response time for the elastic recovery of sheet material at higher tool rotational speeds. By comparing the simulated amplitudes under the same rotational speed of 3000 rpm or 6000 rpm, or under the same vibration frequency of T2 with 6000 rpm and T4 with 3000 rpm, it is found that the vibration amplitudes produced by T4 is always smaller than that of T2.

### Effect of tool design and rotational speed on forming temperature

As observed from Fig. 5, the rotation of both T2 and T4 tools can lead to temperature rise, resulted from the friction heat and plastic work. For all the cases with tool rotation, the forming temperature firstly increases to a peak value, and then keeps decreasing until the end of the process. The possible reason for the decreasing trend of forming temperature at later stage may be the reduced heat gain from friction due to the reduction of forming force at higher drawing angles. The measured forming temperature variations with different tools are shown in Fig. 5(a). With the same rotational speed of 3000 rpm or 6000 rpm, the RV-ISF with T4 tool has higher forming temperature than that with T2 tool, which mainly results from the larger contact area with T4 tool. The temperature difference between RV-ISF with T4 tool and T2 tool was much enlarged when higher rotational speed of 6000 rpm was used. For T4 tool, the effect of rotational speed on the forming temperature is depicted in Fig. 5(b). It is clearly shown that the increase of rotational speed has a positive effect on the temperature rise in the RV-ISF with T4.

<figure><img src="image_1508.png" /><figcaption>Fig. 5. Effect of (a) tool design and (b) rotational speed on the forming temperature from the experimental measurement.</figcaption></figure>

### Effect of tool design and rotational speed on vertical forming force

The effect of tool design on the vertical forming force of the RV-ISF is shown in Fig. 6(a). By comparing with the result of the C-ISF with the conventional tool T0, the sheet vibration induced by tool rotation of T2 and T4 leads to the significant reduction of the vertical forming force, and large force variations especially at the later stage. The force reduction is resulted from the easier dislocation activation under higher vibration energy density $E_v$, which is proportional to the square of vibration amplitude and frequency, respectively, as given by Eq. (4) [12]:

$$E_v = 2 \cdot \pi^2 \cdot \xi^2 \cdot f^2 \cdot \rho \quad (4)$$

where $\xi$ and $f$ are the vibration amplitude and frequency, respectively, and $\rho$ is the material density. With the same rotational speed of 3000 rpm, the use of T2 is more effective in reducing forming force because T2 can provide larger vibration amplitude for the sheet material, as analysed
response time for the elastic recovery of sheet material at higher tool rotational speeds. By comparing the simulated amplitudes under the same rotational speed of 3000 rpm or 6000 rpm, or under the same vibration frequency of T2 with 6000 rpm and T4 with 3000 rpm, it is found that the vibration amplitudes produced by T4 is always smaller than that of T2.

### Effect of tool design and rotational speed on forming temperature

As observed from Fig. 5, the rotation of both T2 and T4 tools can lead to temperature rise, resulted from the friction heat and plastic work. For all the cases with tool rotation, the forming temperature firstly increases to a peak value, and then keeps decreasing until the end of the process. The possible reason for the decreasing trend of forming temperature at later stage may be the reduced heat gain from friction due to the reduction of forming force at higher drawing angles. The measured forming temperature variations with different tools are shown in Fig. 5(a). With the same rotational speed of 3000 rpm or 6000 rpm, the RV-ISF with T4 tool has higher forming temperature than that with T2 tool, which mainly results from the larger contact area with T4 tool. The temperature difference between RV-ISF with T4 tool and T2 tool was much enlarged when higher rotational speed of 6000 rpm was used. For T4 tool, the effect of rotational speed on the forming temperature is depicted in Fig. 5(b). It is clearly shown that the increase of rotational speed has a positive effect on the temperature rise in the RV-ISF with T4.

<figure><img src="image_1512.png" /><figcaption>Fig. 5. Effect of (a) tool design and (b) rotational speed on the forming temperature from the experimental measurement.</figcaption></figure>

### Effect of tool design and rotational speed on vertical forming force

The effect of tool design on the vertical forming force of the RV-ISF is shown in Fig. 6(a). By comparing with the result of the C-ISF with the conventional tool T0, the sheet vibration induced by tool rotation of T2 and T4 leads to the significant reduction of the vertical forming force, and large force variations especially at the later stage. The force reduction is resulted from the easier dislocation activation under higher vibration energy density $E_v$, which is proportional to the square of vibration amplitude and frequency, respectively, as given by Eq. (4) [12]:

$$E_v = 2 \cdot \pi^2 \cdot \xi^2 \cdot f^2 \cdot \rho \quad (4)$$

where $\xi$ and $f$ are the vibration amplitude and frequency, respectively, and $\rho$ is the material density. With the same rotational speed of 3000 rpm, the use of T2 is more effective in reducing forming force because T2 can provide larger vibration amplitude for the sheet material, as analysed
by FE simulation. With the same sheet vibration frequency, i.e., T4 with 3000 rpm and T2 with 6000 rpm, the effect of T2 for the forming force reduction is still greater than that of T4.

<figure><img src="image_1509.png" /><figcaption>Fig. 6. Effect of (a) tool design and (b) rotational speed on the forming force reduction from the experimental measurement.</figcaption></figure>

The effect of rotational speed on vertical forming force by using T4 is studied by comparing the force variation under different rotational speeds, as shown in Fig. 6(b). The vertical forming force is reduced with the increase of rotational speed or vibration frequency. The two possible explanations are: (1) the enhanced temperature rise caused by more friction heating at high rotational speeds; (2) the early dislocation activation due to higher vibration energy density with the increase of rotational speeds, as indicated by Eq. (4).

## Effect of tool design and rotational speed on formability

The effects of tool design and rotational speed on the formability are shown in Fig. 7, in which fracture depth and equivalent fracture strain are used for comparison. By comparing the fracture depth and equivalent fracture strain of parts formed by T0, T2 and T4 at different rotational speeds, it is observed that the parts formed by T4 tool has achieved the highest formability. The reason may be that, compared with the temperature rise by T0 and the low-frequency-high-amplitude vibration with limited temperature rise by T2, T4 can provide both sufficient localized temperature rise and low-frequency-low-amplitude vibration, which may be useful for the improvement of material ductility.

By observing the fracture depth and equivalent fracture strain results by using T4, with the increase of rotational speed, the formability of the formed parts increases first but this is followed by a decreasing trend. The part formed by T4 has the highest formability when the rotational speed is 6000 rpm, and the highest equivalent fracture strain reaches 2.49. The reason for the initial increased formability is resulted from the temperature rise, increased vibration frequency and decreased vibration amplitude. The subsequent decreasing trend with higher rotational speeds may be that the sheet material is excessively softened and the deformation become unstable, so that fracture is initiated and propagated earlier.
by FE simulation. With the same sheet vibration frequency, i.e., T4 with 3000 rpm and T2 with 6000 rpm, the effect of T2 for the forming force reduction is still greater than that of T4.

<figure><img src="image_1513.png" /><figcaption>Fig. 6. Effect of (a) tool design and (b) rotational speed on the forming force reduction from the experimental measurement.</figcaption></figure>

The effect of rotational speed on vertical forming force by using T4 is studied by comparing the force variation under different rotational speeds, as shown in Fig. 6(b). The vertical forming force is reduced with the increase of rotational speed or vibration frequency. The two possible explanations are: (1) the enhanced temperature rise caused by more friction heating at high rotational speeds; (2) the early dislocation activation due to higher vibration energy density with the increase of rotational speeds, as indicated by Eq. (4).

## Effect of tool design and rotational speed on formability

The effects of tool design and rotational speed on the formability are shown in Fig. 7, in which fracture depth and equivalent fracture strain are used for comparison. By comparing the fracture depth and equivalent fracture strain of parts formed by T0, T2 and T4 at different rotational speeds, it is observed that the parts formed by T4 tool has achieved the highest formability. The reason may be that, compared with the temperature rise by T0 and the low-frequency-high-amplitude vibration with limited temperature rise by T2, T4 can provide both sufficient localized temperature rise and low-frequency-low-amplitude vibration, which may be useful for the improvement of material ductility.

By observing the fracture depth and equivalent fracture strain results by using T4, with the increase of rotational speed, the formability of the formed parts increases first but this is followed by a decreasing trend. The part formed by T4 has the highest formability when the rotational speed is 6000 rpm, and the highest equivalent fracture strain reaches 2.49. The reason for the initial increased formability is resulted from the temperature rise, increased vibration frequency and decreased vibration amplitude. The subsequent decreasing trend with higher rotational speeds may be that the sheet material is excessively softened and the deformation become unstable, so that fracture is initiated and propagated earlier.
<figure><img src="image_1510.png" /><figcaption>Fig. 7. Effect of the tool design and rotational speed on the (a) fracture depth and (b) equivalent fracture strain.</figcaption></figure>

## Conclusions

This work analyses the vibration amplitudes and frequencies under different tool designs and rotational speeds in the RV-ISF, and studies the effect of tool design and rotational speed on the forming temperature, forming force, and formability when forming AA3003-O sheets. The main conclusions may be drawn as below:

1. The new design of four-groove rosette tool T4 can lead to the sheet vibration with low frequency and low amplitude, while the double-offset tool T2 causes low frequency and high amplitude vibrations. The increase of tool rotational speed leads to increasing sheet vibration frequency.

2. Higher tool rotational speeds contribute to higher forming temperature increases, and the rotation of T4 leads to higher temperature rise than that by T2 because of larger contact area of T4.

3. Greater reduction of the forming force can be achieved by using T2 than by T4 because of the higher vibration amplitude induced by T2; while the increasing rotational speeds of T4 result in smaller forming forces due to the higher temperature rise and vibration frequency.

4. RV-ISF with T4 can improve the formability as compared with C-ISF with T0, FS-ISF with T0 and RV-ISF with T2, because of T4 producing higher temperature rise and low-frequency-low-amplitude vibration. Applying a rotational speed of T4 below a certain limit has positive effect on the formability improvement. For all the tools studied, the fracture depth and equivalent fracture strain start to decrease when the rotational speed applied is beyond a certain limit.

## Acknowledgements

This work was financially supported by the UK Engineering and Physical Sciences Research Council (EPSRC) through project grants EP/W010089/1 and EP/T005254/1. The authors would like to acknowledge the contribution from Jamie Booth, an Engineering Team Leader of the Department of Mechanical Engineering, the University of Sheffield, for performing all experimental tests.
<figure><img src="image_1514.png" /><figcaption>Fig. 7. Effect of the tool design and rotational speed on the (a) fracture depth and (b) equivalent fracture strain.</figcaption></figure>

## Conclusions

This work analyses the vibration amplitudes and frequencies under different tool designs and rotational speeds in the RV-ISF, and studies the effect of tool design and rotational speed on the forming temperature, forming force, and formability when forming AA3003-O sheets. The main conclusions may be drawn as below:

(1) The new design of four-groove rosette tool T4 can lead to the sheet vibration with low frequency and low amplitude, while the double-offset tool T2 causes low frequency and high amplitude vibrations. The increase of tool rotational speed leads to increasing sheet vibration frequency.

(2) Higher tool rotational speeds contribute to higher forming temperature increases, and the rotation of T4 leads to higher temperature rise than that by T2 because of larger contact area of T4.

(3) Greater reduction of the forming force can be achieved by using T2 than by T4 because of the higher vibration amplitude induced by T2; while the increasing rotational speeds of T4 result in smaller forming forces due to the higher temperature rise and vibration frequency.

(4) RV-ISF with T4 can improve the formability as compared with C-ISF with T0, FS-ISF with T0 and RV-ISF with T2, because of T4 producing higher temperature rise and low-frequency-low-amplitude vibration. Applying a rotational speed of T4 below a certain limit has positive effect on the formability improvement. For all the tools studied, the fracture depth and equivalent fracture strain start to decrease when the rotational speed applied is beyond a certain limit.

## Acknowledgements

This work was financially supported by the UK Engineering and Physical Sciences Research Council (EPSRC) through project grants EP/W010089/1 and EP/T005254/1. The authors would like to acknowledge the contribution from Jamie Booth, an Engineering Team Leader of the Department of Mechanical Engineering, the University of Sheffield, for performing all experimental tests.
## 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, Int. J. Mater. Form. 11 (2018) 743-773. https://doi.org/10.1007/s12289-017-1387-y

[2] M. Otsu, Friction stir incremental forming of aluminum alloy sheets, Proc. Metal forming (2010) 942-945.

[3] Z. Wang, S. Cai, J. Chen, Experimental investigations on friction stir assisted single point incremental forming of low-ductility aluminum alloy sheet for higher formability with reasonable surface quality, J. Mater. Process. Technol. 277 (2020) 116488. https://doi.org/10.1016/j.jmatprotec.2019.116488

[4] X. Zhan, D. An, J. Chen, A novel two-stage friction stir-assisted incremental sheet forming method for uniform microstructure and enhanced properties in aluminum alloys, Int. J. Mach. Tools Manuf. 180 (2022) 103928. https://doi.org/10.1016/j.ijmachtools.2022.103928

[5] Y. Long, Y. Li, J. Sun, I. Ille, J. Li, J. Twiefel, Effects of process parameters on force reduction and temperature variation during ultrasonic assisted incremental sheet forming process, Int. J. Adv. Manuf. Technol. 97 (2018) 13-24. https://doi.org/10.1007/s00170-018-1886-0

[6] S. Amini, A. Hosseinpour Gollo, H. Paktinat, An investigation of conventional and ultrasonic-assisted incremental forming of annealed AA1050 sheet, Int. J. Adv. Manuf. Technol. 90 (2017) 1569-1578. https://doi.org/10.1007/s00170-016-9458-7

[7] S. Amini, F. Nazari, M. Baraheni, A. H. Ghasemi, Investigating the effect of rotation speed and ultrasonic vibrations in the incremental forming process, Int. J. Adv. Design Manuf. Technol. 11(2018) 4 91-97.

[8] H. Long, W. X. Peng, Z. D. Chang, H. Zhu, Y. J. Jiang, Z. H. Li, New Rosette Tools for Developing Rotational Vibration-assisted Incremental Sheet Forming, Journal of Materials Processing Technology 118311 (2024). https://doi.org/10.1016/j.jmatprotec.2024.118311

[9] B. Lu, Z. Li, H. Long, F. Chen, J. Chen, H. Ou, Microstructure refinement by tool rotation-induced vibration in incremental sheet forming, Procedia Eng. 207 (2017) 795-800. https://doi.org/10.1016/j.proeng.2017.10.831

[10] W. Peng, E. Hurtado Molina, F. Solum, J. Booth, H. Long, Surface and Friction Characterisation of Rotational Vibration-Assisted Incremental Sheet Forming, in International Conference on the Technology of Plasticity (2023) Springer 756-765. https://doi.org/10.1007/978-3-031-41023-9_76

[11] A. Saulo, A. Carlos, E. Márcio, Numerical Study on Incremental Sheet Forming Applied on an Impact Attenuator Model, in 22nd International Congress of Mechanical Engineering (COBEM 2013) November 3-7, 2013 Ribeirão Preto, SP, Brazil, (2013) ABCM 6077-6086.

[12] A. Pierce, Basic linear acoustics, Springer Handbook of Acoustics, (2007) 25. https://doi.org/10.1007/978-0-387-30425-0_3
## 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, Int. J. Mater. Form. 11 (2018) 743-773. https://doi.org/10.1007/s12289-017-1387-y

[2] M. Otsu, Friction stir incremental forming of aluminum alloy sheets, Proc. Metal forming (2010) 942-945.

[3] Z. Wang, S. Cai, J. Chen, Experimental investigations on friction stir assisted single point incremental forming of low-ductility aluminum alloy sheet for higher formability with reasonable surface quality, J. Mater. Process. Technol. 277 (2020) 116488. https://doi.org/10.1016/j.jmatprotec.2019.116488

[4] X. Zhan, D. An, J. Chen, A novel two-stage friction stir-assisted incremental sheet forming method for uniform microstructure and enhanced properties in aluminum alloys, Int. J. Mach. Tools Manuf. 180 (2022) 103928. https://doi.org/10.1016/j.ijmachtools.2022.103928

[5] Y. Long, Y. Li, J. Sun, I. Ille, J. Li, J. Twiefel, Effects of process parameters on force reduction and temperature variation during ultrasonic assisted incremental sheet forming process, Int. J. Adv. Manuf. Technol. 97 (2018) 13-24. https://doi.org/10.1007/s00170-018-1886-0

[6] S. Amini, A. Hosseinpour Gollo, H. Paktinat, An investigation of conventional and ultrasonic-assisted incremental forming of annealed AA1050 sheet, Int. J. Adv. Manuf. Technol. 90 (2017) 1569-1578. https://doi.org/10.1007/s00170-016-9458-7

[7] S. Amini, F. Nazari, M. Baraheni, A. H. Ghasemi, Investigating the effect of rotation speed and ultrasonic vibrations in the incremental forming process, Int. J. Adv. Design Manuf. Technol. 11(2018) 4 91-97.

[8] H. Long, W. X. Peng, Z. D. Chang, H. Zhu, Y. J. Jiang, Z. H. Li, New Rosette Tools for Developing Rotational Vibration-assisted Incremental Sheet Forming, Journal of Materials Processing Technology 118311 (2024). https://doi.org/10.1016/j.jmatprotec.2024.118311

[9] B. Lu, Z. Li, H. Long, F. Chen, J. Chen, H. Ou, Microstructure refinement by tool rotation-induced vibration in incremental sheet forming, Procedia Eng. 207 (2017) 795-800. https://doi.org/10.1016/j.proeng.2017.10.831

[10] W. Peng, E. Hurtado Molina, F. Solum, J. Booth, H. Long, Surface and Friction Characterisation of Rotational Vibration-Assisted Incremental Sheet Forming, in International Conference on the Technology of Plasticity (2023) Springer 756-765. https://doi.org/10.1007/978-3-031-41023-9_76

[11] A. Saulo, A. Carlos, E. Márcio, Numerical Study on Incremental Sheet Forming Applied on an Impact Attenuator Model, in 22nd International Congress of Mechanical Engineering (COBEM 2013) November 3-7, 2013 Ribeirão Preto, SP, Brazil, (2013) ABCM 6077-6086.

[12] A. Pierce, Basic linear acoustics, Springer Handbook of Acoustics, (2007) 25. https://doi.org/10.1007/978-0-387-30425-0_3
# Necking detection in stretch-bent materials exhibiting the Portevin-Le Chatelier effect

MARTÍNEZ-DONAIRE Andrés J.<fnref n="1" />,ª, PALOMO David<fnref n="1" />,ᵇ, SÁENZ DE ARGANDOÑA Eneko<fnref n="2" />,ᶜ, VALLELLANO Carpoforo<fnref n="1" />,ᵈ*, MENDIGUREN Joseba<fnref n="2" />,ᵉ*

**Keywords:** Necking, Nakazima, PLC Effect, Stretch-Bending, Time-Dependent, Flat Valley Method

**Abstract.** In recent years, there has been increasing societal awareness of the carbon dioxide (CO₂) footprint resulting from individual actions and lifestyles. One of the research actions is focused on the development of eco-friendly alloys with more recycled scrap material in order to reduce emissions, but this can also result in greater variability of material properties. In this context, accurately characterizing the formability limits of materials is of paramount importance for optimizing manufacturing processes. Although ISO 12004-2:2008 standard is commonly used for necking detection, recent years have seen time-dependent methods yield more accurate predictions. Nevertheless, in materials exhibiting the Portevin-Le Chatelier (PLC) effect, such as some common lightweight alloys used in automotive and aeronautics, necking detection introduces significant challenges, and even more so when the material is subjected to severe local stretch-bending states. In this work, various necking detection techniques were employed to analyze their capabilities in a series of stretch-bending experiments over a 2.94 mm thick AA5754H11 PLC-driven material.

**Introduction**

Given the alarming predictions of reaching the CO₂ capacity threshold in the next 25 years, there is a concerted effort to implement new trends and technologies to reduce emissions. Among others, the most popular are: 1) shift towards more flexible manufacturing processes, including incremental sheet forming methods [1], 2) maximizing the material use, especially in high-CO₂ emission industries such as steel and aluminum production, and 3) developing eco-friendly alloys with more recycled scrap material can reduce emissions [2], although this may result in greater variability of material properties. In this context, the accurate knowledge of the limit strains at necking and at fracture is crucial to optimize the planning of the manufacturing stages.

The use of lightweight materials exhibiting the Portevin-Le Chatelier (PLC) effect, such as 2024-O or 5754-O aluminum alloys, is very common in the industrial manufacturing of sheet metal parts in both the aerospace and automotive industries.

It is well established that the PLC effect is attributed to dynamic interactions between mobile solute atoms and dislocations, termed dynamic strain ageing (DSA) [3]. Plastic flow instabilities associated with dynamic strain ageing (DSA) can be exhibited during plastic straining of Al-Mg alloys [4-6], such as AA5754. These are manifested by the appearance and propagation of localized deformation bands (PLC bands) along the specimen, which not only generates undesirable traces

<figure><img src="image_1511.png" /></figure>

<footnote n="1">¹Department of Mechanical and Manufacturing Engineering, School of Engineering, University of Sevilla, Spain</footnote>

<footnote n="2">²Mondragon Unibertsitatea, Faculty of Engineering, Mechanics and Industrial Production, Mondragon, Spain</footnote>

<footnote>ªajmd@us.es, ᵇdpalomο1@us.es, ᶜesaenzdeargan@mondragon.edu, ᵈcarpofor@us.es,</footnote>

<footnote>ᵉjmendiguren@mondragon.edu</footnote>
# Necking detection in stretch-bent materials exhibiting the Portevin-Le Chatelier effect

MARTÍNEZ-DONAIRE Andrés J.<fnref n="1" />,ª, PALOMO David<fnref n="1" />,ᵇ, SÁENZ DE ARGANDOÑA Eneko<fnref n="2" />,ᶜ, VALLELLANO Carpoforo<fnref n="1" />,ᵈ*, MENDIGUREN Joseba<fnref n="2" />,ᵉ*

**Keywords:** Necking, Nakazima, PLC Effect, Stretch-Bending, Time-Dependent, Flat Valley Method

**Abstract.** In recent years, there has been increasing societal awareness of the carbon dioxide (CO₂) footprint resulting from individual actions and lifestyles. One of the research actions is focused on the development of eco-friendly alloys with more recycled scrap material in order to reduce emissions, but this can also result in greater variability of material properties. In this context, accurately characterizing the formability limits of materials is of paramount importance for optimizing manufacturing processes. Although ISO 12004-2:2008 standard is commonly used for necking detection, recent years have seen time-dependent methods yield more accurate predictions. Nevertheless, in materials exhibiting the Portevin-Le Chatelier (PLC) effect, such as some common lightweight alloys used in automotive and aeronautics, necking detection introduces significant challenges, and even more so when the material is subjected to severe local stretch-bending states. In this work, various necking detection techniques were employed to analyze their capabilities in a series of stretch-bending experiments over a 2.94 mm thick AA5754H11 PLC-driven material.

**Introduction**

Given the alarming predictions of reaching the CO₂ capacity threshold in the next 25 years, there is a concerted effort to implement new trends and technologies to reduce emissions. Among others, the most popular are: 1) shift towards more flexible manufacturing processes, including incremental sheet forming methods [1], 2) maximizing the material use, especially in high-CO₂ emission industries such as steel and aluminum production, and 3) developing eco-friendly alloys with more recycled scrap material can reduce emissions [2], although this may result in greater variability of material properties. In this context, the accurate knowledge of the limit strains at necking and at fracture is crucial to optimize the planning of the manufacturing stages.

The use of lightweight materials exhibiting the Portevin-Le Chatelier (PLC) effect, such as 2024-O or 5754-O aluminum alloys, is very common in the industrial manufacturing of sheet metal parts in both the aerospace and automotive industries.

It is well established that the PLC effect is attributed to dynamic interactions between mobile solute atoms and dislocations, termed dynamic strain ageing (DSA) [3]. Plastic flow instabilities associated with dynamic strain ageing (DSA) can be exhibited during plastic straining of Al-Mg alloys [4-6], such as AA5754. These are manifested by the appearance and propagation of localized deformation bands (PLC bands) along the specimen, which not only generates undesirable traces

<figure><img src="image_1515.png" /></figure>

<footnote n="1">¹Department of Mechanical and Manufacturing Engineering, School of Engineering, University of Sevilla, Spain</footnote>

<footnote n="2">²Mondragon Unibertsitatea, Faculty of Engineering, Mechanics and Industrial Production, Mondragon, Spain</footnote>

<footnote>ªajmd@us.es, ᵇdpalomο1@us.es, ᶜesaenzdeargan@mondragon.edu, ᵈcarpofor@us.es,</footnote>

<footnote>ᵉjmendiguren@mondragon.edu</footnote>
on the product surface due to the discontinuous deformation process but can also reduce the ductility of the alloy [3].

The most common failure mechanism in conventional sheet metal forming, such as stamping or stretching, is the appearance of tensile plastic instability and the subsequent tearing and fracture of the specimen. Many works have focused on accurately determining the onset of necking by proposing different experimental procedures and techniques [7-11]. The standardized position-dependent ISO 12004-2:2008 [7] is commonly used to establish the Forming Limit Curve (FLC) at necking using the Marciniak and Nakazima-type tests, i.e., situations in which the strain/stress gradient does not play an important role. However, in recent years, both time-dependent (t-d) and time-position dependent methods have been pointed out to yield more accurate predictions [8-11], as well as some of these approaches are able to broaden their applicability to more general cases under simultaneous action of stretching and bending [8,12].

These experimental methodologies are mainly based on the temporal analysis of the time derivatives of the major and/or thickness strains over the deformation process. Therefore, their application to materials that exhibit an unsteady strain evolution due to the formation of PLC bands that clearly impact the strain rate computation is complex and has not been thoroughly investigated. Furthermore, when the material is subjected to mild or severe stretch-bending conditions, because the standard ISO 12004-2:2008 cannot be applied, it seems to be a challenge.

The objective of this work consists of applying various existing necking detection techniques to analyze their predictive capabilities in the light of stretch-bending experiments with cylindrical punches of different radii over an AA5754H11 PLC driven material with 2.94 mm thickness. In particular, the radius R20 and R7.5 mm are selected for the analysis presented herein. It is pointed out that time-dependent methods, based on the analysis of time derivatives of the local strains, must be applied carefully to obtain reliable estimations, whereas local curvature evaluation methods, such as the flat-valley method, which focuses on analyzing displacements at the outer sheet surface, accurately assess the onset of necking.

## Necking Detection Techniques

In this section, a summary of the fundamentals of the three experimental approaches analyzed in this contribution is presented. Along with the standardized ISO 12004-2:2008 method [7], the t-d method and a time-position dependent approach known as flat-valley, both previously proposed by the authors [8], have been analyzed. It should be noted that t-d and flat-valley are local approaches, not restricted to the FLC assessment, and therefore also applicable to stretch-bending tests.

### ISO 12004-2:2008 methodology

The ISO12004-2:2008 procedure is a position-dependent method based on the analysis of the major strain profile along a cross section perpendicular to the fracture zone just before the occurrence of the fracture. Briefly, the procedure steps are the following: 1) Defining the crack position, 2) marking the necking region width or the inner limits of the fit windows, 3) obtaining the outer limits of the fit windows, 4) determining the limit major strain by fitting an inverse parabola using the experimental data within the fit windows, 5) determining the limit thickness strain by fitting another inverse parabola and 6) estimating the limit minor strain by applying volume conservation. Further details about the methodology can be found in [7].

### Time-dependent method by Martinez-Donaire et al.

The t-d method [8] is based on the physical fundamentals of the necking process to detect the onset of the plastic instability. This method makes use of the temporal analysis of the major strain distribution and its first derivative over time at various points along a cross section perpendicular to the fracture zone. Fig. 1a shows a schematic of the time evolution of the major strain and its
on the product surface due to the discontinuous deformation process but can also reduce the ductility of the alloy [3].

The most common failure mechanism in conventional sheet metal forming, such as stamping or stretching, is the appearance of tensile plastic instability and the subsequent tearing and fracture of the specimen. Many works have focused on accurately determining the onset of necking by proposing different experimental procedures and techniques [7-11]. The standardized position-dependent ISO 12004-2:2008 [7] is commonly used to establish the Forming Limit Curve (FLC) at necking using the Marciniak and Nakazima-type tests, i.e., situations in which the strain/stress gradient does not play an important role. However, in recent years, both time-dependent (t-d) and time-position dependent methods have been pointed out to yield more accurate predictions [8-11], as well as some of these approaches are able to broaden their applicability to more general cases under simultaneous action of stretching and bending [8,12].

These experimental methodologies are mainly based on the temporal analysis of the time derivatives of the major and/or thickness strains over the deformation process. Therefore, their application to materials that exhibit an unsteady strain evolution due to the formation of PLC bands that clearly impact the strain rate computation is complex and has not been thoroughly investigated. Furthermore, when the material is subjected to mild or severe stretch-bending conditions, because the standard ISO 12004-2:2008 cannot be applied, it seems to be a challenge.

The objective of this work consists of applying various existing necking detection techniques to analyze their predictive capabilities in the light of stretch-bending experiments with cylindrical punches of different radii over an AA5754H11 PLC driven material with 2.94 mm thickness. In particular, the radius R20 and R7.5 mm are selected for the analysis presented herein. It is pointed out that time-dependent methods, based on the analysis of time derivatives of the local strains, must be applied carefully to obtain reliable estimations, whereas local curvature evaluation methods, such as the flat-valley method, which focuses on analyzing displacements at the outer sheet surface, accurately assess the onset of necking.

## Necking Detection Techniques

In this section, a summary of the fundamentals of the three experimental approaches analyzed in this contribution is presented. Along with the standardized ISO 12004-2:2008 method [7], the t-d method and a time-position dependent approach known as flat-valley, both previously proposed by the authors [8], have been analyzed. It should be noted that t-d and flat-valley are local approaches, not restricted to the FLC assessment, and therefore also applicable to stretch-bending tests.

### ISO 12004-2:2008 methodology

The ISO12004-2:2008 procedure is a position-dependent method based on the analysis of the major strain profile along a cross section perpendicular to the fracture zone just before the occurrence of the fracture. Briefly, the procedure steps are the following: 1) Defining the crack position, 2) marking the necking region width or the inner limits of the fit windows, 3) obtaining the outer limits of the fit windows, 4) determining the limit major strain by fitting an inverse parabola using the experimental data within the fit windows, 5) determining the limit thickness strain by fitting another inverse parabola and 6) estimating the limit minor strain by applying volume conservation. Further details about the methodology can be found in [7].

### Time-dependent method by Martinez-Donaire et al.

The t-d method [8] is based on the physical fundamentals of the necking process to detect the onset of the plastic instability. This method makes use of the temporal analysis of the major strain distribution and its first derivative over time at various points along a cross section perpendicular to the fracture zone. Fig. 1a shows a schematic of the time evolution of the major strain and its
time derivative at two points, A and B, located in a section perpendicular to the necking region. The procedure can be divided in the following steps:

1) Obtaining the necking zone width. The last points on both sides of the crack that stop deforming and reach a null strain rate (marked as Point A in Fig. 1a) just before the fracture occurrence are identified. The zone between them defines the area where necking took place.

<figure><img src="image_1512.png" /><figcaption>Fig. 1: Schematic of the t-d method (a) and the flat-valley method (b) to detect the onset of necking and estimate the limit strains [8].</figcaption></figure>

2) Determining the onset of necking ($t_{necking}$). It was established that the plastic instability starts when the “strain rate” at the boundary of the necking zone (Point A) reaches a maximum. This event reveals that the deformation is just about to localize within the necking zone (region A-B-A in Fig. 1a), defining the time at which necking begins.

3) Identifying the fracture point. The point at which the fracture occurs (Point B) can be identified by detecting the highest strain curve during the whole deformation process (see the evolution of $\epsilon_1^B$ in Fig. 1a).

4) Determining the limit major and minor strain at necking $\epsilon_{1,lim}$ and $\epsilon_{2,lim}$. These are defined as the values of major and minor strain at point B at the time $t_{necking}$, respectively.

An analogous method but based on the analysis of the thickness strain ($\epsilon_3$) can be also proposed with similar results [8].

## Flat-valley method

The flat-valley method [8,12] focuses on the direct observation and analysis of the curvature of the topology of the outer surface of the specimen during the test. In particular, it analyzes the time evolution of the displacement perpendicular to the plane of the non-deformed specimen (Z axis) and its spatial derivative along a section perpendicular to the failure region. According to this, the flat-valley can be categorized as a time-position dependent method.

Fig. 1b (top) depicts a schematic of the Z-displacement in a section perpendicular to the fracture region at different times ($t_1$ denotes a time far from necking and $t_4$ denotes a time just before fracture). As can be seen, far from the instant failure, e.g., in $t_1$, the surface of the sheet deforms following the curvature imposed by the tool, in this case a hemispherical or cylindrical punch. Later, this curve begins to flatten in a certain region, see $t_2$, becoming flat at a certain time ($t_3$), until finally a necking valley is observed at time $t_4$.

When the profile becomes flat, the sheet begins to deform independently of the curvature imposed by the punch. This fact physically corresponds to the appearance of the neck, i.e. $t_{necking} = t_3$ in Fig. 1b. This flattening process can be clearly identified by calculating the first spatial derivative of the Z-displacement (Fig. 1b, bottom). Thus, it is hypothesized that necking begins when the slope in the first spatial derivative of the Z-displacement remains constant, not
time derivative at two points, A and B, located in a section perpendicular to the necking region. The procedure can be divided in the following steps:

1) Obtaining the necking zone width. The last points on both sides of the crack that stop deforming and reach a null strain rate (marked as Point A in Fig. 1a) just before the fracture occurrence are identified. The zone between them defines the area where necking took place.

<figure><img src="image_1516.png" /><figcaption>Fig. 1: Schematic of the t-d method (a) and the flat-valley method (b) to detect the onset of necking and estimate the limit strains [8].</figcaption></figure>

2) Determining the onset of necking ($t_{necking}$). It was established that the plastic instability starts when the “strain rate” at the boundary of the necking zone (Point A) reaches a maximum. This event reveals that the deformation is just about to localize within the necking zone (region A-B-A in Fig. 1a), defining the time at which necking begins.

3) Identifying the fracture point. The point at which the fracture occurs (Point B) can be identified by detecting the highest strain curve during the whole deformation process (see the evolution of $\epsilon_1^B$ in Fig. 1a).

4) Determining the limit major and minor strain at necking $\epsilon_{1,lim}$ and $\epsilon_{2,lim}$. These are defined as the values of major and minor strain at point B at the time $t_{necking}$, respectively.

An analogous method but based on the analysis of the thickness strain ($\epsilon_3$) can be also proposed with similar results [8].

## Flat-valley method

The flat-valley method [8,12] focuses on the direct observation and analysis of the curvature of the topology of the outer surface of the specimen during the test. In particular, it analyzes the time evolution of the displacement perpendicular to the plane of the non-deformed specimen (Z axis) and its spatial derivative along a section perpendicular to the failure region. According to this, the flat-valley can be categorized as a time-position dependent method.

Fig. 1b (top) depicts a schematic of the Z-displacement in a section perpendicular to the fracture region at different times ($t_1$ denotes a time far from necking and $t_4$ denotes a time just before fracture). As can be seen, far from the instant failure, e.g., in $t_1$, the surface of the sheet deforms following the curvature imposed by the tool, in this case a hemispherical or cylindrical punch. Later, this curve begins to flatten in a certain region, see $t_2$, becoming flat at a certain time ($t_3$), until finally a necking valley is observed at time $t_4$.

When the profile becomes flat, the sheet begins to deform independently of the curvature imposed by the punch. This fact physically corresponds to the appearance of the neck, i.e. $t_{necking} = t_3$ in Fig. 1b. This flattening process can be clearly identified by calculating the first spatial derivative of the Z-displacement (Fig. 1b, bottom). Thus, it is hypothesized that necking begins when the slope in the first spatial derivative of the Z-displacement remains constant, not
necessarily zero, within the necking region. Finally, as in the t-d method, once the onset of necking is identified, the major and minor limit strains can be obtained at their corresponding strain curve at the most strained point at a time equal to $t_{necking}$.

## Experimental campaign

The material selected for this study was an AA5754H11 with a 2.94 mm thickness. This is an aluminum alloy widely used in automotive industry that exhibits the formation of PLC bands during the plastic deformation.

Several stretch-bending tests using cylindrical punches of two radii, R20 and R7.5 mm, were performed on a universal sheet metal testing machine Erichsen 142-40 (Fig.s 2a and 2b). These punch diameters were selected to achieve two clearly different bending strain gradients: a smooth/mild gradient for R20 mm and a severe gradient for R7.5 mm. These values are very common in stretch-bending operations in industrial practice as well as automotive stamping dies.

The testing parameters were selected according to standard ISO 12004-2:2008. The punch speed was set to 1 mm/s and the tribology system was a combination of PTFE sheets and vaseline between the punch and the metal sheet. A blank-holder force of 400KN and a draw-bead system was used to clamp the specimen.

Fig. 2c shows the geometry of 55 mm width samples tested to reproduce strain paths close to plane strain conditions.

<figure><img src="image_1513.png" /><figcaption>Fig. 2: a) Erichsen machine, DIC system and experimental setup, b) sample with R20 mm radius punch and c) major strain data from DIC over the specimen geometry under near plane strain conditions</figcaption></figure>

Two 5-megapixel digital CCD cameras captured images during the whole deformation process at a rate of 15 frames per second. The strain contour on the outer surface was evaluated via Digital Image Correlation (DIC) technique using ARAMIS® (Fig. 2c). To define the optimum values for each gradient severity, i.e. for each punch radius, a study of sensitivity was performed by using several combinations of facet size and facet step in the DIC software.

## Experimental results and comparative analysis for necking prediction

In this section, the experimental results obtained and the capabilities for predicting the limit strains at the onset of necking are analyzed using the standard ISO 12004-2, the t-d method, and the flat-valley method.

The application of each of these criteria requires measuring by DIC the temporal and spatial distribution of the principal strains on the outer surface of the specimen in the vicinity of the failure
necessarily zero, within the necking region. Finally, as in the t-d method, once the onset of necking is identified, the major and minor limit strains can be obtained at their corresponding strain curve at the most strained point at a time equal to $t_{necking}$.

## Experimental campaign

The material selected for this study was an AA5754H11 with a 2.94 mm thickness. This is an aluminum alloy widely used in automotive industry that exhibits the formation of PLC bands during the plastic deformation.

Several stretch-bending tests using cylindrical punches of two radii, R20 and R7.5 mm, were performed on a universal sheet metal testing machine Erichsen 142-40 (Fig.s 2a and 2b). These punch diameters were selected to achieve two clearly different bending strain gradients: a smooth/mild gradient for R20 mm and a severe gradient for R7.5 mm. These values are very common in stretch-bending operations in industrial practice as well as automotive stamping dies.

The testing parameters were selected according to standard ISO 12004-2:2008. The punch speed was set to 1 mm/s and the tribology system was a combination of PTFE sheets and vaseline between the punch and the metal sheet. A blank-holder force of 400KN and a draw-bead system was used to clamp the specimen.

Fig. 2c shows the geometry of 55 mm width samples tested to reproduce strain paths close to plane strain conditions.

<figure><img src="image_1517.png" /><figcaption>Fig. 2: a) Erichsen machine, DIC system and experimental setup, b) sample with R20 mm radius punch and c) major strain data from DIC over the specimen geometry under near plane strain conditions</figcaption></figure>

Two 5-megapixel digital CCD cameras captured images during the whole deformation process at a rate of 15 frames per second. The strain contour on the outer surface was evaluated via Digital Image Correlation (DIC) technique using ARAMIS® (Fig. 2c). To define the optimum values for each gradient severity, i.e. for each punch radius, a study of sensitivity was performed by using several combinations of facet size and facet step in the DIC software.

## Experimental results and comparative analysis for necking prediction

In this section, the experimental results obtained and the capabilities for predicting the limit strains at the onset of necking are analyzed using the standard ISO 12004-2, the t-d method, and the flat-valley method.

The application of each of these criteria requires measuring by DIC the temporal and spatial distribution of the principal strains on the outer surface of the specimen in the vicinity of the failure
region. Fig. 3a shows the major strain profile along a section perpendicular to the failure region at several stages for a stretch-bending test using R7.5 mm. The evolution of major strain vs. position at the frame immediately before the specimen fracture, i.e. at stage 429 in this case, is the input data for applying ISO 12004-2. Despite being a material, which underwent a severe PLC effect during straining, the major strain profile at the last stage shows the usual distribution for stretch-bending experiments, enabling the a priori application of the ISO procedure. However, it should be noted that the ISO standard was not developed to deal with a mild or severe stress gradient due to bending [8,11,12]. Therefore, it is expected to provide inconsistent predictions.

<figure><img src="image_1514.png" /><figcaption>Fig. 3: Different data along a section perpendicular to the failure zone for a stretch-bending tests using R7.5 mm cylindrical punch: a) major strain profile in different step times; b) temporal evolution of the major strain at different points, and c) Z-displacement on the outer surface in different step times.</figcaption></figure>

Fig. 3b depicts the temporal evolution of major strain for a series of points distributed from the fracture site outward along a section perpendicular to the failure region for a test using the R7.5 mm cylindrical punch. Note that these will be the data required for the application of the t-d procedure. Two remarkable facts can be observed. On the one hand, as expected, the curves are relatively separated from the beginning of the test, in agreement with the existence of a severe strain gradient imposed by the punch along the sheet surface. On the other hand, the PLC banding effect is clearly visible, as manifested in the serrated strain curves (see Fig. 3b), leading to
region. Fig. 3a shows the major strain profile along a section perpendicular to the failure region at several stages for a stretch-bending test using R7.5 mm. The evolution of major strain vs. position at the frame immediately before the specimen fracture, i.e. at stage 429 in this case, is the input data for applying ISO 12004-2. Despite being a material, which underwent a severe PLC effect during straining, the major strain profile at the last stage shows the usual distribution for stretch-bending experiments, enabling the a priori application of the ISO procedure. However, it should be noted that the ISO standard was not developed to deal with a mild or severe stress gradient due to bending [8,11,12]. Therefore, it is expected to provide inconsistent predictions.

<figure><img src="image_1518.png" /><figcaption>Fig. 3: Different data along a section perpendicular to the failure zone for a stretch-bending tests using R7.5 mm cylindrical punch: a) major strain profile in different step times; b) temporal evolution of the major strain at different points, and c) Z-displacement on the outer surface in different step times.</figcaption></figure>

Fig. 3b depicts the temporal evolution of major strain for a series of points distributed from the fracture site outward along a section perpendicular to the failure region for a test using the R7.5 mm cylindrical punch. Note that these will be the data required for the application of the t-d procedure. Two remarkable facts can be observed. On the one hand, as expected, the curves are relatively separated from the beginning of the test, in agreement with the existence of a severe strain gradient imposed by the punch along the sheet surface. On the other hand, the PLC banding effect is clearly visible, as manifested in the serrated strain curves (see Fig. 3b), leading to
significant plastic flow instabilities. This effect is active during most of the test, and seems to reduce its intensity, or even cease, when the plastic instability (necking) begins and develops until the failure of the specimen. At this last stage, the evolution of the strain curves becomes a little bit smoother than in previous frames. However, this material behaviour impact notably the accuracy and the reliability of the time derivatives of the major strain needed for applying the family of t-d methodologies. These methods must be applied carefully and focusing mainly on the last stages to obtain reliable estimations.

For the same test, Fig. 3c shows the evolution in time of the Z-displacement along a section perpendicular to the failure region in different step time until failure. This is the input data required to apply the flat-valley method. As can be observed, a valley that deepens due to the plastic instability process is easily visible in the last stages of the experiment. Moreover, since the processed variables are displacements instead of strains, the curves show a smooth and continuous trend, without noise, independently of the propagation of the PLC band manifested during the plastic straining of the material (see Fig. 3b). As will be seen later, this time and position dependent approach will allow accurate assessing the onset of necking in these materials under stretch-bending states.

Necking prediction using ISO 12004-2:2008

The ISO 12004-2:2008 predictions of major and minor limit strains for both stretch-bending tests, i.e., cylindrical punches of R20 and R7.5 mm, are shown in Table 1. Three specimens have been tested in each case.

<table><caption>Table 1: Forming limit strains predicted by ISO on the outer sheet surface in stretch-bending tests using cylindrical punches of R20 and R7.5mm.</caption><thead><tr><th rowspan="3">ISO 1200-2:2008<br>procedure</th><th colspan="2">Punch<br>R20 mm</th><th colspan="2">Punch<br>R7.5 mm</th></tr><tr><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th></tr><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Test 1</td><td>0.427</td><td>-0.066</td><td>0.220</td><td>0.051</td></tr><tr><td>Test 2</td><td>0.424</td><td>-0.062</td><td>0.307</td><td>0.168</td></tr><tr><td>Test 3</td><td>0.434</td><td>-0.061</td><td>0.655</td><td>0.789</td></tr></tbody></table>

As seen, ISO predictions for the R20 mm punch provide almost identical results for the three experiments performed. In addition, these strain pairs agree with the experimental strain path observed from the DIC, revealing that the process evolved near-plane strain conditions with a slightly negative local strain ratio. However, for the R7.5 mm punch, the results showed a large difference in the necking strains predicted in the three experiments. Moreover, the ISO method estimated positive minor strain values and even minor strains larger than major strains, being completely inconsistent with the experimental strain path around the failure zone. Because the ISO procedure is only tailored for cases with negligible strain gradients, such as Nakazima/Marciniak tests, the predictions for R20 mm are at least consistent. However, when the bending effect becomes severe, e.g., by using a R7.5 mm cylindrical punch, the methodology fails due mainly to the width of the fitting windows proposed in the standard. Similar results have already been discussed by the authors for other metallic materials [8].
significant plastic flow instabilities. This effect is active during most of the test, and seems to reduce its intensity, or even cease, when the plastic instability (necking) begins and develops until the failure of the specimen. At this last stage, the evolution of the strain curves becomes a little bit smoother than in previous frames. However, this material behaviour impact notably the accuracy and the reliability of the time derivatives of the major strain needed for applying the family of t-d methodologies. These methods must be applied carefully and focusing mainly on the last stages to obtain reliable estimations.

For the same test, Fig. 3c shows the evolution in time of the Z-displacement along a section perpendicular to the failure region in different step time until failure. This is the input data required to apply the flat-valley method. As can be observed, a valley that deepens due to the plastic instability process is easily visible in the last stages of the experiment. Moreover, since the processed variables are displacements instead of strains, the curves show a smooth and continuous trend, without noise, independently of the propagation of the PLC band manifested during the plastic straining of the material (see Fig. 3b). As will be seen later, this time and position dependent approach will allow accurate assessing the onset of necking in these materials under stretch-bending states.

## Necking prediction using ISO 12004-2:2008

The ISO 12004-2:2008 predictions of major and minor limit strains for both stretch-bending tests, i.e., cylindrical punches of R20 and R7.5 mm, are shown in Table 1. Three specimens have been tested in each case.

<table><caption>Table 1: Forming limit strains predicted by ISO on the outer sheet surface in stretch-bending tests using cylindrical punches of R20 and R7.5mm.</caption><thead><tr><th rowspan="3">ISO 1200-2:2008<br>procedure</th><th colspan="2">Punch<br>R20 mm</th><th colspan="2">Punch<br>R7.5 mm</th></tr><tr><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th></tr><tr><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Test 1</td><td>0.427</td><td>-0.066</td><td>0.220</td><td>0.051</td></tr><tr><td>Test 2</td><td>0.424</td><td>-0.062</td><td>0.307</td><td>0.168</td></tr><tr><td>Test 3</td><td>0.434</td><td>-0.061</td><td>0.655</td><td>0.789</td></tr></tbody></table>

As seen, ISO predictions for the R20 mm punch provide almost identical results for the three experiments performed. In addition, these strain pairs agree with the experimental strain path observed from the DIC, revealing that the process evolved near-plane strain conditions with a slightly negative local strain ratio. However, for the R7.5 mm punch, the results showed a large difference in the necking strains predicted in the three experiments. Moreover, the ISO method estimated positive minor strain values and even minor strains larger than major strains, being completely inconsistent with the experimental strain path around the failure zone. Because the ISO procedure is only tailored for cases with negligible strain gradients, such as Nakazima/Marciniak tests, the predictions for R20 mm are at least consistent. However, when the bending effect becomes severe, e.g., by using a R7.5 mm cylindrical punch, the methodology fails due mainly to the width of the fitting windows proposed in the standard. Similar results have already been discussed by the authors for other metallic materials [8].
Necking prediction using the time-dependent and flat-valley methods

The application of the t-d method requires the first temporal derivative of the major strain evolution in a series of points along a section perpendicular to the final crack, whereas the flat-valley approach makes use of the Z-displacement in a section perpendicular to the failure region at different time instants up to fracture. Fig. 4 depicts both experimental evolutions corresponding to a R7.5 mm stretch-bending test.

The PLC effect exhibited by the material is clearly visible in Fig. 4a. This leads to an unsteady evolution of the strain rate, manifested by repetitive significant increases and a later sudden drops in the strain rate. This peak-shaped evolution evolves at the different points during the test, experiencing increasing peak amplitudes as the plastic strain increases. Nevertheless, as mentioned before, this effect seems to saturate in the last stages, reducing its intensity, and allows one to observe the usual trend in a necking-controlled failure. This means that differences in the evolutions of the points within and outside the necking region can be identified by analyzing only the last frames (see the square box in Fig. 4a).

This existing noise during mathematical computation makes very difficult to accurately determine the necking zone width and the necking time (steps 1 and 2 numbered in the t-d method session), involving, respectively, the selection of the last point that reaches a zero-strain rate at fracture and the time instant (marked as $t_{\text{necking}}$ in Fig.) at which the strain rate at the boundary of the instability region reaches a local maximum. Therefore, the t-d method must be applied very carefully to obtain reliable limit strain estimates.

<figure><img src="image_1515.png" /><figcaption>Fig. 4: Evolution of a) the first temporal derivative of the major strain at several points along a section perpendicular to the fracture region and b) the Z-displacement rate in a section perpendicular to the failure region at different step times for a R7.5mm stretch-bending tests.</figcaption></figure>

On the other hand, as expected in the light of experimental data shown in Fig. 3c, the flat-valley method provides a straightforward, objective, and direct approach, since the first spatial derivative of Z-displacement shows a smooth and a noise-free trend (see Fig. 4b).

In this method, the onset of necking is established at the step time at which the slope in the spatial derivative of Z-displacement in the plastic instability zone remains constant (not necessarily null), in other words, when this curve becomes flat (identified as step 424 in Fig.). This points out that methodologies based on direct local observation of Z-displacements, such as the flat-valley method, allow an accurate estimate of the onset of necking in PLC-driven material. The flat-valley method, developed and tested for both Nakazima and stretch-bending tests [8], provides an easy
Necking prediction using the time-dependent and flat-valley methods

The application of the t-d method requires the first temporal derivative of the major strain evolution in a series of points along a section perpendicular to the final crack, whereas the flat-valley approach makes use of the Z-displacement in a section perpendicular to the failure region at different time instants up to fracture. Fig. 4 depicts both experimental evolutions corresponding to a R7.5 mm stretch-bending test.

The PLC effect exhibited by the material is clearly visible in Fig. 4a. This leads to an unsteady evolution of the strain rate, manifested by repetitive significant increases and a later sudden drops in the strain rate. This peak-shaped evolution evolves at the different points during the test, experiencing increasing peak amplitudes as the plastic strain increases. Nevertheless, as mentioned before, this effect seems to saturate in the last stages, reducing its intensity, and allows one to observe the usual trend in a necking-controlled failure. This means that differences in the evolutions of the points within and outside the necking region can be identified by analyzing only the last frames (see the square box in Fig. 4a).

This existing noise during mathematical computation makes very difficult to accurately determine the necking zone width and the necking time (steps 1 and 2 numbered in the t-d method session), involving, respectively, the selection of the last point that reaches a zero-strain rate at fracture and the time instant (marked as $t_{\text{necking}}$ in Fig.) at which the strain rate at the boundary of the instability region reaches a local maximum. Therefore, the t-d method must be applied very carefully to obtain reliable limit strain estimates.

<figure><img src="image_1519.png" /><figcaption>Fig. 4: Evolution of a) the first temporal derivative of the major strain at several points along a section perpendicular to the fracture region and b) the Z-displacement rate in a section perpendicular to the failure region at different step times for a R7.5mm stretch-bending tests.</figcaption></figure>

On the other hand, as expected in the light of experimental data shown in Fig. 3c, the flat-valley method provides a straightforward, objective, and direct approach, since the first spatial derivative of Z-displacement shows a smooth and a noise-free trend (see Fig. 4b).

In this method, the onset of necking is established at the step time at which the slope in the spatial derivative of Z-displacement in the plastic instability zone remains constant (not necessarily null), in other words, when this curve becomes flat (identified as step 424 in Fig.). This points out that methodologies based on direct local observation of Z-displacements, such as the flat-valley method, allow an accurate estimate of the onset of necking in PLC-driven material. The flat-valley method, developed and tested for both Nakazima and stretch-bending tests [8], provides an easy
and practical alternative to assess the formability in materials that exhibit the PLC phenomenon. In this work, its predictions are considered a base line to compare the rest of the methods.

<table><caption>Table 2: Forming limit strains predicted by the t-d method on the outer sheet surface in stretch-bending tests using cylindrical punches of R20 and R7.5 mm.</caption><thead><tr><th rowspan="2">t-d<br>method</th><th colspan="2">Punch<br>R20 mm</th><th colspan="2">Punch<br>R7.5 mm</th></tr><tr><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th></tr></thead><tbody><tr><td>Test 1</td><td>0.386</td><td>-0.074</td><td>0.421</td><td>-0.038</td></tr><tr><td>Test 2</td><td>0.397</td><td>-0.074</td><td>0.431</td><td>-0.038</td></tr><tr><td>Test 3</td><td>0.389</td><td>-0.073</td><td>0.415</td><td>-0.038</td></tr></tbody></table>

<table><caption>Table 3: Forming limit strains predicted by the flat-valley method on the outer sheet surface in stretch-bending tests using cylindrical punches of R20 and R7.5mm.</caption><thead><tr><th rowspan="2">flat-valley<br>method</th><th colspan="2">Punch<br>R20 mm</th><th colspan="2">Punch<br>R7.5 mm</th></tr><tr><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th></tr></thead><tbody><tr><td>Test 1</td><td>0.394</td><td>-0.075</td><td>0.469</td><td>-0.039</td></tr><tr><td>Test 2</td><td>0.379</td><td>-0.075</td><td>0.484</td><td>-0.039</td></tr><tr><td>Test 3</td><td>0.386</td><td>-0.073</td><td>0.489</td><td>-0.039</td></tr></tbody></table>

Table 2 and Table 3 show the predictions of strain limits using the t-d and flat valley methods, respectively, for the cylindrical punch stretch-bending tests R20 mm and R7.5 mm. Again, three specimens were tested for each punch radius.

As can be seen, the predictions of both methods for the R20 mm punch are almost coincident, whereas for the R7.5 mm test, the t-d method slightly underestimates the material formability with respect to the flat-valley results, used as reference, due to the difficulties previously discussed associated to the plastic flow instabilities in PLC materials. Generally, the highest difference in mean values was around 14% for the R7.5 mm punch. It can be said that the estimation by both methodologies shows a remarkably small dispersion.

Compared with the ISO standard, although the ISO predictions in the case of smooth strain gradient tests, that is, R20 mm cylindrical punch, slightly overpredict with reference values and yield differences of about 11%. Therefore, it becomes evident its inability to assess the formability limits in cases of severe strain gradient tests, in our case, the R7.5 mm cylindrical punch. As mentioned above, this limitation is related to the fact that this standard was tailored to tests with negligible bending effect (Nakazima/Marciniak tests) and not directly to the fact of the existence of the PLC phenomenon.

## Conclusions

In this study, the formability of a 2.94 mm thick AA5754H11 PLC-driven material is analysed in two different stretch-bending tests using cylindrical punch radii, one R20 mm that generate a smooth/mild strain gradient in the sheet, and another of R7.5 mm, which induces a much severe one. Various necking detection techniques, particularly ISO12004-2:2008 [7], and the time-dependent and flat valley methods proposed in the literature [8], have been employed to obtain and discuss their predictive capabilities to assess the necking limit strains. The primary outcomes of this study are the following:

* ISO 12004-2:2008 is not able to estimate the necking limit strains under mild or severe stretch-bending conditions.
and practical alternative to assess the formability in materials that exhibit the PLC phenomenon. In this work, its predictions are considered a base line to compare the rest of the methods.

<table><caption>Table 2: Forming limit strains predicted by the t-d method on the outer sheet surface in stretch-bending tests using cylindrical punches of R20 and R7.5 mm.</caption><thead><tr><th rowspan="2">t-d<br>method</th><th colspan="2">Punch<br>R20 mm</th><th colspan="2">Punch<br>R7.5 mm</th></tr><tr><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th></tr></thead><tbody><tr><td>Test 1</td><td>0.386</td><td>-0.074</td><td>0.421</td><td>-0.038</td></tr><tr><td>Test 2</td><td>0.397</td><td>-0.074</td><td>0.431</td><td>-0.038</td></tr><tr><td>Test 3</td><td>0.389</td><td>-0.073</td><td>0.415</td><td>-0.038</td></tr></tbody></table>

<table><caption>Table 3: Forming limit strains predicted by the flat-valley method on the outer sheet surface in stretch-bending tests using cylindrical punches of R20 and R7.5mm.</caption><thead><tr><th rowspan="2">flat-valley<br>method</th><th colspan="2">Punch<br>R20 mm</th><th colspan="2">Punch<br>R7.5 mm</th></tr><tr><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th><th>ε<sub>1lim</sub></th><th>ε<sub>2lim</sub></th></tr></thead><tbody><tr><td>Test 1</td><td>0.394</td><td>-0.075</td><td>0.469</td><td>-0.039</td></tr><tr><td>Test 2</td><td>0.379</td><td>-0.075</td><td>0.484</td><td>-0.039</td></tr><tr><td>Test 3</td><td>0.386</td><td>-0.073</td><td>0.489</td><td>-0.039</td></tr></tbody></table>

Table 2 and Table 3 show the predictions of strain limits using the t-d and flat valley methods, respectively, for the cylindrical punch stretch-bending tests R20 mm and R7.5 mm. Again, three specimens were tested for each punch radius.

As can be seen, the predictions of both methods for the R20 mm punch are almost coincident, whereas for the R7.5 mm test, the t-d method slightly underestimates the material formability with respect to the flat-valley results, used as reference, due to the difficulties previously discussed associated to the plastic flow instabilities in PLC materials. Generally, the highest difference in mean values was around 14% for the R7.5 mm punch. It can be said that the estimation by both methodologies shows a remarkably small dispersion.

Compared with the ISO standard, although the ISO predictions in the case of smooth strain gradient tests, that is, R20 mm cylindrical punch, slightly overpredict with reference values and yield differences of about 11%. Therefore, it becomes evident its inability to assess the formability limits in cases of severe strain gradient tests, in our case, the R7.5 mm cylindrical punch. As mentioned above, this limitation is related to the fact that this standard was tailored to tests with negligible bending effect (Nakazima/Marciniak tests) and not directly to the fact of the existence of the PLC phenomenon.

## Conclusions

In this study, the formability of a 2.94 mm thick AA5754H11 PLC-driven material is analysed in two different stretch-bending tests using cylindrical punch radii, one R20 mm that generate a smooth/mild strain gradient in the sheet, and another of R7.5 mm, which induces a much severe one. Various necking detection techniques, particularly ISO12004-2:2008 [7], and the time-dependent and flat valley methods proposed in the literature [8], have been employed to obtain and discuss their predictive capabilities to assess the necking limit strains. The primary outcomes of this study are the following:

* ISO 12004-2:2008 is not able to estimate the necking limit strains under mild or severe stretch-bending conditions.
* PLC-driven materials exhibit significant instability during their plastic deformation and necking development, being not conducive to the application of time-dependent detection methods of necking.

* In general, time-dependent methods, based on the analysis of time derivative of strains, for instance, must be applied very carefully to obtain reliable estimations.

* Local curvature evaluation methods, such as the flat-valley method, seem to be the most suitable to assess the onset of necking in these materials and severe stretch bending conditions.

A more extensive systematic study, using different punch radii, is being carried out to corroborate the previous conclusions more accurately.

## Acknowledgements

The authors acknowledge the funding provided by Grant PID2021-125934OB-I00 financed by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe” (EU). The collaboration of the BSc student Bruno Cruz Luque is also acknowledged.

## References

[1] F. Liu, Y. Li, S. Ghafoor, Z. Chen, F. Li, J. Li, Sustainability assessment of incremental sheet forming: a review, Int. J. Adv. Manuf. Tech. 119 (2022) 1385-1405.
https://doi.org/10.1007/s00170-021-08368-6

[2] C. Wang, S.D.C. Walsh, Z. Weng, M.W. Haynes, D. Summerfield, A. Feitz, Green steel: synergies between the Australian iron ore industry and the production of green hydrogen, Int. J. of Hydrogen Energy 48 (2023) 32277-32293. https://doi.org/10.1016/j.ijhydene.2023.05.041

[3] H. Halim, D.S. Wilkinson, M. Niewczas, The Portevin-Le Chatelier (PLC) effect and shear band formation in an AA5754 alloy, Acta Materialia 55 (2007) 4151-4160.
https://doi.org/10.1016/j.actamat.2007.03.007

[4] X. Feng, G. Fischer, R. Zielke, B. Svendsen, W. Tillmann, Investigation of PLC band nucleation in AA5754, Materials Science & Engineering A 539 (2012) 205-210.
https://doi.org/10.1016/j.msea.2012.01.082

[5] Y. Hou, J. Min, J. Lin, J.E. Carsley, T.B. Stoughton, Plastic instabilities in AA5754-O under various stress states, IOP Conf. Series: Materials Science and Engineering 418 (2018) 012050.
https://doi.org/10.1088/1757-899X/418/1/012050

[6] B. Reyne, P.Y. Manach, N. Moes, Macroscopic consequences of Piobert-Lüders and Portevin-Le Chatelier bands during tensile deformation in Al-Mg alloys, Materials Science & Engineering A 746 (2019) 187-196. https://doi.org/10.1016/j.msea.2019.01.009

[7] ISO 12004-2:2008, Metallic materials-sheet and strip-determination of forming limit curves, Part 2: Determination of forming limit curves in the laboratory, Switzerland, 2008.

[8] A.J. Martínez-Donaire, F.J. García-Lomas, C. Vallellano, New approaches to detect the onset of localised necking in sheets under through-thickness strain gradients, Materials & Design 57 (2014) 135-145. https://doi.org/10.1016/j.matdes.2014.01.012

[9] Q. Situ, M. Jain, D. Metzger, Determination of forming limit diagrams of sheet materials with a hybrid experimental-numerical approach, Int. J. of Mechanical Sciences 53(9) (2011) 707-719.
https://doi.org/10.1016/j.ijmecsci.2011.06.003

[10] J. Min, T.B. Stoughton, J.E. Carsley, J. Lin, An improved curvature method of detecting the onset of localized necking in Marciniak tests and its extension to Nakazima tests, Int. J. of Mechanical Sciences 123 (2017) 238-252. https://doi.org/10.1016/j.ijmecsci.2017.02.011
* PLC-driven materials exhibit significant instability during their plastic deformation and necking development, being not conducive to the application of time-dependent detection methods of necking.

* In general, time-dependent methods, based on the analysis of time derivative of strains, for instance, must be applied very carefully to obtain reliable estimations.

* Local curvature evaluation methods, such as the flat-valley method, seem to be the most suitable to assess the onset of necking in these materials and severe stretch bending conditions.

A more extensive systematic study, using different punch radii, is being carried out to corroborate the previous conclusions more accurately.

## Acknowledgements

The authors acknowledge the funding provided by Grant PID2021-125934OB-I00 financed by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe” (EU). The collaboration of the BSc student Bruno Cruz Luque is also acknowledged.

## References

[1] F. Liu, Y. Li, S. Ghafoor, Z. Chen, F. Li, J. Li, Sustainability assessment of incremental sheet forming: a review, Int. J. Adv. Manuf. Tech. 119 (2022) 1385-1405.
https://doi.org/10.1007/s00170-021-08368-6

[2] C. Wang, S.D.C. Walsh, Z. Weng, M.W. Haynes, D. Summerfield, A. Feitz, Green steel: synergies between the Australian iron ore industry and the production of green hydrogen, Int. J. of Hydrogen Energy 48 (2023) 32277-32293. https://doi.org/10.1016/j.ijhydene.2023.05.041

[3] H. Halim, D.S. Wilkinson, M. Niewczas, The Portevin-Le Chatelier (PLC) effect and shear band formation in an AA5754 alloy, Acta Materialia 55 (2007) 4151-4160.
https://doi.org/10.1016/j.actamat.2007.03.007

[4] X. Feng, G. Fischer, R. Zielke, B. Svendsen, W. Tillmann, Investigation of PLC band nucleation in AA5754, Materials Science & Engineering A 539 (2012) 205-210.
https://doi.org/10.1016/j.msea.2012.01.082

[5] Y. Hou, J. Min, J. Lin, J.E. Carsley, T.B. Stoughton, Plastic instabilities in AA5754-O under various stress states, IOP Conf. Series: Materials Science and Engineering 418 (2018) 012050.
https://doi.org/10.1088/1757-899X/418/1/012050

[6] B. Reyne, P.Y. Manach, N. Moes, Macroscopic consequences of Piobert-Lüders and Portevin-Le Chatelier bands during tensile deformation in Al-Mg alloys, Materials Science & Engineering A 746 (2019) 187-196. https://doi.org/10.1016/j.msea.2019.01.009

[7] ISO 12004-2:2008, Metallic materials-sheet and strip-determination of forming limit curves, Part 2: Determination of forming limit curves in the laboratory, Switzerland, 2008.

[8] A.J. Martínez-Donaire, F.J. García-Lomas, C. Vallellano, New approaches to detect the onset of localised necking in sheets under through-thickness strain gradients, Materials & Design 57 (2014) 135-145. https://doi.org/10.1016/j.matdes.2014.01.012

[9] Q. Situ, M. Jain, D. Metzger, Determination of forming limit diagrams of sheet materials with a hybrid experimental-numerical approach, Int. J. of Mechanical Sciences 53(9) (2011) 707-719.
https://doi.org/10.1016/j.ijmecsci.2011.06.003

[10] J. Min, T.B. Stoughton, J.E. Carsley, J. Lin, An improved curvature method of detecting the onset of localized necking in Marciniak tests and its extension to Nakazima tests, Int. J. of Mechanical Sciences 123 (2017) 238-252. https://doi.org/10.1016/j.ijmecsci.2017.02.011
[11] M.M. Shahzamanian, M. Parsazadeh, P.D. Wu, Numerical and analytical analyses of the formability and fracture of AA7075-O aluminum sheets in hemispherical punch tests, Int. J. of Solids and Structures 286-287 (2024) 112558. https://doi.org/10.1016/j.ijsolstr.2023.112558

[12] G. Centeno, A.J. Martínez-Donaire, D. Morales, C. Vallellano, M.B. Silva, P.A.F. Martins, Novel experimental techniques for the determination of the forming limits at necking and fracture, in: J.P. Davim (Ed.), Materials Forming and Machining, Woodhead Publishing Elsevier Ltd., 2015, pp. 1-24. https://doi.org/10.1016/B978-0-85709-483-4.00001-6
[11] M.M. Shahzamanian, M. Parsazadeh, P.D. Wu, Numerical and analytical analyses of the formability and fracture of AA7075-O aluminum sheets in hemispherical punch tests, Int. J. of Solids and Structures 286-287 (2024) 112558. https://doi.org/10.1016/j.ijsolstr.2023.112558

[12] G. Centeno, A.J. Martínez-Donaire, D. Morales, C. Vallellano, M.B. Silva, P.A.F. Martins, Novel experimental techniques for the determination of the forming limits at necking and fracture, in: J.P. Davim (Ed.), Materials Forming and Machining, Woodhead Publishing Elsevier Ltd., 2015, pp. 1-24. https://doi.org/10.1016/B978-0-85709-483-4.00001-6
# Heat-assisted single point incremental forming of Mg-Zn-Zr alloy

OZDEN Ecem¹, a*, VANHOVE Hans¹, b, BRAEM Annabel², c and DUFLOU Joost R.¹, d

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

²KU Leuven, Department of Materials Engineering, Kasteelpark Arenberg 44, 3001 Leuven, Belgium

aecem.ozden@kuleuven.be; bhans.vanhove@kuleuven.be; cannabel.braem@kuleuven.be;

djoost.duflou@kuleuven.be

**Keywords:** Single Point Incremental Forming, Heat-Assisted Forming, Magnesium Alloy

**Abstract.** Magnesium-zinc-zirconium (Mg-Zn-Zr) alloys, with their biocompatibility and biodegradability, exhibit great potential for biomedical applications. However, forming complex geometries poses a challenge due to the low formability of magnesium. Single Point Incremental Forming (SPIF) has emerged as a promising rapid manufacturing technique capable of producing complex-shaped, high-quality products. This preliminary study aims to investigate the feasibility of heat-assisted SPIF for Mg-Zn-Zr alloys, addressing formability at elevated temperatures, and the outcomes in terms of geometrical accuracy and material properties. Systematic parameter variations revealed that elevated temperatures and multistage toolpath strategies significantly improved formability, surpassing a 60° maximum wall angle. Nevertheless, this enhancement led to increased surface defects and reduced strength during forming at elevated temperatures. The key finding highlights the need for a balanced combination of elevated temperature and maximum wall angle to optimize surface quality and strength in complex geometries.

**Introduction**

Magnesium is one of the most promising lightweight metals in terms of strength-to-weight ratio, making it essential for manufacturing lightweight products and contributing to reducing CO₂ emissions [1]. Beyond its numerous advantages, specific magnesium alloys have potential in biomedical applications due to their biocompatibility, biodegradability, and similarity to bone structure with respect to mechanical properties. However, advanced biomedical applications usually require complex geometries, which can be challenging to form due to the limited number of deformation modes of magnesium alloys at room temperature. Forming at elevated temperatures is often suggested to activate slip systems of the hexagonal close-packed (HCP) structure of magnesium alloys. Conventional manufacturing methods such as die casting often compromise endurance strength and ductility. Mg-Zn-Zr is considered highly promising due to its excellent combination of biocompatibility, formability-to-strength balance, and corrosion resistance. Single Point Incremental Forming (SPIF) is a rapid manufacturing technique capable of producing a wide array of geometries with enhanced mechanical properties [2]. However, there exists a gap in the literature regarding incremental sheet forming applications of Mg-Zn-Zr alloys.

In this study, the heat-assisted SPIF process was investigated at varying temperatures, with a specific emphasis on the commercial Mg-Zn-Zr alloy, ZK61. The influence of process parameters on the deformation mechanism and material properties was examined through formability and product properties. Additionally, the evaluation of formability limits encompassed a comprehensive examination of both geometric and material aspects. Overall, this study provides valuable insights and guidelines for the SPIF manufacturing of thin-walled Mg-Zn-Zr products, highlighting their potential in biomedical applications.

<figure><img src="image_1516.png" /></figure>
# Heat-assisted single point incremental forming of Mg-Zn-Zr alloy

OZDEN Ecem¹, a*, VANHOVE Hans¹, b, BRAEM Annabel², c and DUFLOU Joost R.¹, d

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

²KU Leuven, Department of Materials Engineering, Kasteelpark Arenberg 44, 3001 Leuven, Belgium

aecem.ozden@kuleuven.be; bhans.vanhove@kuleuven.be; cannabel.braem@kuleuven.be;

djoost.duflou@kuleuven.be

**Keywords:** Single Point Incremental Forming, Heat-Assisted Forming, Magnesium Alloy

**Abstract.** Magnesium-zinc-zirconium (Mg-Zn-Zr) alloys, with their biocompatibility and biodegradability, exhibit great potential for biomedical applications. However, forming complex geometries poses a challenge due to the low formability of magnesium. Single Point Incremental Forming (SPIF) has emerged as a promising rapid manufacturing technique capable of producing complex-shaped, high-quality products. This preliminary study aims to investigate the feasibility of heat-assisted SPIF for Mg-Zn-Zr alloys, addressing formability at elevated temperatures, and the outcomes in terms of geometrical accuracy and material properties. Systematic parameter variations revealed that elevated temperatures and multistage toolpath strategies significantly improved formability, surpassing a 60° maximum wall angle. Nevertheless, this enhancement led to increased surface defects and reduced strength during forming at elevated temperatures. The key finding highlights the need for a balanced combination of elevated temperature and maximum wall angle to optimize surface quality and strength in complex geometries.

**Introduction**

Magnesium is one of the most promising lightweight metals in terms of strength-to-weight ratio, making it essential for manufacturing lightweight products and contributing to reducing CO₂ emissions [1]. Beyond its numerous advantages, specific magnesium alloys have potential in biomedical applications due to their biocompatibility, biodegradability, and similarity to bone structure with respect to mechanical properties. However, advanced biomedical applications usually require complex geometries, which can be challenging to form due to the limited number of deformation modes of magnesium alloys at room temperature. Forming at elevated temperatures is often suggested to activate slip systems of the hexagonal close-packed (HCP) structure of magnesium alloys. Conventional manufacturing methods such as die casting often compromise endurance strength and ductility. Mg-Zn-Zr is considered highly promising due to its excellent combination of biocompatibility, formability-to-strength balance, and corrosion resistance. Single Point Incremental Forming (SPIF) is a rapid manufacturing technique capable of producing a wide array of geometries with enhanced mechanical properties [2]. However, there exists a gap in the literature regarding incremental sheet forming applications of Mg-Zn-Zr alloys.

In this study, the heat-assisted SPIF process was investigated at varying temperatures, with a specific emphasis on the commercial Mg-Zn-Zr alloy, ZK61. The influence of process parameters on the deformation mechanism and material properties was examined through formability and product properties. Additionally, the evaluation of formability limits encompassed a comprehensive examination of both geometric and material aspects. Overall, this study provides valuable insights and guidelines for the SPIF manufacturing of thin-walled Mg-Zn-Zr products, highlighting their potential in biomedical applications.

<figure><img src="image_1520.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
## Materials and Methods

In this study, commercially available ZK61 in sheet form was chosen for the examination of the SPIF process on Mg-Zn-Zr alloys. The production background is derived from the as-cast ZK61 magnesium alloy, which underwent homogenization at 480°C. Subsequently, it was subjected to hot rolling, followed by multiple warm-rolling processes at 380°C and 280°C, respectively, with final annealing. The as-received sheet had a chemical composition comprising Mg-5.3Zn-0.5Zr (in wt%). The dimensions of the sheets were 225x225 mm with a thickness of 1.6 mm.

For the SPIF process, a KUKA KR500MT robotic arm equipped with a non-rotating hemispherical tipped tool was employed. Air heating was implemented as the heating method, using a Nabertherm N210E chamber kiln with a power rating of 11 kW. The sheet assembly was attached to the side of the heating equipment, which functions as a clampable door (Fig. 1a). This setup ensures secure attachment of the sheet to the heating source when locked and allows easy removal without unclamping the sheet assembly, which primarily comprises the sheet, its backing plate, and the rig. Consequently, this facilitates precise control of the holding time, immediate air cooling post-forming, and continuous clamping during geometrical accuracy measurements.

<figure><img src="image_1517.png" /><figcaption>Fig 1. Configuration of the heat-assisted SPIF setup (a) pre-clamping stage, (b) forming stage after clamping.</figcaption></figure>

The typical forming temperature range practice of magnesium alloys falls between 150°C and 350°C with slight allowances, aiming to activate the basal texture without exceeding grain growth [1,3,4]. Therefore, here, the sheets were heated at specific temperatures of 150°C, 200, 250, 300, and 325°C, respectively. To ensure temperature control of the setup prior to forming, sheet surface temperatures were measured by type K thermocouples that were spot-welded onto spare ZK61 sheets. The thermocouples were strategically placed on both surfaces of the sheet: the heating contact surface, which directly contacts the heat source (Fig. 3a), and the tool forming surface, which directly contacts the air at room temperature (Fig. 3b). As depicted in Fig. 3a and 3b, two thermocouples were positioned on each surface. Temperature readings were recorded throughout the analyses.

Heat-assisted SPIF was investigated through two main geometries: first, a truncated hyperboloid geometry was employed for formability studies to depict the achievable forming range and optimal process parameters, followed by a straight pyramid geometry to explore the temperature impact on material properties.

Table 1 indicates the investigated process parameters in formability studies. A truncated hyperboloid geometry, shown by its start and end angles, incorporates high wall angles to facilitate the investigation of potential fractures. This allows for an examination of fracture positions and surface defects, and consequently, estimating the formability range for each parameter. Besides the temperature, the effect of step size, tool diameter, and the presence of a MoS2 lacquer as a lubricant were investigated through the production of defined geometries. The hyperboloid geometries have an initial diameter of 90 mm, allowing the deformation of four trials per sheet.
## Materials and Methods

In this study, commercially available ZK61 in sheet form was chosen for the examination of the SPIF process on Mg-Zn-Zr alloys. The production background is derived from the as-cast ZK61 magnesium alloy, which underwent homogenization at 480°C. Subsequently, it was subjected to hot rolling, followed by multiple warm-rolling processes at 380°C and 280°C, respectively, with final annealing. The as-received sheet had a chemical composition comprising Mg-5.3Zn-0.5Zr (in wt%). The dimensions of the sheets were 225x225 mm with a thickness of 1.6 mm.

For the SPIF process, a KUKA KR500MT robotic arm equipped with a non-rotating hemispherical tipped tool was employed. Air heating was implemented as the heating method, using a Nabertherm N210E chamber kiln with a power rating of 11 kW. The sheet assembly was attached to the side of the heating equipment, which functions as a clampable door (Fig. 1a). This setup ensures secure attachment of the sheet to the heating source when locked and allows easy removal without unclamping the sheet assembly, which primarily comprises the sheet, its backing plate, and the rig. Consequently, this facilitates precise control of the holding time, immediate air cooling post-forming, and continuous clamping during geometrical accuracy measurements.

<figure><img src="image_1521.png" /><figcaption>Fig 1. Configuration of the heat-assisted SPIF setup (a) pre-clamping stage, (b) forming stage after clamping.</figcaption></figure>

The typical forming temperature range practice of magnesium alloys falls between 150°C and 350°C with slight allowances, aiming to activate the basal texture without exceeding grain growth [1,3,4]. Therefore, here, the sheets were heated at specific temperatures of 150°C, 200, 250, 300, and 325°C, respectively. To ensure temperature control of the setup prior to forming, sheet surface temperatures were measured by type K thermocouples that were spot-welded onto spare ZK61 sheets. The thermocouples were strategically placed on both surfaces of the sheet: the heating contact surface, which directly contacts the heat source (Fig. 3a), and the tool forming surface, which directly contacts the air at room temperature (Fig. 3b). As depicted in Fig. 3a and 3b, two thermocouples were positioned on each surface. Temperature readings were recorded throughout the analyses.

Heat-assisted SPIF was investigated through two main geometries: first, a truncated hyperboloid geometry was employed for formability studies to depict the achievable forming range and optimal process parameters, followed by a straight pyramid geometry to explore the temperature impact on material properties.

Table 1 indicates the investigated process parameters in formability studies. A truncated hyperboloid geometry, shown by its start and end angles, incorporates high wall angles to facilitate the investigation of potential fractures. This allows for an examination of fracture positions and surface defects, and consequently, estimating the formability range for each parameter. Besides the temperature, the effect of step size, tool diameter, and the presence of a MoS2 lacquer as a lubricant were investigated through the production of defined geometries. The hyperboloid geometries have an initial diameter of 90 mm, allowing the deformation of four trials per sheet.
<table><caption>Table 1. Heat-assisted SPIF process variables in formability studies.</caption><thead><tr><td>Process parameter</td><td>Applied variable</td></tr></thead><tbody><tr><td>Cone start-end angle [deg]</td><td>40-65, 55-80</td></tr><tr><td>Sheet temperature [°C]</td><td>150, 200, 250, 300, 325</td></tr><tr><td>Step size [mm]</td><td>0.1, 0.5, 1.1, 1.5</td></tr><tr><td>Tool diameter [mm]</td><td>5, 10,15</td></tr><tr><td>Lubricant use</td><td>Yes, No</td></tr></tbody></table>

Pyramidal geometries were formed with a set of process parameters determined by the outcome of the formability study. The width of the pyramid was maintained at 184 mm for each experiment, and the tool forces during SPIF were recorded throughout the process. A constant feed rate of 2000 mm/min was consistently maintained throughout all trials in both truncated hyperboloid and pyramid geometries.

After the forming process, sheets were scanned using a Coord3 MC16 CMM equipped with an LC60Dx Laser Line Scanner. The acquired scans were then subjected to post-processing using GOM Inspect 2019 commercial software to observe the thickness distribution for both geometries and the wall angle at their forming heights for truncated hyperboloid geometry.

Further investigations into material properties involved conducting hardness and microstructural analyses on the produced pyramids. Hardness measurements were performed through the thickness of the sheet. First, at a 15-mm distance from the top of the formed pyramid along the vertical axis-z for each process, samples were cut perpendicular to the forming angle (Fig. 2a). Subsequently, Vickers hardness (HV) tests were conducted on both embedded and ground specimens using a Vickers hardness tester (FV-700, Future-Tech) with an applied load of 0.5 kgf for a dwelling time of 15 s. The measuring lines were defined considering the tool forming surface of the sheet, referred to as the tool contact surface, with the distance proportioned accordingly (Fig. 2b). As depicted in Fig. 2b, five measuring lines were defined with their distances, and for each line, five measurements were taken.

<figure><img src="image_1518.png" /><figcaption>Fig 2. Hardness measurement on the pyramids (a) cutting position and (b) hardness measurement lines.</figcaption></figure>

Microstructural analyses were carried out using optical light microscopy (Leica Laborlux) with image analysis software (AxioCam MRc5, Zeiss). The cutting process remained consistent with that in the hardness characterization, performed through the sheet thickness from the same position
<table><caption>Table 1. Heat-assisted SPIF process variables in formability studies.</caption><thead><tr><td>Process parameter</td><td>Applied variable</td></tr></thead><tbody><tr><td>Cone start-end angle [deg]</td><td>40-65, 55-80</td></tr><tr><td>Sheet temperature [°C]</td><td>150, 200, 250, 300, 325</td></tr><tr><td>Step size [mm]</td><td>0.1, 0.5, 1.1, 1.5</td></tr><tr><td>Tool diameter [mm]</td><td>5, 10,15</td></tr><tr><td>Lubricant use</td><td>Yes, No</td></tr></tbody></table>

Pyramidal geometries were formed with a set of process parameters determined by the outcome of the formability study. The width of the pyramid was maintained at 184 mm for each experiment, and the tool forces during SPIF were recorded throughout the process. A constant feed rate of 2000 mm/min was consistently maintained throughout all trials in both truncated hyperboloid and pyramid geometries.

After the forming process, sheets were scanned using a Coord3 MC16 CMM equipped with an LC60Dx Laser Line Scanner. The acquired scans were then subjected to post-processing using GOM Inspect 2019 commercial software to observe the thickness distribution for both geometries and the wall angle at their forming heights for truncated hyperboloid geometry.

Further investigations into material properties involved conducting hardness and microstructural analyses on the produced pyramids. Hardness measurements were performed through the thickness of the sheet. First, at a 15-mm distance from the top of the formed pyramid along the vertical axis-z for each process, samples were cut perpendicular to the forming angle (Fig. 2a). Subsequently, Vickers hardness (HV) tests were conducted on both embedded and ground specimens using a Vickers hardness tester (FV-700, Future-Tech) with an applied load of 0.5 kgf for a dwelling time of 15 s. The measuring lines were defined considering the tool forming surface of the sheet, referred to as the tool contact surface, with the distance proportioned accordingly (Fig. 2b). As depicted in Fig. 2b, five measuring lines were defined with their distances, and for each line, five measurements were taken.

<figure><img src="image_1522.png" /><figcaption>Fig 2. Hardness measurement on the pyramids (a) cutting position and (b) hardness measurement lines.</figcaption></figure>

Microstructural analyses were carried out using optical light microscopy (Leica Laborlux) with image analysis software (AxioCam MRc5, Zeiss). The cutting process remained consistent with that in the hardness characterization, performed through the sheet thickness from the same position
of the pyramid, as depicted in Fig. 2a. After grinding and polishing steps, samples were etched with an acetic-glycol solution consisting of 20% acetic acid, 1% nitric acid, 60% ethylene glycol, and 19% water for 30 s.

## Results and discussion

Temperature profile of the sheet. A series of experimental analyses were conducted to determine three key parameters: the required holding time for the sheet to achieve the desired temperature and steady-state condition, the temperature difference between the front and back surfaces of the sheet, and the temperature distribution on each surface. Fig. 3c depicts the temperature readings for the minimum and maximum temperature studied, with the average sheet temperatures of 150°C and 325°C, respectively, from the moment the sheet was inserted into the heat source. After approximately 50 min of clamping the sheet into the heat source, temperature values remained almost consistent with a slight increase of less than 10°C in 45 min, indicating the attainment of a steady-state condition. The temperature recordings of thermocouples A, B, and C are depicted in Fig. 3c, where they were recorded simultaneously. However, thermocouple D was measured in a separate experiment, maintaining consistent values throughout. Therefore, 50 min was considered the holding time for the sheet at all specific temperature values until the initiation of the forming process. Subsequent to completing the forming, the sheet was immediately subjected to air cooling.

In line with Leonhardt et al. [3], a positive correlation was observed between the forming temperature and the thermal gradient under steady-state conditions. Specifically, a thermal gradient of 24°C was observed across the average sheet temperature of 325°C, while a difference of 15°C was measured at 150°C (Fig. 3c).

<figure><img src="image_1519.png" /><figcaption>Fig 3. Experimental investigation of temperature control for the sheet (a) thermocouple positions on the heating contact surface, (b) thermocouple positions on the tool contact surface, and (c) time-dependent thermocouple recordings.</figcaption></figure>

Formability analysis under varying SPIF parameters. The maximum wall angle plays a pivotal role in the incremental sheet forming process, acting as a key indicator of material formability, and thus limiting the process window [5]. For conventional single-step SPIF, the final thickness can be approximated by the sine law, which is a working factor in stretching, shear, and plane strain deformation. However, as described by Jackson et al. [6], SPIF processes, involving shear in parallel and perpendicular to the forming direction, stretching perpendicular to the forming direction, result in significantly lower thickness values compared to the sine law. The maximum
of the pyramid, as depicted in Fig. 2a. After grinding and polishing steps, samples were etched with an acetic-glycol solution consisting of 20% acetic acid, 1% nitric acid, 60% ethylene glycol, and 19% water for 30 s.

## Results and discussion

Temperature profile of the sheet. A series of experimental analyses were conducted to determine three key parameters: the required holding time for the sheet to achieve the desired temperature and steady-state condition, the temperature difference between the front and back surfaces of the sheet, and the temperature distribution on each surface. Fig. 3c depicts the temperature readings for the minimum and maximum temperature studied, with the average sheet temperatures of 150°C and 325°C, respectively, from the moment the sheet was inserted into the heat source. After approximately 50 min of clamping the sheet into the heat source, temperature values remained almost consistent with a slight increase of less than 10°C in 45 min, indicating the attainment of a steady-state condition. The temperature recordings of thermocouples A, B, and C are depicted in Fig. 3c, where they were recorded simultaneously. However, thermocouple D was measured in a separate experiment, maintaining consistent values throughout. Therefore, 50 min was considered the holding time for the sheet at all specific temperature values until the initiation of the forming process. Subsequent to completing the forming, the sheet was immediately subjected to air cooling.

In line with Leonhardt et al. [3], a positive correlation was observed between the forming temperature and the thermal gradient under steady-state conditions. Specifically, a thermal gradient of 24°C was observed across the average sheet temperature of 325°C, while a difference of 15°C was measured at 150°C (Fig. 3c).

<figure><img src="image_1523.png" /><figcaption>Fig 3. Experimental investigation of temperature control for the sheet (a) thermocouple positions on the heating contact surface, (b) thermocouple positions on the tool contact surface, and (c) time-dependent thermocouple recordings.</figcaption></figure>

Formability analysis under varying SPIF parameters. The maximum wall angle plays a pivotal role in the incremental sheet forming process, acting as a key indicator of material formability, and thus limiting the process window [5]. For conventional single-step SPIF, the final thickness can be approximated by the sine law, which is a working factor in stretching, shear, and plane strain deformation. However, as described by Jackson et al. [6], SPIF processes, involving shear in parallel and perpendicular to the forming direction, stretching perpendicular to the forming direction, result in significantly lower thickness values compared to the sine law. The maximum
wall angle accordingly could be improved through multistage toolpath strategies, forming at elevated temperatures, as well as material properties [2, 5, 7]. In line with a multi-stage toolpath strategy, the achievable forming range was investigated in this study with a truncated hyperboloid geometry with a starting and ending angle, aiming to explore the limits at elevated temperatures. However, it is important to note that the chosen approach is useful for identifying the feasible formability range rather than determining the maximum angle which in this regard, cone tests are more suitable. Here, the results may yield slightly higher values than those expected from ultimate formability tests using actual cones.

Initially, the examination focused on temperature, employing truncated hyperboloid geometries while keeping other parameters constant (10 mm tool diameter and a 0.5 mm step size, without the use of lubricant). At a temperature of 150°C, the maximum wall angle was reported as 44°, increasing to 50° and 65° for temperatures of 200°C and 250°C, respectively. Fig. 4 illustrates two trial examples, showcasing incremental wall angles from the start to the end, along with resulting thickness values at each forming distance. Notably, at 300°C, a wall angle of 65° (Fig. 4b), and at 325°C, a wall angle of 80° was reached without any fractures (Fig. 4a).

Despite achieving high wall angles, the surface quality of the sheets exhibited limitations. Forming at lower temperatures of 150°C and 200°C led to debris formation during SPIF. Forming temperatures exceeding 200°C showed incrementally improved surface quality in terms of debris formation and loss of tearing. However, at 300°C and 325°C, oxidation/contamination on the tool-sheet interface was observed, indicated by a noticeable color change. Although orange peel defects were observed in all trials, higher wall angles show an increase in orange peel defects, both parallel to and perpendicular to the forming direction.

<figure><img src="image_1520.png" /><figcaption>Fig 4. Thickness distribution and wall angles of the produced truncated hyperboloid geometry
(a) 55-80° wall angle, at 325°C (b) 40-65° wall angle, at 300°C.</figcaption></figure>

Furthermore, in the investigation of the step size, a smaller step size led to higher achievable wall angles with improved surface qualities. While the difference between 0.1 mm and 0.5 mm in the range of formability did not yield a significant impact, better surface quality was observed at 0.1 mm. Conversely, larger step sizes of 1.1 mm and 1.5 mm, resulted in increased surface cracks, especially with sharp cracks at the step-down location, mainly due to the presence of excessive in-plane compressive stress in a higher contact area of the sheet-tool surface [2].

The use of a hemispherical tool with a diameter of 15 mm led to a considerable reduction in formability, resulting in the occurrence of a bulging defect at the process onset, even at the highest temperature of 325°C.

The application of a MoS₂ lacquer lubricant had a positive impact on surface quality, which will be further investigated especially at temperatures above 300°C, to remove contamination on the tool-sheet interface observed completely.
wall angle accordingly could be improved through multistage toolpath strategies, forming at elevated temperatures, as well as material properties [2, 5, 7]. In line with a multi-stage toolpath strategy, the achievable *forming range* was investigated in this study with a truncated hyperboloid geometry with a starting and ending angle, aiming to explore the limits at elevated temperatures. However, it is important to note that the chosen approach is useful for identifying the feasible formability range rather than determining the maximum angle which in this regard, cone tests are more suitable. Here, the results may yield slightly higher values than those expected from ultimate formability tests using actual cones.

Initially, the examination focused on temperature, employing truncated hyperboloid geometries while keeping other parameters constant (10 mm tool diameter and a 0.5 mm step size, without the use of lubricant). At a temperature of 150°C, the maximum wall angle was reported as 44°, increasing to 50° and 65° for temperatures of 200°C and 250°C, respectively. Fig. 4 illustrates two trial examples, showcasing incremental wall angles from the start to the end, along with resulting thickness values at each forming distance. Notably, at 300°C, a wall angle of 65° (Fig. 4b), and at 325°C, a wall angle of 80° was reached without any fractures (Fig. 4a).

Despite achieving high wall angles, the surface quality of the sheets exhibited limitations. Forming at lower temperatures of 150°C and 200°C led to debris formation during SPIF. Forming temperatures exceeding 200°C showed incrementally improved surface quality in terms of debris formation and loss of tearing. However, at 300°C and 325°C, oxidation/contamination on the tool-sheet interface was observed, indicated by a noticeable color change. Although orange peel defects were observed in all trials, higher wall angles show an increase in orange peel defects, both parallel to and perpendicular to the forming direction.

<figure><img src="image_1524.png" /><figcaption>Fig 4. Thickness distribution and wall angles of the produced truncated hyperboloid geometry
(a) 55-80° wall angle, at 325°C (b) 40-65° wall angle, at 300°C.</figcaption></figure>

Furthermore, in the investigation of the step size, a smaller step size led to higher achievable wall angles with improved surface qualities. While the difference between 0.1 mm and 0.5 mm in the range of formability did not yield a significant impact, better surface quality was observed at 0.1 mm. Conversely, larger step sizes of 1.1 mm and 1.5 mm, resulted in increased surface cracks, especially with sharp cracks at the step-down location, mainly due to the presence of excessive in-plane compressive stress in a higher contact area of the sheet-tool surface [2].

The use of a hemispherical tool with a diameter of 15 mm led to a considerable reduction in formability, resulting in the occurrence of a bulging defect at the process onset, even at the highest temperature of 325°C.

The application of a MoS₂ lacquer lubricant had a positive impact on surface quality, which will be further investigated especially at temperatures above 300°C, to remove contamination on the tool-sheet interface observed completely.
Effect of temperature on product geometry and material properties. Initially, the aim was to produce a set wall angle for comparing a broad temperature range. Since the maximum wall angle was obtained in the formability analysis at the lowest temperature (44° at 150°C), a comparable wall angle of 40° was determined for a fixed pyramid wall angle. Considering the formability investigations as well as time efficiency, a 10 mm tool diameter, and a 0.5 mm step size were chosen as default parameters.

During forming at 150°C and 200°C debris formation was observed, along with distinct linear grooves, galling marks, and tearing. At 250°C, tearing was also observed in a relatively shorter area of the sheet. At 325°C, a color change on the tool surface was noted, as previously observed.

As shown in Fig. 5, forming at 325°C, the average sheet thickness on the formed areas was 1.23 mm, 0.94, and 0.8 mm for the wall angles of 40°, 50°, and 60°, respectively. Forming forces increased with increasing wall angle but decreased with increasing temperature for straight wall angles.

<figure><img src="image_1521.png" /><figcaption>Fig 5. Thickness distribution of straight pyramids formed at 325°C with wall angles of (a) 40°, (b) 50°, and (c) 60°.</figcaption></figure>

Fig. 6 indicates the average hardness values along with their corresponding standard deviations for each line. It was noted that as the forming temperature decreases, there is an increase in the hardness values of the sheets with a 40° wall angle. Moreover, the hardness distribution from the tool contact surface to the heating contact surface decreases, as the shear being predominant on the tool surface. However, upon closer inspection of the 40° angle at 325°C, the hardness values are close to those of the as-received sheet, which has undergone extensive annealing. When comparing these values to those obtained from the 60° wall angle at 325°C, similar hardness values were observed. Despite experiencing different strains, both sheets exhibit similar hardness values to each other and closely resemble the hardness of the as-received sheet. This suggests that recrystallization has taken place at the forming temperature of 325°C.
Effect of temperature on product geometry and material properties. Initially, the aim was to produce a set wall angle for comparing a broad temperature range. Since the maximum wall angle was obtained in the formability analysis at the lowest temperature (44° at 150°C), a comparable wall angle of 40° was determined for a fixed pyramid wall angle. Considering the formability investigations as well as time efficiency, a 10 mm tool diameter, and a 0.5 mm step size were chosen as default parameters.

During forming at 150°C and 200°C debris formation was observed, along with distinct linear grooves, galling marks, and tearing. At 250°C, tearing was also observed in a relatively shorter area of the sheet. At 325°C, a color change on the tool surface was noted, as previously observed.

As shown in Fig. 5, forming at 325°C, the average sheet thickness on the formed areas was 1.23 mm, 0.94, and 0.8 mm for the wall angles of 40°, 50°, and 60°, respectively. Forming forces increased with increasing wall angle but decreased with increasing temperature for straight wall angles.

<figure><img src="image_1525.png" /><figcaption>Fig 5. Thickness distribution of straight pyramids formed at 325°C with wall angles of (a) 40°, (b) 50°, and (c) 60°.</figcaption></figure>

Fig. 6 indicates the average hardness values along with their corresponding standard deviations for each line. It was noted that as the forming temperature decreases, there is an increase in the hardness values of the sheets with a 40° wall angle. Moreover, the hardness distribution from the tool contact surface to the heating contact surface decreases, as the shear being predominant on the tool surface. However, upon closer inspection of the 40° angle at 325°C, the hardness values are close to those of the as-received sheet, which has undergone extensive annealing. When comparing these values to those obtained from the 60° wall angle at 325°C, similar hardness values were observed. Despite experiencing different strains, both sheets exhibit similar hardness values to each other and closely resemble the hardness of the as-received sheet. This suggests that recrystallization has taken place at the forming temperature of 325°C.
<figure><img src="image_1522.png" /><figcaption>Fig 6. *Through-thickness hardness of the pyramids formed under heat-assisted SPIF in varying wall angles and temperatures.*</figcaption></figure>

Fig. 7 presents microstructural images of through-thickness samples formed at temperatures of 250°C and 325°C. At 250°C, distinct forming stripes aligned with the forming direction were evident (Fig. 7c), while at 325°C, a weakened texture was observed (Fig. 7d). This weakened texture was attributed to significant recrystallization, evolving the formation of equiaxed fine grains. These observations are consistent with the typical ZK60/61 microstructure following severe forming at elevated temperatures, wherein coarse grains are broken under shear stress and subsequently evolve into finer equiaxed grains [8, 9]. This finding is also in line with the hardness values observed in formed sheets at 325°C for wall angles of 40° and 60° (refer to Fig. 6), indicating the weakened texture due to recrystallization.

Additionally, grains in the microstructure are typically surrounded by visible precipitates along the grain boundaries, appearing as black dots, which are more obvious in Fig. 7d. These precipitates, often associated with the MgZn₂, play a significant role in dynamic recrystallization and hinder grain growth by acting as effective nucleation sites [10].
<figure><img src="image_1526.png" /><figcaption>Fig 6. Through-thickness hardness of the pyramids formed under heat-assisted SPIF in varying wall angles and temperatures.</figcaption></figure>

Fig. 7 presents microstructural images of through-thickness samples formed at temperatures of 250°C and 325°C. At 250°C, distinct forming stripes aligned with the forming direction were evident (Fig. 7c), while at 325°C, a weakened texture was observed (Fig. 7d). This weakened texture was attributed to significant recrystallization, evolving the formation of equiaxed fine grains. These observations are consistent with the typical ZK60/61 microstructure following severe forming at elevated temperatures, wherein coarse grains are broken under shear stress and subsequently evolve into finer equiaxed grains [8, 9]. This finding is also in line with the hardness values observed in formed sheets at 325°C for wall angles of 40° and 60° (refer to Fig. 6), indicating the weakened texture due to recrystallization.

Additionally, grains in the microstructure are typically surrounded by visible precipitates along the grain boundaries, appearing as black dots, which are more obvious in Fig. 7d. These precipitates, often associated with the MgZn₂, play a significant role in dynamic recrystallization and hinder grain growth by acting as effective nucleation sites [10].
<figure><img src="image_1523.png" /><figcaption>Fig 7. Microstructures of pyramids at a wall angle of 40°, near the tool touching surface formed at (a, c) 250°C and (b, d) 325°C.</figcaption></figure>

## Conclusions and Outlook

In this study, the novel use of heat-assisted SPIF on Mg-Zn-Zr magnesium alloy sheets was explored, providing valuable insights into how forming temperatures and toolpath strategies affect the resulting product geometry and material properties. Throughout the forming process, precise temperature control was facilitated by the integrated heating system.

Significant improvement was observed at elevated temperatures, along with the implementation of multistage toolpath strategies. Notably, maximum wall angles exceeding 60° were achieved without cracks above 250°C. However, an increase in the orange peel defect was noted, both parallel and perpendicular to the forming direction, when wall angles exceeded 60°. Furthermore, a noticeable decrease in strength, primarily attributed to a weakened texture due to recrystallization, was observed, particularly at a temperature of 325°C. Conversely, at forming temperatures of 150 and 200°C, surface defects characterized by distinct linear grooves and debris formation were evident on the product surface. Consequently, the findings of this study suggest a crucial temperature range for achieving high-quality SPIF-formed Mg-Zn-Zr products, ranging from above 200°C to below 325°C. Within this range, establishing a precise process window is essential for the attainment of enhanced product quality in the production of complex geometries.

## Acknowledgment

This research was conducted within the framework of the STIFF project, facilitated by the KU Leuven C2 research fund (C24E/21/030). The authors express their gratitude to the master's thesis student, Tomas De Langhe for his contributions during the experimental work in the heat-assisted SPIF process.
<figure><img src="image_1527.png" /><figcaption>Fig 7. Microstructures of pyramids at a wall angle of 40°, near the tool touching surface formed at (a, c) 250°C and (b, d) 325°C.</figcaption></figure>

## Conclusions and Outlook

In this study, the novel use of heat-assisted SPIF on Mg-Zn-Zr magnesium alloy sheets was explored, providing valuable insights into how forming temperatures and toolpath strategies affect the resulting product geometry and material properties. Throughout the forming process, precise temperature control was facilitated by the integrated heating system.

Significant improvement was observed at elevated temperatures, along with the implementation of multistage toolpath strategies. Notably, maximum wall angles exceeding 60° were achieved without cracks above 250°C. However, an increase in the orange peel defect was noted, both parallel and perpendicular to the forming direction, when wall angles exceeded 60°. Furthermore, a noticeable decrease in strength, primarily attributed to a weakened texture due to recrystallization, was observed, particularly at a temperature of 325°C. Conversely, at forming temperatures of 150 and 200°C, surface defects characterized by distinct linear grooves and debris formation were evident on the product surface. Consequently, the findings of this study suggest a crucial temperature range for achieving high-quality SPIF-formed Mg-Zn-Zr products, ranging from above 200°C to below 325°C. Within this range, establishing a precise process window is essential for the attainment of enhanced product quality in the production of complex geometries.

## Acknowledgment

This research was conducted within the framework of the STIFF project, facilitated by the KU Leuven C2 research fund (C24E/21/030). The authors express their gratitude to the master's thesis student, Tomas De Langhe for his contributions during the experimental work in the heat-assisted SPIF process.
## References

[1] Chen, M., Ma, C., Liu, Q., Cheng, M., Wang, H., & Hu, X. (2023). Plastic Deformation Mechanism of High Strength and Toughness ZK61 Magnesium Alloy Plate by Multipass Horizontal Continuous Rolling. Materials, 16(3). https://doi.org/10.3390/ma16031320

[2] Duflou, J. R., Habraken, A. M., Cao, J., Malhotra, R., Bambach, M., Adams, D., Vanhove, H., Mohammadi, A., & Jeswiet, J. (2018). Single point incremental forming: state-of-the-art and prospects. In International Journal of Material Forming (Vol. 11, Issue 6). https://doi.org/10.1007/s12289-017-1387-y

[3] Leonhardt, A., Kurz, G., Victoria-Hernández, J., Kräusel, V., Landgrebe, D., & Letzig, D. (2018). Experimental study on incremental sheet forming of magnesium alloy AZ31 with hot air heating. Procedia Manufacturing, 15. https://doi.org/10.1016/j.promfg.2018.07.369

[4] Ambrogio, G., Filice, L., & Manco, G. L. (2008). Warm incremental forming of magnesium alloy AZ31. CIRP Annals - Manufacturing Technology, 57(1). https://doi.org/10.1016/j.cirp.2008.03.066

[5] Duflou, J. R., Verbert, J., Belkassem, B., Gu, J., Sol, H., Henrard, C., & Habraken, A. M. (2008). Process window enhancement for single point incremental forming through multi-step toolpaths. CIRP Annals - Manufacturing Technology, 57(1). https://doi.org/10.1016/j.cirp.2008.03.030

[6] Jackson, K., & Allwood, J. (2009). The mechanics of incremental sheet forming. Journal of Materials Processing Technology, 209(3). https://doi.org/10.1016/j.jmatprotec.2008.03.025

[7] Hirt, G., Bambach, M., Bleck, W., Prahl, U., & Stollenwerk, J. (2015). The Development of Incremental Sheet Forming from Flexible Forming to Fully Integrated Production of Sheet Metal Parts. https://doi.org/10.1007/978-3-319-12304-2_9

[8] Shi, F., Piao, N., Wang, H., Wang, J., Zang, Q., Guo, Y., Chen, C., & Zhang, L. (2023). Investigation of microstructure and mechanical properties of ZK60 magnesium alloy achieved by extrusion-shearing process. Journal of Materials Research and Technology, 25. https://doi.org/10.1016/j.jmrt.2023.05.256

[9] Mansoor, B., Mukhcrjee, S., & Ghosh, A. (2010). High strength ZK60 Mg plate produced by grain refinement and precipitation during alternate biaxial reverse corrugation (ABRC) process and friction stir process (FSP). Magnesium Technology.

[10] Xue, Y., Pang, X., Karparvarfard, S. M. H., Jahed, H., Luo, S., & Shen, Y. (2022). Corrosion Protection of ZK60 Wrought Magnesium Alloys by Micro-Arc Oxidation. Metals, 12(3). https://doi.org/10.3390/met12030449
## References

[1] Chen, M., Ma, C., Liu, Q., Cheng, M., Wang, H., & Hu, X. (2023). Plastic Deformation Mechanism of High Strength and Toughness ZK61 Magnesium Alloy Plate by Multipass Horizontal Continuous Rolling. Materials, 16(3). https://doi.org/10.3390/ma16031320

[2] Duflou, J. R., Habraken, A. M., Cao, J., Malhotra, R., Bambach, M., Adams, D., Vanhove, H., Mohammadi, A., & Jeswiet, J. (2018). Single point incremental forming: state-of-the-art and prospects. In International Journal of Material Forming (Vol. 11, Issue 6). https://doi.org/10.1007/s12289-017-1387-y

[3] Leonhardt, A., Kurz, G., Victoria-Hernández, J., Kräusel, V., Landgrebe, D., & Letzig, D. (2018). Experimental study on incremental sheet forming of magnesium alloy AZ31 with hot air heating. Procedia Manufacturing, 15. https://doi.org/10.1016/j.promfg.2018.07.369

[4] Ambrogio, G., Filice, L., & Manco, G. L. (2008). Warm incremental forming of magnesium alloy AZ31. CIRP Annals - Manufacturing Technology, 57(1). https://doi.org/10.1016/j.cirp.2008.03.066

[5] Duflou, J. R., Verbert, J., Belkassem, B., Gu, J., Sol, H., Henrard, C., & Habraken, A. M. (2008). Process window enhancement for single point incremental forming through multi-step toolpaths. CIRP Annals - Manufacturing Technology, 57(1). https://doi.org/10.1016/j.cirp.2008.03.030

[6] Jackson, K., & Allwood, J. (2009). The mechanics of incremental sheet forming. Journal of Materials Processing Technology, 209(3). https://doi.org/10.1016/j.jmatprotec.2008.03.025

[7] Hirt, G., Bambach, M., Bleck, W., Prahl, U., & Stollenwerk, J. (2015). The Development of Incremental Sheet Forming from Flexible Forming to Fully Integrated Production of Sheet Metal Parts. https://doi.org/10.1007/978-3-319-12304-2_9

[8] Shi, F., Piao, N., Wang, H., Wang, J., Zang, Q., Guo, Y., Chen, C., & Zhang, L. (2023). Investigation of microstructure and mechanical properties of ZK60 magnesium alloy achieved by extrusion-shearing process. Journal of Materials Research and Technology, 25. https://doi.org/10.1016/j.jmrt.2023.05.256

[9] Mansoor, B., Mukhcrjee, S., & Ghosh, A. (2010). High strength ZK60 Mg plate produced by grain refinement and precipitation during alternate biaxial reverse corrugation (ABRC) process and friction stir process (FSP). Magnesium Technology.

[10] Xue, Y., Pang, X., Karparvarfard, S. M. H., Jahed, H., Luo, S., & Shen, Y. (2022). Corrosion Protection of ZK60 Wrought Magnesium Alloys by Micro-Arc Oxidation. Metals, 12(3). https://doi.org/10.3390/met12030449
# Stretch forming of isotropic materials: Influence of the ratio between yielding in pure shear and uniaxial tension on the stress state

REVIL-BAUDARD Benoit<sup>1,a*</sup>, GODOY Hernan<sup>2,b</sup> and CAZACU Oana<sup>1,c</sup>

<sup>1</sup>University of Arizona, Department of Materials Science and Engineering, 235 James E. Rogers Way, Tucson, AZ 85719, USA

<sup>2</sup>University of Florida/REEF, 1350 N. Poquito Rd, Shalimar, FL3279, USA

<sup>a</sup>revil@arizona.edu, <sup>b</sup>herncg@ufl.edu, <sup>c</sup>oana.cazacu@arizona.edu

**Keywords:** Isotropic Material, Bulging, Metal Forming, Stress Path

**Abstract.** The formability of metallic sheets could be assessed by performing a hydraulic bulge test. For isotropic materials, interpretation of the bulge test is usually done using the von Mises yield criterion. F.E. simulations of bulge tests were conducted to study how the specificities of the plastic behavior of an isotropic material influences the strain paths and stress paths experienced during hemispherical and elliptical bulging. In this paper, we investigate the role played by the ratio between the yield stresses in pure shear and uniaxial tension of the material, $\tau_y/\sigma_T$ on the mechanical behavior during hydraulic bulging. To this end, we make used of the Drucker yield criterion, which can describe the mechanical behavior of isotropic materials with different ratios $\tau_y/\sigma_T$ with a unique parameter $c$. For $c = 0$, the Drucker criterion reduces to the von Mises yield criterion (i.e. $\tau_y/\sigma_T = 1/\sqrt{3}$) while for $c \neq 0$, it involves dependence on the third invariant of the stress deviator, $J_3$. Finite element predictions using the yield criterion reveals a correlation between the ratio $\tau_y/\sigma_T$ which is uniquely defined in terms of the parameter $c$ and the level of plastic strains that develop in the dome, the thickness reduction at the top of the apex, and the strain paths achieved in an elliptical bulge test.

**Introduction**

Several experiments could be used to extract stress-strain relationship of the metallic materials, the most common one being the tensile test. However, in a tensile test, the range of strains that can be achieved prior to the onset of necking could be limited. A major advantage of the bulge test is that the nature of deformation is such that in most materials, non-uniform strains in the gage area, occurs for larger deformation compared to the tensile test. Therefore, the hardening characteristics of the material may be established over a larger range of effective strain than in the tensile test [1].

The hydraulic bulge test, with either hemispheric or elliptic dies, is also a common method to evaluate the formability of metallic sheets experimentally, due to relatively simplicity of the testing setup (e.g. see [2]–[4]). In a bulge test, a sheet sample is held between two dies and subjected to a hydraulic pressure, which makes the material bulge out of the die opening and form a dome. Since radial deformation is prevented, the sample thickness decreases until it bursts and fails.

A great deal of efforts has been devoted to the theoretical and numerical analysis of the plastic strains and instabilities that develop under hydrostatic bulging. Generally, the test results are interpreted assuming isotropic behavior and yielding governed by the von Mises yield criterion [5]. However, the assumption of plastic behavior governed by the von Mises yield criterion [5] may lead to an overestimation of how much a specimen can bulge before it fails. For isotropic materials that display tension-compression asymmetry on yielding, a new solution to the problem and a correction to the classical relation generally used to determine the stress-strain curve have been derived by Cazacu and Revil-Baudard [6]. However, for textured sheets, the stress state at the top of the bulge can only be determined with the use of a constitutive model for the plastic

<figure><img src="image_1524.png" /></figure>
# Stretch forming of isotropic materials: Influence of the ratio between yielding in pure shear and uniaxial tension on the stress state

REVIL-BAUDARD Benoit<sup>1,a*</sup>, GODOY Hernan<sup>2,b</sup> and CAZACU Oana<sup>1,c</sup>

<sup>1</sup>University of Arizona, Department of Materials Science and Engineering, 235 James E. Rogers Way, Tucson, AZ 85719, USA

<sup>2</sup>University of Florida/REEF, 1350 N. Poquito Rd, Shalimar, FL3279, USA

<sup>a</sup>revil@arizona.edu, <sup>b</sup>herncg@ufl.edu, <sup>c</sup>oana.cazacu@arizona.edu

**Keywords:** Isotropic Material, Bulging, Metal Forming, Stress Path

**Abstract.** The formability of metallic sheets could be assessed by performing a hydraulic bulge test. For isotropic materials, interpretation of the bulge test is usually done using the von Mises yield criterion. F.E. simulations of bulge tests were conducted to study how the specificities of the plastic behavior of an isotropic material influences the strain paths and stress paths experienced during hemispherical and elliptical bulging. In this paper, we investigate the role played by the ratio between the yield stresses in pure shear and uniaxial tension of the material, $\tau_y/\sigma_T$ on the mechanical behavior during hydraulic bulging. To this end, we make used of the Drucker yield criterion, which can describe the mechanical behavior of isotropic materials with different ratios $\tau_y/\sigma_T$ with a unique parameter $c$. For $c = 0$, the Drucker criterion reduces to the von Mises yield criterion (i.e. $\tau_y/\sigma_T = 1/\sqrt{3}$) while for $c \neq 0$, it involves dependence on the third invariant of the stress deviator, $J_3$. Finite element predictions using the yield criterion reveals a correlation between the ratio $\tau_y/\sigma_T$ which is uniquely defined in terms of the parameter $c$ and the level of plastic strains that develop in the dome, the thickness reduction at the top of the apex, and the strain paths achieved in an elliptical bulge test.

**Introduction**

Several experiments could be used to extract stress-strain relationship of the metallic materials, the most common one being the tensile test. However, in a tensile test, the range of strains that can be achieved prior to the onset of necking could be limited. A major advantage of the bulge test is that the nature of deformation is such that in most materials, non-uniform strains in the gage area, occurs for larger deformation compared to the tensile test. Therefore, the hardening characteristics of the material may be established over a larger range of effective strain than in the tensile test [1].

The hydraulic bulge test, with either hemispheric or elliptic dies, is also a common method to evaluate the formability of metallic sheets experimentally, due to relatively simplicity of the testing setup (e.g. see [2]–[4]). In a bulge test, a sheet sample is held between two dies and subjected to a hydraulic pressure, which makes the material bulge out of the die opening and form a dome. Since radial deformation is prevented, the sample thickness decreases until it bursts and fails.

A great deal of efforts has been devoted to the theoretical and numerical analysis of the plastic strains and instabilities that develop under hydrostatic bulging. Generally, the test results are interpreted assuming isotropic behavior and yielding governed by the von Mises yield criterion [5]. However, the assumption of plastic behavior governed by the von Mises yield criterion [5] may lead to an overestimation of how much a specimen can bulge before it fails. For isotropic materials that display tension-compression asymmetry on yielding, a new solution to the problem and a correction to the classical relation generally used to determine the stress-strain curve have been derived by Cazacu and Revil-Baudard [6]. However, for textured sheets, the stress state at the top of the bulge can only be determined with the use of a constitutive model for the plastic

<figure><img src="image_1528.png" /></figure>
behavior of the material. Furthermore, for orthotropic metallic sheets, the assumptions of an equibiaxial stress state at the top of a hemispherical bulge could lead to significant error in the analysis of the test.

In this paper, an analysis of the bulge test for isotropic materials with the same yielding response in tension and compression, but different ratios between the yield stresses in pure shear and uniaxial tension of the material, $\tau_y/\sigma_T$ is provided. F.E. simulations of bulge tests were conducted to study how the specificities of the plastic behavior of an isotropic material influences the strain paths and stress paths experienced during hemispherical and elliptical bulging. Specifically, the yielding of the isotropic material is described by the Drucker [7] yield criterion that involve a dependence of the yielding on the third invariant of the stress deviator, $J_3$. This yield criterion involves a unique parameter $c$, which is solely expressible in terms of the ratio between the yield stress in simple tension and that in shear $\tau_y/\sigma_T$. Only in the case when $c = 0$ (i.e. $\tau_y/\sigma_T = 1/\sqrt{3}$), the Drucker yield criterion reduces to the von Mises criterion while for $c > 0$, the material yielding is characterized by $\tau_y/\sigma_T < 1/\sqrt{3}$, while for $c < 0$, the material is characterized by a ratio in yielding between pure shear and uniaxial tension $\tau_y/\sigma_T > 1/\sqrt{3}$. Simulations of bulging tests of various die geometries were conducted for isotropic materials characterized by various values of the ratio $\tau_y/\sigma_T$. The paper is organized as follows. In section 2, we present the constitutive model, while the F.E. predictions of bulge tests for a variety of materials to provide the evolution of plastic strains and their distribution during the test and better understanding of the state of stress and deformation developing during elliptical bulge tests, are given in section 3. A summary and concluding remarks are presented in section 4. As concerns the notations and conventions used, tensile stresses and strains are positive; the double-contracted product between any two second-order tensors is defined as $M:N=M_{ij}N_{ij}$, i, j = 1...3.

## Constitutive modeling

Our goal is to study how the yielding description affects the data analysis and the overall behavior of isotropic materials under biaxial stretching deformation. For this purpose, we use an elastic-plastic model with yielding based on Drucker's [7] yield criterion in conjunction with an associated flow rule, and isotropic hardening law. The total rate of deformation $D$ is the sum of the elastic part $D^e$ and the plastic part $D^p$. Linear elastic and isotropic behavior governed by the Hooke's law is used to describe the elastic behavior. The stress-strain relation is given by

$$ \sigma = C^e(D - D^p) \quad (1) $$

where $\sigma$ is the Cauchy stress tensor. In Eq. (1), $C^e$ is the fourth-order stiffness tensor, which, for an isotropic material, is expressed with respect to any coordinate system as

$$ C_{ijkl}^{e} = 2G\delta_{ik}\delta_{jl} - (K - 2G/3)\delta_{ij}\delta_{kl} \quad (2) $$

where $i, j, k, l = 1...3$, $\delta_{ij}$ is the Kronecker unit delta tensor, and $G$ and $K$ are the shear and bulk moduli, respectively. The plastic part of the strain tensor is determined using the associated flow rule:

$$ D^p = \dot{\lambda} \frac{\partial F}{\partial \sigma} \quad (3) $$

where $F$ is the yield function and $\dot{\lambda}$ is the plastic multiplier. Therefore, the plastic potential $F$ in Eq. (3) is of the general form:

$$ F(\sigma, \bar{\epsilon}^p) = \bar{\sigma}(\sigma) - Y(\bar{\epsilon}^p) \quad (4) $$
behavior of the material. Furthermore, for orthotropic metallic sheets, the assumptions of an equibiaxial stress state at the top of a hemispherical bulge could lead to significant error in the analysis of the test.

In this paper, an analysis of the bulge test for isotropic materials with the same yielding response in tension and compression, but different ratios between the yield stresses in pure shear and uniaxial tension of the material, $\tau_y/\sigma_T$ is provided. F.E. simulations of bulge tests were conducted to study how the specificities of the plastic behavior of an isotropic material influences the strain paths and stress paths experienced during hemispherical and elliptical bulging. Specifically, the yielding of the isotropic material is described by the Drucker [7] yield criterion that involve a dependence of the yielding on the third invariant of the stress deviator, $J_3$. This yield criterion involves a unique parameter $c$, which is solely expressible in terms of the ratio between the yield stress in simple tension and that in shear $\tau_y/\sigma_T$. Only in the case when $c = 0$ (i.e. $\tau_y/\sigma_T = 1/\sqrt{3}$), the Drucker yield criterion reduces to the von Mises criterion while for $c > 0$, the material yielding is characterized by $\tau_y/\sigma_T < 1/\sqrt{3}$, while for $c < 0$, the material is characterized by a ratio in yielding between pure shear and uniaxial tension $\tau_y/\sigma_T > 1/\sqrt{3}$. Simulations of bulging tests of various die geometries were conducted for isotropic materials characterized by various values of the ratio $\tau_y/\sigma_T$. The paper is organized as follows. In section 2, we present the constitutive model, while the F.E. predictions of bulge tests for a variety of materials to provide the evolution of plastic strains and their distribution during the test and better understanding of the state of stress and deformation developing during elliptical bulge tests, are given in section 3. A summary and concluding remarks are presented in section 4. As concerns the notations and conventions used, tensile stresses and strains are positive; the double-contracted product between any two second-order tensors is defined as $M:N=M_{ij}N_{ij}$, i, j = 1...3.

## Constitutive modeling

Our goal is to study how the yielding description affects the data analysis and the overall behavior of isotropic materials under biaxial stretching deformation. For this purpose, we use an elastic-plastic model with yielding based on Drucker's [7] yield criterion in conjunction with an associated flow rule, and isotropic hardening law. The total rate of deformation $D$ is the sum of the elastic part $D^e$ and the plastic part $D^p$. Linear elastic and isotropic behavior governed by the Hooke's law is used to describe the elastic behavior. The stress-strain relation is given by

$$ \sigma = C^e(D - D^p) \qquad (1) $$

where $\sigma$ is the Cauchy stress tensor. In Eq.(1), $C^e$ is the fourth-order stiffness tensor, which, for an isotropic material, is expressed with respect to any coordinate system as

$$ C_{ijkl}^{e} = 2G\delta_{ik}\delta_{jl} - (K - 2G/3)\delta_{ij}\delta_{kl} \qquad (2) $$

where $i, j, k, l = 1...3$, $\delta_{ij}$ is the Kronecker unit delta tensor, and $G$ and $K$ are the shear and bulk moduli, respectively. The plastic part of the strain tensor is determined using the associated flow rule:

$$ D^p = \dot{\lambda} \frac{\partial F}{\partial \sigma} \qquad (3) $$

where $F$ is the yield function and $\dot{\lambda}$ is the plastic multiplier. Therefore, the plastic potential $F$ in Eq. (3) is of the general form:

$$ F(\sigma, \bar{\epsilon}^p) = \bar{\sigma}(\sigma) - Y(\bar{\epsilon}^p) \qquad (4) $$
where $\bar{\sigma}$ is the effective stress associated with the Drucker [7] yield criterion, $\bar{\epsilon}^p$ the corresponding work-equivalent accumulated plastic strain and $Y(\bar{\epsilon}^p)$ is the isotropic hardening law. The effective stress associated with the Drucker [7] yield criterion is given by

$$ \bar{\sigma} = B[(J_2^0)^3 - c(J_3^0)^2]^{1/6} \quad (5) $$

In the above equation, $J_2 = 1/2 tra(\sigma^2)$ and $J_3 = 1/3 tra(\sigma^3)$ are the second and third invariant of the stress deviator, respectively. $B$ is a constant defined such that the effective stress reduces to the yield stress for uniaxial tension

$$ B = \frac{3}{(1 - 4c/27)^{1/6}} \quad (6) $$

In Eq. (5) and Eq. (6), $c$ is a parameter that is solely expressible in terms of the yield stress in the uniaxial tension, $\sigma_T$ and the yield stress in pure shear, $\tau_Y$, as

$$ c = \frac{27}{4} \left[ 1 - \left( \frac{\tau_Y \sqrt{3}}{\sigma_T} \right)^6 \right] \quad (7) $$

<figure><img src="image_1525.png" /><figcaption>Fig. 1. Projection in the $(\sigma_{xx}, \sigma_{yy})$ plane of the yield loci according to the isotropic Drucker [7] criterion corresponding to $\tau_Y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_Y/\sigma_T = 1/\sqrt{3}$ ($c = 0$, von Mises material) and $\tau_Y/\sigma_T = 0.60$ ($c = -2$), respectively. Stresses are normalized by the yield stress in uniaxial tension, $\sigma_T$</figcaption></figure>

The convexity of the yield surface for the Drucker yield criterion is ensured for $27/8 \le c \le 2.25$. Hence, the Drucker [7] yield criterion can be used to describe the yielding behavior of isotropic material for which the ratio $\tau_Y/\sigma_T$ is in between $0.54 \le \tau_Y/\sigma_T \le 0.62$ (see Eq. (7)). Note that for $c = 0$, $\tau_Y/\sigma_T = 1/\sqrt{3}$ and the Drucker criterion reduces to the von Mises criterion. The yield surfaces for isotropic materials with their yielding behavior characterized by $c > 0$ (i.e. $\tau_Y/\sigma_T < 1/\sqrt{3}$) are interior to the von Mises one, while for isotropic materials characterized by
where $\bar{\sigma}$ is the effective stress associated with the Drucker [7] yield criterion, $\bar{\epsilon}^p$ the corresponding work-equivalent accumulated plastic strain and $Y(\bar{\epsilon}^p)$ is the isotropic hardening law. The effective stress associated with the Drucker [7] yield criterion is given by

$$ \bar{\sigma} = B[(J_2^0)^3 - c(J_3^0)^2]^{1/6} \quad (5) $$

In the above equation, $J_2 = 1/2 tra(\sigma^2)$ and $J_3 = 1/3 tra(\sigma^3)$ are the second and third invariant of the stress deviator, respectively. $B$ is a constant defined such that the effective stress reduces to the yield stress for uniaxial tension

$$ B = \frac{3}{(1 - 4c/27)^{1/6}} \quad (6) $$

In Eq. (5) and Eq. (6), $c$ is a parameter that is solely expressible in terms of the yield stress in the uniaxial tension, $\sigma_T$ and the yield stress in pure shear, $\tau_Y$, as

$$ c = \frac{27}{4} \left[ 1 - \left( \frac{\tau_Y \sqrt{3}}{\sigma_T} \right)^6 \right] \quad (7) $$

<figure><img src="image_1529.png" /><figcaption>Fig. 1. Projection in the $(\sigma_{xx}, \sigma_{yy})$ plane of the yield loci according to the isotropic Drucker [7] criterion corresponding to $\tau_Y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_Y/\sigma_T = 1/\sqrt{3}$ ($c = 0$, von Mises material) and $\tau_Y/\sigma_T = 0.60$ ($c = -2$), respectively. Stresses are normalized by the yield stress in uniaxial tension, $\sigma_T$</figcaption></figure>

The convexity of the yield surface for the Drucker yield criterion is ensured for $27/8 \le c \le 2.25$. Hence, the Drucker [7] yield criterion can be used to describe the yielding behavior of isotropic material for which the ratio $\tau_Y/\sigma_T$ is in between $0.54 \le \tau_Y/\sigma_T \le 0.62$ (see Eq. (7)). Note that for $c = 0$, $\tau_Y/\sigma_T = 1/\sqrt{3}$ and the Drucker criterion reduces to the von Mises criterion. The yield surfaces for isotropic materials with their yielding behavior characterized by $c > 0$ (i.e. $\tau_Y/\sigma_T < 1/\sqrt{3}$) are interior to the von Mises one, while for isotropic materials characterized by
$c < 0$ (i.e. $\tau_y/\sigma_T > 1/\sqrt{3}$), their yield loci are exterior to the von Mises one. As an illustration, the projections of the yield loci in the biaxial plane ($\sigma_{xx}, \sigma_{yy}$) for materials characterized by $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_y/\sigma_T = 1/\sqrt{3}$ ($c = 0$, von Mises material) and $\tau_y/\sigma_T = 0.60$ ($c = -2$) are plotted in Fig. 1. The isotropic hardening of the material is characterized by a Swift hardening law, defined as

$$Y(\bar{\epsilon}^p) = K_0(\epsilon_0 + \bar{\epsilon}^p)^n \quad (8)$$

Where $K_0$, $\epsilon_0$, and $n$ are material parameters.

## F.E. results for hemispheric and elliptic bulge tests

The main goal of our study is to investigate how yielding behavior characteristics of isotropic materials influence the finite element (F.E.) predictions of the distribution of plastic strain and thickness reduction in a bulge specimen for both hemispheric and elliptic die geometries. For this purpose, numerical simulations of the bulge test were conducted with our in-house UMAT developed for the constitutive model given by the Eq. (1)-(8). Due to the materials' isotropy and the symmetry of the bulging process, only one-quarter of the blank is meshed using 4557 hexahedral element with reduced integration (Abaqus C3D8R elements [8]). The same type of mesh for the blank was used for both hemispherical and elliptical bulge tests (see also Fig. 2(a)), i.e. only the geometries of the die was changed. The blank radius is of 100 mm and the blank's thickness is 1.8 mm, the drawbeads are modeled by radially pinning the nodes at the boundary of the blank. The applied bulging pressure is evolving linearly with time. The values of the elastic parameters considered were: Young's modulus, E=110 GPa and Poisson ratio, v= 0.3. The parameters in the hardening law given by Eq.(8) were set to $K_0=1050$ MPa, $\epsilon_0= 0.001$ and $n= 0.2$. As an example, in Fig. 2(b) is shown the F.E. prediction of the isocontours of the thickness plastic strain for a hemispherical bulge test of a material characterized by a stress ratio $\tau_y/\sigma_T = 0.55$ ($c = 1.5$) for a bulge height of 34.6 mm.

Simulation results are presented for both hemispheric and elliptic bulge test of three isotropic materials characterized by $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_y/\sigma_T = 1/\sqrt{3}$ ($c = 0$, von Mises material) and $\tau_y/\sigma_T = 0.60$ ($c = -2$) and both hemispherical and elliptical dies. The predicted thickness at the pole vs. height of the apex for the hemispheric bulging of the three isotropic materials is shown in Fig. 3. Note that even for hemispherical bulging, for which all the materials have the same yield stress under equibiaxial stretching (see also Fig.1), it is predicted that the thickness at the pole depends on the yielding ratio $\tau_y/\sigma_T$ of the material. For a same bulge height, the lower is the value of the ratio $\tau_y/\sigma_T$, the lower is the thickness reduction. As an example, for the apex experienced a thickness reduction of 25%, it is predicted that the bulge height for a von Mises material is of 31.93 mm, while for a material characterized by $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), the bulge height is of 33.24 mm and for an isotropic material characterized by $\tau_y/\sigma_T = 0.60$ ($c = -2$), the predicted bulge height is of 30.67 mm. This suggests that for a material characterized by a ratio $\tau_y/\sigma_T > 1/\sqrt{3}$ (i.e. a material with a yield locus exterior to the von Mises one), neglecting the specificity of the yielding behavior and modeling its plastic behavior using the von Mises criterion could results into underestimating the thinning of the material under hemispherical bulging.
$c < 0$ (i.e. $\tau_y/\sigma_T > 1/\sqrt{3}$), their yield loci are exterior to the von Mises one. As an illustration, the projections of the yield loci in the biaxial plane $(\sigma_{xx}, \sigma_{yy})$ for materials characterized by $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_y/\sigma_T = 1/\sqrt{3}$ ($c = 0$, von Mises material) and $\tau_y/\sigma_T = 0.60$ ($c = -2$) are plotted in Fig. 1. The isotropic hardening of the material is characterized by a Swift hardening law, defined as

$$Y(\bar{\epsilon}^p) = K_0(\epsilon_0 + \bar{\epsilon}^p)^n \qquad (8)$$

Where $K_0$, $\epsilon_0$, and $n$ are material parameters.

## F.E. results for hemispheric and elliptic bulge tests

The main goal of our study is to investigate how yielding behavior characteristics of isotropic materials influence the finite element (F.E.) predictions of the distribution of plastic strain and thickness reduction in a bulge specimen for both hemispheric and elliptic die geometries. For this purpose, numerical simulations of the bulge test were conducted with our in-house UMAT developed for the constitutive model given by the Eq. (1)-(8). Due to the materials' isotropy and the symmetry of the bulging process, only one-quarter of the blank is meshed using 4557 hexahedral element with reduced integration (Abaqus C3D8R elements [8]). The same type of mesh for the blank was used for both hemispherical and elliptical bulge tests (see also Fig. 2(a)), i.e. only the geometries of the die was changed. The blank radius is of 100 mm and the blank's thickness is 1.8 mm, the drawbeads are modeled by radially pinning the nodes at the boundary of the blank. The applied bulging pressure is evolving linearly with time. The values of the elastic parameters considered were: Young's modulus, E=110 GPa and Poisson ratio, v= 0.3. The parameters in the hardening law given by Eq.(8) were set to $K_0=1050$ MPa, $\epsilon_0= 0.001$ and $n= 0.2$. As an example, in Fig. 2(b) is shown the F.E. prediction of the isocontours of the thickness plastic strain for a hemispherical bulge test of a material characterized by a stress ratio $\tau_y/\sigma_T = 0.55$ ($c = 1.5$) for a bulge height of 34.6 mm.

Simulation results are presented for both hemispheric and elliptic bulge test of three isotropic materials characterized by $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_y/\sigma_T = 1/\sqrt{3}$ ($c = 0$, von Mises material) and $\tau_y/\sigma_T = 0.60$ ($c = -2$) and both hemispherical and elliptical dies. The predicted thickness at the pole vs. height of the apex for the hemispheric bulging of the three isotropic materials is shown in Fig. 3. Note that even for hemispherical bulging, for which all the materials have the same yield stress under equibiaxial stretching (see also Fig.1), it is predicted that the thickness at the pole depends on the yielding ratio $\tau_y/\sigma_T$ of the material. For a same bulge height, the lower is the value of the ratio $\tau_y/\sigma_T$, the lower is the thickness reduction. As an example, for the apex experienced a thickness reduction of 25%, it is predicted that the bulge height for a von Mises material is of 31.93 mm, while for a material characterized by $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), the bulge height is of 33.24 mm and for an isotropic material characterized by $\tau_y/\sigma_T = 0.60$ ($c = -2$), the predicted bulge height is of 30.67 mm. This suggests that for a material characterized by a ratio $\tau_y/\sigma_T > 1/\sqrt{3}$ (i.e. a material with a yield locus exterior to the von Mises one), neglecting the specificity of the yielding behavior and modeling its plastic behavior using the von Mises criterion could results into underestimating the thinning of the material under hemispherical bulging.
<figure><img src="image_1526.png" /><figcaption>Fig. 2. (a) F.E. mesh of the specimen used for the bulge test with circular die. (b) Hemispherical bulging: F.E. predictions of the isocontours of thickness strain for a material by characterized by $\tau_y/\sigma_T = 0.55$.</figcaption></figure>

<figure><img src="image_1527.png" /><figcaption>Fig. 3. F.E. Predictions according to the Drucker [7] yield criterion of the thickness vs. height at the apex of the hemispherical bulge for isotropic materials having various ratios between the yield in shear and uniaxial tension $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_y/\sigma_T = 1/\sqrt{3}$ ($c = 0$, von Mises material) and $\tau_y/\sigma_T = 0.60$ ($c = -2$).</figcaption></figure>

Same conclusions could be drawn from the predicted thickness at the pole vs. height of the apex for elliptical bulging, i.e for a same height of the apex, the thinning will be delay for a material characterized by a ratio $\tau_y/\sigma_T < 1/\sqrt{3}$ compare to a von Mises material. On the other hand, thinning will be occurring for a lower apex height for a material characterized by a ratio $\tau_y/\sigma_T > 1/\sqrt{3}$ than a von Mises material. As an example, Fig. 4 shows the F.E. results according to the Drucker yield criterion for the thickness vs. height at the apex of the elliptic bulge with die ratio $a/b = 1.29$ ($a$ and $b$ denoting the long axis and short axis of the elliptic die, respectively) for
<figure><img src="image_1530.png" /><figcaption>Fig. 2. (a) F.E. mesh of the specimen used for the bulge test with circular die. (b) Hemispherical bulging: F.E. predictions of the isocontours of thickness strain for a material by characterized by $\tau_y/\sigma_T = 0.55$.</figcaption></figure>

<figure><img src="image_1531.png" /><figcaption>Fig. 3. F.E. Predictions according to the Drucker [7] yield criterion of the thickness vs. height at the apex of the hemispherical bulge for isotropic materials having various ratios between the yield in shear and uniaxial tension $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_y/\sigma_T = 1/\sqrt{3}$ ($c = 0$, von Mises material) and $\tau_y/\sigma_T = 0.60$ ($c = -2$).</figcaption></figure>

Same conclusions could be drawn from the predicted thickness at the pole vs. height of the apex for elliptical bulging, i.e for a same height of the apex, the thinning will be delay for a material characterized by a ratio $\tau_y/\sigma_T < 1/\sqrt{3}$ compare to a von Mises material. On the other hand, thinning will be occurring for a lower apex height for a material characterized by a ratio $\tau_y/\sigma_T > 1/\sqrt{3}$ than a von Mises material. As an example, Fig. 4 shows the F.E. results according to the Drucker yield criterion for the thickness vs. height at the apex of the elliptic bulge with die ratio $a/b = 1.29$ ($a$ and $b$ denoting the long axis and short axis of the elliptic die, respectively) for
three isotropic materials. It is worth mentioning the 5.1 % difference in bulge height between the considered isotropic materials to reach the same thickness reduction (25% thickness reduction).

<figure><img src="image_1528.png" /><figcaption>Fig. 4. F.E. Predictions according to the Drucker [7] yield criterion of the thickness vs. height at the apex of the elliptic bulge (die ratio a/b = 1.29) for isotropic materials having various ratios between the yield in shear and uniaxial tension $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_y/\sigma_T = 1/\sqrt{3}$ ($c = 0$), von Mises material) and $\tau_y/\sigma_T = 0.60$ ($c = -2$).</figcaption></figure>

It is worth further examining the influence of the material yielding characteristics on the predicted ratio between the two in-plane strains developing at the apex of the bulge. As expected, under hemispherical bulging, irrespective of the material, the strains ratio is about 1 (see also the yield loci plotted in Fig. 1). However, for an elliptic bulge test, the strain paths achieved during the same experiment are strongly dependent on the plastic properties of the material, namely the ratio $\tau_y/\sigma_T$. Let denote major strain by the strain developing in the direction of the short axis of the elliptic die and minor strain, the strain occurring in the direction of the long axis of the elliptic die.

As an example, it is shown in Fig. 5 the evolution of the major to minor plastic strain ratio developing in an elliptic bulging test with an die aspect ratio of 1.29 for three isotropic materials. The F.E. results indicated that for the same forming process, the strain ratio occurring at the apex of the elliptic bulge is dependent on the yielding properties of the material. For a die aspect ratio of 1.29, the plastic strain ratio is 1.37 for the von Mises material, 1.42 for the material characterized by $\tau_y/\sigma_T = 0.55$ and 1.29 for a material characterized by $\tau_y/\sigma_T = 0.60$. Therefore, for the same bulge test, the difference in strain ratio between the three isotropic materials is about ~10%.
three isotropic materials. It is worth mentioning the 5.1 % difference in bulge height between the considered isotropic materials to reach the same thickness reduction (25% thickness reduction).

<figure><img src="image_1532.png" /><figcaption>Fig. 4. F.E. Predictions according to the Drucker [7] yield criterion of the thickness vs. height at the apex of the elliptic bulge (die ratio a/b = 1.29) for isotropic materials having various ratios between the yield in shear and uniaxial tension $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_y/\sigma_T = 1/\sqrt{3}$ ($c = 0$), von Mises material) and $\tau_y/\sigma_T = 0.60$ ($c = -2$).</figcaption></figure>

It is worth further examining the influence of the material yielding characteristics on the predicted ratio between the two in-plane strains developing at the apex of the bulge. As expected, under hemispherical bulging, irrespective of the material, the strains ratio is about 1 (see also the yield loci plotted in Fig. 1). However, for an elliptic bulge test, the strain paths achieved during the same experiment are strongly dependent on the plastic properties of the material, namely the ratio $\tau_y/\sigma_T$. Let denote major strain by the strain developing in the direction of the short axis of the elliptic die and minor strain, the strain occurring in the direction of the long axis of the elliptic die.

As an example, it is shown in Fig. 5 the evolution of the major to minor plastic strain ratio developing in an elliptic bulging test with an die aspect ratio of 1.29 for three isotropic materials. The F.E. results indicated that for the same forming process, the strain ratio occurring at the apex of the elliptic bulge is dependent on the yielding properties of the material. For a die aspect ratio of 1.29, the plastic strain ratio is 1.37 for the von Mises material, 1.42 for the material characterized by $\tau_y/\sigma_T = 0.55$ and 1.29 for a material characterized by $\tau_y/\sigma_T = 0.60$. Therefore, for the same bulge test, the difference in strain ratio between the three isotropic materials is about ~10%.
<figure><img src="image_1529.png" /><figcaption>Fig. 5. Influence of the yielding ratio $\tau_y/\sigma_T$ on the major to minor plastic strains ratio at the pole for isotropic materials according to Drucker [7] and von Mises criterion for materials with various ratios between the yield in shear and uniaxial tension $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_y/\sigma_T = 1/\sqrt{3}$ ($c = 0$, von Mises material) and $\tau_y/\sigma_T = 0.60$ ($c = -2$).</figcaption></figure>

## Conclusions

A study of the influence of the ratio between the yield stress in uniaxial tension and in pure shear on the behavior under bulging of isotropic materials has been performed. F.E. simulations of bulge tests were conducted to study how the specificities of the plastic behavior of an isotropic material influences the strain paths and stress paths experienced during elliptical bulging. The plastic behavior of the isotropic material was described by the Drucker's [7] yield criterion. This yield criterion involves a unique parameter $c$, which is solely expressible in terms of the ratio between the yield stress in simple tension and that in shear $\tau_y/\sigma_T$. F.E. simulations of bulge tests were performed for various isotropic materials with different ratios $\tau_y/\sigma_T$ to study how the plastic behavior of an isotropic material influences the strain paths and stress paths experienced during biaxial stretching. The F.E. results show that for both hemispheric and elliptic bulge tests, the greatest thickness reduction is predicted for a material characterized by a yield locus exterior to the von Mises yield surface (or $\tau_y/\sigma_T > 1/\sqrt{3}$) while for material whose yield surface is interior to the von Mises one, a lowest thickness reduction is predicted. Moreover, for elliptical bulging, the plastic strain ratio achieved during the bulge test is larger for a material characterized by $\tau_y/\sigma_T < 1/\sqrt{3}$ ($c > 0$) than in the case of a von Mises material. On the other hand, for a material characterized by a yielding ratio $\tau_y/\sigma_T > 1/\sqrt{3}$ ($c < 0$), the plastic strain ratio is less than in the case of a von Mises material.

F.E. simulations of bulge tests were conducted to study how the specificities of the plastic behavior of an isotropic material influences the strain paths and stress paths experienced during elliptical bulging.
<figure><img src="image_1533.png" /><figcaption>Fig. 5. Influence of the yielding ratio $\tau_y/\sigma_T$ on the major to minor plastic strains ratio at the pole for isotropic materials according to Drucker [7] and von Mises criterion for materials with various ratios between the yield in shear and uniaxial tension $\tau_y/\sigma_T = 0.55$ ($c = 1.5$), $\tau_y/\sigma_T = 1/\sqrt{3}$ ($c = 0$, von Mises material) and $\tau_y/\sigma_T = 0.60$ ($c = -2$).</figcaption></figure>

## Conclusions

A study of the influence of the ratio between the yield stress in uniaxial tension and in pure shear on the behavior under bulging of isotropic materials has been performed. F.E. simulations of bulge tests were conducted to study how the specificities of the plastic behavior of an isotropic material influences the strain paths and stress paths experienced during elliptical bulging. The plastic behavior of the isotropic material was described by the Drucker's [7] yield criterion. This yield criterion involves a unique parameter $c$, which is solely expressible in terms of the ratio between the yield stress in simple tension and that in shear $\tau_y/\sigma_T$. F.E. simulations of bulge tests were performed for various isotropic materials with different ratios $\tau_y/\sigma_T$ to study how the plastic behavior of an isotropic material influences the strain paths and stress paths experienced during biaxial stretching. The F.E. results show that for both hemispheric and elliptic bulge tests, the greatest thickness reduction is predicted for a material characterized by a yield locus exterior to the von Mises yield surface (or $\tau_y/\sigma_T > 1/\sqrt{3}$) while for material whose yield surface is interior to the von Mises one, a lowest thickness reduction is predicted. Moreover, for elliptical bulging, the plastic strain ratio achieved during the bulge test is larger for a material characterized by $\tau_y/\sigma_T < 1/\sqrt{3}$ ($c > 0$) than in the case of a von Mises material. On the other hand, for a material characterized by a yielding ratio $\tau_y/\sigma_T > 1/\sqrt{3}$ ($c < 0$), the plastic strain ratio is less than in the case of a von Mises material.

F.E. simulations of bulge tests were conducted to study how the specificities of the plastic behavior of an isotropic material influences the strain paths and stress paths experienced during elliptical bulging.
## References

[1] R. Young, J. Bird, and J. Duncan, “An automated hydraulic bulge tester,” *J. Appl. Metalwork.*, vol. 2, no. 1, pp. 11–18, 1981. https://doi.org/10.1007/BF02833994

[2] D. Banabic, *Formability of metallic materials: plastic anisotropy, formability testing, forming limits*. Springer Science & Business Media, 2000. https://doi.org/10.1007/978-3-662-04013-3

[3] P. Mellor, “Stretch forming under fluid pressure,” *J. Mech. Phys. Solids*, vol. 5, no. 1, pp. 41–56, 1956. https://doi.org/10.1016/0022-5096(56)90006-0

[4] J. L. Duncan, J. Kolodziejski, and G. Glover, “Bulge Testing as an Aid to Formability Assessment,” in *Proc of the 9th Biennial Congress of International Deep Drawing Research Group*, 1976.

[5] R. von Mises, “Mechanik der plastischen Formänderung von Kristallen,” *ZAMM-J. Appl. Math. Mech. Für Angew. Math. Mech.*, vol. 8, no. 3, pp. 161–185, 1928. https://doi.org/10.1002/zamm.19280080302

[6] O. Cazacu and B. Revil-Baudard, *Plasticity of Metallic Materials: Modeling and Applications to Forming*. Elsevier, 2020.

[7] D. C. Drucker, *The Relation of Experiments to Mathematical Theories of Plasticity*. Division of Applied Mathematics, Brown University, 1948.

[8] Abaqus, *Abaqus - Version 6.14-1*. Dassault Systemes Simulia Corp., Providence, RI, 2014.
## References

[1] R. Young, J. Bird, and J. Duncan, “An automated hydraulic bulge tester,” *J. Appl. Metalwork.*, vol. 2, no. 1, pp. 11–18, 1981. https://doi.org/10.1007/BF02833994

[2] D. Banabic, *Formability of metallic materials: plastic anisotropy, formability testing, forming limits*. Springer Science & Business Media, 2000. https://doi.org/10.1007/978-3-662-04013-3

[3] P. Mellor, “Stretch forming under fluid pressure,” *J. Mech. Phys. Solids*, vol. 5, no. 1, pp. 41–56, 1956. https://doi.org/10.1016/0022-5096(56)90006-0

[4] J. L. Duncan, J. Kolodziejski, and G. Glover, “Bulge Testing as an Aid to Formability Assessment,” in *Proc of the 9th Biennial Congress of International Deep Drawing Research Group*, 1976.

[5] R. von Mises, “Mechanik der plastischen Formänderung von Kristallen,” *ZAMM-J. Appl. Math. Mech. Für Angew. Math. Mech.*, vol. 8, no. 3, pp. 161–185, 1928. https://doi.org/10.1002/zamm.19280080302

[6] O. Cazacu and B. Revil-Baudard, *Plasticity of Metallic Materials: Modeling and Applications to Forming*. Elsevier, 2020.

[7] D. C. Drucker, *The Relation of Experiments to Mathematical Theories of Plasticity*. Division of Applied Mathematics, Brown University, 1948.

[8] Abaqus, *Abaqus - Version 6.14-1*. Dassault Systemes Simulia Corp., Providence, RI, 2014.
# Experimental testing and numerical modelling of ductile fracture of PEEK in incremental sheet forming process

GATEA Shakir¹,²,a, and OU Hengan¹,b*

¹Department of Mechanical, Materials and Manufacturing Engineering, Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, UK

²Department of Materials Engineering, Faculty of Engineering, University of Kufa, Al-Najaf, Iraq

ashakir.gatea1@nottingham.ac.uk, bh.ou@nottingham.ac.uk

**Keywords:** PEEK, Ductile Fracture, SPIF, Stress Triaxiality, Hooputra Damage Model

**Abstract.** This research investigates the use of Hooputra-based damage model to predict fracture of polyether-ether-ketone (PEEK) material and its application to single point incremental forming (SPIF) process under different conditions. Flat PEEK sheets are used to examine the influence of temperature changes on the Hooputra fracture strain-stress triaxiality damage curve and to verify the capability of the Hooputra ductile damage model in predicting fracture in uniaxial tensile tests and the SPIF process. The research showed that the formulas used to calculate stress triaxiality based on geometric dimensions are insufficient with notched samples because the radius of the notched sample is no longer circular after plastic deformation. Temperature alters the Hooputra damage curve; therefore, the Hooputra ductile damage formula should be developed to consider the effect of temperature. When the Hooputra damage curve is established according to the temperature effect, the uniaxial fracture is precisely predicted at different temperatures. Hooputra ductile damage model could be developed to capture the fracture initiation and propagation in SPIF process.

**Introduction**

Polyether-ether-ketone (PEEK) is a versatile thermoplastic polymeric material commonly used in industrial applications due to its excellent combination of physical and mechanical properties [1]. In addition, PEEK presents a viable option for medical implants since it has a comparable elastic modulus to cortical bone and is biocompatible [2]. PEEK's industrial and medical applications could be expanded by using novel forming processes such as single point incremental forming (SPIF). SPIF is a potential technique that could be developed as an alternative production solution for PEEK materials with significant lead time and cost benefits. To employ the SPIF to produce high-quality parts of PEEK, precise prediction of mechanical response and fracture behaviour of PEEK materials is important.

The stress triaxiality and plastic strain are considered to be the two most significant variables in crack initiation, growth, and coalescence in ductile fracture of materials [3]. As a result, various ductile fracture models based on stress triaxiality, and fracture strain were established. Johnson and Cook established a model to predict ductile fractures at high strain rates and temperatures, and the fracture strain is related to stress triaxiality [4]. A void growth model was developed by Kanvinde et al. to evaluate the ductile fracture in the range of high stress triaxiality [5]. Brünig et al. established a damage model as a function of stress triaxiality [6]. Peng et al. suggested an uncoupled ductile fracture model based on stress triaxiality and lode angle [7].

Based on the macroscopic strains and stresses a ductile fracture model was presented by Hooputra et al. [8]. The fracture strain in this model is a function of stress triaxiality. Many numerical investigations have been conducted based on Hooputra model. To select and calibrate the appropriate fracture criterion for a specific application, seven different fracture models were

<figure><img src="image_1530.png" /></figure>
# Experimental testing and numerical modelling of ductile fracture of PEEK in incremental sheet forming process

GATEA Shakir<fnref n="1" />,<fnref n="2" />,ᵃ, and OU Hengan<fnref n="1" />,ᵇ*

**Keywords:** PEEK, Ductile Fracture, SPIF, Stress Triaxiality, Hooputra Damage Model

**Abstract.** This research investigates the use of Hooputra-based damage model to predict fracture of polyether-ether-ketone (PEEK) material and its application to single point incremental forming (SPIF) process under different conditions. Flat PEEK sheets are used to examine the influence of temperature changes on the Hooputra fracture strain-stress triaxiality damage curve and to verify the capability of the Hooputra ductile damage model in predicting fracture in uniaxial tensile tests and the SPIF process. The research showed that the formulas used to calculate stress triaxiality based on geometric dimensions are insufficient with notched samples because the radius of the notched sample is no longer circular after plastic deformation. Temperature alters the Hooputra damage curve; therefore, the Hooputra ductile damage formula should be developed to consider the effect of temperature. When the Hooputra damage curve is established according to the temperature effect, the uniaxial fracture is precisely predicted at different temperatures. Hooputra ductile damage model could be developed to capture the fracture initiation and propagation in SPIF process.

**Introduction**

Polyether-ether-ketone (PEEK) is a versatile thermoplastic polymeric material commonly used in industrial applications due to its excellent combination of physical and mechanical properties [1]. In addition, PEEK presents a viable option for medical implants since it has a comparable elastic modulus to cortical bone and is biocompatible [2]. PEEK's industrial and medical applications could be expanded by using novel forming processes such as single point incremental forming (SPIF). SPIF is a potential technique that could be developed as an alternative production solution for PEEK materials with significant lead time and cost benefits. To employ the SPIF to produce high-quality parts of PEEK, precise prediction of mechanical response and fracture behaviour of PEEK materials is important.

The stress triaxiality and plastic strain are considered to be the two most significant variables in crack initiation, growth, and coalescence in ductile fracture of materials [3]. As a result, various ductile fracture models based on stress triaxiality, and fracture strain were established. Johnson and Cook established a model to predict ductile fractures at high strain rates and temperatures, and the fracture strain is related to stress triaxiality [4]. A void growth model was developed by Kanvinde et al. to evaluate the ductile fracture in the range of high stress triaxiality [5]. Brünig et al. established a damage model as a function of stress triaxiality [6]. Peng et al. suggested an uncoupled ductile fracture model based on stress triaxiality and lode angle [7].

Based on the macroscopic strains and stresses a ductile fracture model was presented by Hooputra et al. [8]. The fracture strain in this model is a function of stress triaxiality. Many numerical investigations have been conducted based on Hooputra model. To select and calibrate the appropriate fracture criterion for a specific application, seven different fracture models were

<figure><img src="image_1534.png" /></figure>

<footnote n="1">¹Department of Mechanical, Materials and Manufacturing Engineering, Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, UK</footnote>

<footnote n="2">²Department of Materials Engineering, Faculty of Engineering, University of Kufa, Al-Najaf, Iraq</footnote>

<footnote>ᵃshakir.gatea1@nottingham.ac.uk, ᵇh.ou@nottingham.ac.uk</footnote>
employed. Hooputra damage model was one of them. To achieve the goal, 15 different
experimental tests on 2024-T351 aluminium covering a stress triaxiality range of -0.3 to 0.9 were
performed. The Hooputra damage model was discovered to be very promising because it
accurately predicts fracture in all types of experimental tests [9]. To determine which model could
be used to predict the forming limits of 2024-T3 aluminium alloy, eight ductile fracture models
were considered and calibrated. The Hooputra ductile damage model was shown to be more robust
against over-fitting and could be used when the number of calibration data points is small [10].
The experimental tests required to determine the parameters of Hooputra ductile damage model of
St14 steel could be neglected, and the parameters of Hooputra ductile damage model could be
predicated using a forming limit diagram [11]. To analyse the impact of mechanical discontinuities
on the crashworthiness performance of aluminium, the Hooputra damage model was combined
with the Müschenborn-Sonne forming limit diagram. It was discovered that incorporating it led to
more reliable crash simulations [12]. To predict the ductile fracture of metal alloys, a model based
on stress triaxiality, and shear stress ratio was proposed. The proposed model was compared to the
Hooputra ductile damage model, and there was a good correlation [13].

There are some research publications in the literature that illustrate the SPIF process's capacity to successfully deform PEEK material [14]. A novel incremental sheet forming (ISF) process was proposed to analyze the thermal formability of PEEK and investigate the geometric accuracy, twisting, and fracture mechanisms. The results showed that as the temperature increased, the accuracy of ISF part decreased and twisting angle increased. Striations marking is a common fracture mode in hot PEEK ISF [15]. An alternate spiral toolpath based on linear interpolation was developed to enhance the accuracy of SPIF of PEEK parts by reducing twisting in the deformed parts [16]. In comparison to traditional processes, the ISF process could be used to manufacture a customised product from PEEK, such as a cranial plate, with technological and economic advantages [2]. Cold SPIF of PEEK sheets was performed to evaluate the effect of ISF parameters on formability and to determine the best parameters for manufacturing the cranial plate and cheekbone. The findings of this study demonstrated the capability of the SPIF process in the biomedical field [17].

In this scientific context, precise selection of fracture strain and stress triaxiality values can
improve the prediction of ductile fracture models that were established based on the plastic strain
and stress triaxiality. The main motivation for this work is to evaluate the effect of temperature on
the fracture strain and stress triaxiality of PEEK and to implement these values in the Hooputra
ductile damage model to verify its accuracy to predict the ductile fracture of PEEK under different
strain states e.g., uniaxial tensile test and the plane strain of SPIF process.

**Material and experimental procedures**

PEEK 450G extruded sheets with a thickness of 3 mm were used in this study's experimental
testing. The glass transition and melting temperatures were determined using differential scanning
calorimetry. PEEK has glass transition and melting temperatures of 149 °C and 340 °C,
respectively. Tensile testing of standard samples (smooth) was performed by INSTRON testing
machine in accordance with ASTM/D638/sample type V to determine the mechanical properties
of PEEK sheets. The dimensions of the standard PEEK tensile sample were set to 63.5 mm total
length, 9.53 mm gauge length, and 3.18 mm the narrow section width. All tests were performed at
a constant strain rate of 1.75 ×10⁻³ S⁻¹. Four flat notched samples with varying notch radiuses (0.5,
1, 3, and 5 mm) were machined and tested to obtain different stress triaxialities. The gauge length
width in smooth tensile sample and at notch root was designed to be 3.18 mm for each sample.
Tensile tests on smooth and notched samples were performed at room temperature, 80 °C, 120 °C,
and 150 °C to examine the effect of temperature on fracture strain and stress triaxiality. Tensile
samples were examined in a temperature chamber. Tensile samples were painted with a white and
employed. Hooputra damage model was one of them. To achieve the goal, 15 different
experimental tests on 2024-T351 aluminium covering a stress triaxiality range of -0.3 to 0.9 were
performed. The Hooputra damage model was discovered to be very promising because it
accurately predicts fracture in all types of experimental tests [9]. To determine which model could
be used to predict the forming limits of 2024-T3 aluminium alloy, eight ductile fracture models
were considered and calibrated. The Hooputra ductile damage model was shown to be more robust
against over-fitting and could be used when the number of calibration data points is small [10].
The experimental tests required to determine the parameters of Hooputra ductile damage model of
St14 steel could be neglected, and the parameters of Hooputra ductile damage model could be
predicated using a forming limit diagram [11]. To analyse the impact of mechanical discontinuities
on the crashworthiness performance of aluminium, the Hooputra damage model was combined
with the Müschenborn-Sonne forming limit diagram. It was discovered that incorporating it led to
more reliable crash simulations [12]. To predict the ductile fracture of metal alloys, a model based
on stress triaxiality, and shear stress ratio was proposed. The proposed model was compared to the
Hooputra ductile damage model, and there was a good correlation [13].

There are some research publications in the literature that illustrate the SPIF process's capacity to successfully deform PEEK material [14]. A novel incremental sheet forming (ISF) process was proposed to analyze the thermal formability of PEEK and investigate the geometric accuracy, twisting, and fracture mechanisms. The results showed that as the temperature increased, the accuracy of ISF part decreased and twisting angle increased. Striations marking is a common fracture mode in hot PEEK ISF [15]. An alternate spiral toolpath based on linear interpolation was developed to enhance the accuracy of SPIF of PEEK parts by reducing twisting in the deformed parts [16]. In comparison to traditional processes, the ISF process could be used to manufacture a customised product from PEEK, such as a cranial plate, with technological and economic advantages [2]. Cold SPIF of PEEK sheets was performed to evaluate the effect of ISF parameters on formability and to determine the best parameters for manufacturing the cranial plate and cheekbone. The findings of this study demonstrated the capability of the SPIF process in the biomedical field [17].

In this scientific context, precise selection of fracture strain and stress triaxiality values can
improve the prediction of ductile fracture models that were established based on the plastic strain
and stress triaxiality. The main motivation for this work is to evaluate the effect of temperature on
the fracture strain and stress triaxiality of PEEK and to implement these values in the Hooputra
ductile damage model to verify its accuracy to predict the ductile fracture of PEEK under different
strain states e.g., uniaxial tensile test and the plane strain of SPIF process.

**Material and experimental procedures**

PEEK 450G extruded sheets with a thickness of 3 mm were used in this study's experimental
testing. The glass transition and melting temperatures were determined using differential scanning
calorimetry. PEEK has glass transition and melting temperatures of 149 °C and 340 °C,
respectively. Tensile testing of standard samples (smooth) was performed by INSTRON testing
machine in accordance with ASTM/D638/sample type V to determine the mechanical properties
of PEEK sheets. The dimensions of the standard PEEK tensile sample were set to 63.5 mm total
length, 9.53 mm gauge length, and 3.18 mm the narrow section width. All tests were performed at
a constant strain rate of 1.75 ×10⁻³ S⁻¹. Four flat notched samples with varying notch radiuses (0.5,
1, 3, and 5 mm) were machined and tested to obtain different stress triaxialities. The gauge length
width in smooth tensile sample and at notch root was designed to be 3.18 mm for each sample.
Tensile tests on smooth and notched samples were performed at room temperature, 80 °C, 120 °C,
and 150 °C to examine the effect of temperature on fracture strain and stress triaxiality. Tensile
samples were examined in a temperature chamber. Tensile samples were painted with a white and
black stochastic pattern, and strains were captured using video measurement during the tensile tests. Table 1 reports the mechanical properties of PEEK sheet at room temperature.

To validate the accuracy of the Hooputra model in predicting ductile fracture in SPIF under plane strain conditions, an SPIF test of PEEK sheets was performed using a HURCO CNC milling machine equipped with a heating system to produce a hyperbolic truncated cone with varying angle (from 40° to 90°) (see Fig. 1). The PEEK blank has a size of 150 × 150 mm and a thickness of 3 mm. To improve the surface quality of the produced part, the ROCOL RDT grease compound was used as a lubricant between the high-speed steel forming tool and the PEEK sheet. The SPIF test was carried out with a tool diameter of 10 mm, a step size of 0.5 mm, a feed rate of 1000 mm/min, and a spindle speed of 100 rpm. An alternate spiral toolpath was used to reduce the twisting phenomenon in PEEK SPIF.

<figure><img src="image_1531.png" /><figcaption>Fig. 1. (a) Geometric shape of the hyperbolic cone, and (b) SPIF experimental fixture.</figcaption></figure>

<table><caption>Table 1. Mechanical properties of PEEK 450G at room temperature.</caption><thead><tr><td>Elastic Modulus [GPa]</td><td>Poisson's ratio</td><td>Yield stress [MPa]</td><td>Density [kg/m³]</td></tr></thead><tbody><tr><td>3.6</td><td>0.38</td><td>84</td><td>1300</td></tr></tbody></table>

Finite element modeling

Abaqus/Explicit is used for estimating stress triaxiality and predicting crack initiation and propagation in uniaxial tensile samples and in the SPIF process. Tensile samples and the PEEK sheets in the SPIF process were modelled as deformable bodies, with isotropic elastic and plastic yielding, whereas the forming tool, blank holder, and backing plate were modelled as analytical rigid bodies. To predict stress triaxiality and fracture in the FE model, a 3D hexahedral element with 8 nodes and reduced integration (C3D8RT) was used. Due to the considerable deformation of PEEK materials at high temperatures, a small element size should be employed. The element size in this study is 0.15 mm in the tensile test and 0.25 mm in the SPIF test. In FE simulation, several friction coefficient values were investigated in order to determine the optimal friction coefficient. The results revealed that there is a good correlation between the FE and the experimental forming force with a coefficient of friction of 0.05.
black stochastic pattern, and strains were captured using video measurement during the tensile tests. Table 1 reports the mechanical properties of PEEK sheet at room temperature.

To validate the accuracy of the Hooputra model in predicting ductile fracture in SPIF under plane strain conditions, an SPIF test of PEEK sheets was performed using a HURCO CNC milling machine equipped with a heating system to produce a hyperbolic truncated cone with varying angle (from 40° to 90°) (see Fig. 1). The PEEK blank has a size of 150 × 150 mm and a thickness of 3 mm. To improve the surface quality of the produced part, the ROCOL RDT grease compound was used as a lubricant between the high-speed steel forming tool and the PEEK sheet. The SPIF test was carried out with a tool diameter of 10 mm, a step size of 0.5 mm, a feed rate of 1000 mm/min, and a spindle speed of 100 rpm. An alternate spiral toolpath was used to reduce the twisting phenomenon in PEEK SPIF.

<figure><img src="image_1535.png" /><figcaption>Fig. 1. (a) Geometric shape of the hyperbolic cone, and (b) SPIF experimental fixture.</figcaption></figure>

<table><caption>Table 1. Mechanical properties of PEEK 450G at room temperature.</caption><thead><tr><td>Elastic Modulus [GPa]</td><td>Poisson's ratio</td><td>Yield stress [MPa]</td><td>Density [kg/m³]</td></tr></thead><tbody><tr><td>3.6</td><td>0.38</td><td>84</td><td>1300</td></tr></tbody></table>

Finite element modeling

Abaqus/Explicit is used for estimating stress triaxiality and predicting crack initiation and propagation in uniaxial tensile samples and in the SPIF process. Tensile samples and the PEEK sheets in the SPIF process were modelled as deformable bodies, with isotropic elastic and plastic yielding, whereas the forming tool, blank holder, and backing plate were modelled as analytical rigid bodies. To predict stress triaxiality and fracture in the FE model, a 3D hexahedral element with 8 nodes and reduced integration (C3D8RT) was used. Due to the considerable deformation of PEEK materials at high temperatures, a small element size should be employed. The element size in this study is 0.15 mm in the tensile test and 0.25 mm in the SPIF test. In FE simulation, several friction coefficient values were investigated in order to determine the optimal friction coefficient. The results revealed that there is a good correlation between the FE and the experimental forming force with a coefficient of friction of 0.05.
The Hooputra ductile damage model is utilized in this study to predict fractures in both tensile and SPIF tests. The onset of fracture in the Hooputra ductile damage model is based on the nucleation, growth, and coalescence of voids. In this concept, fracture strain is only a function of stress triaxiality. The following is the Hooputra ductile damage model.

$$ \epsilon_{eq}(\eta) = a e^{-c\eta} + b e^{c\eta} \qquad (1) $$

where the $\epsilon_{eq}$ is equivalent plastic strain at fracture; $\eta = \sigma_H/\sigma_{eq}$ is stress triaxiality; $\sigma_H$ is the hydrostatic stress; $\sigma_{eq}$ Von Mises equivalent stress; a, b, and c are materials parameters to be found from the tests. When the following condition is met in the FE model, the crack is initiated:

$$ D = \int_{0}^{\epsilon_{eq}} \frac{d\epsilon_{eq}}{\epsilon_{eq}(\eta)} = 1 \qquad (2) $$

where D is the damage variable that ranges from 0 (virgin material) to 1 (fractured material). Crack initiation and propagation are modelled using element deletion. The element is eliminated from the FE model when the damage variable equals 1.

## Results and discussion

### Effect of plastic deformation and temperature on stress triaxiality

There are two approaches for determining the stress triaxiality of notched tensile samples. The first approach is based on the geometric dimensions of the specimen, and certain formulas were developed to compute the stress triaxiality, such as the Bridgman formula for rounded tensile samples and the Yuanli Bai formula [18] for flat samples. However, these formulas are no longer strictly relevant because the dimensions of tensile samples do not remain constant during the tensile test. Another approach is to use numerical simulation, where stress triaxiality is defined as the ratio of hydrostatic stress to Von Mises equivalent stress. The stress triaxiality in uniaxial tensile samples and in the SPIF process of a hyperbolic truncated cone is calculated using FE simulation in this work.

The position of stress triaxiality was measured using the minimum cross section of a flat tensile sample. As a result, the FE approach was used to examine the distribution of stress triaxiality on the minimum cross section through the thickness. To demonstrate that the empirical formulas that was used to capture stress triaxiality due to geometric dimensions are no longer strictly applicable, the FE simulation of smooth and notched samples (notch radius (R)=0.5, 1, 3, 5 mm) was run under the assumption that the material is elastic, then the simulation with elastic plastic properties was run. The stress triaxiality values are constant under the elastic condition because the dimensions of the tensile sample are not changed during the simulation and the sample returns to its original shape when the load is removed. As a result, the stress triaxiality under elastic condition is 0.33, 0.4, 0.43, 0.53, and 0.58 for smooth, R5 mm, R3 mm, R1 mm, and R0.5 mm samples, respectively. Fig. 2 (a) depicts the influence of plastic deformation at room temperature on the stress triaxiality distribution at the centre of an R5 mm notched sample. The stress triaxiality is uniform under elastic conditions because the dimensions of the notched sample are not changed, whereas under elastic-plastic deformation the radius of the notched sample is no longer circular after plastic deformation, so the stress triaxiality changes through the thickness of the sample and reaches its maximum value at the centre. The behaviour of all notched samples is the same, and the values of stress triaxiality at the centre of the other samples are presented in Table 2. The plastic deformation has no effect on the stress triaxiality of the smooth sample because in the tensile test of the PEEK smooth sample, the necking begins at the beginning of the test and extends throughout the gauge length of the tensile sample as shown in Fig. 2 (b), which means that the
The Hooputra ductile damage model is utilized in this study to predict fractures in both tensile and SPIF tests. The onset of fracture in the Hooputra ductile damage model is based on the nucleation, growth, and coalescence of voids. In this concept, fracture strain is only a function of stress triaxiality. The following is the Hooputra ductile damage model.

$$ \varepsilon_{eq}(\eta) = a e^{-c\eta} + b e^{c\eta} \qquad (1) $$

where the $\varepsilon_{eq}$ is equivalent plastic strain at fracture; $\eta = \sigma_H/\sigma_{eq}$ is stress triaxiality; $\sigma_H$ is the hydrostatic stress; $\sigma_{eq}$ Von Mises equivalent stress; a, b, and c are materials parameters to be found from the tests. When the following condition is met in the FE model, the crack is initiated:

$$ D = \int_{0}^{\varepsilon_{eq}} \frac{d\varepsilon_{eq}}{\varepsilon_{eq}(\eta)} = 1 \qquad (2) $$

where D is the damage variable that ranges from 0 (virgin material) to 1 (fractured material). Crack initiation and propagation are modelled using element deletion. The element is eliminated from the FE model when the damage variable equals 1.

## Results and discussion

### Effect of plastic deformation and temperature on stress triaxiality

There are two approaches for determining the stress triaxiality of notched tensile samples. The first approach is based on the geometric dimensions of the specimen, and certain formulas were developed to compute the stress triaxiality, such as the Bridgman formula for rounded tensile samples and the Yuanli Bai formula [18] for flat samples. However, these formulas are no longer strictly relevant because the dimensions of tensile samples do not remain constant during the tensile test. Another approach is to use numerical simulation, where stress triaxiality is defined as the ratio of hydrostatic stress to Von Mises equivalent stress. The stress triaxiality in uniaxial tensile samples and in the SPIF process of a hyperbolic truncated cone is calculated using FE simulation in this work.

The position of stress triaxiality was measured using the minimum cross section of a flat tensile sample. As a result, the FE approach was used to examine the distribution of stress triaxiality on the minimum cross section through the thickness. To demonstrate that the empirical formulas that was used to capture stress triaxiality due to geometric dimensions are no longer strictly applicable, the FE simulation of smooth and notched samples (notch radius (R)=0.5, 1, 3, 5 mm) was run under the assumption that the material is elastic, then the simulation with elastic plastic properties was run. The stress triaxiality values are constant under the elastic condition because the dimensions of the tensile sample are not changed during the simulation and the sample returns to its original shape when the load is removed. As a result, the stress triaxiality under elastic condition is 0.33, 0.4, 0.43, 0.53, and 0.58 for smooth, R5 mm, R3 mm, R1 mm, and R0.5 mm samples, respectively. Fig. 2 (a) depicts the influence of plastic deformation at room temperature on the stress triaxiality distribution at the centre of an R5 mm notched sample. The stress triaxiality is uniform under elastic conditions because the dimensions of the notched sample are not changed, whereas under elastic-plastic deformation the radius of the notched sample is no longer circular after plastic deformation, so the stress triaxiality changes through the thickness of the sample and reaches its maximum value at the centre. The behaviour of all notched samples is the same, and the values of stress triaxiality at the centre of the other samples are presented in Table 2. The plastic deformation has no effect on the stress triaxiality of the smooth sample because in the tensile test of the PEEK smooth sample, the necking begins at the beginning of the test and extends throughout the gauge length of the tensile sample as shown in Fig. 2 (b), which means that the