaption>Fig. 7 – Predicted profile compared with the desired shape</figcaption></figure>
<figure><img src="image_1438.png" /><figcaption>Fig. 8 – Final predicted forming part obtained with local heating at 100°C</figcaption></figure>

**Conclusion**

The results of the experiment and numerical study of the warm robotized incremental forming of high impact polystyrene show that the heating system is a parameter having important impact on the final geometry. However, the desired final shape could not be achieved. The experimental results showed that an optimization of the tool path is necessary. Further research is also needed to improve the numerical model and obtain a more predictive tool.

**Reference**

[1] Liu Z., Cheng K., Peng K., Exploring the deformation potential of composite materials processed by incremental sheet forming: a review, International Journal of Advances Manufacturing Technology, 2021.

[2] Rosca, N.; Oleksik, M.; Rosca, L.; Avrigean, E.; Trzepiecinski,T.; Najm, S.M.; Oleksik, V. Minimizing the Main Strains and Thickness Reduction in the Single Point Incremental Forming Process of Polyamide and High-Density Polyethylene Sheets. Materials 2023, 16, 1644. https://doi.org/10.3390/ma16041644

[3] Atmani O., Abbès F., Li Y., Batkam S., Abbès B., Experimental investigation and constitutive modelling of the deformation behaviour of high impact polystyrene for plug-assisted thermoforming, (2020) Mechanics and Industry, 21 (6), art. no. 607. https://doi.org/10.1051/meca/2020084
<figure><img src="image_1442.png" /><figcaption>Fig. 8 – Final predicted forming part obtained with local heating at 100°C</figcaption></figure>

**Conclusion**

The results of the experiment and numerical study of the warm robotized incremental forming of high impact polystyrene show that the heating system is a parameter having important impact on the final geometry. However, the desired final shape could not be achieved. The experimental results showed that an optimization of the tool path is necessary. Further research is also needed to improve the numerical model and obtain a more predictive tool.

**Reference**

[1] Liu Z., Cheng K., Peng K., Exploring the deformation potential of composite materials processed by incremental sheet forming: a review, International Journal of Advances Manufacturing Technology, 2021.

[2] Rosca, N.; Oleksik, M.; Rosca, L.; Avrigean, E.; Trzepiecinski,T.; Najm, S.M.; Oleksik, V. Minimizing the Main Strains and Thickness Reduction in the Single Point Incremental Forming Process of Polyamide and High-Density Polyethylene Sheets. Materials 2023, 16, 1644. https://doi.org/10.3390/ma16041644

[3] Atmani O., Abbès F., Li Y., Batkam S., Abbès B., Experimental investigation and constitutive modelling of the deformation behaviour of high impact polystyrene for plug-assisted thermoforming, (2020) Mechanics and Industry, 21 (6), art. no. 607. https://doi.org/10.1051/meca/2020084
# Study on the effects of tool design and process parameters on the robustness of deep drawing

HEINZEL Christine<fnref n="1" />,ª*, THIERY Sebastian<fnref n="1" />,ᵇ and BEN KHALIFA Noomane<fnref n="1" />,<fnref n="2" />,ᶜ

**Keywords:** Deep Drawing, Robustness, Modular Tool, Material Draw-In

**Abstract.** In metal forming manufacturing processes, parameter fluctuations and an incomplete understanding of the process can lead to an undesirable deviation of the product properties from the required specifications and, therefore, affect the robustness of the process. In deep drawing, defects such as cracks and wrinkling can be linked to uncertainties within the process- and tool design parameters. To investigate the combined effects of these parameters on the quality of the finished product, simulation models are used to study the effects of parameter changes on the product quality. With the purpose of studying the effects of significant process and tool design parameters on the deep drawing process, a numerical parameter study is carried out based on a modular tool which allows for an investigation of process parameter variations within an adjustable parameter range of tool radii. While the material draw-in and the maximum punch force are used as quality indicators of the deep-drawing process, it could be shown that the elongation of the absolute length of the finished part can be used as an additional indicator for material thinning when observing the effects of the punch shoulder and die shoulder radii on the process robustness.

**Introduction**

The deep drawing process is the primary manufacturing process used in the automotive industry for the production of sheet metal components [1]. However, fluctuations of the process parameters as well as an incomplete understanding of the process can lead to deviations in the product properties from the required specifications [2]. Combined with the increasingly complex geometry in modern applications, the requirements on the robustness of the deep drawing process have become more challenging in recent years [3]. To define a safe process window and avoid wasting resources, it is necessary to understand the relationship between parameter fluctuations and defects such as cracks and wrinkles. While typical defects such as cracks and wrinkles are usually related to process parameters like blank holder force and the wear of the tool surfaces, the tool design parameters play a significant role within the development of defects and the thinning of the metal sheet [4, 5]. A thorough investigation of tool design and process parameters and their interactions can, therefore, help to achieve an increased robustness of the deep drawing process.

The effects of the process parameters and tool design parameters can be analyzed both experimentally and using finite element models. Experimental studies have the advantage of fully capturing the complexity of the forming process, whereas simulation models often rely on assumptions and simplifications, such as the material model or the modeling of the friction conditions. However, numerical investigations are significantly more time-efficient and allow for precise and flexible adjustment of the analyzed parameters as well as an easy extraction of parameters such as stresses and strains. [6, 7]

<figure><img src="image_1439.png" /></figure>

<footnote n="1">¹Institute for Production Technology and Systems, Leuphana University Lüneburg, Universitätsallee 1, 21335 Lüneburg, Germany</footnote>

<footnote n="2">²Institute of Materials and Process Design, Helmholtz-Zentrum Hereon, Max-Planck-Str. 1, 21502 Geesthacht, Germany</footnote>

<footnote>ªchristine.heinzel@leuphana.de, ᵇsebastian.thiery@leuphana.de,</footnote>

<footnote>ᶜnoomane.ben_khalifa@leuphana.de</footnote>
# Study on the effects of tool design and process parameters on the robustness of deep drawing

HEINZEL Christine<fnref n="1" />,ª*, THIERY Sebastian<fnref n="1" />,ᵇ and BEN KHALIFA Noomane<fnref n="1" />,<fnref n="2" />,ᶜ

**Keywords:** Deep Drawing, Robustness, Modular Tool, Material Draw-In

**Abstract.** In metal forming manufacturing processes, parameter fluctuations and an incomplete understanding of the process can lead to an undesirable deviation of the product properties from the required specifications and, therefore, affect the robustness of the process. In deep drawing, defects such as cracks and wrinkling can be linked to uncertainties within the process- and tool design parameters. To investigate the combined effects of these parameters on the quality of the finished product, simulation models are used to study the effects of parameter changes on the product quality. With the purpose of studying the effects of significant process and tool design parameters on the deep drawing process, a numerical parameter study is carried out based on a modular tool which allows for an investigation of process parameter variations within an adjustable parameter range of tool radii. While the material draw-in and the maximum punch force are used as quality indicators of the deep-drawing process, it could be shown that the elongation of the absolute length of the finished part can be used as an additional indicator for material thinning when observing the effects of the punch shoulder and die shoulder radii on the process robustness.

**Introduction**

The deep drawing process is the primary manufacturing process used in the automotive industry for the production of sheet metal components [1]. However, fluctuations of the process parameters as well as an incomplete understanding of the process can lead to deviations in the product properties from the required specifications [2]. Combined with the increasingly complex geometry in modern applications, the requirements on the robustness of the deep drawing process have become more challenging in recent years [3]. To define a safe process window and avoid wasting resources, it is necessary to understand the relationship between parameter fluctuations and defects such as cracks and wrinkles. While typical defects such as cracks and wrinkles are usually related to process parameters like blank holder force and the wear of the tool surfaces, the tool design parameters play a significant role within the development of defects and the thinning of the metal sheet [4, 5]. A thorough investigation of tool design and process parameters and their interactions can, therefore, help to achieve an increased robustness of the deep drawing process.

The effects of the process parameters and tool design parameters can be analyzed both experimentally and using finite element models. Experimental studies have the advantage of fully capturing the complexity of the forming process, whereas simulation models often rely on assumptions and simplifications, such as the material model or the modeling of the friction conditions. However, numerical investigations are significantly more time-efficient and allow for precise and flexible adjustment of the analyzed parameters as well as an easy extraction of parameters such as stresses and strains. [6, 7]

<figure><img src="image_1443.png" /></figure>

<footnote n="1">¹Institute for Production Technology and Systems, Leuphana University Lüneburg, Universitätsallee 1, 21335 Lüneburg, Germany</footnote>

<footnote n="2">²Institute of Materials and Process Design, Helmholtz-Zentrum Hereon, Max-Planck-Str. 1, 21502 Geesthacht, Germany</footnote>

<footnote>ªchristine.heinzel@leuphana.de, ᵇsebastian.thiery@leuphana.de,</footnote>

<footnote>ᶜnoomane.ben_khalifa@leuphana.de</footnote>
Instead of looking directly into the material thinning or the defects that occur during deep drawing, investigations are often based on quality indices such as the punch force or the flange draw-in, which are accessible parameters during the deep drawing experiment. The flange draw-in is directly related to the material flow and, therefore, failure modes such as wrinkles or cracks [1]. It is also suitable for inline investigations such as closed-loop control systems in deep drawing [8]. Phenomena such as flange wrinkles and ironing can be identified and analyzed via the punch force curve, while the maximum punch force is one of the most relevant output parameters and commonly used to describe the deep drawing process. [9, 10]

The effects of process parameters on the deep drawing process are commonly studied using finite element models. The blank holder force has been identified to have a significant influence on the formation of wrinkles and cracks, as it can be used to control the material flow and thus influence the thinning of the material. Additional process parameters such as blank thickness, the friction coefficient and the temperature have been found to influence the deep drawing process, with the temperature affecting the effective stresses and the yield strength as well as the lubrication conditions [9, 11]. In reality, metal forming processes are subjected to uncertainties such as changing process conditions and varying process parameters. Changes in temperature, lubrications conditions and process parameters affect the process window and can therefore lead to failure of the finished product. To ensure that the deep drawing process is not negatively affected by the process uncertainties, the effects of both, tool design and process parameters need to be explained such that the parameters can be chosen in a way that ensures the quality of the produced parts, even if the process is subjected to uncertainties. [12]

The shoulder radii of the punch and die of the tool have a significant impact on the material flow, the thickness distribution of the blank and can even affect the earring height in cylindrical aluminum cups as well as wrinkle formation [11, 13]. Other studies were conducted on the influence of the tool geometry on the process robustness in micro deep-drawing [10, 14]. Besides cylindrical cup geometries, Chaimongkon et al. [15] studied the formability of aluminum sheet using a cross-die geometry which is used to resemble the complex strain paths and forming conditions of geometries formed in industrial applications. Kardan et al. [16] conducted both numerical and experimental studies on the influences of the shoulder radii of the tool with carbon steel as blank material. While he found that increase in the die shoulder radius decreases the punch force, an increase of the punch shoulder radius can either increase or decrease the punch force depending on the parameter space chosen for the punch shoulder radius. While the values for the shoulder radii have to be chosen with care, i.e. the die radius not being smaller than the die shoulder radius and the die shoulder radius being large enough to avoid the die cutting through the metal sheet, the optimal design of the tool depends on parameters such as the material, size and shape of the drawn part [4]. The mentioned significant parameters of the deep drawing process are shown in Fig. 1, where the geometry of the punch, die and finished part are demonstrated on a cylindrical cup geometry for visualization purposes.
Instead of looking directly into the material thinning or the defects that occur during deep drawing, investigations are often based on quality indices such as the punch force or the flange draw-in, which are accessible parameters during the deep drawing experiment. The flange draw-in is directly related to the material flow and, therefore, failure modes such as wrinkles or cracks [1]. It is also suitable for inline investigations such as closed-loop control systems in deep drawing [8]. Phenomena such as flange wrinkles and ironing can be identified and analyzed via the punch force curve, while the maximum punch force is one of the most relevant output parameters and commonly used to describe the deep drawing process. [9, 10]

The effects of process parameters on the deep drawing process are commonly studied using finite element models. The blank holder force has been identified to have a significant influence on the formation of wrinkles and cracks, as it can be used to control the material flow and thus influence the thinning of the material. Additional process parameters such as blank thickness, the friction coefficient and the temperature have been found to influence the deep drawing process, with the temperature affecting the effective stresses and the yield strength as well as the lubrication conditions [9, 11]. In reality, metal forming processes are subjected to uncertainties such as changing process conditions and varying process parameters. Changes in temperature, lubrications conditions and process parameters affect the process window and can therefore lead to failure of the finished product. To ensure that the deep drawing process is not negatively affected by the process uncertainties, the effects of both, tool design and process parameters need to be explained such that the parameters can be chosen in a way that ensures the quality of the produced parts, even if the process is subjected to uncertainties. [12]

The shoulder radii of the punch and die of the tool have a significant impact on the material flow, the thickness distribution of the blank and can even affect the earring height in cylindrical aluminum cups as well as wrinkle formation [11, 13]. Other studies were conducted on the influence of the tool geometry on the process robustness in micro deep-drawing [10, 14]. Besides cylindrical cup geometries, Chaimongkon et al. [15] studied the formability of aluminum sheet using a cross-die geometry which is used to resemble the complex strain paths and forming conditions of geometries formed in industrial applications. Kardan et al. [16] conducted both numerical and experimental studies on the influences of the shoulder radii of the tool with carbon steel as blank material. While he found that increase in the die shoulder radius decreases the punch force, an increase of the punch shoulder radius can either increase or decrease the punch force depending on the parameter space chosen for the punch shoulder radius. While the values for the shoulder radii have to be chosen with care, i.e. the die radius not being smaller than the die shoulder radius and the die shoulder radius being large enough to avoid the die cutting through the metal sheet, the optimal design of the tool depends on parameters such as the material, size and shape of the drawn part [4]. The mentioned significant parameters of the deep drawing process are shown in Fig. 1, where the geometry of the punch, die and finished part are demonstrated on a cylindrical cup geometry for visualization purposes.
<figure><img src="image_1440.png" /><figcaption>Fig. 1: Deep drawing process and parameters (based on [17])</figcaption></figure>

Even if the influences of the parameters and the tool geometry on the deep drawing process are known, process failure often occurs in real production operations due to process fluctuations. Explainable modeling approaches of the stochastic process noise are needed in the design process of deep drawing tools to optimize the tool geometry. To pursue this goal, a modular deep drawing tool is developed in this paper to investigate the effects of the effective surfaces. This tool allows for the investigation of the robustness of the deep drawing process within a specified range of punch and die shoulder radius parameters, considering varying blank holder force values and variations of process conditions such as temperature and lubrication. A finite element model is used to analyze and explain the combined effects and interactions of the tool design parameters and the blank holder force on two quality indicators (maximum punch force and material draw-in). These numerical investigations form the basis for studying the influences of parameter fluctuations on the process limits associated with the die surface geometries and serve as a baseline investigation for future studies with the aim of developing guidelines for robust tool design. The study is carried out on a sheet metal cross die geometry to capture the complex nature of geometries in industrial applications.

## Methodology

Modular tool. The numerical design of experiments with varying punch shoulder and die shoulder radii on a cross-die geometry is conducted based on a modular tool which allows for an independent exchange of the punch shoulder radii and die shoulder radii. Additionally, the punch, die, and blank holder are equipped with a temperature control system, which enables the analysis of temperature fluctuations on the deep drawing process in future studies. Modular tools have already successfully been developed making use of exchangeable inserts. Rigas and Merklein [18] used this approach to study different tool coatings while Özkaya [19] investigated oxidized inserts in dry deep drawing.

The tool used in this study is composed of three levels: the punch (bottom), the blank holder (center), and the die (top). Fig. 2 displays the top and bottom parts of the modular tool. The variation in the shoulder radius of the die is made possible by the modular design of the upper part of the tool, in which the dies are designed as inlays. In addition, holes for heating cartridges as well as thermocouples were provided in the punch, die, and blank holder. The developed tool can be extended by cross-punch inserts with any punch and die edge radii.
<figure><img src="image_1444.png" /><figcaption>Fig. 1: Deep drawing process and parameters (based on [17])</figcaption></figure>

Even if the influences of the parameters and the tool geometry on the deep drawing process are known, process failure often occurs in real production operations due to process fluctuations. Explainable modeling approaches of the stochastic process noise are needed in the design process of deep drawing tools to optimize the tool geometry. To pursue this goal, a modular deep drawing tool is developed in this paper to investigate the effects of the effective surfaces. This tool allows for the investigation of the robustness of the deep drawing process within a specified range of punch and die shoulder radius parameters, considering varying blank holder force values and variations of process conditions such as temperature and lubrication. A finite element model is used to analyze and explain the combined effects and interactions of the tool design parameters and the blank holder force on two quality indicators (maximum punch force and material draw-in). These numerical investigations form the basis for studying the influences of parameter fluctuations on the process limits associated with the die surface geometries and serve as a baseline investigation for future studies with the aim of developing guidelines for robust tool design. The study is carried out on a sheet metal cross die geometry to capture the complex nature of geometries in industrial applications.

## Methodology

Modular tool. The numerical design of experiments with varying punch shoulder and die shoulder radii on a cross-die geometry is conducted based on a modular tool which allows for an independent exchange of the punch shoulder radii and die shoulder radii. Additionally, the punch, die, and blank holder are equipped with a temperature control system, which enables the analysis of temperature fluctuations on the deep drawing process in future studies. Modular tools have already successfully been developed making use of exchangeable inserts. Rigas and Merklein [18] used this approach to study different tool coatings while Özkaya [19] investigated oxidized inserts in dry deep drawing.

The tool used in this study is composed of three levels: the punch (bottom), the blank holder (center), and the die (top). Fig. 2 displays the top and bottom parts of the modular tool. The variation in the shoulder radius of the die is made possible by the modular design of the upper part of the tool, in which the dies are designed as inlays. In addition, holes for heating cartridges as well as thermocouples were provided in the punch, die, and blank holder. The developed tool can be extended by cross-punch inserts with any punch and die edge radii.
<figure><img src="image_1441.png" /><figcaption>Fig. 2: Punch (left) and die (right) design of the modular tool</figcaption></figure>

Finite Element Model. Finite element analysis was used in ABAQUS/Explicit to investigate the effects and interactions of the die shoulder and punch shoulder radii, as well as the blank holder force. The study is conducted on a steel cross-die geometry to examine strain states resembling the complex geometries used in the automotive industry [15]. In the finite element model, only a quarter of the cross-die geometry was used due to its symmetry across the X and Y planes, on which symmetric boundary conditions were applied in the model. The punch, die, and blank holder are modeled as discrete rigid bodies, while the blank is modeled as a flexible body consisting of approximately 25,000 solid elements (C3D8R). Material parameters were determined using a tensile test and implemented with the anisotropic behavior being modeled using the Hill stress potential, with R<sub>22</sub>= 1.0182, R<sub>33</sub>= 1.2976 and R<sub>12</sub> = 1.0723. The contact between the tool and the blank is modeled with kinematic contact and finite sliding conditions. Fig. 3 displays the finite element model parts and the meshed blank in its final state.

<figure><img src="image_1442.png" /><figcaption>Fig. 3: Finite element model components (left) and mesh in final state (right)</figcaption></figure>

The friction coefficient of the initial model is used as a tuning parameter for fitting the punch forces determined by the simulation model and the experiment, which results in a friction coefficient of μ=0.05. The model is then validated using the draw-in in the x-direction, with the numerical and the experimental result showing a deviation of less than 5%. Moving forward, the punch force and the draw-in in x-direction are used as output parameters for the Design of Experiments (DoE) campaign.

Effect analysis. To further investigate the influences of the tool design on the deep drawing process, two design parameters (punch shoulder and die shoulder radius) were varied alongside one process parameter (blank holder force), which was chosen for having a high influence on the punch force and the material thinning and, therefore, on the failure modes such as cracks and
<figure><img src="image_1445.png" /><figcaption>Fig. 2: Punch (left) and die (right) design of the modular tool</figcaption></figure>

Finite Element Model. Finite element analysis was used in ABAQUS/Explicit to investigate the effects and interactions of the die shoulder and punch shoulder radii, as well as the blank holder force. The study is conducted on a steel cross-die geometry to examine strain states resembling the complex geometries used in the automotive industry [15]. In the finite element model, only a quarter of the cross-die geometry was used due to its symmetry across the X and Y planes, on which symmetric boundary conditions were applied in the model. The punch, die, and blank holder are modeled as discrete rigid bodies, while the blank is modeled as a flexible body consisting of approximately 25,000 solid elements (C3D8R). Material parameters were determined using a tensile test and implemented with the anisotropic behavior being modeled using the Hill stress potential, with R<sub>22</sub>= 1.0182, R<sub>33</sub>= 1.2976 and R<sub>12</sub> = 1.0723. The contact between the tool and the blank is modeled with kinematic contact and finite sliding conditions. Fig. 3 displays the finite element model parts and the meshed blank in its final state.

<figure><img src="image_1446.png" /><figcaption>Fig. 3: Finite element model components (left) and mesh in final state (right)</figcaption></figure>

The friction coefficient of the initial model is used as a tuning parameter for fitting the punch forces determined by the simulation model and the experiment, which results in a friction coefficient of μ=0.05. The model is then validated using the draw-in in the x-direction, with the numerical and the experimental result showing a deviation of less than 5%. Moving forward, the punch force and the draw-in in x-direction are used as output parameters for the Design of Experiments (DoE) campaign.

Effect analysis. To further investigate the influences of the tool design on the deep drawing process, two design parameters (punch shoulder and die shoulder radius) were varied alongside one process parameter (blank holder force), which was chosen for having a high influence on the punch force and the material thinning and, therefore, on the failure modes such as cracks and
wrinkling. The effects on the output parameters as well as the interactions are investigated using three levels for every parameter to account for possible nonlinear relationships. Using a full factorial DoE, i.e. a set of experiments, in which every level of a parameter is combined with each level of the remaining parameters, results in a total of 27 simulations and ensures that nonlinear effects can be observed [20]. The levels for each of the three parameters are chosen by setting the values used in the initial, validated model as the center point and then ensuring an equidistant spacing between the minimum and maximum level. The resulting levels for the input parameter can be found in Table 1.

<table><caption>Table 1: Levels for die shoulder radius, punch shoulder radius and blank holder force</caption><thead><tr><td>Parameter</td><td>Min</td><td>Center</td><td>Max</td></tr></thead><tbody><tr><td>Die radius r<sub>d</sub></td><td>8 mm</td><td>10 mm</td><td>12 mm</td></tr><tr><td>Punch radius r<sub>p</sub></td><td>16 mm</td><td>20 mm</td><td>24 mm</td></tr><tr><td>Blank holder force F<sub>BH</sub></td><td>278 kN</td><td>309 kN</td><td>340 kN</td></tr></tbody></table>

## Results and discussion

Results. After the simulations are conducted the effects of the varying parameters, i.e. factors, on the draw-in and the maximum punch force are determined. The effect is calculated from the change in the average of the output parameters at a certain level of each input parameter under consideration. For example, the effect of the die shoulder radius changing from 8 mm to 10 mm on the draw-in is calculated from the change of the mean draw-in values resulting from all experiments conducted with r<sub>d</sub>=10 mm and all experiments conducted with r<sub>d</sub>=8 mm [20]. Fig. 4 shows the effect plots of the die and punch shoulder and the blank holder force on the draw-in of the flange and the punch force.
wrinkling. The effects on the output parameters as well as the interactions are investigated using three levels for every parameter to account for possible nonlinear relationships. Using a full factorial DoE, i.e. a set of experiments, in which every level of a parameter is combined with each level of the remaining parameters, results in a total of 27 simulations and ensures that nonlinear effects can be observed [20]. The levels for each of the three parameters are chosen by setting the values used in the initial, validated model as the center point and then ensuring an equidistant spacing between the minimum and maximum level. The resulting levels for the input parameter can be found in Table 1.

<table><caption>Table 1: Levels for die shoulder radius, punch shoulder radius and blank holder force</caption><thead><tr><td>Parameter</td><td>Min</td><td>Center</td><td>Max</td></tr></thead><tbody><tr><td>Die radius r<sub>d</sub></td><td>8 mm</td><td>10 mm</td><td>12 mm</td></tr><tr><td>Punch radius r<sub>p</sub></td><td>16 mm</td><td>20 mm</td><td>24 mm</td></tr><tr><td>Blank holder force F<sub>BH</sub></td><td>278 kN</td><td>309 kN</td><td>340 kN</td></tr></tbody></table>

## Results and discussion

Results. After the simulations are conducted the effects of the varying parameters, i.e. factors, on the draw-in and the maximum punch force are determined. The effect is calculated from the change in the average of the output parameters at a certain level of each input parameter under consideration. For example, the effect of the die shoulder radius changing from 8 mm to 10 mm on the draw-in is calculated from the change of the mean draw-in values resulting from all experiments conducted with r<sub>d</sub>=10 mm and all experiments conducted with r<sub>d</sub>=8 mm [20]. Fig. 4 shows the effect plots of the die and punch shoulder and the blank holder force on the draw-in of the flange and the punch force.
<figure><img src="image_1443.png" /><figcaption>Fig. 4: Effects on draw-in and maximum punch force</figcaption></figure>

It can be observed that the punch shoulder radius has a significant, linear influence on the draw-in, where increasing the punch shoulder radius leads to a smaller draw-in which indicates less material flow. A similar relation can be observed regarding the blank holder force, but with a smaller impact on the draw-in. The die shoulder radius tends to affect the draw-in nonlinearly, by increasing the draw-in when increased to 10 mm but decreasing it when again increased to 12 mm. The effect itself is overall insignificant and more datapoints are needed to further investigate it. Furthermore, an increase in the blank holder force increases the punch force linearly to a moderate extent. An increased die shoulder radius leads to a significant drop of the punch force, where the effect is more dominant for lower radii and, therefore shows a nonlinear relation between the punch force and the die shoulder radius. The punch radius has a negligible influence on the punch force with a linear trend. In summary, the die shoulder radius affects the punch force notably, while the effects of the punch shoulder radius are way less pronounced.

While the main effects show the relation between the input and output parameters, it should also be investigated whether the parameters influence each other, i.e. if the effects of the input parameters are cross-dependent. For that purpose, the interactions of the parameters are shown in Fig. 5 wherein each column shows the effects of one of the three input parameters on the corresponding output parameter (maximum punch force and draw-in).
<figure><img src="image_1447.png" /><figcaption>Fig. 4: Effects on draw-in and maximum punch force</figcaption></figure>

It can be observed that the punch shoulder radius has a significant, linear influence on the draw-in, where increasing the punch shoulder radius leads to a smaller draw-in which indicates less material flow. A similar relation can be observed regarding the blank holder force, but with a smaller impact on the draw-in. The die shoulder radius tends to affect the draw-in nonlinearly, by increasing the draw-in when increased to 10 mm but decreasing it when again increased to 12 mm. The effect itself is overall insignificant and more datapoints are needed to further investigate it. Furthermore, an increase in the blank holder force increases the punch force linearly to a moderate extent. An increased die shoulder radius leads to a significant drop of the punch force, where the effect is more dominant for lower radii and, therefore shows a nonlinear relation between the punch force and the die shoulder radius. The punch radius has a negligible influence on the punch force with a linear trend. In summary, the die shoulder radius affects the punch force notably, while the effects of the punch shoulder radius are way less pronounced.

While the main effects show the relation between the input and output parameters, it should also be investigated whether the parameters influence each other, i.e. if the effects of the input parameters are cross-dependent. For that purpose, the interactions of the parameters are shown in Fig. 5 wherein each column shows the effects of one of the three input parameters on the corresponding output parameter (maximum punch force and draw-in).
<figure><img src="image_1444.png" /><figcaption>Fig. 5: Interaction plot of draw-in and the maximum punch force</figcaption></figure>

Upon examining the interaction plot for the draw-in, it is evident that there are no significant interactions between the parameters. The effects of each parameter remain consistent when the remaining two parameters are varied, and the absolute deviations of the lines correspond to the main effects of the respective parameters. It can be noted that, within the examined parameter range, the punch shoulder radius has no significant effects or interactions reactions regarding the punch force.

Discussion. Since the parameters show no cross-dependencies within the examined parameter range, their impacts on the process robustness are discussed with assumed independence from each other. The effect analysis shows that higher punch shoulder radii lead to a lower draw-in, while an increase in die shoulder radii result in a decrease in the punch force. In summary, the die shoulder radius affects the punch force notably, while the effects of the punch shoulder radius are much less pronounced. During the forming process, the sheet is being subjected to continuous double bending stresses at the die curvature, which explains the die shoulder radius having a significant influence on the punch force. An increase of the blank holder leads to a lower draw-in as well as a higher punch force. The effects of the blank holder force can be explained by controlling the material flow. The friction coefficient between the blank holder, die and blank creates a tangential force that counteracts the flow of material. High blank holder force values increase the risk of tearing, while low values can cause wrinkles. [1, 4]

The observed effects of the punch shoulder radius on the draw-in and the maximum punch force are supported by studies conducted by Lee [21] as well as Moshksar [22]. Behrens et al. [14] found an increase in die shoulder radius to decrease the maximum punch force, and explains the effect with the increased back bending, with the back bending force being inversely proportional to the die radius, which also leads to reduced stresses in back bending, with larger die radii. However, the risk of wrinkling formations should be taken into account with increased die radii, since the contact area of the blank holder on the blank is reduced [23]. Furthermore, Qin et al. [5] explains that the change in the surface area of the finished part influences the effect of the die shoulder radius on material thinning and should, therefore, be taken into account when evaluating the robustness of the process. For these reasons, the influence of the punch shoulder and die shoulder radii on the robustness of the deep drawing process based on the chosen quality indices is investigated by examining the elongation of the absolute length $l_a$ of the blank as well as the maximum stresses and equivalent plastic strains in the finished part, which are depicted in Fig. 6. The elongation of half of the absolute length in x-direction ($l_{ax}$) is taken as a measure for the increase of the surface area of the finished part, whereas the stresses and equivalent plastic strains are assumed to be an indication of an increased risk of material thinning.
<figure><img src="image_1448.png" /><figcaption>Fig. 5: Interaction plot of draw-in and the maximum punch force</figcaption></figure>

Upon examining the interaction plot for the draw-in, it is evident that there are no significant interactions between the parameters. The effects of each parameter remain consistent when the remaining two parameters are varied, and the absolute deviations of the lines correspond to the main effects of the respective parameters. It can be noted that, within the examined parameter range, the punch shoulder radius has no significant effects or interactions reactions regarding the punch force.

Discussion. Since the parameters show no cross-dependencies within the examined parameter range, their impacts on the process robustness are discussed with assumed independence from each other. The effect analysis shows that higher punch shoulder radii lead to a lower draw-in, while an increase in die shoulder radii result in a decrease in the punch force. In summary, the die shoulder radius affects the punch force notably, while the effects of the punch shoulder radius are much less pronounced. During the forming process, the sheet is being subjected to continuous double bending stresses at the die curvature, which explains the die shoulder radius having a significant influence on the punch force. An increase of the blank holder leads to a lower draw-in as well as a higher punch force. The effects of the blank holder force can be explained by controlling the material flow. The friction coefficient between the blank holder, die and blank creates a tangential force that counteracts the flow of material. High blank holder force values increase the risk of tearing, while low values can cause wrinkles. [1, 4]

The observed effects of the punch shoulder radius on the draw-in and the maximum punch force are supported by studies conducted by Lee [21] as well as Moshksar [22]. Behrens et al. [14] found an increase in die shoulder radius to decrease the maximum punch force, and explains the effect with the increased back bending, with the back bending force being inversely proportional to the die radius, which also leads to reduced stresses in back bending, with larger die radii. However, the risk of wrinkling formations should be taken into account with increased die radii, since the contact area of the blank holder on the blank is reduced [23]. Furthermore, Qin et al. [5] explains that the change in the surface area of the finished part influences the effect of the die shoulder radius on material thinning and should, therefore, be taken into account when evaluating the robustness of the process. For these reasons, the influence of the punch shoulder and die shoulder radii on the robustness of the deep drawing process based on the chosen quality indices is investigated by examining the elongation of the absolute length $l_a$ of the blank as well as the maximum stresses and equivalent plastic strains in the finished part, which are depicted in Fig. 6. The elongation of half of the absolute length in x-direction ($l_{ax}$) is taken as a measure for the increase of the surface area of the finished part, whereas the stresses and equivalent plastic strains are assumed to be an indication of an increased risk of material thinning.
<figure><img src="image_1445.png" /><figcaption>Fig. 6: Effect of punch radius on maximum stresses, equivalent plastic strain and elongation of the absolute length</figcaption></figure>

It can be shown that an increase in punch shoulder radii decreases the absolute length of the finished part and decreases the maximum stress and plastic strain within the metal sheet, which is found near the area of back bending in the finished part. The same trend can be observed with the increase of the die shoulder radii. It can be shown that, while the reduction in the maximum punch force for higher die shoulder radii serves as a reliable indicator for a lowered risk of material thinning, i.e. lower stresses and plastic strain, the increase of draw-in for lower die radii indicates a more robust process, while the stresses and plastic strains increase in the finished part. Therefore, the absolute length and thus the surface area of the finished part should be considered when accessing the robustness of the deep drawing process.

## Summary

The aim of this study was to investigate the dependence of the deep drawing process on the variation of the blank holder force in combination with the effects of a varying range of tool radii. For that purpose, a simulation model was developed based on a modular deep drawing tool to conduct a parameter study and study the effects and interactions of the tool shoulder radii and blank holder force. It could be demonstrated that the punch shoulder radius has a dominant effect on the material draw-in within the investigated parameter range of the tool, while the die shoulder radius has a significant influence on the punch force. Smaller punch shoulder radii result in increased material draw-in, while smaller die shoulder radii result in an increased punch force. An increase of the shoulder radii has been found to increase the risk of material thinning, which is
<figure><img src="image_1449.png" /><figcaption>Fig. 6: Effect of punch radius on maximum stresses, equivalent plastic strain and elongation of the absolute length</figcaption></figure>

It can be shown that an increase in punch shoulder radii decreases the absolute length of the finished part and decreases the maximum stress and plastic strain within the metal sheet, which is found near the area of back bending in the finished part. The same trend can be observed with the increase of the die shoulder radii. It can be shown that, while the reduction in the maximum punch force for higher die shoulder radii serves as a reliable indicator for a lowered risk of material thinning, i.e. lower stresses and plastic strain, the increase of draw-in for lower die radii indicates a more robust process, while the stresses and plastic strains increase in the finished part. Therefore, the absolute length and thus the surface area of the finished part should be considered when accessing the robustness of the deep drawing process.

## Summary

The aim of this study was to investigate the dependence of the deep drawing process on the variation of the blank holder force in combination with the effects of a varying range of tool radii. For that purpose, a simulation model was developed based on a modular deep drawing tool to conduct a parameter study and study the effects and interactions of the tool shoulder radii and blank holder force. It could be demonstrated that the punch shoulder radius has a dominant effect on the material draw-in within the investigated parameter range of the tool, while the die shoulder radius has a significant influence on the punch force. Smaller punch shoulder radii result in increased material draw-in, while smaller die shoulder radii result in an increased punch force. An increase of the shoulder radii has been found to increase the risk of material thinning, which is
indicated by increased maximum stresses and equivalent plastic strains. The material thinning is suggested to be related to an increase of the absolute length which is used as a measure for in the surface area of the finished part. It is found that the absolute length of the finished part, which is related to its surface area, can be used as an indicator for material thinning and should therefore be taken into account when relating the tool geometry to the robustness of the deep drawing process. The presented results will be used as a basis for experimental studies with varying process conditions with the goal to determine guidelines for a robust tool design.

## Acknowledgments

The authors would like to thank the German Research Foundation (DFG) for funding the project "Development of a data-driven model for the evaluation and optimisation of the process robustness in the die face design of deep-drawing tools" (project number: 520204466), which is part of the priority program "Data-driven process modelling in metal forming technology" (SPP 2422). The authors would also like to thank Mr. Hauke Hoppe for his contributions in designing the modular tool and his involvement in preparing the parameter study.

## References

[1] Schuler GmbH, Metal forming handbook, Springer-Verlag, Berlin, New York, 1998.

[2] J.M. Allwood, S.R. Duncan, J. Cao, P. Groche, G. Hirt, B. Kinsey, T. Kuboki, M. Liewald, A. Sterzing, A.E. Tekkaya, Closed-loop control of product properties in metal forming, CIRP Annals 65 (2016) 573–596. https://doi.org/10.1016/j.cirp.2016.06.002

[3] F. Henning, E. Moeller (Eds.), Handbuch Leichtbau: Methoden, Werkstoffe, Fertigung, second., überarbeitete und erweiterte Auflage, Hanser, München, 2020.

[4] F. Klocke, Manufacturing Processes 4, Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. https://doi.org/10.1007/978-3-642-36772-4

[5] S. Qin, H.M. Shang, C.J. Tay, J.X. Mo, On the extent of stretching during deep drawing, Journal of Materials Processing Technology 59 (1996) 386–390. https://doi.org/10.1016/0924-0136(95)02169-8

[6] A. Birkert, S. Haage, M. Straub, Umformtechnische Herstellung komplexer Karosserieteile, Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. https://doi.org/10.1007/978-3-642-34670-5

[7] Z. Marciniak, J.L. Dunca, S.J. Hu, Mechanics of sheet metal forming, secondnd ed., Elsevier Butterworth-Heinemann, Oxford, 2002.

[8] K. Siegert, E. Dannenmann, S. Wagner, A. Galaiko, Closed-Loop Control System for Blank Holder Forces in Deep Drawing, CIRP Annals 44 (1995) 251–254. https://doi.org/10.1016/S0007-8506(07)62319-1

[9] K. Chen, A. Breunig, J. Ha, B.L. Kinsey, P. Groche, Y.P. Korkolis, Robustness of deep-drawing finite-element simulations to process variations, Int J Mater Form 15 (2022). https://doi.org/10.1007/s12289-022-01695-3

[10] G. Behrens, F. Vollertsen, Influence of Tool Geometry Variation on the Punch Force in Micro Deep Drawing, KEM 554-557 (2013) 1306–1311. https://doi.org/10.4028/www.scientific.net/KEM.554-557.1306

[11] P.R. Tiwari, A. Rathore, M.G. Bodkhe, Factors affecting the deep drawing process – A review, Materials Today: Proceedings 56 (2022) 2902–2908. https://doi.org/10.1016/j.matpr.2021.10.189

[12] M. Liewald, T. Bergs, P. Groche, B.-A. Behrens, D. Briesenick, M. Müller, P. Niemietz, C. Kubik, F. Müller, Perspectives on data-driven models and its potentials in metal forming and
indicated by increased maximum stresses and equivalent plastic strains. The material thinning is suggested to be related to an increase of the absolute length which is used as a measure for in the surface area of the finished part. It is found that the absolute length of the finished part, which is related to its surface area, can be used as an indicator for material thinning and should therefore be taken into account when relating the tool geometry to the robustness of the deep drawing process. The presented results will be used as a basis for experimental studies with varying process conditions with the goal to determine guidelines for a robust tool design.

## Acknowledgments

The authors would like to thank the German Research Foundation (DFG) for funding the project "Development of a data-driven model for the evaluation and optimisation of the process robustness in the die face design of deep-drawing tools" (project number: 520204466), which is part of the priority program "Data-driven process modelling in metal forming technology" (SPP 2422). The authors would also like to thank Mr. Hauke Hoppe for his contributions in designing the modular tool and his involvement in preparing the parameter study.

## References

[1] Schuler GmbH, Metal forming handbook, Springer-Verlag, Berlin, New York, 1998.

[2] J.M. Allwood, S.R. Duncan, J. Cao, P. Groche, G. Hirt, B. Kinsey, T. Kuboki, M. Liewald, A. Sterzing, A.E. Tekkaya, Closed-loop control of product properties in metal forming, CIRP Annals 65 (2016) 573–596. https://doi.org/10.1016/j.cirp.2016.06.002

[3] F. Henning, E. Moeller (Eds.), Handbuch Leichtbau: Methoden, Werkstoffe, Fertigung, second., überarbeitete und erweiterte Auflage, Hanser, München, 2020.

[4] F. Klocke, Manufacturing Processes 4, Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. https://doi.org/10.1007/978-3-642-36772-4

[5] S. Qin, H.M. Shang, C.J. Tay, J.X. Mo, On the extent of stretching during deep drawing, Journal of Materials Processing Technology 59 (1996) 386–390. https://doi.org/10.1016/0924-0136(95)02169-8

[6] A. Birkert, S. Haage, M. Straub, Umformtechnische Herstellung komplexer Karosserieteile, Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. https://doi.org/10.1007/978-3-642-34670-5

[7] Z. Marciniak, J.L. Dunca, S.J. Hu, Mechanics of sheet metal forming, secondnd ed., Elsevier Butterworth-Heinemann, Oxford, 2002.

[8] K. Siegert, E. Dannenmann, S. Wagner, A. Galaiko, Closed-Loop Control System for Blank Holder Forces in Deep Drawing, CIRP Annals 44 (1995) 251–254. https://doi.org/10.1016/S0007-8506(07)62319-1

[9] K. Chen, A. Breunig, J. Ha, B.L. Kinsey, P. Groche, Y.P. Korkolis, Robustness of deep-drawing finite-element simulations to process variations, Int J Mater Form 15 (2022). https://doi.org/10.1007/s12289-022-01695-3

[10] G. Behrens, F. Vollertsen, Influence of Tool Geometry Variation on the Punch Force in Micro Deep Drawing, KEM 554-557 (2013) 1306–1311. https://doi.org/10.4028/www.scientific.net/KEM.554-557.1306

[11] P.R. Tiwari, A. Rathore, M.G. Bodkhe, Factors affecting the deep drawing process – A review, Materials Today: Proceedings 56 (2022) 2902–2908. https://doi.org/10.1016/j.matpr.2021.10.189

[12] M. Liewald, T. Bergs, P. Groche, B.-A. Behrens, D. Briesenick, M. Müller, P. Niemietz, C. Kubik, F. Müller, Perspectives on data-driven models and its potentials in metal forming and
blanking technologies, Prod. Eng. Res. Devel. 16 (2022) 607-625. https://doi.org/10.1007/s11740-022-01115-0

[13] K. Bouchaâla, M.F. Ghanameh, M. Faqir, M. Mada, E. Essadiqi, Numerical investigation of the effect of punch corner radius and die shoulder radius on the flange earrings for AA1050 and AA1100 aluminum alloys in cylindrical deep drawing process, Heliyon 7 (2021) e06662. https://doi.org/10.1016/j.heliyon.2021.e06662

[14] G. Behrens, M. Ruhe, H. Tetzel, F. Vollertsen, Effect of tool geometry variations on the punch force in micro deep drawing of rectangular components, Prod. Eng. Res. Devel. 9 (2015) 195-201. https://doi.org/10.1007/s11740-015-0604-0

[15] T. Chaimongkon, S. Panich, V. Uthaisangsuk, Anisotropic fracture forming limit curve and its applications for sheet metal forming with complex strain paths of aluminum sheet, Int J Adv Manuf Technol 115 (2021) 3553-3577. https://doi.org/10.1007/s00170-021-07357-z

[16] M. Kardan, A. Parvizi, A. Askari, Experimental and Finite Element Results for Optimization of Punch Force and Thickness Distribution in Deep Drawing Process, Arab J Sci Eng 43 (2018) 1165-1175. https://doi.org/10.1007/s13369-017-2783-9

[17] E. Doege, B.-A. Behrens, Handbuch Umformtechnik: Grundlagen, Technologien, Maschinen, third. Aufl. twentithsixteenth, Springer Berlin Heidelberg, Berlin, Heidelberg, 2016. https://doi.org/10.1007/978-3-662-43891-6

[18] N. Rigas, M. Merklein, Characterization of the Tribological Behavior of Different Tool Coatings and Dry Lubricant for High-Strength Aluminum Alloys at Elevated Temperatures, Adv Eng Mater 25 (2023). https://doi.org/10.1002/adem.202201650

[19] F. Özkaya, Development of a tool concept with selectively oxidised inserts for dry deep drawing (2019).

[20] K. Siebertz, D. van Bebber, T. Hochkirchen, Statistische Versuchsplanung, Springer Berlin Heidelberg, Berlin, Heidelberg, 2017. https://doi.org/10.1007/978-3-662-55743-3

[21] P.K.-S. Lee, An Analysis of the Effect of Process Parameters on the Formability of Sheet Metal. Dissertation, Birmingham, 1972.

[22] M. Moshksar, A. Zamanian, Optimization of the tool geometry in the deep drawing of aluminium, Journal of Materials Processing Technology 72 (1997) 363-370. https://doi.org/10.1016/S0924-0136(97)00196-9

[23] G. Behrens, F.O. Trier, H. Tetzel, F. Vollertsen, Influence of tool geometry variations on the limiting drawing ratio in micro deep drawing, Int J Mater Form 9 (2016) 253-258. https://doi.org/10.1007/s12289-015-1228-9
blanking technologies, Prod. Eng. Res. Devel. 16 (2022) 607-625. https://doi.org/10.1007/s11740-022-01115-0

[13] K. Bouchaâla, M.F. Ghanameh, M. Faqir, M. Mada, E. Essadiqi, Numerical investigation of the effect of punch corner radius and die shoulder radius on the flange earrings for AA1050 and AA1100 aluminum alloys in cylindrical deep drawing process, Heliyon 7 (2021) e06662. https://doi.org/10.1016/j.heliyon.2021.e06662

[14] G. Behrens, M. Ruhe, H. Tetzel, F. Vollertsen, Effect of tool geometry variations on the punch force in micro deep drawing of rectangular components, Prod. Eng. Res. Devel. 9 (2015) 195-201. https://doi.org/10.1007/s11740-015-0604-0

[15] T. Chaimongkon, S. Panich, V. Uthaisangsuk, Anisotropic fracture forming limit curve and its applications for sheet metal forming with complex strain paths of aluminum sheet, Int J Adv Manuf Technol 115 (2021) 3553-3577. https://doi.org/10.1007/s00170-021-07357-z

[16] M. Kardan, A. Parvizi, A. Askari, Experimental and Finite Element Results for Optimization of Punch Force and Thickness Distribution in Deep Drawing Process, Arab J Sci Eng 43 (2018) 1165-1175. https://doi.org/10.1007/s13369-017-2783-9

[17] E. Doege, B.-A. Behrens, Handbuch Umformtechnik: Grundlagen, Technologien, Maschinen, third. Aufl. twentieithsixteenth, Springer Berlin Heidelberg, Berlin, Heidelberg, 2016. https://doi.org/10.1007/978-3-662-43891-6

[18] N. Rigas, M. Merklein, Characterization of the Tribological Behavior of Different Tool Coatings and Dry Lubricant for High-Strength Aluminum Alloys at Elevated Temperatures, Adv Eng Mater 25 (2023). https://doi.org/10.1002/adem.202201650

[19] F. Özkaya, Development of a tool concept with selectively oxidised inserts for dry deep drawing (2019).

[20] K. Siebertz, D. van Bebber, T. Hochkirchen, Statistische Versuchsplanung, Springer Berlin Heidelberg, Berlin, Heidelberg, 2017. https://doi.org/10.1007/978-3-662-55743-3

[21] P.K.-S. Lee, An Analysis of the Effect of Process Parameters on the Formability of Sheet Metal. Dissertation, Birmingham, 1972.

[22] M. Moshksar, A. Zamanian, Optimization of the tool geometry in the deep drawing of aluminium, Journal of Materials Processing Technology 72 (1997) 363-370. https://doi.org/10.1016/S0924-0136(97)00196-9

[23] G. Behrens, F.O. Trier, H. Tetzel, F. Vollertsen, Influence of tool geometry variations on the limiting drawing ratio in micro deep drawing, Int J Mater Form 9 (2016) 253-258. https://doi.org/10.1007/s12289-015-1228-9
# Influence of toolpath strategies on the final accuracy and thickness distributions in multi-stage incremental forming

VANHULST Marthe¹,ᵃ*, VANHOVE Hans¹,ᵇ and DUFLOU Joost R.¹,ᶜ

¹Department of Mechanical Engineering, KU Leuven / Flanders Make, Celestijnenlaan 300B, B-3001 Leuven, Belgium

ᵃmarthe.vanhulst@kuleuven.be, ᵇhans.vanhove@kuleuven.be, ᶜjoost.duflou@kuleuven.be

**Keywords:** Single Point Incremental Forming, Multi-Stage Forming, Process Control

**Abstract.** This study focuses on finding a toolpath strategy for accurately forming geometric details on a preshaped sheet metal part by incremental forming in multiple steps. The final thickness distributions and geometrical accuracy are analyzed for spiraling and dedicated feature toolpath strategies. The results are compared to forming the full part (base shape with details) in a conventional single stage manner. Forming the part in multiple steps did improve the accuracy of the part, by decreasing the underforming of the base shape compared to single stage forming. The observed overforming was highly influenced by the location of the detail. In terms of thickness distributions, the toolpath strategy highly influenced the location of the minimal thickness inside each detail. Here, the dedicated feature toolpath proved to be effective for achieving a more uniform thickness distribution.

**Introduction**

Incremental sheet forming (ISF) is a versatile and innovative manufacturing process that enables the production of complex 3D shapes through the gradual deformation of sheet metal. Unlike traditional forming methods that use dies or molds to shape entire components in one pass, ISF employs a more localized approach by incrementally deforming the material in each pass of a generic tool. This process offers several advantages, including reduced tooling costs and increased flexibility in design. In Single Point Incremental Forming (SPIF), a single tool, mostly hemispherical in shape, moves along a toolpath and thereby locally deforms the sheet [1].

One of the advantages of SPIF lies in its flexibility to form complex shapes. However, this flexibility also introduces challenges in controlling unwanted deformations, primarily due to the absence of sheet support. Proposed solutions for mitigating these unwanted deformations include compensation methods that adjust the Computer Aided Design (CAD) model of the desired shape, or adapting the toolpath [1]. Many studies show the influence of the toolpath strategy on the final feasibility and geometric accuracy [2,3]. Typically used toolpath strategies are spiral or contour toolpaths, with spirals offering superior surface finish, but also posing challenges in implementation on more complex geometries [1]. Another important toolpath parameter is the stepdown, where a higher value results in a reduced surface quality, but a too low stepdown might result in early failure due to fatigue or cyclic loading [1]. Additional strategies to enhance geometric accuracy include inclination of the plane for the toolpath [4,5], separation of features [6] and multi-stage forming [7,8,9]. For the feature separation, Lu et al. [6] suggested identifying regions based on edges such that the toolpath can be adapted properly and the geometric accuracy improved. Here, the features are considered as different areas divided from each other by edges.

This paper delves into the adaptation of toolpaths in combination with multi-stage forming, focusing on complex shapes that comprise a base shape with intricate details superimposed, or on enhancing preformed parts with additional details. Particularly this last category is promising, since hybrid techniques, that combine ISF with mass sheet forming methods, might improve efficiency and can be a good solution for customized parts that have a common base shape. Here,

<figure><img src="image_1446.png" /></figure>
# Influence of toolpath strategies on the final accuracy and thickness distributions in multi-stage incremental forming

VANHULST Marthe¹,ᵃ*, VANHOVE Hans¹,ᵇ and DUFLOU Joost R.¹,ᶜ

¹Department of Mechanical Engineering, KU Leuven / Flanders Make, Celestijnenlaan 300B, B-3001 Leuven, Belgium

ᵃmarthe.vanhulst@kuleuven.be, ᵇhans.vanhove@kuleuven.be, ᶜjoost.duflou@kuleuven.be

**Keywords:** Single Point Incremental Forming, Multi-Stage Forming, Process Control

**Abstract.** This study focuses on finding a toolpath strategy for accurately forming geometric details on a preshaped sheet metal part by incremental forming in multiple steps. The final thickness distributions and geometrical accuracy are analyzed for spiraling and dedicated feature toolpath strategies. The results are compared to forming the full part (base shape with details) in a conventional single stage manner. Forming the part in multiple steps did improve the accuracy of the part, by decreasing the underforming of the base shape compared to single stage forming. The observed overforming was highly influenced by the location of the detail. In terms of thickness distributions, the toolpath strategy highly influenced the location of the minimal thickness inside each detail. Here, the dedicated feature toolpath proved to be effective for achieving a more uniform thickness distribution.

**Introduction**

Incremental sheet forming (ISF) is a versatile and innovative manufacturing process that enables the production of complex 3D shapes through the gradual deformation of sheet metal. Unlike traditional forming methods that use dies or molds to shape entire components in one pass, ISF employs a more localized approach by incrementally deforming the material in each pass of a generic tool. This process offers several advantages, including reduced tooling costs and increased flexibility in design. In Single Point Incremental Forming (SPIF), a single tool, mostly hemispherical in shape, moves along a toolpath and thereby locally deforms the sheet [1].

One of the advantages of SPIF lies in its flexibility to form complex shapes. However, this flexibility also introduces challenges in controlling unwanted deformations, primarily due to the absence of sheet support. Proposed solutions for mitigating these unwanted deformations include compensation methods that adjust the Computer Aided Design (CAD) model of the desired shape, or adapting the toolpath [1]. Many studies show the influence of the toolpath strategy on the final feasibility and geometric accuracy [2,3]. Typically used toolpath strategies are spiral or contour toolpaths, with spirals offering superior surface finish, but also posing challenges in implementation on more complex geometries [1]. Another important toolpath parameter is the stepdown, where a higher value results in a reduced surface quality, but a too low stepdown might result in early failure due to fatigue or cyclic loading [1]. Additional strategies to enhance geometric accuracy include inclination of the plane for the toolpath [4,5], separation of features [6] and multi-stage forming [7,8,9]. For the feature separation, Lu et al. [6] suggested identifying regions based on edges such that the toolpath can be adapted properly and the geometric accuracy improved. Here, the features are considered as different areas divided from each other by edges.

This paper delves into the adaptation of toolpaths in combination with multi-stage forming, focusing on complex shapes that comprise a base shape with intricate details superimposed, or on enhancing preformed parts with additional details. Particularly this last category is promising, since hybrid techniques, that combine ISF with mass sheet forming methods, might improve efficiency and can be a good solution for customized parts that have a common base shape. Here,

<figure><img src="image_1450.png" /></figure>
traditional techniques can be used for producing the base shape in mass and personalized details can be added later using ISF. One example of such hybrid forming is the combination of stretch forming with SPIF, where the stretch forming can be used to avoid noncompliance with process limits by producing a preshape with more uniform thickness distributions on which SPIF can be applied in a later stage [10]. By combining the strengths of both conventional and incremental forming, manufacturers can meet the growing demand for customized, high-detail components across diverse industries.

This paper dives further into identifying an appropriate toolpath strategy for details when a preformed base shape is available, offering insights into optimizing the incremental forming process.

## Experiments and Methods

Using a single stage strategy in Single Point Incremental Forming has limitations in terms of geometric accuracy and the ability to produce complex shapes consisting of intricate details. This study focuses on applying details on an already preformed part, thereby comparing multi-stage forming strategies with conventional single stage forming of the full detailed part in only one step. Two different toolpath strategies are applied to form the details on top of an already formed base shape. This exploration includes both regular spiral toolpaths and specialized feature toolpaths, the last one proposed by Carette et al. [11], for the individual details. Subsequently, all resulting parts are compared in terms of geometric accuracy and final thickness distributions for optimizing the incremental forming process.

The experiments are conducted on 225x225 mm sheets of pure commercial Zinc with a thickness of 1.5 mm on a Maho600 CNC machine. The shape to be considered consists of a hemispherical base shape with multiple spherical details, at different locations on the base shape (see Fig. 1). The locations are chosen by dividing the base shape in three areas: zone 1 close to the backing plate (details A, D and F), zone 3 at the top (detail C) and zone 2 in the middle between zones 1 and 3 (details B and E). The midpoints of the spheres are located in these zones, which have the same meridional distance *s* from each other and from the backing plate, measured along the hemispherical surface. The spherical details have diameters of 20, 30 or 40 mm, and are inserted such that their maximal angle with the blank (the *xy*-plane) is 90 degrees. The sheet is clamped along its circumference with a circular backing plate of 180 mm diameter.

<figure><img src="image_1447.png" /><figcaption>Figure 1: Geometric shape used for the experimental campaign: a spherical base shape with multiple spherical details.</figcaption></figure>

After lubricating the sheet with Nuto 46 oil, toolpaths are applied with a 10 mm diameter tool at a feedrate of 2000 mm/min, following either spiral or feature toolpaths for the details with a scallop width of 1 mm. The base shape is formed with a spiral toolpath with the same scallop
traditional techniques can be used for producing the base shape in mass and personalized details can be added later using ISF. One example of such hybrid forming is the combination of stretch forming with SPIF, where the stretch forming can be used to avoid noncompliance with process limits by producing a preshape with more uniform thickness distributions on which SPIF can be applied in a later stage [10]. By combining the strengths of both conventional and incremental forming, manufacturers can meet the growing demand for customized, high-detail components across diverse industries.

This paper dives further into identifying an appropriate toolpath strategy for details when a preformed base shape is available, offering insights into optimizing the incremental forming process.

## Experiments and Methods

Using a single stage strategy in Single Point Incremental Forming has limitations in terms of geometric accuracy and the ability to produce complex shapes consisting of intricate details. This study focuses on applying details on an already preformed part, thereby comparing multi-stage forming strategies with conventional single stage forming of the full detailed part in only one step. Two different toolpath strategies are applied to form the details on top of an already formed base shape. This exploration includes both regular spiral toolpaths and specialized feature toolpaths, the last one proposed by Carette et al. [11], for the individual details. Subsequently, all resulting parts are compared in terms of geometric accuracy and final thickness distributions for optimizing the incremental forming process.

The experiments are conducted on 225x225 mm sheets of pure commercial Zinc with a thickness of 1.5 mm on a Maho600 CNC machine. The shape to be considered consists of a hemispherical base shape with multiple spherical details, at different locations on the base shape (see Fig. 1). The locations are chosen by dividing the base shape in three areas: zone 1 close to the backing plate (details A, D and F), zone 3 at the top (detail C) and zone 2 in the middle between zones 1 and 3 (details B and E). The midpoints of the spheres are located in these zones, which have the same meridional distance *s* from each other and from the backing plate, measured along the hemispherical surface. The spherical details have diameters of 20, 30 or 40 mm, and are inserted such that their maximal angle with the blank (the *xy*-plane) is 90 degrees. The sheet is clamped along its circumference with a circular backing plate of 180 mm diameter.

<figure><img src="image_1451.png" /><figcaption>Figure 1: Geometric shape used for the experimental campaign: a spherical base shape with multiple spherical details.</figcaption></figure>

After lubricating the sheet with Nuto 46 oil, toolpaths are applied with a 10 mm diameter tool at a feedrate of 2000 mm/min, following either spiral or feature toolpaths for the details with a scallop width of 1 mm. The base shape is formed with a spiral toolpath with the same scallop
width. Both the spiral and feature toolpaths perform a spiraling type of movement, with the difference that for the conventional spiral toolpath these are oriented parallel to the blank (the xy-plane), whereas the feature toolpath starts by following the edges of the part (see Fig. 2). For this shape, the feature toolpaths of the spherical details correspond to tilted spiral toolpaths. The details are formed in the following sequence: A-D-B-C-F-E. The single stage forming is done with a contour toolpath, since the spiral toolpath cannot form details if they are deeper than the base shape. In that case, when taking a z-slice, multiple sections can be extracted and the spiral toolpath cannot be formed anymore.

In between each step and after production, the sheet is measured on a Coord3 MC16 CMM with LC60Dx Laser Line Scanner to obtain the geometric accuracy and thickness distributions.

<figure><img src="image_1448.png" /><figcaption>Figure 2: (a) Contour toolpath to form the part in one step and (b) spiral and (c) feature toolpaths for forming details A-F in multiple stages (see Fig. 1). Closeups of the (d) contour, (e) spiral and (f) feature toolpath for detail D.</figcaption></figure>

## Results and Discussion

*Geometric Accuracy.* Fig. 3 shows that forming the details on top of a preshaped part (multi-stage forming with spiral or contour toolpaths for the details, Fig. 3.b and c) significantly affects the geometric accuracy compared to forming the full shape with details in only one step. In Fig. 3.a, it is clear that the underforming (orange areas) of the base shape close to the backing plate is much higher in single stage than when the details are formed in a later stage. Table 1 confirms these observations, with a maximal underforming of the base shape reducing from 1.65 mm in the single stage case to less than 1 mm when applying the details separately. The underformed areas in the single stage case are located within the xy-plane of a toolpath contour, surrounding the formed details. In each single stage contour, the toolpath extends outwards while shaping the details, as can be seen in the toolpath in Fig. 2.d. Close to the backing plate, the three details A, D and F influence the base shape significantly. This is due to the outward tool movement, resulting in a flattening of the already formed areas in the base shape. Consequently, this phenomenon leads to an underforming of the part. This effect might be less outspoken in the details further away from the backing plate (details B, C and E in zones 2 and 3) due to a higher geometrical stiffness of the already formed areas.

The highest geometrical deviations occur due to overforming at the edges between the base shape and the details (blue areas in Fig. 3). Table 2 shows that the overforming is maximal at the edge of detail C, located at the top of the hemispherical base, with a geometrical deviation of 4.09
width. Both the spiral and feature toolpaths perform a spiraling type of movement, with the difference that for the conventional spiral toolpath these are oriented parallel to the blank (the xy-plane), whereas the feature toolpath starts by following the edges of the part (see Fig. 2). For this shape, the feature toolpaths of the spherical details correspond to tilted spiral toolpaths. The details are formed in the following sequence: A-D-B-C-F-E. The single stage forming is done with a contour toolpath, since the spiral toolpath cannot form details if they are deeper than the base shape. In that case, when taking a z-slice, multiple sections can be extracted and the spiral toolpath cannot be formed anymore.

In between each step and after production, the sheet is measured on a Coord3 MC16 CMM with LC60Dx Laser Line Scanner to obtain the geometric accuracy and thickness distributions.

<figure><img src="image_1452.png" /><figcaption>Figure 2: (a) Contour toolpath to form the part in one step and (b) spiral and (c) feature toolpaths for forming details A-F in multiple stages (see Fig. 1). Closeups of the (d) contour, (e) spiral and (f) feature toolpath for detail D.</figcaption></figure>

## Results and Discussion

*Geometric Accuracy.* Fig. 3 shows that forming the details on top of a preshaped part (multi-stage forming with spiral or contour toolpaths for the details, Fig. 3.b and c) significantly affects the geometric accuracy compared to forming the full shape with details in only one step. In Fig. 3.a, it is clear that the underforming (orange areas) of the base shape close to the backing plate is much higher in single stage than when the details are formed in a later stage. Table 1 confirms these observations, with a maximal underforming of the base shape reducing from 1.65 mm in the single stage case to less than 1 mm when applying the details separately. The underformed areas in the single stage case are located within the xy-plane of a toolpath contour, surrounding the formed details. In each single stage contour, the toolpath extends outwards while shaping the details, as can be seen in the toolpath in Fig. 2.d. Close to the backing plate, the three details A, D and F influence the base shape significantly. This is due to the outward tool movement, resulting in a flattening of the already formed areas in the base shape. Consequently, this phenomenon leads to an underforming of the part. This effect might be less outspoken in the details further away from the backing plate (details B, C and E in zones 2 and 3) due to a higher geometrical stiffness of the already formed areas.

The highest geometrical deviations occur due to overforming at the edges between the base shape and the details (blue areas in Fig. 3). Table 2 shows that the overforming is maximal at the edge of detail C, located at the top of the hemispherical base, with a geometrical deviation of 4.09
mm in single stage forming. Here, multi-stage forming significantly decreases this maximal deviation to 3.46 mm when using spiral toolpaths and 2.89 mm for feature toolpaths.

<figure><img src="image_1449.png" /><figcaption>Figure 3: Geometrical accuracy for (a) the contour single stage strategy, (b) the spiral multi-stage strategy and (c) the feature multi-stage strategy. Negative values indicate overforming, positive underforming.</figcaption></figure>

<table><caption>Table 1: Geometric deviations for each strategy, presenting the maximal underforming (positive values in Fig. 3).</caption><thead><tr><th colspan="5">Maximal underforming [mm]</th></tr><tr><th>Detail</th><th>Detail position</th><th>Single stage spiral / contour</th><th>Multi-stage spiral</th><th>Multi-stage feature</th></tr></thead><tbody><tr><td>base shape</td><td>NA</td><td>1.65</td><td>0.97</td><td>0.99</td></tr><tr><td>A</td><td>1</td><td>1.08</td><td>0.49</td><td>0.66</td></tr><tr><td>B</td><td>2</td><td>0.68</td><td>0.70</td><td>0.95</td></tr><tr><td>C</td><td>3</td><td>0.73</td><td>0.62</td><td>0.85</td></tr><tr><td>D</td><td>1</td><td>1.24</td><td>0.53</td><td>0.57</td></tr><tr><td>E</td><td>2</td><td>0.90</td><td>0.41</td><td>0.51</td></tr><tr><td>F</td><td>1</td><td>1.13</td><td>0.64</td><td>0.79</td></tr></tbody></table>

<table><caption>Table 2: Geometric deviations for each strategy, presenting the maximal overforming (negative values in Fig. 3). The location of this maximal overforming is always at the transition between the detail and the base shape.</caption><thead><tr><th colspan="5">Maximal overforming [mm]</th></tr><tr><th>Detail</th><th>Detail position</th><th>Single stage spiral / contour</th><th>Multi-stage spiral</th><th>Multi-stage feature</th></tr></thead><tbody><tr><td>A</td><td>1</td><td>0.67</td><td>0.66</td><td>0.71</td></tr><tr><td>B</td><td>2</td><td>1.46</td><td>1.66</td><td>1.72</td></tr><tr><td>C</td><td>3</td><td>4.09</td><td>3.46</td><td>2.89</td></tr><tr><td>D</td><td>1</td><td>0.45</td><td>0.59</td><td>0.60</td></tr><tr><td>E</td><td>2</td><td>1.89</td><td>1.81</td><td>1.92</td></tr><tr><td>F</td><td>1</td><td>0.84</td><td>0.71</td><td>0.73</td></tr></tbody></table>

Fig. 4.a visualizes the maximal overforming of each detail, as shown in Table 2. Interesting to observe here are consistent patterns that can be connected to the location of the detail. As can be
mm in single stage forming. Here, multi-stage forming significantly decreases this maximal deviation to 3.46 mm when using spiral toolpaths and 2.89 mm for feature toolpaths.

<figure><img src="image_1453.png" /><figcaption>Figure 3: Geometrical accuracy for (a) the contour single stage strategy, (b) the spiral multi-stage strategy and (c) the feature multi-stage strategy. Negative values indicate overforming, positive underforming.</figcaption></figure>

<table><caption>Table 1: Geometric deviations for each strategy, presenting the maximal underforming (positive values in Fig. 3).</caption><thead><tr><th colspan="5">Maximal underforming [mm]</th></tr><tr><th>Detail</th><th>Detail position</th><th>Single stage spiral / contour</th><th>Multi-stage spiral</th><th>Multi-stage feature</th></tr></thead><tbody><tr><td>base shape</td><td>NA</td><td>1.65</td><td>0.97</td><td>0.99</td></tr><tr><td>A</td><td>1</td><td>1.08</td><td>0.49</td><td>0.66</td></tr><tr><td>B</td><td>2</td><td>0.68</td><td>0.70</td><td>0.95</td></tr><tr><td>C</td><td>3</td><td>0.73</td><td>0.62</td><td>0.85</td></tr><tr><td>D</td><td>1</td><td>1.24</td><td>0.53</td><td>0.57</td></tr><tr><td>E</td><td>2</td><td>0.90</td><td>0.41</td><td>0.51</td></tr><tr><td>F</td><td>1</td><td>1.13</td><td>0.64</td><td>0.79</td></tr></tbody></table>

<table><caption>Table 2: Geometric deviations for each strategy, presenting the maximal overforming (negative values in Fig. 3). The location of this maximal overforming is always at the transition between the detail and the base shape.</caption><thead><tr><th colspan="5">Maximal overforming [mm]</th></tr><tr><th>Detail</th><th>Detail position</th><th>Single stage spiral / contour</th><th>Multi-stage spiral</th><th>Multi-stage feature</th></tr></thead><tbody><tr><td>A</td><td>1</td><td>0.67</td><td>0.66</td><td>0.71</td></tr><tr><td>B</td><td>2</td><td>1.46</td><td>1.66</td><td>1.72</td></tr><tr><td>C</td><td>3</td><td>4.09</td><td>3.46</td><td>2.89</td></tr><tr><td>D</td><td>1</td><td>0.45</td><td>0.59</td><td>0.60</td></tr><tr><td>E</td><td>2</td><td>1.89</td><td>1.81</td><td>1.92</td></tr><tr><td>F</td><td>1</td><td>0.84</td><td>0.71</td><td>0.73</td></tr></tbody></table>

Fig. 4.a visualizes the maximal overforming of each detail, as shown in Table 2. Interesting to observe here are consistent patterns that can be connected to the location of the detail. As can be
seen in this graph, the details closest to the backing plate (A, D and F) show the least overforming at their edges with the base shape. This maximal overforming seems very consistent, even though the detail size and proximity to neighboring features are not the same. Details in zone 2 (B and E, see Fig. 1) also show similar overforming, higher than the overforming in zone 1, close to the backing plate. Detail C, the one in the center of the spherical base, shows the highest overforming for all different strategies. Here, the influence of the strategy is more clear than for the other details, with higher deviations from the desired CAD model for the single stage strategy, and the best performance for the multi-stage feature toolpath.

<figure><img src="image_1450.png" /><figcaption>Figure 4: Comparison between the different strategies of all details for (a) the maximal overforming from Table 2 and (b) the minimal thickness from Table 3.</figcaption></figure>

A sectional view of the geometric accuracy at different locations is shown in Fig. 5. Here, the overforming at the edge between the detail and the base shape is even more clear. The section at $Z=h_1$ shows the underforming of the base shape within the $xy$-plane of a toolpath contour. Interesting to note here is that the underforming is much higher for single stage forming, but the curvature at the edge of the detail is more outspoken in a section parallel to the blank.
seen in this graph, the details closest to the backing plate (A, D and F) show the least overforming at their edges with the base shape. This maximal overforming seems very consistent, even though the detail size and proximity to neighboring features are not the same. Details in zone 2 (B and E, see Fig. 1) also show similar overforming, higher than the overforming in zone 1, close to the backing plate. Detail C, the one in the center of the spherical base, shows the highest overforming for all different strategies. Here, the influence of the strategy is more clear than for the other details, with higher deviations from the desired CAD model for the single stage strategy, and the best performance for the multi-stage feature toolpath.

<figure><img src="image_1454.png" /><figcaption>Figure 4: Comparison between the different strategies of all details for (a) the maximal overforming from Table 2 and (b) the minimal thickness from Table 3.</figcaption></figure>

A sectional view of the geometric accuracy at different locations is shown in Fig. 5. Here, the overforming at the edge between the detail and the base shape is even more clear. The section at $Z=h_1$ shows the underforming of the base shape within the $xy$-plane of a toolpath contour. Interesting to note here is that the underforming is much higher for single stage forming, but the curvature at the edge of the detail is more outspoken in a section parallel to the blank.
<figure><img src="image_1451.png" /><figcaption>Figure 5: Comparison of the measured shape and desired shape for all strategies at section A, B and C in Fig. 1.</figcaption></figure>

**Thickness distributions.** Fig. 6 illustrates that the thickness distributions vary significantly among the three strategies. Table 3 also reveals that the minimal thickness after forming is considerably lower in the single stage case compared to multi-stage forming. Depending on the toolpath strategy, the location of minimal thickness within each detail also differs. Multi-stage spiral and single stage contour toolpaths yield lower thicknesses in areas with higher wall angles, while feature toolpaths avoid such local thinning by tilting the toolpath, thereby decreasing the maximal wall angles compared to the forming direction. Table 3 highlights the influence of the
<figure><img src="image_1455.png" /><figcaption>Figure 5: Comparison of the measured shape and desired shape for all strategies at section A, B and C in Fig. 1.</figcaption></figure>

*Thickness distributions.* Fig. 6 illustrates that the thickness distributions vary significantly among the three strategies. Table 3 also reveals that the minimal thickness after forming is considerably lower in the single stage case compared to multi-stage forming. Depending on the toolpath strategy, the location of minimal thickness within each detail also differs. Multi-stage spiral and single stage contour toolpaths yield lower thicknesses in areas with higher wall angles, while feature toolpaths avoid such local thinning by tilting the toolpath, thereby decreasing the maximal wall angles compared to the forming direction. Table 3 highlights the influence of the
strategy on the thickness distributions. Although, in theory, none of the details should be feasible according to the sine rule, in practice the angles do not reach 90 degrees due to unwanted overforming at the edges (see Fig. 5). For the detail located at the top of the base shape (detail C), no significant difference in thickness distributions across the three strategies can be observed. This is logical, since the three strategies have a very similar toolpath for this part of the shape. Fig. 4.b also shows a comparison of the minimal thickness of each detail for all different strategies. Here, it is also clear that the multi-stage strategies significantly increase the minimal thickness for almost all details, with the feature toolpath showing the best performance.

<figure><img src="image_1452.png" /><figcaption>Figure 6: Thickness distributions of (a) the contour single stage strategy, (b) the spiral multi-stage strategy and (c) the feature multi-stage strategy.</figcaption></figure>

<table><caption>Table 3: Minimal thickness for each strategy.</caption><thead><tr><th rowspan="2">Detail</th><th colspan="3">Minimal thickness [mm]</th></tr><tr><th>Single stage spiral / contour</th><th>Multi-stage spiral</th><th>Multi-stage feature</th></tr></thead><tbody><tr><td>A</td><td>0.38</td><td>0.50</td><td>0.58</td></tr><tr><td>B</td><td>0.35</td><td>0.45</td><td>0.55</td></tr><tr><td>C</td><td>0.36</td><td>0.40</td><td>0.41</td></tr><tr><td>D</td><td>0.37</td><td>0.50</td><td>0.60</td></tr><tr><td>E</td><td>0.29</td><td>0.37</td><td>0.52</td></tr><tr><td>F</td><td>0.21</td><td>0.46</td><td>0.56</td></tr></tbody></table>

## Conclusion

This study investigated toolpath strategies in combination with multi-stage forming for improving the geometric accuracy and thickness distributions of complex shapes consisting of a base shape with one or more details. The following conclusions can be made:

* Both a spiral and a feature toolpath decreased the deviations for the proposed shape. The maximal overforming at the edges of the details showed to be connected to the location of the detail, where details further from the backing plate showed larger overforming. The single stage contour toolpath resulted in higher deviations in the base shape compared to the multi-stage toolpaths, with much higher underforming in the *xy*-plane around the details close to the backing plate.

* The thickness distributions were significantly influenced by the toolpath strategy. Both of the proposed multi-stage strategies resulted in a higher minimal thickness compared to single stage forming with a contour toolpath. The multi-stage feature toolpath showed to be the best choice in terms of thickness distributions when forming details on steeper walls,
strategy on the thickness distributions. Although, in theory, none of the details should be feasible according to the sine rule, in practice the angles do not reach 90 degrees due to unwanted overforming at the edges (see Fig. 5). For the detail located at the top of the base shape (detail C), no significant difference in thickness distributions across the three strategies can be observed. This is logical, since the three strategies have a very similar toolpath for this part of the shape. Fig. 4.b also shows a comparison of the minimal thickness of each detail for all different strategies. Here, it is also clear that the multi-stage strategies significantly increase the minimal thickness for almost all details, with the feature toolpath showing the best performance.

<figure><img src="image_1456.png" /><figcaption>Figure 6: Thickness distributions of (a) the contour single stage strategy, (b) the spiral multi-stage strategy and (c) the feature multi-stage strategy.</figcaption></figure>

<table><caption>Table 3: Minimal thickness for each strategy.</caption><thead><tr><th rowspan="2">Detail</th><th colspan="3">Minimal thickness [mm]</th></tr><tr><th>Single stage spiral / contour</th><th>Multi-stage spiral</th><th>Multi-stage feature</th></tr></thead><tbody><tr><td>A</td><td>0.38</td><td>0.50</td><td>0.58</td></tr><tr><td>B</td><td>0.35</td><td>0.45</td><td>0.55</td></tr><tr><td>C</td><td>0.36</td><td>0.40</td><td>0.41</td></tr><tr><td>D</td><td>0.37</td><td>0.50</td><td>0.60</td></tr><tr><td>E</td><td>0.29</td><td>0.37</td><td>0.52</td></tr><tr><td>F</td><td>0.21</td><td>0.46</td><td>0.56</td></tr></tbody></table>

## Conclusion

This study investigated toolpath strategies in combination with multi-stage forming for improving the geometric accuracy and thickness distributions of complex shapes consisting of a base shape with one or more details. The following conclusions can be made:

* Both a spiral and a feature toolpath decreased the deviations for the proposed shape. The maximal overforming at the edges of the details showed to be connected to the location of the detail, where details further from the backing plate showed larger overforming. The single stage contour toolpath resulted in higher deviations in the base shape compared to the multi-stage toolpaths, with much higher underforming in the *xy*-plane around the details close to the backing plate.

* The thickness distributions were significantly influenced by the toolpath strategy. Both of the proposed multi-stage strategies resulted in a higher minimal thickness compared to single stage forming with a contour toolpath. The multi-stage feature toolpath showed to be the best choice in terms of thickness distributions when forming details on steeper walls,
where the feature toolpath inclination decreased the wall angles compared to the forming direction.

These results show that the choice of the toolpath strategy plays an important role in complex parts with details. The study gives more insight in adding details to preshaped parts, which is a promising direction for applying ISF as a hybrid process in industry. The findings show that multi-stage forming is promising for forming the proposed details. However, a tailored approach, depending on the formed shape and location of the details, might be needed.

Future research may explore a broader range of parts, both for the base shape and for the shape and placement of the details. Applying the found strategies on real-life examples can then be enhanced by automatic separation of the detail from the base shape, such that the base and detail toolpaths can be quickly subtracted and merged with each other. Subtracting the detail can be done by detection of the edges, for example based on the feature detection proposed by Behera et al. [9].

## Acknowledgements

Marthe Vanhulst was supported by a Predoctoral Strategic Basic Research Fellowship of the Research Foundation—Flanders (FWO) with project 1S47622N.

## References

[1] J. R. Duflou, A. M. Habraken, J. Cao, R. Malhotra, M. Bambach, D. Adams, H. Vanhove, A. Mohammadi, J. Jeswiet, Single point incremental forming: state-of-the-art and prospects, International Journal of Material Forming. Vol. 11, no. 6 (2018) 743-773.
https://doi.org/10.1007/s12289-017-1387-y

[2] J. M. Allwood, O. Music, A. Raithathna, S.R. Duncan, Closed-loop feedback control of product properties in flexible metal forming processes with mobile tools, CIRP Annals. Vol. 58, no. 1 (2009) 287-290.
https://doi.org/10.1016/J.CIRP.2009.03.065

[3] L. Ben Said, The incremental sheet forming; technology, modeling and formability: A brief review, Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering. Vol. 236, no. 6 (2022) 2729-2755.
https://doi.org/10.1177/09544089221093306

[4] H. Vanhove, J. Gu, H. Sol, J. R. Dulfou, Process Window Extension for Incremental Forming through Optimal Work Plane Rotation. In the 10th anniversary of the International Conference on Technology of Plasticity, Aachen, Germany on September 25th-30th, 2011.

[5] S. Tanaka, Incremental sheet forming with direction control of path planes, ICTP2011. Vol. 53, no. 619 (2011) 503-507.

[6] B. Lu, J. Chen, H. Ou, J. Cao, Feature-based tool path generation approach for incremental sheet forming process, J Mater Process Technol. Vol. 213, no. 7 (2013) 1221-1233.
https://doi.org/10.1016/j.jmatprotec.2013.01.023

[7] Z. Yan, H. Hassanin, M. A. El-Sayed, H. M. Eldessouky, J. R. P. Djuansjah, N. A. Alsaleh, K. Essa, M. A. Ahmadein, Multistage Tool Path Optimisation of Single-Point Incremental Forming Process, Materials 2021. Vol. 14, no. 22 (2021) 6794.
https://doi.org/10.3390/MA14226794

[8] K. S. Jung, J. H. Yu, W. J. Chung, C. W. Lee, Tool Path Design of the Counter Single Point Incremental Forming Process to Decrease Shape Error, Materials 2020. Vol. 13, no. 21 (2020) 4719.
https://doi.org/10.3390/MA13214719
where the feature toolpath inclination decreased the wall angles compared to the forming direction.

These results show that the choice of the toolpath strategy plays an important role in complex parts with details. The study gives more insight in adding details to preshaped parts, which is a promising direction for applying ISF as a hybrid process in industry. The findings show that multi-stage forming is promising for forming the proposed details. However, a tailored approach, depending on the formed shape and location of the details, might be needed.

Future research may explore a broader range of parts, both for the base shape and for the shape and placement of the details. Applying the found strategies on real-life examples can then be enhanced by automatic separation of the detail from the base shape, such that the base and detail toolpaths can be quickly subtracted and merged with each other. Subtracting the detail can be done by detection of the edges, for example based on the feature detection proposed by Behera et al. [9].

## Acknowledgements

Marthe Vanhulst was supported by a Predoctoral Strategic Basic Research Fellowship of the Research Foundation—Flanders (FWO) with project 1S47622N.

## References

[1] J. R. Duflou, A. M. Habraken, J. Cao, R. Malhotra, M. Bambach, D. Adams, H. Vanhove, A. Mohammadi, J. Jeswiet, Single point incremental forming: state-of-the-art and prospects, International Journal of Material Forming. Vol. 11, no. 6 (2018) 743-773.
https://doi.org/10.1007/s12289-017-1387-y

[2] J. M. Allwood, O. Music, A. Raithathna, S.R. Duncan, Closed-loop feedback control of product properties in flexible metal forming processes with mobile tools, CIRP Annals. Vol. 58, no. 1 (2009) 287-290.
https://doi.org/10.1016/J.CIRP.2009.03.065

[3] L. Ben Said, The incremental sheet forming; technology, modeling and formability: A brief review, Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering. Vol. 236, no. 6 (2022) 2729-2755.
https://doi.org/10.1177/09544089221093306

[4] H. Vanhove, J. Gu, H. Sol, J. R. Dulfou, Process Window Extension for Incremental Forming through Optimal Work Plane Rotation. In the 10th anniversary of the International Conference on Technology of Plasticity, Aachen, Germany on September 25th-30th, 2011.

[5] S. Tanaka, Incremental sheet forming with direction control of path planes, ICTP2011. Vol. 53, no. 619 (2011) 503-507.

[6] B. Lu, J. Chen, H. Ou, J. Cao, Feature-based tool path generation approach for incremental sheet forming process, J Mater Process Technol. Vol. 213, no. 7 (2013) 1221-1233.
https://doi.org/10.1016/j.jmatprotec.2013.01.023

[7] Z. Yan, H. Hassanin, M. A. El-Sayed, H. M. Eldessouky, J. R. P. Djuansjah, N. A. Alsaleh, K. Essa, M. A. Ahmadein, Multistage Tool Path Optimisation of Single-Point Incremental Forming Process, Materials 2021. Vol. 14, no. 22 (2021) 6794.
https://doi.org/10.3390/MA14226794

[8] K. S. Jung, J. H. Yu, W. J. Chung, C. W. Lee, Tool Path Design of the Counter Single Point Incremental Forming Process to Decrease Shape Error, Materials 2020. Vol. 13, no. 21 (2020) 4719.
https://doi.org/10.3390/MA13214719
[9] A. K. Behera, B. Lauwers, J. R. Duflou, Tool path generation framework for accurate manufacture of complex 3D sheet metal parts using single point incremental forming, Comput Ind. Vol. 65, no. 4 (2014) 563–584. https://doi.org/10.1016/J.COMPIND.2014.01.002

[10] R. Jagtap, S. Kumar, An experimental investigation on thinning and formability in hybrid incremental sheet forming process, Procedia Manuf. Vol. 30 (2019) 71–76. https://doi.org/10.1016/j.promfg.2019.02.011

[11] Y. Carette, H. Vanhove, J.R. Duflou, Multi-step incremental forming using local feature based toolpaths, Procedia Manuf. Vol. 29 (2019) 28–35. https://doi.org/10.1016/J.PROMFG.2019.02.102
[9] A. K. Behera, B. Lauwers, J. R. Duflou, Tool path generation framework for accurate manufacture of complex 3D sheet metal parts using single point incremental forming, Comput Ind. Vol. 65, no. 4 (2014) 563–584. https://doi.org/10.1016/J.COMPIND.2014.01.002

[10] R. Jagtap, S. Kumar, An experimental investigation on thinning and formability in hybrid incremental sheet forming process, Procedia Manuf. Vol. 30 (2019) 71–76. https://doi.org/10.1016/j.promfg.2019.02.011

[11] Y. Carette, H. Vanhove, J.R. Duflou, Multi-step incremental forming using local feature based toolpaths, Procedia Manuf. Vol. 29 (2019) 28–35. https://doi.org/10.1016/J.PROMFG.2019.02.102
# Characterization of the wrinkling limit curve using in-plane compression tests

MAGRINHO João P.G.¹,ª*, SANTOS João A.O.¹,ᵇ and SILVA M. Beatriz¹,ᶜ

¹IDME, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal

ªjoao.magrinho@tecnico.ulisboa.pt, ᵇjoao.oliveira.santos@tecnico.ulisboa.pt,

ᶜbeatriz.silva@tecnico.ulisboa.pt

**Keywords:** Wrinkling, Wrinkling Limit Curve (WLC), Onset of Wrinkling

**Abstract.** The global need for a green economy and the ongoing challenge of climate change have spurred demand for lightweight construction in various industries. This transition involves redesigning components with intricate geometries and advanced materials to reduce sheet metal thickness and the overall part weight. However, this pursuit of lightweight structures introduces challenges, particularly in wrinkling, raising the need to develop accurate methodologies for determining formability limits by wrinkling. This paper focuses on in-plane compression tests, specifically addressing the compression stress state associated with wrinkling in the flange area in deep drawing or a strip during flexible roll forming. Three physically-based methodologies (A, B, and C) are proposed to detect the onset of wrinkling and characterize the wrinkling limit curve (WLC) through experimental tests. Methodology A involves the analysis of the Z displacement, methodology B focuses on the in-plane minor strain rate evolution, and methodology C employs the analysis of the force-displacement evolution. The experimental work involves the mechanical characterization of AA1050-O aluminum sheets and the in-plane compression tests of rectangular specimens with different lengths. The results reveal distinct behaviors in wrinkling initiation for the different specimen lengths. The critical strain pairs obtained from each methodology are utilized to construct the WLCs in the principal strain space, providing insights into the formability limits by wrinkling.

**Introduction**

The transition to a green economy and the climate change challenge is fueling a trend toward lightweight construction in the transportation sector, i.e., automotive, aerospace, and aeronautical industries, general engineering, machine-tools, and architecture [1]. Redesigning parts with more complex geometries and new materials, with higher mechanical properties, allows the reduction of sheet metal thickness and the final weight of the parts. However, these factors make the material more prone to failure by wrinkling.

The methodologies for determining the onset and formability limits by wrinkling are not consolidated. The reason for this is attributed to the fact that wrinkling is influenced by a wide range of parameters that include the mechanical properties of the material, the geometry of the sheet part, the contact conditions imparted by the tools, and the applied level of stresses and strains [2].

The concept of the wrinkling-limit curve (WLC), which proposes the existence of a locus of in-plane strains defining the initiation of wrinkling in sheet metal forming, is credited to Havranek [3]. The precise position of the WLC typically occurs in the lower left-hand section of the second quadrant within the principal strain space, as schematically represented in Fig. 1. However, its specific location is often determined on a case-by-case basis, contingent upon the areas of focus within the sheet metal forming processes under analysis. Nevertheless, a minimal variation in the strain path can give rise to wrinkling in sheet metal forming [5], it is essential to conduct

<figure><img src="image_1453.png" /></figure>
# Characterization of the wrinkling limit curve using in-plane compression tests

MAGRINHO João P.G.¹,ª*, SANTOS João A.O.¹,ᵇ and SILVA M. Beatriz¹,ᶜ

¹IDME, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal

ªjoao.magrinho@tecnico.ulisboa.pt, ᵇjoao.oliveira.santos@tecnico.ulisboa.pt,

ᶜbeatriz.silva@tecnico.ulisboa.pt

**Keywords:** Wrinkling, Wrinkling Limit Curve (WLC), Onset of Wrinkling

**Abstract.** The global need for a green economy and the ongoing challenge of climate change have spurred demand for lightweight construction in various industries. This transition involves redesigning components with intricate geometries and advanced materials to reduce sheet metal thickness and the overall part weight. However, this pursuit of lightweight structures introduces challenges, particularly in wrinkling, raising the need to develop accurate methodologies for determining formability limits by wrinkling. This paper focuses on in-plane compression tests, specifically addressing the compression stress state associated with wrinkling in the flange area in deep drawing or a strip during flexible roll forming. Three physically-based methodologies (A, B, and C) are proposed to detect the onset of wrinkling and characterize the wrinkling limit curve (WLC) through experimental tests. Methodology A involves the analysis of the Z displacement, methodology B focuses on the in-plane minor strain rate evolution, and methodology C employs the analysis of the force-displacement evolution. The experimental work involves the mechanical characterization of AA1050-O aluminum sheets and the in-plane compression tests of rectangular specimens with different lengths. The results reveal distinct behaviors in wrinkling initiation for the different specimen lengths. The critical strain pairs obtained from each methodology are utilized to construct the WLCs in the principal strain space, providing insights into the formability limits by wrinkling.

**Introduction**

The transition to a green economy and the climate change challenge is fueling a trend toward lightweight construction in the transportation sector, i.e., automotive, aerospace, and aeronautical industries, general engineering, machine-tools, and architecture [1]. Redesigning parts with more complex geometries and new materials, with higher mechanical properties, allows the reduction of sheet metal thickness and the final weight of the parts. However, these factors make the material more prone to failure by wrinkling.

The methodologies for determining the onset and formability limits by wrinkling are not consolidated. The reason for this is attributed to the fact that wrinkling is influenced by a wide range of parameters that include the mechanical properties of the material, the geometry of the sheet part, the contact conditions imparted by the tools, and the applied level of stresses and strains [2].

The concept of the wrinkling-limit curve (WLC), which proposes the existence of a locus of in-plane strains defining the initiation of wrinkling in sheet metal forming, is credited to Havranek [3]. The precise position of the WLC typically occurs in the lower left-hand section of the second quadrant within the principal strain space, as schematically represented in Fig. 1. However, its specific location is often determined on a case-by-case basis, contingent upon the areas of focus within the sheet metal forming processes under analysis. Nevertheless, a minimal variation in the strain path can give rise to wrinkling in sheet metal forming [5], it is essential to conduct

<figure><img src="image_1457.png" /></figure>
experimental characterizations of WLCs under these representative loading conditions. This approach ensures that preventive measures can be implemented to avoid wrinkling.

<figure><img src="image_1454.png" /><figcaption>Fig. 1. Schematic representation of the formability limits by necking, fracture, and wrinkling in sheet metal forming [4].</figcaption></figure>

Considering the compression state of stress directly related to the wrinkling of the flange area in a stamping operation or the compression stress state stems from the wrinkling registered in the flange area of a strip during flexible roll forming, this paper is focused on the in-plane compression test originally proposed by Kasaei et al. [6]. Three different methodologies to determine the onset of wrinkling and characterize the WLC are considered to evaluate their potential for successfully predicting the occurrence of wrinkling in sheet metal forming processes.

## Experimentation

Mechanical Characterization. The investigation involved the mechanical characterization of commercially available AA1050-O aluminum sheets with 1.5 mm thickness. Tensile tests were conducted using an INSTRON 5900 universal testing machine, following the ASTM standard E8/E8M-16 [7], at room temperature. Test specimens were machined from the provided sheets at 0, 45, and 90° angles with respect to the rolling direction (RD) to characterize the anisotropy.

The mechanical properties obtained from the tensile tests are presented in Table 1, such as the modulus of elasticity, *E*, the yield stress, $\sigma_Y$, the ultimate tensile stress, $\sigma_{UTS}$, the elongation at break, *A*, and the planar anisotropy coefficient, *r*.

<table><caption>Table 1. Mechanical properties of aluminum AA1050-O sheet with 1.5 mm thickness [8].</caption><thead><tr><th>Specimen Direction</th><th>Modulus of elasticity E (GPa)</th><th>Yield strength σ<sub>Y</sub> (MPa)</th><th>Ultimate strength σ<sub>UTS</sub> (MPa)</th><th>Elongation at break A (%)</th><th>Anisotropy coefficient r</th><th>K (MPa)</th><th>n</th></tr></thead><tbody><tr><td>0°</td><td>78.63</td><td>24.78</td><td>72.74</td><td>45.98</td><td>1.08</td><td>136.94</td><td>0.27</td></tr><tr><td>45°</td><td>79.40</td><td>25.64</td><td>82.12</td><td>52.86</td><td>3.20</td><td>159.07</td><td>0.30</td></tr><tr><td>90°</td><td>78.36</td><td>24.11</td><td>71.48</td><td>47.26</td><td>0.81</td><td>134.44</td><td>0.27</td></tr><tr><td>Average</td><td>78.95</td><td>25.04</td><td>77.11</td><td>49.74</td><td>2.08</td><td>147.38</td><td>0.29</td></tr></tbody></table>

The average values included in Table 1 were determined as follows (where *x* denotes a mechanical property),
experimental characterizations of WLCs under these representative loading conditions. This approach ensures that preventive measures can be implemented to avoid wrinkling.

<figure><img src="image_1458.png" /><figcaption>Fig. 1. Schematic representation of the formability limits by necking, fracture, and wrinkling in sheet metal forming [4].</figcaption></figure>

Considering the compression state of stress directly related to the wrinkling of the flange area in a stamping operation or the compression stress state stems from the wrinkling registered in the flange area of a strip during flexible roll forming, this paper is focused on the in-plane compression test originally proposed by Kasaei et al. [6]. Three different methodologies to determine the onset of wrinkling and characterize the WLC are considered to evaluate their potential for successfully predicting the occurrence of wrinkling in sheet metal forming processes.

## Experimentation

Mechanical Characterization. The investigation involved the mechanical characterization of commercially available AA1050-O aluminum sheets with 1.5 mm thickness. Tensile tests were conducted using an INSTRON 5900 universal testing machine, following the ASTM standard E8/E8M-16 [7], at room temperature. Test specimens were machined from the provided sheets at 0, 45, and 90° angles with respect to the rolling direction (RD) to characterize the anisotropy.

The mechanical properties obtained from the tensile tests are presented in Table 1, such as the modulus of elasticity, *E*, the yield stress, $\sigma_Y$, the ultimate tensile stress, $\sigma_{UTS}$, the elongation at break, *A*, and the planar anisotropy coefficient, *r*.

<table><caption>Table 1. Mechanical properties of aluminum AA1050-O sheet with 1.5 mm thickness [8].</caption><thead><tr><th>Specimen Direction</th><th>Modulus of elasticity E (GPa)</th><th>Yield strength σ<sub>Y</sub> (MPa)</th><th>Ultimate strength σ<sub>UTS</sub> (MPa)</th><th>Elongation at break A (%)</th><th>Anisotropy coefficient r</th><th>K (MPa)</th><th>n</th></tr></thead><tbody><tr><td>0°</td><td>78.63</td><td>24.78</td><td>72.74</td><td>45.98</td><td>1.08</td><td>136.94</td><td>0.27</td></tr><tr><td>45°</td><td>79.40</td><td>25.64</td><td>82.12</td><td>52.86</td><td>3.20</td><td>159.07</td><td>0.30</td></tr><tr><td>90°</td><td>78.36</td><td>24.11</td><td>71.48</td><td>47.26</td><td>0.81</td><td>134.44</td><td>0.27</td></tr><tr><td>Average</td><td>78.95</td><td>25.04</td><td>77.11</td><td>49.74</td><td>2.08</td><td>147.38</td><td>0.29</td></tr></tbody></table>

The average values included in Table 1 were determined as follows (where *x* denotes a mechanical property),
$$ \bar{x} = \frac{x_0+2x_{45}+x_{90}}{4}. \qquad (1) $$

The anisotropy coefficient derived from the tensile tests on sheets aligned at a 45° angle to the rolling direction exhibits a significantly greater value compared to sheets aligned at 0° and 90°. This phenomenon was previously recognized by Sharma et al. [9] in sheets resulting from the accumulative roll-bonding manufacturing process. The average stress-strain curve can be approximated by the following Ludwik-Hollomon's equation,

$$ \bar{x} = 147.38 \bar{\epsilon}^{0.29}. \qquad (2) $$

Experimental Plane. Table 2 provides a summary of the in-plane compression tests employed for characterizing the WLC of the aluminum AA1050-O sheet. A minimum of five repetitions were conducted for each dimension among the four different lengths. The in-plane compression tests were executed under room temperature conditions and quasi-static operational settings, involving the monotonous application of a tensile load at a velocity of 5 mm/min. Specimens were cut from the supplied sheets perpendicular to the rolling direction of the sheet.

<table><caption>Table 2. Summary of the experimental work plan for the in-plane compression tests.</caption><thead><tr><th>Geometry</th><th>Thickness t (mm)</th><th>Length L (mm)</th><th>Compression Specimen (CS)</th></tr></thead><tbody><tr><td rowspan="4">[Image of geometry]</td><td rowspan="4">1.5</td><td>20</td><td>CS20</td></tr><tr><td>35</td><td>CS35</td></tr><tr><td>50</td><td>CS50</td></tr><tr><td>100</td><td>CS100</td></tr></tbody></table>

The necessary testing equipment for the in-plane compression tests includes an Instron universal testing machine, model SATEC 1200 KN, and a double-action tool system (Fig. 2a), that converts the vertical displacement of the Instron punch to a horizontal displacement [2]. Additionally, these experimental tests also require the tool to fix the in-plane compression test specimen to the double-action tool (Fig. 2b). Steel shims were also used to ensure uniform pressure distribution when tightening and create different boundary conditions when intended.

**Displacement and Strain Measurements.** The determination of the onset of wrinkling in the aluminum sheet material of in-plane compression test specimens involved a three-dimensional measurement of displacement and strain on the sheet surface, conducted using a DIC system, the Dantec Dynamics Model Q-400 3D. The specimen surfaces were coated with a stochastic black speckle pattern against a uniformly matte white background. Illumination was provided by a single spotlight directed at the test specimens. The DIC system is equipped with two cameras boasting a 6-megapixel resolution, a 50.2 mm focal length, and an aperture of f/11 to enhance depth of focus. Image acquisition occurred at a frequency of 10 frames per second, and the correlation algorithm utilized the INSTRA 4D software (version 4.6, Dantec Dynamics, Skovlunde, Denmark). A facet size of 17 pixels, along with a spacing grid of 13 pixels, was employed in the analysis.
$$ \bar{x} = \frac{x_0+2x_{45}+x_{90}}{4}. \qquad (1) $$

The anisotropy coefficient derived from the tensile tests on sheets aligned at a 45° angle to the rolling direction exhibits a significantly greater value compared to sheets aligned at 0° and 90°. This phenomenon was previously recognized by Sharma et al. [9] in sheets resulting from the accumulative roll-bonding manufacturing process. The average stress-strain curve can be approximated by the following Ludwik-Hollomon's equation,

$$ \bar{x} = 147.38 \bar{\epsilon}^{0.29}. \qquad (2) $$

Experimental Plane. Table 2 provides a summary of the in-plane compression tests employed for characterizing the WLC of the aluminum AA1050-O sheet. A minimum of five repetitions were conducted for each dimension among the four different lengths. The in-plane compression tests were executed under room temperature conditions and quasi-static operational settings, involving the monotonous application of a tensile load at a velocity of 5 mm/min. Specimens were cut from the supplied sheets perpendicular to the rolling direction of the sheet.

<table><caption>Table 2. Summary of the experimental work plan for the in-plane compression tests.</caption><thead><tr><th>Geometry</th><th>Thickness t (mm)</th><th>Length L (mm)</th><th>Compression Specimen (CS)</th></tr></thead><tbody><tr><td rowspan="4">[Image of geometry]</td><td rowspan="4">1.5</td><td>20</td><td>CS20</td></tr><tr><td>35</td><td>CS35</td></tr><tr><td>50</td><td>CS50</td></tr><tr><td>100</td><td>CS100</td></tr></tbody></table>

The necessary testing equipment for the in-plane compression tests includes an Instron universal testing machine, model SATEC 1200 KN, and a double-action tool system (Fig. 2a), that converts the vertical displacement of the Instron punch to a horizontal displacement [2]. Additionally, these experimental tests also require the tool to fix the in-plane compression test specimen to the double-action tool (Fig. 2b). Steel shims were also used to ensure uniform pressure distribution when tightening and create different boundary conditions when intended.

**Displacement and Strain Measurements.** The determination of the onset of wrinkling in the aluminum sheet material of in-plane compression test specimens involved a three-dimensional measurement of displacement and strain on the sheet surface, conducted using a DIC system, the Dantec Dynamics Model Q-400 3D. The specimen surfaces were coated with a stochastic black speckle pattern against a uniformly matte white background. Illumination was provided by a single spotlight directed at the test specimens. The DIC system is equipped with two cameras boasting a 6-megapixel resolution, a 50.2 mm focal length, and an aperture of f/11 to enhance depth of focus. Image acquisition occurred at a frequency of 10 frames per second, and the correlation algorithm utilized the INSTRA 4D software (version 4.6, Dantec Dynamics, Skovlunde, Denmark). A facet size of 17 pixels, along with a spacing grid of 13 pixels, was employed in the analysis.
<figure><img src="image_1455.png" /><figcaption>Fig. 2. Photograph of (a) the double-action tool system [2] and (b) the tool to fix the in-plane compression test specimens: 1) die shoe; 2) fixation bolts; 3) sheet holder block; 4) sheet holder bolts; 5) sheet holder block slot.</figcaption></figure>

Methodology

This paper proposes three physically-based methodologies to detect the onset of wrinkling and to determine the formability limits by wrinkling in sheet metal forming through experimental in-plane compression tests.

Methodology A. Methodology A relies on directly analyzing the Z displacement (perpendicular to the sheet surface) along the in-plane compression test specimen (section AA'), as depicted in Fig. 3a. The analysis of Z displacement along section AA' enables the determination of the amplitude of the plastic instability wave corresponding to the maximum Z displacement of section AA' (Fig. 3b). The schematic presentation of the evolution of the plastic instability wave over time along the in-plane compression test is illustrated in Fig. 3c. A critical amplitude of the plastic instability wave $h_{crit}$ can be defined as 5-10% of the sheet thickness, as initially proposed by Du et al. [10] for Yoshida buckling tests.

The instant when the amplitude of the plastic instability wave equals the critical amplitude marks the point in time for determining the critical strain pairs along the strain loading path of a point localized in the central zone of section AA', as schematically shown in Fig. 3d. These critical strain pairs, obtained for in-plane compression test specimens with different dimensions, are then utilized to construct the WLC in the principal strain space.

Methodology B. Methodology B involves the direct analysis of the in-plane minor strain within the central zone of the in-plane compression test specimen, as depicted in Fig. 3a as the measuring area. This area, where strains are measured by the DIC system, corresponds to the zone with the maximum Z displacement of section AA'. Before the onset of instability, the evolution of the in-plane minor strain in the central zone of the in-plane compression test specimen is influenced by the specimen's geometry. However, after the occurrence of plastic instability, the direction of the evolution of in-plane minor strain changes. The analysis of the strain rate (the first-time derivative of in-plane minor strain) enables the detection of this phenomenon, and the minimum point corresponds to the critical time when buckling is initiated, as shown in Fig. 4a. The critical time signifies the moment when critical strain pairs can be determined from the strain loading path of a
<figure><img src="image_1459.png" /><figcaption>Fig. 2. Photograph of (a) the double-action tool system [2] and (b) the tool to fix the in-plane compression test specimens: 1) die shoe; 2) fixation bolts; 3) sheet holder block; 4) sheet holder bolts; 5) sheet holder block slot.</figcaption></figure>

## Methodology

This paper proposes three physically-based methodologies to detect the onset of wrinkling and to determine the formability limits by wrinkling in sheet metal forming through experimental in-plane compression tests.

### Methodology A.
Methodology A relies on directly analyzing the Z displacement (perpendicular to the sheet surface) along the in-plane compression test specimen (section AA'), as depicted in Fig. 3a. The analysis of Z displacement along section AA' enables the determination of the amplitude of the plastic instability wave corresponding to the maximum Z displacement of section AA' (Fig. 3b). The schematic presentation of the evolution of the plastic instability wave over time along the in-plane compression test is illustrated in Fig. 3c. A critical amplitude of the plastic instability wave $h_{crit}$ can be defined as 5-10% of the sheet thickness, as initially proposed by Du et al. [10] for Yoshida buckling tests.

The instant when the amplitude of the plastic instability wave equals the critical amplitude marks the point in time for determining the critical strain pairs along the strain loading path of a point localized in the central zone of section AA', as schematically shown in Fig. 3d. These critical strain pairs, obtained for in-plane compression test specimens with different dimensions, are then utilized to construct the WLC in the principal strain space.

### Methodology B.
Methodology B involves the direct analysis of the in-plane minor strain within the central zone of the in-plane compression test specimen, as depicted in Fig. 3a as the measuring area. This area, where strains are measured by the DIC system, corresponds to the zone with the maximum Z displacement of section AA'. Before the onset of instability, the evolution of the in-plane minor strain in the central zone of the in-plane compression test specimen is influenced by the specimen's geometry. However, after the occurrence of plastic instability, the direction of the evolution of in-plane minor strain changes. The analysis of the strain rate (the first-time derivative of in-plane minor strain) enables the detection of this phenomenon, and the minimum point corresponds to the critical time when buckling is initiated, as shown in Fig. 4a. The critical time signifies the moment when critical strain pairs can be determined from the strain loading path of a
point located in the central zone of section AA', as depicted in Fig. 4b. These critical strain pairs, obtained for in-plane compression test specimens with different geometries, are then utilized to construct the WLC in the principal strain space.

<figure><img src="image_1456.png" /><figcaption>Fig. 3. Summary of methodology A: (a) schematic representation of the in-plane compression test; (b) identification of wrinkling amplitude; (c) evolution of the wrinkling amplitude over time; (d) representation of the strain loading path and identification of the critical strains in principal strain space.</figcaption></figure>

<figure><img src="image_1457.png" /><figcaption>Fig. 4. Summary of methodology B: (a) evolution of the in-plane minor strain and strain rate; (b) representation of the strain loading path and identification of the critical strains in principal strain space.</figcaption></figure>

Methodology C. Methodology C is based on the direct analyses of the force-displacement curves of the in-plane compression tests. The onset of wrinkling defined by this methodology occurs when a bifurcation point is reached in the force-displacement curve, as seen in Fig. 5a. The bifurcation
point located in the central zone of section AA', as depicted in Fig. 4b. These critical strain pairs, obtained for in-plane compression test specimens with different geometries, are then utilized to construct the WLC in the principal strain space.

<figure><img src="image_1460.png" /><figcaption>Fig. 3. Summary of methodology A: (a) schematic representation of the in-plane compression test; (b) identification of wrinkling amplitude; (c) evolution of the wrinkling amplitude over time; (d) representation of the strain loading path and identification of the critical strains in principal strain space.</figcaption></figure>

<figure><img src="image_1461.png" /><figcaption>Fig. 4. Summary of methodology B: (a) evolution of the in-plane minor strain and strain rate; (b) representation of the strain loading path and identification of the critical strains in principal strain space.</figcaption></figure>

Methodology C. Methodology C is based on the direct analyses of the force-displacement curves of the in-plane compression tests. The onset of wrinkling defined by this methodology occurs when a bifurcation point is reached in the force-displacement curve, as seen in Fig. 5a. The bifurcation
point occurs due to the existence of a secondary path $E_S$ with lower deformation energy than in the primary path $E_P$, as explained by Kim and Yang [11]. The bifurcation point corresponds to the critical time and at this instant, the critical strain pairs can be determined from the strain loading path of a point located in the central zone of section AA', as depicted in Fig. 5b. These critical strain pairs, obtained for in-plane compression test specimens with different geometries, are then utilized to construct the WLC in the principal strain space.

<figure><img src="image_1458.png" /><figcaption>Fig. 5. Summary of methodology C: (a) evolution of the forced displacement; (b) representation of the strain loading path and identification of the critical strains in principal strain space.</figcaption></figure>

## Results and Discussion

Methodology A. The position-dependent methodology defines the onset of wrinkling when the wrinkle amplitude is 5% of the sheet metal thickness. Since the thickness of the sheet metal is 1.5 mm, this means that the onset of wrinkling occurs when the wrinkle amplitude is 0.075 mm. This amplitude height is represented in Fig. 6a as the dashed-black line. Fig. 6a also presents the Z-displacement for the central point of each set of compression specimens, as represented in Fig. 6b. The rate at which the wrinkle amplitude increases is related to the increase in the length of the in-plane compression specimen, as is possible to verify the different rates of the wrinkle amplitude increase between the specimen with 20 mm and the specimen with 100 mm of length. The time difference between the onset of wrinkling defined by methodology A for the in-plane compression specimen with 20 and 100 mm of length is approximately 10s. Kasaei et al. [6] and Magrinho et al. [2] also registered that the onset of wrinkling occurs sooner for specimens with a higher length.

The critical instant corresponds to the instant of time that Z-displacement evolution intercepts the dashed-black line. For this instant, the critical strain pairs are determined to draw the WLC in the principal strain space.
point occurs due to the existence of a secondary path $E_S$ with lower deformation energy than in the primary path $E_P$, as explained by Kim and Yang [11]. The bifurcation point corresponds to the critical time and at this instant, the critical strain pairs can be determined from the strain loading path of a point located in the central zone of section AA', as depicted in Fig. 5b. These critical strain pairs, obtained for in-plane compression test specimens with different geometries, are then utilized to construct the WLC in the principal strain space.

<figure><img src="image_1462.png" /><figcaption>Fig. 5. Summary of methodology C: (a) evolution of the forced displacement; (b) representation of the strain loading path and identification of the critical strains in principal strain space.</figcaption></figure>

## Results and Discussion

Methodology A. The position-dependent methodology defines the onset of wrinkling when the wrinkle amplitude is 5% of the sheet metal thickness. Since the thickness of the sheet metal is 1.5 mm, this means that the onset of wrinkling occurs when the wrinkle amplitude is 0.075 mm. This amplitude height is represented in Fig. 6a as the dashed-black line. Fig. 6a also presents the Z-displacement for the central point of each set of compression specimens, as represented in Fig. 6b. The rate at which the wrinkle amplitude increases is related to the increase in the length of the in-plane compression specimen, as is possible to verify the different rates of the wrinkle amplitude increase between the specimen with 20 mm and the specimen with 100 mm of length. The time difference between the onset of wrinkling defined by methodology A for the in-plane compression specimen with 20 and 100 mm of length is approximately 10s. Kasaei et al. [6] and Magrinho et al. [2] also registered that the onset of wrinkling occurs sooner for specimens with a higher length.

The critical instant corresponds to the instant of time that Z-displacement evolution intercepts the dashed-black line. For this instant, the critical strain pairs are determined to draw the WLC in the principal strain space.
<figure><img src="image_1459.png" /><figcaption>Fig. 6. (a) Z-displacement of the center of each set of in-plane compression test specimens with a length of 20, 35, 50, and 100 mm; (b) Point A located in the center of the in-plane compression specimen.</figcaption></figure>

Methodology B. The physically-based methodology evaluates the in-plane minor strain rate evolution at the center of the in-plane compression test specimens, represented by the middle point along section AA' in Fig.s 7b and 7d. The in-plane minor strain and in-plane minor strain rate evolution over time at the center of test specimens with 20 and 100 mm of length are represented in Fig.s 7a and 7d, respectively.

By applying the same color scale to all specimens, it is possible to access the minor strain distribution on the surface of the in-plane compression test specimen. In Fig.s 7b and 7d it is possible to verify that there is a slight concentration of the in-plane minor strain in the direction perpendicular to the center of section AA'. Since all specimens wrinkled to the same side, away from the DIC system, the DIC cameras measured the bottom face of the wrinkle, which is the reason for the in-plane minor strain to have compressive values along the whole length of the specimen. The critical time instants for the onset of wrinkling defined by methodology B, show that when the length of the specimen increases, wrinkling occurs sooner. In addition, the critical in-plane minor strain continuously decreases as the length of the specimen increases. For this instant, the critical strain pairs are determined to draw the WLC in the principal strain space.

Methodology C. The onset of wrinkling determined by methodology C is defined by the bifurcation point, the maximum point, in the force-displacement curves presented in Fig. 8. The same procedure was also applied by Kasaei et al. [6] and Magrinho et al. [2] to determine the onset of wrinkling and the obtained results were similar. The force and displacement at which the bifurcation point occurs increases with the decrease in the specimen length.

Wrinkling Limit Curve. Fig. 9 represents the formability limits by wrinkling determined by in-plane compression tests in conjunction with the three different methodologies: A, B, and C. The WLCs were obtained by using the average critical strain pairs obtained from each methodology and imposing a polynomial regression of degree 2, which is the curve that best fits the behavior of the material in these experimental tests. Due to the difference in the average critical strain pairs obtained from methodology A when compared to those obtained from methodologies B and C, a 4x zoom is added to Fig. 9 to provide a more detailed view of the WLCs obtained from the latter methodologies.
<figure><img src="image_1463.png" /><figcaption>Fig. 6. (a) Z-displacement of the center of each set of in-plane compression test specimens with a length of 20, 35, 50, and 100 mm; (b) Point A located in the center of the in-plane compression specimen.</figcaption></figure>

Methodology B. The physically-based methodology evaluates the in-plane minor strain rate evolution at the center of the in-plane compression test specimens, represented by the middle point along section AA' in Fig.s 7b and 7d. The in-plane minor strain and in-plane minor strain rate evolution over time at the center of test specimens with 20 and 100 mm of length are represented in Fig.s 7a and 7d, respectively.

By applying the same color scale to all specimens, it is possible to access the minor strain distribution on the surface of the in-plane compression test specimen. In Fig.s 7b and 7d it is possible to verify that there is a slight concentration of the in-plane minor strain in the direction perpendicular to the center of section AA'. Since all specimens wrinkled to the same side, away from the DIC system, the DIC cameras measured the bottom face of the wrinkle, which is the reason for the in-plane minor strain to have compressive values along the whole length of the specimen. The critical time instants for the onset of wrinkling defined by methodology B, show that when the length of the specimen increases, wrinkling occurs sooner. In addition, the critical in-plane minor strain continuously decreases as the length of the specimen increases. For this instant, the critical strain pairs are determined to draw the WLC in the principal strain space.

Methodology C. The onset of wrinkling determined by methodology C is defined by the bifurcation point, the maximum point, in the force-displacement curves presented in Fig. 8. The same procedure was also applied by Kasaei et al. [6] and Magrinho et al. [2] to determine the onset of wrinkling and the obtained results were similar. The force and displacement at which the bifurcation point occurs increases with the decrease in the specimen length.

Wrinkling Limit Curve. Fig. 9 represents the formability limits by wrinkling determined by in-plane compression tests in conjunction with the three different methodologies: A, B, and C. The WLCs were obtained by using the average critical strain pairs obtained from each methodology and imposing a polynomial regression of degree 2, which is the curve that best fits the behavior of the material in these experimental tests. Due to the difference in the average critical strain pairs obtained from methodology A when compared to those obtained from methodologies B and C, a 4x zoom is added to Fig. 9 to provide a more detailed view of the WLCs obtained from the latter methodologies.
<figure><img src="image_1460.png" /></figure>

In-plane minor strain and in-plane minor strain rate evolution over time for specimens with lengths of (a) 20 and (c) 100 mm. In-plane minor strain rate distribution at the onset of wrinkling obtained from the DIC according to methodology B for specimens with lengths of (b) 20 and (d) 100 mm.

However, there is a common behavior in all specimens registered in all the methodologies, which corresponds to the following: the critical in-plane minor strain always increases in module with the decrease in the specimen length, while the in-plane major strain follows the same behavior except between the compression specimens with 35 and 20 mm of length, where it decreases. This behavior was also registered by Magrinho et al. [2] for aluminum alloy AA1050-O with thicknesses of 1, 2, 3, and 4 mm.

In Fig. 9 it is possible to verify that the ratio between in-plane minor and major strains for the critical strain pairs determined by methodology B and C for the specimens with a length of 100, 50, and 35 mm is approximately -1, while the specimens with a length of 20 mm have a $\beta = \varepsilon_2/\varepsilon_1 = -2$. For this reason, the WLC obtained from these methodologies initially has an approximate slope corresponding to pure shear and at the end tends to deviate towards the uniaxial compression deformation.
<figure><img src="image_1464.png" /></figure>

In-plane minor strain and in-plane minor strain rate evolution over time for specimens with lengths of (a) 20 and (c) 100 mm. In-plane minor strain rate distribution at the onset of wrinkling obtained from the DIC according to methodology B for specimens with lengths of (b) 20 and (d) 100 mm.

However, there is a common behavior in all specimens registered in all the methodologies, which corresponds to the following: the critical in-plane minor strain always increases in module with the decrease in the specimen length, while the in-plane major strain follows the same behavior except between the compression specimens with 35 and 20 mm of length, where it decreases. This behavior was also registered by Magrinho et al. [2] for aluminum alloy AA1050-O with thicknesses of 1, 2, 3, and 4 mm.

In Fig. 9 it is possible to verify that the ratio between in-plane minor and major strains for the critical strain pairs determined by methodology B and C for the specimens with a length of 100, 50, and 35 mm is approximately -1, while the specimens with a length of 20 mm have a $\beta = \varepsilon_2/\varepsilon_1 = -2$. For this reason, the WLC obtained from these methodologies initially has an approximate slope corresponding to pure shear and at the end tends to deviate towards the uniaxial compression deformation.
<figure><img src="image_1461.png" /><figcaption>Fig. 8. Evolution of the force with displacement and identification of the bifurcation point for the in-plane compression test specimens with lengths of 20, 35, 50, and 100 mm.</figcaption></figure>

<figure><img src="image_1462.png" /><figcaption>Fig. 9. Formability limits by wrinkling defined by methodology A, B, and C for AA1050-O aluminum sheet with 1.5 mm thickness.</figcaption></figure>

In contrast, the critical strain pairs obtained from methodology A at the onset of wrinkling for the specimens with a length of 100 mm have an approximate ratio of -1. A $\beta = -2$ is obtained for the specimens with lengths of 50 and 35 mm. The in-plane strains ratio proceeds to decrease for the specimens with 20 mm length, reaching a value of $\beta = -8$. Hence, the WLC determined by methodology A continuously tends towards slopes of increasing compression state as the length of the specimen decreases.

## Conclusions

In conclusion, this study introduces three robust methodologies (A, B, and C) for detecting the onset of wrinkling and characterizing the WLC in sheet metal forming processes. Through comprehensive experimental work, including mechanical characterization and in-plane compression tests on AA1050-O aluminum sheets with 1.5 mm thickness, the proposed methodologies exhibit their effectiveness in predicting wrinkling initiation under varying conditions.

Methodology A, focusing on Z displacement analysis, determines the onset of wrinkling by identifying the amplitude of the plastic instability wave. Methodology B, centered on in-plane minor strain rate evolution, utilizes DIC system measurements to detect critical times and strain pairs. Methodology C employs force-displacement curve analysis, identifying the bifurcation point to determine critical strain pairs.
<figure><img src="image_1465.png" /><figcaption>Fig. 8. Evolution of the force with displacement and identification of the bifurcation point for the in-plane compression test specimens with lengths of 20, 35, 50, and 100 mm.</figcaption></figure>

<figure><img src="image_1466.png" /><figcaption>Fig. 9. Formability limits by wrinkling defined by methodology A, B, and C for AA1050-O aluminum sheet with 1.5 mm thickness.</figcaption></figure>

In contrast, the critical strain pairs obtained from methodology A at the onset of wrinkling for the specimens with a length of 100 mm have an approximate ratio of -1. A $\beta = -2$ is obtained for the specimens with lengths of 50 and 35 mm. The in-plane strains ratio proceeds to decrease for the specimens with 20 mm length, reaching a value of $\beta = -8$. Hence, the WLC determined by methodology A continuously tends towards slopes of increasing compression state as the length of the specimen decreases.

## Conclusions

In conclusion, this study introduces three robust methodologies (A, B, and C) for detecting the onset of wrinkling and characterizing the WLC in sheet metal forming processes. Through comprehensive experimental work, including mechanical characterization and in-plane compression tests on AA1050-O aluminum sheets with 1.5 mm thickness, the proposed methodologies exhibit their effectiveness in predicting wrinkling initiation under varying conditions.

Methodology A, focusing on Z displacement analysis, determines the onset of wrinkling by identifying the amplitude of the plastic instability wave. Methodology B, centered on in-plane minor strain rate evolution, utilizes DIC system measurements to detect critical times and strain pairs. Methodology C employs force-displacement curve analysis, identifying the bifurcation point to determine critical strain pairs.
The study highlights the influence of specimen length on wrinkling initiation, with distinct behaviors observed in the three methodologies. The obtained critical strain pairs contribute to constructing WLCs, offering valuable insights into formability limits by wrinkling. This research advances our understanding of wrinkling phenomena, aiding in developing preventive measures and design strategies for lightweight structures in various industries.

## References

[1] M. Kleiner, M. Geiger, A. Klaus, Manufacturing of Lightweight Components by Metal Forming. CIRP Annals 52(2) (2003) 521-542. https://doi.org/10.1016/S0007-8506(07)60202-9

[2] J.P.G. Magrinho, C.M.A. Silva, M.B. Silva, P.A.F. Martins, Formability limits by wrinkling in sheet metal forming, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications 232(8) (2016) 681-692. https://doi.org/10.1177/1464420716642794

[3] J. Havranek, Wrinkling limit of tapered pressing, Journal of the Australian Institute of Metals 20 (1975) 114-119.

[4] P.A.F. Martins, N. Bay, A.E. Tekkaya, A.G. Atkins, Characterization of fracture loci in metal forming. International Journal of Mechanical Sciences 83 (2014) 112-123. https://doi.org/10.1016/j.ijmecsci.2014.04.003

[5] A.M. Szacinski, P.F. Thompson, Investigation of the existence of a wrinkling-limit curve in plastically deforming metal sheet, Journal of Materials Processing Technology 25(2) (1991) 125-137. https://doi.org/10.1016/0924-0136(91)90085-S

[6] M.M. Kasaei, H. Moslemi Naeini, G.H. Liaghat, C.M.A. Silva, M.B. Silva, P.A.F. Martins, Revisiting the wrinkling limits in flexible roll forming, The Journal of Strain Analysis for Engineering Design 50(7) (2015) 528-541. https://doi.org/10.1177/0309324715590956

[7] ASTM Standard E8/E8M, Standard Test Methods for Tension Testing of Metallic Materials, West Conshohocken, USA: ASTM International. 2016. https://doi.org/10.1520/E0008_E0008M-16A

[8] J.A.O. Santos, J.P.G. Magrinho, M.B. Silva, Physically-Based Methodology for the Characterization of Wrinkling Limit Curve Validated by Yoshida Tests, Metals 13(4) (2023) 746. https://doi.org/10.3390/met13040746

[9] S. Sharma, R.P. Singh, S. Kumar, Mechanical Anisotropy of Aluminium AA1050 and Aluminium Alloy AA6016 produced by Accumulative Roll Bonding, International Journal of Innovations Engineering and Technology 7 (2016) 350-356.

[10] B. Du, J. Xie, H. Li, C. Zhao, X. Zhang, X. Yuan, Determining factors affecting sheet metal plastic wrinkling in response to nonuniform tension using wrinkling limit diagrams, Thin-Walled Structures 147 (2020) 106535. https://doi.org/10.1016/j.tws.2019.106535

[11] J.B. Kim, D.Y. Yang, Prediction of wrinkling initiation in sheet metal forming processes, Engineering Computations 20(1) (2003) 6-39. https://doi.org/10.1108/02644400310458810
The study highlights the influence of specimen length on wrinkling initiation, with distinct behaviors observed in the three methodologies. The obtained critical strain pairs contribute to constructing WLCs, offering valuable insights into formability limits by wrinkling. This research advances our understanding of wrinkling phenomena, aiding in developing preventive measures and design strategies for lightweight structures in various industries.

## References

[1] M. Kleiner, M. Geiger, A. Klaus, Manufacturing of Lightweight Components by Metal Forming. CIRP Annals 52(2) (2003) 521-542. https://doi.org/10.1016/S0007-8506(07)60202-9

[2] J.P.G. Magrinho, C.M.A. Silva, M.B. Silva, P.A.F. Martins, Formability limits by wrinkling in sheet metal forming, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications 232(8) (2016) 681-692. https://doi.org/10.1177/1464420716642794

[3] J. Havranek, Wrinkling limit of tapered pressing, Journal of the Australian Institute of Metals 20 (1975) 114-119.

[4] P.A.F. Martins, N. Bay, A.E. Tekkaya, A.G. Atkins, Characterization of fracture loci in metal forming. International Journal of Mechanical Sciences 83 (2014) 112-123. https://doi.org/10.1016/j.ijmecsci.2014.04.003

[5] A.M. Szacinski, P.F. Thompson, Investigation of the existence of a wrinkling-limit curve in plastically deforming metal sheet, Journal of Materials Processing Technology 25(2) (1991) 125-137. https://doi.org/10.1016/0924-0136(91)90085-S

[6] M.M. Kasaei, H. Moslemi Naeini, G.H. Liaghat, C.M.A. Silva, M.B. Silva, P.A.F. Martins, Revisiting the wrinkling limits in flexible roll forming, The Journal of Strain Analysis for Engineering Design 50(7) (2015) 528-541. https://doi.org/10.1177/0309324715590956

[7] ASTM Standard E8/E8M, Standard Test Methods for Tension Testing of Metallic Materials, West Conshohocken, USA: ASTM International. 2016. https://doi.org/10.1520/E0008_E0008M-16A

[8] J.A.O. Santos, J.P.G. Magrinho, M.B. Silva, Physically-Based Methodology for the Characterization of Wrinkling Limit Curve Validated by Yoshida Tests, Metals 13(4) (2023) 746. https://doi.org/10.3390/met13040746

[9] S. Sharma, R.P. Singh, S. Kumar, Mechanical Anisotropy of Aluminium AA1050 and Aluminium Alloy AA6016 produced by Accumulative Roll Bonding, International Journal of Innovations Engineering and Technology 7 (2016) 350-356.

[10] B. Du, J. Xie, H. Li, C. Zhao, X. Zhang, X. Yuan, Determining factors affecting sheet metal plastic wrinkling in response to nonuniform tension using wrinkling limit diagrams, Thin-Walled Structures 147 (2020) 106535. https://doi.org/10.1016/j.tws.2019.106535

[11] J.B. Kim, D.Y. Yang, Prediction of wrinkling initiation in sheet metal forming processes, Engineering Computations 20(1) (2003) 6-39. https://doi.org/10.1108/02644400310458810
# Experimental and finite element investigation on the effect of process parameters in incremental forming of polymeric materials

RAGAI Ihab¹, a*, BUFFA Gianluca², b, VANDALINI Andrea², c and FRATINI Livan², d

¹School of Engineering, The Pennsylvania State University, Erie, PA 16563, USA

²Department of Engineering, University of Palermo, Viale Delle Scienze, 90128 Palermo, Italy

**Keywords:** Incremental Forming, Polymers, Formability, FE Analysis, Isothermal Process

**Abstract.** The purpose of this paper is to conduct finite element simulations of the incremental forming process of polycarbonate material. The process was simulated using the commercial software LS-Dyna. An explicit loading-implicit unloading approach for springback was set up, reproducing the experimental conditions. The shell element type and the number of integration points along the thickness were defined through a preliminary numerical approach. A spiral 3D surface was generated and measurements of cone geometry were conducted. The model was validated by comparing experimental and numerical thickness in a cross section, and finally used to investigate the effect of step size on the process mechanics. It was found that a decrease in step size would yield to a more uniform thinning along the profile cross section and produces larger thickness reduction.

## Introduction

Single-Point Incremental Forming (SPIF) is a merging process that proved to be economical for low-batch production and prototyping purposes. It involves securing the sheet material into a generic fixture, typically a frame-like plate. The forming process takes place by moving a hemispheric tool installed in a CNC machine running a specific code containing the desired geometry. With each pass, the sheet is deformed incrementally, each increment in this process is called the step size. The final shape is achieved by modifying the tool path and the step size. The process is gaining attraction in the forming field due to its low cost, high flexibility, and its capability of forming axisymmetric and asymmetric geometries [1].

There is a significant amount of SPIF-related research conducted on sheet metals [2-7]. However, there is still limited knowledge regarding the use of SPIF with polymeric materials and, therefore, it has been attracting more interest in recent years, due to the nature of failure mechanism, for example [8-11]. Previous research reported that process parameters have significant effect on formability of polymeric materials. Martins et al. [12] conducted research on five different materials, namely polyoxymethylene (POM), polyethylene (PE), polyamide (PA), polyvinyl chloride (PVC), and polycarbonate (PC). Sheets were formed at room temperature, constant spindle speed, and a conical toolpath was utilized. They highlighted the promising possibilities of forming thermoplastic materials. However, they also highlighted the influence of process parameters on formability.

In SPIF, formability can be defined as the maximum wall angle and depth that can be reached prior to failure. In addition to altering process parameters to increase formability, heating the specimens will also enable greater elongation due to the induced thermal softening of the material. Other methods such as tool modification [13,14], vibration-assisted [15,16], and axially-offset rotating tool [17,18] were also reported to increase formability; however, they are not as effective

<figure><img src="image_1463.png" /></figure>

<footnote>a ihab.ragai@psu.edu, b bgianluca.buffa@unipa.it</footnote>

<footnote>c andrea.vandalini@unipa.it, d livan.fratini@unipa.it</footnote>
# Experimental and finite element investigation on the effect of process parameters in incremental forming of polymeric materials

RAGAI Ihab¹, a*, BUFFA Gianluca², b, VANDALINI Andrea², c and FRATINI Livan², d

¹School of Engineering, The Pennsylvania State University, Erie, PA 16563, USA

²Department of Engineering, University of Palermo, Viale Delle Scienze, 90128 Palermo, Italy

**Keywords:** Incremental Forming, Polymers, Formability, FE Analysis, Isothermal Process

**Abstract.** The purpose of this paper is to conduct finite element simulations of the incremental forming process of polycarbonate material. The process was simulated using the commercial software LS-Dyna. An explicit loading-implicit unloading approach for springback was set up, reproducing the experimental conditions. The shell element type and the number of integration points along the thickness were defined through a preliminary numerical approach. A spiral 3D surface was generated and measurements of cone geometry were conducted. The model was validated by comparing experimental and numerical thickness in a cross section, and finally used to investigate the effect of step size on the process mechanics. It was found that a decrease in step size would yield to a more uniform thinning along the profile cross section and produces larger thickness reduction.

## Introduction

Single-Point Incremental Forming (SPIF) is a merging process that proved to be economical for low-batch production and prototyping purposes. It involves securing the sheet material into a generic fixture, typically a frame-like plate. The forming process takes place by moving a hemispheric tool installed in a CNC machine running a specific code containing the desired geometry. With each pass, the sheet is deformed incrementally, each increment in this process is called the step size. The final shape is achieved by modifying the tool path and the step size. The process is gaining attraction in the forming field due to its low cost, high flexibility, and its capability of forming axisymmetric and asymmetric geometries [1].

There is a significant amount of SPIF-related research conducted on sheet metals [2-7]. However, there is still limited knowledge regarding the use of SPIF with polymeric materials and, therefore, it has been attracting more interest in recent years, due to the nature of failure mechanism, for example [8-11]. Previous research reported that process parameters have significant effect on formability of polymeric materials. Martins et al. [12] conducted research on five different materials, namely polyoxymethylene (POM), polyethylene (PE), polyamide (PA), polyvinyl chloride (PVC), and polycarbonate (PC). Sheets were formed at room temperature, constant spindle speed, and a conical toolpath was utilized. They highlighted the promising possibilities of forming thermoplastic materials. However, they also highlighted the influence of process parameters on formability.

In SPIF, formability can be defined as the maximum wall angle and depth that can be reached prior to failure. In addition to altering process parameters to increase formability, heating the specimens will also enable greater elongation due to the induced thermal softening of the material. Other methods such as tool modification [13,14], vibration-assisted [15,16], and axially-offset rotating tool [17,18] were also reported to increase formability; however, they are not as effective

<figure><img src="image_1467.png" /></figure>

<footnote>a ihab.ragai@psu.edu, b bgianluca.buffa@unipa.it</footnote>

<footnote>c andrea.vandalini@unipa.it, d livan.fratini@unipa.it</footnote>
as heat-assisted forming. Furthermore, it was reported that toolpath modifications and multi-pass will also result in an increase in formability [19,20].

One variable that has been investigated in previous studies is the friction between the tool and the blank and its effect on formability. Bagudanch et al. [21] reported that formability will generally increase due to the increased temperature caused by friction at higher spindle speeds. However, this observation is not necessarily true in all cases. Materials melting and glass-transition temperatures ($T_g$) played a significant role. When forming temperatures increased, materials with lower melting temperature, such as polypropylene (PP), experienced premature failures; on the other hand, materials with higher $T_g$, such as PC and PVC, had better formability and experienced a ductile failure. Ragai et al. [1] investigated the effect of tool-workpiece interaction. In their work, they used three different tool materials, namely stainless steel (SS), copper 110 (CU-110), and beryllium-copper (BeCu) as well as three different sheet materials, namely PP, PVC, and PC. They reported that the higher the spindle speed (temperature also increases) the higher the surface roughness of the workpiece, which led to lower formability due to surface deterioration. They also observed that BeCu and Cu-110 tools showed the most promising results in terms of surface condition and formability.

Furthermore, Upcraft et al. [22] conducted a full-factorial 3 × 3 design of experiment test matrix to investigate the effect of feed rate, starting angle, and step size on formability of PP, PVC, and PC materials. It was reported that higher springback would occur with the increase of wall angle. Also, a decrease in wall angle yields a more uniform thinning along the profile cross-section. It was also mentioned that feed rates have no significant effect on material thinning, particularly for 10° wall angle.

In the study presented herein, the SPIF process of polycarbonate sheets is both experimentally and numerically investigated. The purpose of this work is to examine the possibility of simulating thermoplastic materials undergoing isothermal forming processes. Feed rate and wall angle were kept constant while the step size varied. The FE simulations are validated with the experimental results to determine the models' accuracy and validity.

## Materials and Methods

The material tested in the current study is extruded PC sheets. The sheets were received with initial dimensions of 304.8 mm × 304.8 mm [12 in × 12 in] and a nominal thickness of 1.60 mm [0.0625 in], with a tolerance of ±0.076 mm [0.003 in]. Material properties for the sheets are listed in Table 1 [23]. The specimens were then cut to the desired dimensions of 152.4 mm × 152.4 mm and 8 screw holes were drilled along the outer perimeter to help secure the sheets in the fixture.

The forming tool used in this work was BeCu, since it provided better tool-sheet interaction in SPIF, particularly with PC [1]. The geometry of the tool consists of a hemispherical pin with 12.7 mm [0.50 in] diameter and 114 mm [4.50 in] in total length. Table 1 lists the typical mechanical properties for the tool materials [24]. Experiments took place in a HAAS TM 1P CNC milling machine. The square sheets were placed in an aluminum fixture that has a 115 mm [4.5 in] square pocket and provided a work depth of 50 mm [2 in]. Figure 1 shows the experimental setup.

<table><caption>Table 1. Nominal material properties</caption><thead><tr><th>Material</th><th>Yield Strength MPa [psi]</th><th>Ultimate Strength MPa [psi]</th><th>Elongation [%]</th><th>Young's Modulus GPa [ksi]</th><th>Shear Modulus MPa [psi]</th><th>Rockwell Hardness</th><th>Density g/cm³ [lb/in³]</th></tr></thead><tbody><tr><td>PC</td><td>62 [9,000]</td><td>66 [9,500]</td><td>110</td><td>2.34 [340]</td><td>786 [114,000]</td><td>M70/R118</td><td>1.19 [0.043]</td></tr><tr><td>BeCu</td><td>1000 [145]</td><td>1170 [170]</td><td>5</td><td>131 [19,000]</td><td>-</td><td>C40</td><td>8.36 [0.302]</td></tr></tbody></table>
as heat-assisted forming. Furthermore, it was reported that toolpath modifications and multi-pass will also result in an increase in formability [19,20].

One variable that has been investigated in previous studies is the friction between the tool and the blank and its effect on formability. Bagudanch et al. [21] reported that formability will generally increase due to the increased temperature caused by friction at higher spindle speeds. However, this observation is not necessarily true in all cases. Materials melting and glass-transition temperatures ($T_g$) played a significant role. When forming temperatures increased, materials with lower melting temperature, such as polypropylene (PP), experienced premature failures; on the other hand, materials with higher $T_g$, such as PC and PVC, had better formability and experienced a ductile failure. Ragai et al. [1] investigated the effect of tool-workpiece interaction. In their work, they used three different tool materials, namely stainless steel (SS), copper 110 (CU-110), and beryllium-copper (BeCu) as well as three different sheet materials, namely PP, PVC, and PC. They reported that the higher the spindle speed (temperature also increases) the higher the surface roughness of the workpiece, which led to lower formability due to surface deterioration. They also observed that BeCu and Cu-110 tools showed the most promising results in terms of surface condition and formability.

Furthermore, Upcraft et al. [22] conducted a full-factorial 3 × 3 design of experiment test matrix to investigate the effect of feed rate, starting angle, and step size on formability of PP, PVC, and PC materials. It was reported that higher springback would occur with the increase of wall angle. Also, a decrease in wall angle yields a more uniform thinning along the profile cross-section. It was also mentioned that feed rates have no significant effect on material thinning, particularly for 10° wall angle.

In the study presented herein, the SPIF process of polycarbonate sheets is both experimentally and numerically investigated. The purpose of this work is to examine the possibility of simulating thermoplastic materials undergoing isothermal forming processes. Feed rate and wall angle were kept constant while the step size varied. The FE simulations are validated with the experimental results to determine the models' accuracy and validity.

## Materials and Methods

The material tested in the current study is extruded PC sheets. The sheets were received with initial dimensions of 304.8 mm × 304.8 mm [12 in × 12 in] and a nominal thickness of 1.60 mm [0.0625 in], with a tolerance of ±0.076 mm [0.003 in]. Material properties for the sheets are listed in Table 1 [23]. The specimens were then cut to the desired dimensions of 152.4 mm × 152.4 mm and 8 screw holes were drilled along the outer perimeter to help secure the sheets in the fixture.

The forming tool used in this work was BeCu, since it provided better tool-sheet interaction in SPIF, particularly with PC [1]. The geometry of the tool consists of a hemispherical pin with 12.7 mm [0.50 in] diameter and 114 mm [4.50 in] in total length. Table 1 lists the typical mechanical properties for the tool materials [24]. Experiments took place in a HAAS TM 1P CNC milling machine. The square sheets were placed in an aluminum fixture that has a 115 mm [4.5 in] square pocket and provided a work depth of 50 mm [2 in]. Figure 1 shows the experimental setup.

<table><caption>Table 1. Nominal material properties</caption><thead><tr><th>Material</th><th>Yield Strength MPa [psi]</th><th>Ultimate Strength MPa [psi]</th><th>Elongation [%]</th><th>Young's Modulus GPa [ksi]</th><th>Shear Modulus MPa [psi]</th><th>Rockwell Hardness</th><th>Density g/cm<sup>3</sup> [lb/in<sup>3</sup>]</th></tr></thead><tbody><tr><td>PC</td><td>62 [9,000]</td><td>66 [9,500]</td><td>110</td><td>2.34 [340]</td><td>786 [114,000]</td><td>M70/R118</td><td>1.19 [0.043]</td></tr><tr><td>BeCu</td><td>1000 [145]</td><td>1170 [170]</td><td>5</td><td>131 [19,000]</td><td>-</td><td>C40</td><td>8.36 [0.302]</td></tr></tbody></table>
<figure><img src="image_1464.png" /><figcaption>Fig. 1. Experimental setup</figcaption></figure>

The formed part was a cone-shaped geometry that was generated using Mastercam. The theoretical geometry of the formed shape is illustrated in Fig. 2. For the research presented herein, the feed rate was kept constant at 2000 mm/min, the starting angle at 20 degrees, and the step size took values of 0.1, 0.5, and 1.0 mm. The program ran until completion or sample failure.

<figure><img src="image_1465.png" /><figcaption>Fig. 2. Cone geometry</figcaption></figure>

After the cones were formed, a Sharpe and Gage 2000 CMM was utilized to measure the profile at 14 points taken across the profile, as shown in Fig. 3. The horizontal (X-) and vertical (Z-) axes of the coordinate machine were free to move while the transverse (Y-) axis of the machine was locked. This ensures accurate measurements along the X-Z plane. These points are then used to represent the formed geometry. Additionally, a Magna-Mike 8600, with a 1.5875 mm [0.0625 in] ball, was utilized to measure the thickness of the profile at the aforementioned datum points.
<figure><img src="image_1468.png" /><figcaption>Fig. 1. Experimental setup</figcaption></figure>

The formed part was a cone-shaped geometry that was generated using Mastercam. The theoretical geometry of the formed shape is illustrated in Fig. 2. For the research presented herein, the feed rate was kept constant at 2000 mm/min, the starting angle at 20 degrees, and the step size took values of 0.1, 0.5, and 1.0 mm. The program ran until completion or sample failure.

<figure><img src="image_1469.png" /><figcaption>Fig. 2. Cone geometry</figcaption></figure>

After the cones were formed, a Sharpe and Gage 2000 CMM was utilized to measure the profile at 14 points taken across the profile, as shown in Fig. 3. The horizontal (X-) and vertical (Z-) axes of the coordinate machine were free to move while the transverse (Y-) axis of the machine was locked. This ensures accurate measurements along the X-Z plane. These points are then used to represent the formed geometry. Additionally, a Magna-Mike 8600, with a 1.5875 mm [0.0625 in] ball, was utilized to measure the thickness of the profile at the aforementioned datum points.
<figure><img src="image_1466.png" /><figcaption>Fig. 3. Profile measurement points</figcaption></figure>

Commercial finite-element-analysis software (ABAQUS/Explicit) was utilized to conduct the numerical simulations. The workpiece was modeled using S4R shell elements with a thickness of 1.6 mm, featuring 5 integration points through the thickness and a mesh size of 1 mm in the tool movement zone. Outside this zone, a coarser mesh characterized by an average size of 2mm was used and a transition zone between the finer and coarser mesh areas was assigned. The tool was treated as a rigid body, and a friction coefficient of 0.05 was set for the interaction between the tool and the workpiece. The experimental fixture effect was modeled by applying encastre boundary conditions to all the nodes at the edge of the fixture. As the tool movement is regarded, this was derived from the G-code used for the experiments and provided, for each process condition, in tabular form. Figure 4 shows the model ready for the simulation.

<figure><img src="image_1467.png" /><figcaption>Fig. 4. Numerical model mesh at the beginning of the process</figcaption></figure>

The material's elastic behavior was characterized by a Young's modulus (E) of 2.34 GPa and a Poisson's ratio (v) of 0.38. Plastic behavior was defined in the form of tabular data according to the information provided by the supplier, as listed in Table 1. For the springback prediction, an implicit analysis was carried out.

## Experimental Results

In the current study, the starting angle of 20° is considered a moderate angle compared to a shallow 10° angle, for example. Therefore, the formed geometry produced a U-shaped profile, as opposed to a “V-” shape for smaller angles. Additionally, all experiments ran until program completion as no failures were detected. Springback can be defined as the deviation of the incrementally formed
<figure><img src="image_1470.png" /><figcaption>Fig. 3. Profile measurement points</figcaption></figure>

Commercial finite-element-analysis software (ABAQUS/Explicit) was utilized to conduct the numerical simulations. The workpiece was modeled using S4R shell elements with a thickness of 1.6 mm, featuring 5 integration points through the thickness and a mesh size of 1 mm in the tool movement zone. Outside this zone, a coarser mesh characterized by an average size of 2mm was used and a transition zone between the finer and coarser mesh areas was assigned. The tool was treated as a rigid body, and a friction coefficient of 0.05 was set for the interaction between the tool and the workpiece. The experimental fixture effect was modeled by applying encastre boundary conditions to all the nodes at the edge of the fixture. As the tool movement is regarded, this was derived from the G-code used for the experiments and provided, for each process condition, in tabular form. Figure 4 shows the model ready for the simulation.

<figure><img src="image_1471.png" /><figcaption>Fig. 4. Numerical model mesh at the beginning of the process</figcaption></figure>

The material's elastic behavior was characterized by a Young's modulus (E) of 2.34 GPa and a Poisson's ratio (v) of 0.38. Plastic behavior was defined in the form of tabular data according to the information provided by the supplier, as listed in Table 1. For the springback prediction, an implicit analysis was carried out.

## Experimental Results

In the current study, the starting angle of 20° is considered a moderate angle compared to a shallow 10° angle, for example. Therefore, the formed geometry produced a U-shaped profile, as opposed to a “V-” shape for smaller angles. Additionally, all experiments ran until program completion as no failures were detected. Springback can be defined as the deviation of the incrementally formed
shape from the desired, theoretical, geometry. Figure 5 shows the profile for each step size with respect to the horizontal, x-position. It is observed that all samples reached an average height of approximately 29 mm. As mentioned earlier, no failure was detected along the walls of the profile. Additionally, it is shown that step size has some minor effect on both formability and shape deviation or springback. For the given values of step sizes (0.1, 0.5, 1.0 mm), it is shown that the decrease in step size will result in a slight increase in formability. This could be attributed to the multiple passes a location would experience in the case of smaller step size. In other words, with a tool diameter of 12.7 mm, a step size of 1.0 mm will not have the same number of overlapping spiral passes as the 0.1 mm step size, it will have less passes. Thus, the material will deform less at these locations. This is also observed in the run time, the 0.1 mm step size required over 2 hours of machine operation versus slightly over 20 minutes for the 1.0 mm step size. Also, springback is observed, as shown by the offset of the profiles in Figure 5. However, it was not possible to experimentally quantify the effect of this process parameter since the overall geometries overlapped or crossed over at certain locations, particularly for the 0.5 mm step size. The offset of the righthand side of the profile could be attributed to the way the sample is mounted in the fixture during measurements and is worth investigating in future studies.

<figure><img src="image_1468.png" /><figcaption>Fig. 5. Shape profiles for incrementally formed PC sheets at different step sizes
(Process parameters: feed = 2000 mm/min, wall angle = 20°, spindle = free spin)</figcaption></figure>

Thickness at the datum points was recorded and plotted against the horizontal position, as shown in Figure 6. It is expected that since the decrease in step size results in increased formability, the side walls of the profile would experience uniform thinning as the material stretches. In the smaller step size experiments, the material experiences larger thickness reduction. The purpose of reporting the full profile, rather than only one half for axisymmetry, is to highlight possible slight differences in the profile due to the way the part is fixed. To quantitatively capture the amount of thickness reduction along the profile wall, the per cent reduction (compared to 1.6 mm initial thickness) was calculated at the datum points previously presented in Fig. 3. Taking into consideration that the shape is axisymmetric and the differences along the profile can be neglected, Fig. 7 shows that the maximum thickness reduction for all samples is averaged at 52%. It can be generalized that the decrease in step size would result in an increase in thickness reduction of the material. Points (7,8) and (2,13) are at the transition sections and therefore experienced the minimum forming and minimum change in thickness.
shape from the desired, theoretical, geometry. Figure 5 shows the profile for each step size with respect to the horizontal, x-position. It is observed that all samples reached an average height of approximately 29 mm. As mentioned earlier, no failure was detected along the walls of the profile. Additionally, it is shown that step size has some minor effect on both formability and shape deviation or springback. For the given values of step sizes (0.1, 0.5, 1.0 mm), it is shown that the decrease in step size will result in a slight increase in formability. This could be attributed to the multiple passes a location would experience in the case of smaller step size. In other words, with a tool diameter of 12.7 mm, a step size of 1.0 mm will not have the same number of overlapping spiral passes as the 0.1 mm step size, it will have less passes. Thus, the material will deform less at these locations. This is also observed in the run time, the 0.1 mm step size required over 2 hours of machine operation versus slightly over 20 minutes for the 1.0 mm step size. Also, springback is observed, as shown by the offset of the profiles in Figure 5. However, it was not possible to experimentally quantify the effect of this process parameter since the overall geometries overlapped or crossed over at certain locations, particularly for the 0.5 mm step size. The offset of the righthand side of the profile could be attributed to the way the sample is mounted in the fixture during measurements and is worth investigating in future studies.

<figure><img src="image_1472.png" /><figcaption>Fig. 5. Shape profiles for incrementally formed PC sheets at different step sizes
(Process parameters: feed = 2000 mm/min, wall angle = 20°, spindle = free spin)</figcaption></figure>

Thickness at the datum points was recorded and plotted against the horizontal position, as shown in Figure 6. It is expected that since the decrease in step size results in increased formability, the side walls of the profile would experience uniform thinning as the material stretches. In the smaller step size experiments, the material experiences larger thickness reduction. The purpose of reporting the full profile, rather than only one half for axisymmetry, is to highlight possible slight differences in the profile due to the way the part is fixed. To quantitatively capture the amount of thickness reduction along the profile wall, the per cent reduction (compared to 1.6 mm initial thickness) was calculated at the datum points previously presented in Fig. 3. Taking into consideration that the shape is axisymmetric and the differences along the profile can be neglected, Fig. 7 shows that the maximum thickness reduction for all samples is averaged at 52%. It can be generalized that the decrease in step size would result in an increase in thickness reduction of the material. Points (7,8) and (2,13) are at the transition sections and therefore experienced the minimum forming and minimum change in thickness.
<figure><img src="image_1469.png" /><figcaption>Fig. 6. Thickness distribution for incrementally formed PC sheets at different step sizes
(Process parameters: feed = 2000 mm/min, wall angle = 20°, spindle = free spin)</figcaption></figure>

<figure><img src="image_1470.png" /><figcaption>Fig. 7. Thickness reduction % for incrementally formed PC sheets at different step sizes
(Process parameters: feed = 2000 mm/min, wall angle = 20°, spindle = free spin)</figcaption></figure>

Simulation Results

First the numerical model was validated against part profile and thickness distribution in a cross section. Figure 8 shows the comparison between measured and calculated data for the case study characterized by step size of 1mm. From the figure, it can be seen that good agreement is obtained for both thickness distribution (Fig. 8a) and profile height (Fig. 8b), although a few discrepancies can be observed especially in profile height. This is due to the springback step simulation which will be fine-tuned (ongoing work) in order to produce more accurate results.
<figure><img src="image_1473.png" /><figcaption>Fig. 6. Thickness distribution for incrementally formed PC sheets at different step sizes
(Process parameters: feed = 2000 mm/min, wall angle = 20°, spindle = free spin)</figcaption></figure>

<figure><img src="image_1474.png" /><figcaption>Fig. 7. Thickness reduction % for incrementally formed PC sheets at different step sizes
(Process parameters: feed = 2000 mm/min, wall angle = 20°, spindle = free spin)</figcaption></figure>

Simulation Results

First the numerical model was validated against part profile and thickness distribution in a cross section. Figure 8 shows the comparison between measured and calculated data for the case study characterized by step size of 1mm. From the figure, it can be seen that good agreement is obtained for both thickness distribution (Fig. 8a) and profile height (Fig. 8b), although a few discrepancies can be observed especially in profile height. This is due to the springback step simulation which will be fine-tuned (ongoing work) in order to produce more accurate results.
<figure><img src="image_1471.png" /><figcaption>Fig. 8. Comparison of experimental and numerical (a) thickness distribution and (b) profile height for a step size = 1.0 mm</figcaption></figure>

The model was then used to predict the equivalent plastic strain that the material experiences during the process. Figure 9a shows the evolution of the plastic strain during the simulation, while Figure 9b shows the trends of the same variable at the end of the simulation, for the three considered case studies.

Due to the increasing wall angle of the cone geometry, strain increases to a maximum value, corresponding to the fillet area, before dropping to zero at the bottom of the cone. This trend is common for the three step-down values. However, a major difference is found in the peak value in the case of 0.1 mm step size. As experimentally observed, in these process conditions a larger thickness reduction is measured, particularly in the control points closer to the fillet zone, i.e., points 6-9. As to the numerical prediction, similar maximum strain values are calculated for step sizes of 1.0 and 0.5 mm, corresponding to about 2, while a slightly larger value is predicted for the step size of 0.1 mm condition, for which maximum plastic strain reaches a value of about 2.5. For the latter case, the area interested by the larger strain values and the largest difference with respect to the other two cases, corresponds to a distance from the edge ranging from about 25 mm to about 35 mm. This result is in good agreement with the ones shown in Figs. 5 and 7.
<figure><img src="image_1475.png" /><figcaption>Fig. 8. Comparison of experimental and numerical (a) thickness distribution and (b) profile height for a step size = 1.0 mm</figcaption></figure>

The model was then used to predict the equivalent plastic strain that the material experiences during the process. Figure 9a shows the evolution of the plastic strain during the simulation, while Figure 9b shows the trends of the same variable at the end of the simulation, for the three considered case studies.

Due to the increasing wall angle of the cone geometry, strain increases to a maximum value, corresponding to the fillet area, before dropping to zero at the bottom of the cone. This trend is common for the three step-down values. However, a major difference is found in the peak value in the case of 0.1 mm step size. As experimentally observed, in these process conditions a larger thickness reduction is measured, particularly in the control points closer to the fillet zone, i.e., points 6-9. As to the numerical prediction, similar maximum strain values are calculated for step sizes of 1.0 and 0.5 mm, corresponding to about 2, while a slightly larger value is predicted for the step size of 0.1 mm condition, for which maximum plastic strain reaches a value of about 2.5. For the latter case, the area interested by the larger strain values and the largest difference with respect to the other two cases, corresponds to a distance from the edge ranging from about 25 mm to about 35 mm. This result is in good agreement with the ones shown in Figs. 5 and 7.