St900_1 grade, which has the lowest strain-hardening. This indicates that the contact point shift and multi-breakage happen faster for materials with lower strain-hardening. This conclusion is also supported by the fact that the decrease of the Z-displacement after 40° is also fastest for the St900_1 grade, meaning that the punch-sheet separation is largest, due to the pronounced multi-breakage effect.

Fig. 6 presents the peak strain, cross-section moment, Z-displacement and their first derivatives measured from 500 MPa and 700 MPa steels bent using punch radii $R_p = 0.66 \text{ t}$, $R_p = 2 \text{ t}$, and $R_p = 4 \text{ t}$. As can be seen from Fig. 6a and Fig. 6d, the development of the peak strain and strain rate for the presented tests start out roughly identical, as most of the deformation is still elastic, but diverge after around 10°–20° bend angle when the work-hardening properties come into effect. The peak strain increases at a higher rate for the St700 grade in all tool setups. For the St700 grade bent with the largest punch ($R_p = 4 \text{ t}$) the strain rate starts to gradually decrease after 30° bend angle, and a steady state is reached at around 50°. For the St500 grade bent with the largest punch, the decrease of the strain rate starts already at around 15°, but the steady state is still reached at around 50°. For both materials bent with the medium punch ($R_p = 2 \text{ t}$), the strain rate starts to gradually decrease after 40°, reaching a steady state around after 80°. With the smallest punch ($R_p = 0.66 \text{ t}$), the difference between the materials is noticeable. For the St700 grade, the strain rate keeps increasing until around 85°. Whereas for the St500 grade, the strain rate decreases between 15–20°, then remains roughly constant before starting to decrease again after around 75°.

<figure><img src="image_987.png" /><figcaption>Fig. 6. Development of a) the peak strain, b) cross-section moment, c) Z-displacement, d) peak strain derivative, e) moment derivative and f) Z-displacement derivative, from St500 and St700 grades bent in transverse (TD) orientation with punch radii $R_p = 0.66 \text{ t}$, $R_p = 2 \text{ t}$ and $R_p = 4 \text{ t}$.</figcaption></figure>

From Fig. 6e it is seen that for both materials, the moment derivative $dM/dα$ reaches a minimum and starts to increase again at around 30°, 40° and 80° for the largest punch ($R_p = 4 \text{ t}$), medium punch ($R_p = 2 \text{ t}$) and smallest punch ($R_p = 0.66 \text{ t}$), respectively. For the St700 grade, these angles correspond to the angles at which the strain rate first starts to decrease. For the St500 grade, the connection between the strain rate curve and the moment derivative curve is not as clear, due to the changes in the strain rate being more gradual and due to the initial decrease in strain rate after 15°. However, it is still assumed that the wrap-around angle coincides with the $dM/dα$ minimum for both materials. It can also be seen here that for all three tool setups, the St700 grade has a lower $dM/dα$ value at the wrap-around angle compared to the St500 grade, which again demonstrates that materials with lower strain-hardening, that produce more localized strain distribution, will also produce a lower $dM/dα$ minimum.
Fig. 6c and Fig. 6f show the development of the Z-displacement and its derivative for the two grades and three setups. It is observed that the Z-displacement increases at a constant rate at the beginning for all tests but starts to decrease after a certain point. The start of the decrease of the Z-displacement derivative seems to coincide with the wrap-around angle determined from the moment derivative curves. The exception for this is the St500 grade with the smallest punch ($R_p$ = 0.66 t), for which the changes in the Z-displacement are too gradual to discern from the measurement noise.

As discussed earlier, the start of the increase of the moment derivative $dM/d\alpha$, as well as the decreases in the Z-displacement and strain rate, can be attributed to the wrap-around and multi-breakage effects, where the loading scheme changes, and the contact points shift away from the centerline. The wrap-around angle seems to decrease when punch radius is increased, which seems logical, as the wrap-around happens when the inside radius becomes equal to the punch radius.

Fig. 7 shows the correlation between the moment derivative $dM/d\alpha$ minimum value and two global formability measures: the uniform elongation $A_{gt}$ from tensile tests and the peak bending strain $\epsilon_{b\_max}$* at 70° bend angle with punch radius $R_p$ = 2 t. The strong correlation indicates that the moment derivative minimum is connected to the global formability properties of a material and that it may be possible to predict the peak bending strain from the moment derivative minimum.

The peak bending strain at a given angle has been deemed a decent measure for quantifying the strain distribution [17]. The connection between the peak strain and the cross-section moment implies that it may be possible to predict the development of the strain distribution using only the force-displacement data of a press brake, circumventing the need for strain measurement. This would also mean that the effects of the strain distribution, such as the shape of the bent sheet, inner radius, thinning, bend allowance, multi-breakage and punch-sheet separation could be predicted from the measured cross-section moment.

<figure><img src="image_981.png" /><figcaption>Fig. 7. Moment derivative $dM/d\alpha$ minimum value as a measure of global formability. a) Correlation with the peak strain $\epsilon_{b\_max}$* measured at $\alpha$ = 70° from specimens bent using punch radius $R_p$ = 2t; b) Correlation with the uniform elongation measured from tensile tests.
*corrected for thickness variation [1].</figcaption></figure>

In other words, by measuring punch travel and force with sufficient accuracy during the first 30–40°, it may be possible to predict the behavior and shape of the bent workpiece during the rest of the bending process. This information could be used for adjusting the process parameters on the fly (so-called “adaptive control”), which could potentially lead to increased accuracy in the dimensions of the final product, fewer failed parts, as well as decreasing the time and resources spent on determining the correct process parameters through test bends.
Fig. 6c and Fig. 6f show the development of the Z-displacement and its derivative for the two grades and three setups. It is observed that the Z-displacement increases at a constant rate at the beginning for all tests but starts to decrease after a certain point. The start of the decrease of the Z-displacement derivative seems to coincide with the wrap-around angle determined from the moment derivative curves. The exception for this is the St500 grade with the smallest punch ($R_p$ = 0.66 t), for which the changes in the Z-displacement are too gradual to discern from the measurement noise.

As discussed earlier, the start of the increase of the moment derivative $dM/d\alpha$, as well as the decreases in the Z-displacement and strain rate, can be attributed to the wrap-around and multi-breakage effects, where the loading scheme changes, and the contact points shift away from the centerline. The wrap-around angle seems to decrease when punch radius is increased, which seems logical, as the wrap-around happens when the inside radius becomes equal to the punch radius.

Fig. 7 shows the correlation between the moment derivative $dM/d\alpha$ minimum value and two global formability measures: the uniform elongation $A_{gt}$ from tensile tests and the peak bending strain $\epsilon_{b\_max}$* at 70° bend angle with punch radius $R_p$ = 2 t. The strong correlation indicates that the moment derivative minimum is connected to the global formability properties of a material and that it may be possible to predict the peak bending strain from the moment derivative minimum.

The peak bending strain at a given angle has been deemed a decent measure for quantifying the strain distribution [17]. The connection between the peak strain and the cross-section moment implies that it may be possible to predict the development of the strain distribution using only the force-displacement data of a press brake, circumventing the need for strain measurement. This would also mean that the effects of the strain distribution, such as the shape of the bent sheet, inner radius, thinning, bend allowance, multi-breakage and punch-sheet separation could be predicted from the measured cross-section moment.

<figure><img src="image_988.png" /><figcaption>Fig. 7. Moment derivative $dM/d\alpha$ minimum value as a measure of global formability. a) Correlation with the peak strain $\epsilon_{b\_max}$* measured at $\alpha$ = 70° from specimens bent using punch radius $R_p$ = 2t; b) Correlation with the uniform elongation measured from tensile tests.
*corrected for thickness variation [1].</figcaption></figure>

In other words, by measuring punch travel and force with sufficient accuracy during the first 30–40°, it may be possible to predict the behavior and shape of the bent workpiece during the rest of the bending process. This information could be used for adjusting the process parameters on the fly (so-called “adaptive control”), which could potentially lead to increased accuracy in the dimensions of the final product, fewer failed parts, as well as decreasing the time and resources spent on determining the correct process parameters through test bends.
## Summary

In this paper, the formability of nine hot rolled steel grades with various strengths and ductility properties were investigated by air-bending tests using three different setups. The developments of the strain distribution, Z-displacement, and cross-section moment were measured throughout the bending process. The aim of the paper was to investigate the connection between the cross-section moment and other phenomena related to air-bending, such as the strain distribution development and the multi-breakage effect. The main observations and conclusions are summarized as follows:

* The wrap-around and multi-breakage can be detected from the derivative curves of the peak strain, cross-section moment and Z-displacement.

* Materials with lower strain-hardening are found to have lower minimum value of the cross-section moment derivative dM/dα at the wrap-around angle, as well as higher peak strain on the outer surface.

* The wrap-around angle is smaller for materials with low strain hardening, as well as when a larger punch radius is used.

* A connection between the peak strain and the cross-section moment implies that it may be possible to estimate the strain distribution in bending from the force-displacement curve of the punch.

## Acknowledgments

Financial assistance of the Business Finland, project FOSSA- Fossil-Free Steel Applications, is acknowledged.

## References

[1] A.-P. Pokka, V. Kesti, A. Kaijalainen, Global formability and bendability of ultra-high strength steels: Effect of mechanical properties on the strain distribution and behaviour in air-bending, Mater. Today Commun. 37 (2023), 107081. https://doi.org/10.1016/j.mtcomm.2023.107081

[2] A.-P. Pokka, A.-M. Arola, A. Kaijalainen, V. Kesti, J. Larkiola, Strain distribution during air bending of ultra-high strength steels, ESAFORM 2021 (2021). https://doi.org/10.25518/esaform21.2509

[3] S. D. Benson, Press Brake Technology: A Guide to precision sheet metal bending, Society of Manufacturing Engineers, Dearborn, Michigan, 1997.

[4] V. Vorkov, R. Aerens, D. Vandepitte, J. R. Duflou, The Multi-Breakage Phenomenon in Air Bending Process, Key Eng. Mater. 611-612 (2014) 1047-1053. https://doi.org/10.4028/www.scientific.net/KEM.611-612.1047

[5] A. Väisänen, K. Mäntyjärvi, J. A. Karjalainen, Bendability of Ultra-High-Strength Steel, Key Eng. Mater. 410-411 (2009) 611-620. https://doi.org/10.4028/www.scientific.net/KEM.410-411.611

[6] J. Heikkala, A. Väisänen, Usability Testing of Ultra High-Strength Steels, Proc. ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis 4 (2012) 163-173. https://doi.org/10.1115/ESDA2012-82770

[7] V. Kesti, A. Kaijalainen, A. Väisänen, A. Järvenpää, A. Määttä, A.-M. Arola, K. Mäntyjärvi, R. Ruoppa, Bendability and Microstructure of direct quenched Optim® 960QC, Mater. Sci. Forum 783-786 (2014) 818-824. https://doi.org/10.4028/www.scientific.net/MSF.783-786.818

[8] A.-M. Arola, V. Kesti, R. Ruoppa, The effect of punch radius on the deformation of ultra-high strength steel in bending, Key Eng. Mater. 639 (2015) 139-146.
## Summary

In this paper, the formability of nine hot rolled steel grades with various strengths and ductility properties were investigated by air-bending tests using three different setups. The developments of the strain distribution, Z-displacement, and cross-section moment were measured throughout the bending process. The aim of the paper was to investigate the connection between the cross-section moment and other phenomena related to air-bending, such as the strain distribution development and the multi-breakage effect. The main observations and conclusions are summarized as follows:

* The wrap-around and multi-breakage can be detected from the derivative curves of the peak strain, cross-section moment and Z-displacement.

* Materials with lower strain-hardening are found to have lower minimum value of the cross-section moment derivative dM/dα at the wrap-around angle, as well as higher peak strain on the outer surface.

* The wrap-around angle is smaller for materials with low strain hardening, as well as when a larger punch radius is used.

* A connection between the peak strain and the cross-section moment implies that it may be possible to estimate the strain distribution in bending from the force-displacement curve of the punch.

## Acknowledgments

Financial assistance of the Business Finland, project FOSSA- Fossil-Free Steel Applications, is acknowledged.

## References

[1] A.-P. Pokka, V. Kesti, A. Kaijalainen, Global formability and bendability of ultra-high strength steels: Effect of mechanical properties on the strain distribution and behaviour in air-bending, Mater. Today Commun. 37 (2023), 107081. https://doi.org/10.1016/j.mtcomm.2023.107081

[2] A.-P. Pokka, A.-M. Arola, A. Kaijalainen, V. Kesti, J. Larkiola, Strain distribution during air bending of ultra-high strength steels, ESAFORM 2021 (2021). https://doi.org/10.25518/esaform21.2509

[3] S. D. Benson, Press Brake Technology: A Guide to precision sheet metal bending, Society of Manufacturing Engineers, Dearborn, Michigan, 1997.

[4] V. Vorkov, R. Aerens, D. Vandepitte, J. R. Duflou, The Multi-Breakage Phenomenon in Air Bending Process, Key Eng. Mater. 611-612 (2014) 1047-1053. https://doi.org/10.4028/www.scientific.net/KEM.611-612.1047

[5] A. Väisänen, K. Mäntyjärvi, J. A. Karjalainen, Bendability of Ultra-High-Strength Steel, Key Eng. Mater. 410-411 (2009) 611-620. https://doi.org/10.4028/www.scientific.net/KEM.410-411.611

[6] J. Heikkala, A. Väisänen, Usability Testing of Ultra High-Strength Steels, Proc. ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis 4 (2012) 163-173. https://doi.org/10.1115/ESDA2012-82770

[7] V. Kesti, A. Kaijalainen, A. Väisänen, A. Järvenpää, A. Määttä, A.-M. Arola, K. Mäntyjärvi, R. Ruoppa, Bendability and Microstructure of direct quenched Optim® 960QC, Mater. Sci. Forum 783-786 (2014) 818-824. https://doi.org/10.4028/www.scientific.net/MSF.783-786.818

[8] A.-M. Arola, V. Kesti, R. Ruoppa, The effect of punch radius on the deformation of ultra-high strength steel in bending, Key Eng. Mater. 639 (2015) 139-146.
[9] L. Wagner, H. Schauer, H. Pauli, J. Hinterdorfer, Improved bendability characterization of UHSS sheets, IOP Conf. Ser. Mater. Sci. Eng. 651 (2019) 012019. https://doi.org/10.1088/1757-899X/651/1/012019

[10] K. Cheong, J. Noder, A. Zhumagulov, C. Butcher, Characterization and Prediction of Plane Strain Bendability in Advanced High-Strength Steels, Metals 13 (2023) 1711. https://doi.org/10.3390/met13101711

[11] M. Kaupper, M. Merklein, Bendability of advanced high strength steels - A new evaluation procedure, CIRP Ann. - Manuf. Technol. 62 (2013) 247-250. https://doi.org/10.1016/j.cirp.2013.03.049

[12] K. Cheong, K. Omer, C. Butcher, R. George, J. Dykeman, Evaluation of the VDA 238-100 Tight Radius Bending Test using Digital Image Correlation Strain Measurement, J. Phys. Conf. Ser. 896 (2017) 12075. https://doi.org/10.1088/1742-6596/896/1/012075

[13] A.-M. Arola, A. Kaijalainen, V. Kesti, A.-P. Pokka, J. Larkiola, Digital image correlation and optical strain measuring in bendability assessment of ultra-high strength structural steels, Procedia Manuf. 29 (2019) 398-405. https://doi.org/10.1016/j.promfg.2019.02.154

[14] S. Gothivarekar, S. Coppieters, A. Van de Velde, D. Debruyne, Advanced FE model validation of cold-forming process using DIC: Air bending of high strength steel, Int. J. Mater. Form. 13 (2020) 409-421. https://doi.org/10.1007/s12289-020-01536-1

[15] R. Ruoppa, R. Toppila, V. Kesti, A. Arola, Bendability tests for ultra-high-strength steels with optical strain analysis and prediction of bending force, Proc. METNET Seminar 2014 Moscow (2014) 68-78.

[16] C. Suppan, T. Hebesberger, A. Pichler, J. Rehrl, O. Kolednik, On the microstructure control of the bendability of advanced high strength steels, Mater. Sci. Eng. A 735 (2018) 89-98. https://doi.org/10.1016/j.msea.2018.07.080

[17] A.-P. Pokka, V. Kesti, A. Kaijalainen, Total strain on the outer surface of steel sheets in air bending, Mater. Res. Proc. 28, (2023) 695-704. https://doi.org/10.21741/9781644902479-75

[18] L. Troive, New method for evaluation of bendability based on three-point-bending and the evolution of the cross-section moment, J. Phys. Conf. Ser. 896 (2017) 12006. https://doi.org/10.1088/1742-6596/896/1/012006

[19] C. Wang, G. Kinzel, T. Altan, Mathematical modeling of plane-strain bending of sheet and plate, J. Mater. Process. Technol. 39 (1993) 279-304. https://doi.org/10.1016/0924-0136(93)90164-2

[20] SFS-EN ISO 7438:2016: Metallic materials – Bend test.
[9] L. Wagner, H. Schauer, H. Pauli, J. Hinterdorfer, Improved bendability characterization of UHSS sheets, IOP Conf. Ser. Mater. Sci. Eng. 651 (2019) 012019. https://doi.org/10.1088/1757-899X/651/1/012019

[10] K. Cheong, J. Noder, A. Zhumagulov, C. Butcher, Characterization and Prediction of Plane Strain Bendability in Advanced High-Strength Steels, Metals 13 (2023) 1711. https://doi.org/10.3390/met13101711

[11] M. Kaupper, M. Merklein, Bendability of advanced high strength steels - A new evaluation procedure, CIRP Ann. - Manuf. Technol. 62 (2013) 247-250. https://doi.org/10.1016/j.cirp.2013.03.049

[12] K. Cheong, K. Omer, C. Butcher, R. George, J. Dykeman, Evaluation of the VDA 238-100 Tight Radius Bending Test using Digital Image Correlation Strain Measurement, J. Phys. Conf. Ser. 896 (2017) 12075. https://doi.org/10.1088/1742-6596/896/1/012075

[13] A.-M. Arola, A. Kaijalainen, V. Kesti, A.-P. Pokka, J. Larkiola, Digital image correlation and optical strain measuring in bendability assessment of ultra-high strength structural steels, Procedia Manuf. 29 (2019) 398-405. https://doi.org/10.1016/j.promfg.2019.02.154

[14] S. Gothivarekar, S. Coppieters, A. Van de Velde, D. Debruyne, Advanced FE model validation of cold-forming process using DIC: Air bending of high strength steel, Int. J. Mater. Form. 13 (2020) 409-421. https://doi.org/10.1007/s12289-020-01536-1

[15] R. Ruoppa, R. Toppila, V. Kesti, A. Arola, Bendability tests for ultra-high-strength steels with optical strain analysis and prediction of bending force, Proc. METNET Seminar 2014 Moscow (2014) 68-78.

[16] C. Suppan, T. Hebesberger, A. Pichler, J. Rehrl, O. Kolednik, On the microstructure control of the bendability of advanced high strength steels, Mater. Sci. Eng. A 735 (2018) 89-98. https://doi.org/10.1016/j.msea.2018.07.080

[17] A.-P. Pokka, V. Kesti, A. Kaijalainen, Total strain on the outer surface of steel sheets in air bending, Mater. Res. Proc. 28, (2023) 695-704. https://doi.org/10.21741/9781644902479-75

[18] L. Troive, New method for evaluation of bendability based on three-point-bending and the evolution of the cross-section moment, J. Phys. Conf. Ser. 896 (2017) 12006. https://doi.org/10.1088/1742-6596/896/1/012006

[19] C. Wang, G. Kinzel, T. Altan, Mathematical modeling of plane-strain bending of sheet and plate, J. Mater. Process. Technol. 39 (1993) 279-304. https://doi.org/10.1016/0924-0136(93)90164-2

[20] SFS-EN ISO 7438:2016: Metallic materials – Bend test.
# Characterization of the superplastic behavior of a Ti6Al4V-ELI alloy bilayer sheet

GUGLIELMI Pasquale¹,ᵃ*, PICCININNI Antonio¹,ᵇ, CUSANNO Angela¹,ᶜ, VAIANI Lorenzo¹,ᵈ, UVA Antonio Emmanuele¹,ᵉ, PALUMBO Gianfranco¹,ᵉ

¹DMMM - Politecnico di Bari, Viale Japigia 182, Bari 70126, Italy

ᵃpasquale.guglielmi@poliba.it, ᵇantonio.piccininni@poliba.it, ᶜangela.cusanno@poliba.it,

ᵈlorenzo.vaiani@poliba.it, ᶜantonio.uva@poliba.it, ᵈgianfranco.palumbo@poliba.it

**Keywords:** Ti Alloy, Material Characterization, Superplasticity, Finite Element, Inverse Analysis

**Abstract.** The aim of the present work is to investigate the superplastic behaviour of a bilayer Ti6Al4V-ELI to numerically design a full custom prosthesis manufacturing process by means of SuperPlastic Forming (SPF); the bilayer was composed by: (i) a monolithic Titanium blank (cut by a rolled sheet); (b) a porous blank cut by a billet produced by Hot Isostatic Pressing (HIP) and subjected to a Solid State Foaming (SSF) heat treatment. Experimental bulge tests aimed to evaluate the effects of both the initial porosity level of the HIPed layer and the temperature (850 and 900 °C) of the SPF process. The bilayer sample was deformed by a constant argon gas pressure and the time evolution of the dome height was recorded during the test in order to be used as target in the IA to determine the constants of the rheological model of the porous layer, being the behavior of the monolithic layer already known. A 2D FE model of the free inflation test was integrated in an automatic optimization procedure with the aim to minimize the error between the calculated and the acquired dome height vs time curve. Finally, the obtained material constants determined were used to design the SPF process by means of the numerical simulation, identifying as a case study a zygomatic full custom prosthesis. The initial porosity level resulted to poorly affect the superplastic behavior of the bilayer at 850 °C; on the contrary, when increasing the temperature up to 900 °C, an evident reduction of the forming time was obtained. Numerical simulations showed that at 900 °C it is possible to obtain the very complex geometry of the adopted case study in about 50000 seconds.

**Introduction**

Titanium (Ti) alloys are characterized by a wide range of applications because of their physical and chemical properties. High corrosion resistance properties and high strength combined with weight savings are highly valued by the chemistry and aerospace industries. In recent years, these properties together with biocompatibility with body fluids have motivated their adoption in the biomedical field for manufacturing prosthetic implants. In this regard, one of the most important aspects is the durability of the implant without any infectious risk to the host patient. Not surprisingly, large differences in terms of Young's modulus between the human bone and the prosthesis material cause stress-shielding phenomena. To avoid this problem, one possible solution is the adoption of porous structures capable of modifying the elastic response; the presence of porous structures also promotes bone tissue regeneration. In fact, porous-structure materials are increasingly used because of their attractive physical and mechanical properties, such as high specific strength directly related to the amount and size of pores. In addition, when optimally designed, they are characterized by a significant increase in energy dissipation and vibration absorption capacity, not disdaining appreciable thermal as well as acoustic properties. According to this, one possible solution is to adopt Ti alloys obtained by Hot Isostatic Pressing (HIP) [1] subsequently subjected to a Solid State Foaming (SSF) heat treatment [2] aimed at changing the

<figure><img src="image_982.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.
# Characterization of the superplastic behavior of a Ti6Al4V-ELI alloy bilayer sheet

GUGLIELMI Pasquale¹,ᵃ*, PICCININNI Antonio¹,ᵇ, CUSANNO Angela¹,ᶜ, VAIANI Lorenzo¹,ᵈ, UVA Antonio Emmanuele¹,ᵉ, PALUMBO Gianfranco¹,*f

¹DMMM - Politecnico di Bari, Viale Japigia 182, Bari 70126, Italy

ᵃpasquale.guglielmi@poliba.it, ᵇantonio.piccininni@poliba.it, ᶜangela.cusanno@poliba.it,

ᵈlorenzo.vaiani@poliba.it, ᶜantonio.uva@poliba.it, ᶠgianfranco.palumbo@poliba.it

**Keywords:** Ti Alloy, Material Characterization, Superplasticity, Finite Element, Inverse Analysis

**Abstract.** The aim of the present work is to investigate the superplastic behaviour of a bilayer Ti6Al4V-ELI to numerically design a full custom prosthesis manufacturing process by means of SuperPlastic Forming (SPF); the bilayer was composed by: (i) a monolithic Titanium blank (cut by a rolled sheet); (b) a porous blank cut by a billet produced by Hot Isostatic Pressing (HIP) and subjected to a Solid State Foaming (SSF) heat treatment. Experimental bulge tests aimed to evaluate the effects of both the initial porosity level of the HIPed layer and the temperature (850 and 900 °C) of the SPF process. The bilayer sample was deformed by a constant argon gas pressure and the time evolution of the dome height was recorded during the test in order to be used as target in the IA to determine the constants of the rheological model of the porous layer, being the behavior of the monolithic layer already known. A 2D FE model of the free inflation test was integrated in an automatic optimization procedure with the aim to minimize the error between the calculated and the acquired dome height vs time curve. Finally, the obtained material constants determined were used to design the SPF process by means of the numerical simulation, identifying as a case study a zygomatic full custom prosthesis. The initial porosity level resulted to poorly affect the superplastic behavior of the bilayer at 850 °C; on the contrary, when increasing the temperature up to 900 °C, an evident reduction of the forming time was obtained. Numerical simulations showed that at 900 °C it is possible to obtain the very complex geometry of the adopted case study in about 50000 seconds.

**Introduction**

Titanium (Ti) alloys are characterized by a wide range of applications because of their physical and chemical properties. High corrosion resistance properties and high strength combined with weight savings are highly valued by the chemistry and aerospace industries. In recent years, these properties together with biocompatibility with body fluids have motivated their adoption in the biomedical field for manufacturing prosthetic implants. In this regard, one of the most important aspects is the durability of the implant without any infectious risk to the host patient. Not surprisingly, large differences in terms of Young's modulus between the human bone and the prosthesis material cause stress-shielding phenomena. To avoid this problem, one possible solution is the adoption of porous structures capable of modifying the elastic response; the presence of porous structures also promotes bone tissue regeneration. In fact, porous-structure materials are increasingly used because of their attractive physical and mechanical properties, such as high specific strength directly related to the amount and size of pores. In addition, when optimally designed, they are characterized by a significant increase in energy dissipation and vibration absorption capacity, not disdaining appreciable thermal as well as acoustic properties. According to this, one possible solution is to adopt Ti alloys obtained by Hot Isostatic Pressing (HIP) [1] subsequently subjected to a Solid State Foaming (SSF) heat treatment [2] aimed at changing the

<figure><img src="image_989.png" /></figure>
porosity level [3]. In addition, the growing demand for fully customized prostheses has increased the geometric complexity of the implants. Among others, SPF could be an appropriate choice to ensure complex shapes [4]. The correct process design requires a reliable Finite Element (FE) model, based on an accurate material modelling. In addition to the economic savings due to the reduction of the experiments, thanks to the possibility of coupling a numerical model to an optimization loop, the adoption of the numerical modeling represents an increasingly more essential solution. Regarding the SPF process, the FE method is usually adopted to predict both the final thickness distribution and the optimal pressure profile. In this regard, a robust characterization of the material is necessary [5]. To this should also be added the significant role of the type of material model adopted to reproduce in the FE environment the mechanical and deformation properties of the material. To confirm this, several research activities are based on this objective. According to the evidence obtained by Enikeev and Kruglov [6], it is possible to determine the constants of the material model exclusively by means of bulge tests. In order to reduce as much as possible, the number of experimental tests required for the material characterization, a highly valuable tool is represented by IA [7]. The IA aims at the determination of the objective function(s) through the minimization of the error between the experimental data and that of the prediction model.

The present work aims at investigating the superplastic behaviour of a bilayer Ti6Al4V-ELI to numerically design a full custom prosthesis manufacturing process by means of SuperPlastic Forming (SPF); the bilayer was composed by: (i) a monolithic Titanium blank (cut by a rolled sheet); (b) a porous blank cut by a billet produced by Hot Isostatic Pressing (HIP) and subjected to a Solid State Foaming (SSF) heat treatment. Experimental bulge tests aimed to evaluate the effects of both the initial porosity level of the HIPed layer and the temperature (850 and 900 °C) of the SPF process. The bilayer sample was deformed by a constant argon gas pressure and the time evolution of the dome height was recorded during the test in order to be used as target in the IA to determine the constants of the rheological model of the porous layer, being the behavior of the monolithic layer already known. A 2D FE model of the free inflation test was integrated in an automatic optimization procedure with the aim to minimize the error between the calculated and the acquired dome height vs time curve. Finally, the obtained material constants determined were used to design the SPF process by means of the numerical simulation, identifying as a case study a zygomatic full custom prosthesis.

## Material and Methods

The experimental campaign aimed to evaluate the effectiveness of this approach to reduce the final component production time without disdaining the mechanical and functional performance of the final component. Unfortunately, as demonstrated in a previous work [8], it is not possible to characterize under superplastic conditions individually the HIPed layer because of the coalescence of porosities resulting in gas leakage during the test, which makes the blank deformation not possible. Based on this, experimental tests were performed directly on the bilayer by placing the monolithic blank in direct contact with Argon gas. Then, by implementing the data in the IA, it was possible to obtain the HIPed material's constants under superplastic conditions, considering also the constants referred to the BULK layer determined separately. Finally, the obtained values were implemented in a FE numerical model to design a hybrid full custom prosthesis.

## Material

The experimental activity was conducted using a 2 layers titanium blank obtained by rolling (named BULK) and by HIP process. Regarding the HIPed material, the manufacturing process was characterized by (i) Argon pressure of 0.2 MPa and (ii) HIP pressure equal to 80 MPa. Ti6Al4V-ELI powders with a diameter ranging from 50 to 100 µm were used. From a chemical point of view, the composition of the investigated materials is the same as listed in Table 1.
porosity level [3]. In addition, the growing demand for fully customized prostheses has increased the geometric complexity of the implants. Among others, SPF could be an appropriate choice to ensure complex shapes [4]. The correct process design requires a reliable Finite Element (FE) model, based on an accurate material modelling. In addition to the economic savings due to the reduction of the experiments, thanks to the possibility of coupling a numerical model to an optimization loop, the adoption of the numerical modeling represents an increasingly more essential solution. Regarding the SPF process, the FE method is usually adopted to predict both the final thickness distribution and the optimal pressure profile. In this regard, a robust characterization of the material is necessary [5]. To this should also be added the significant role of the type of material model adopted to reproduce in the FE environment the mechanical and deformation properties of the material. To confirm this, several research activities are based on this objective. According to the evidence obtained by Enikeev and Kruglov [6], it is possible to determine the constants of the material model exclusively by means of bulge tests. In order to reduce as much as possible, the number of experimental tests required for the material characterization, a highly valuable tool is represented by IA [7]. The IA aims at the determination of the objective function(s) through the minimization of the error between the experimental data and that of the prediction model.

The present work aims at investigating the superplastic behaviour of a bilayer Ti6Al4V-ELI to numerically design a full custom prosthesis manufacturing process by means of SuperPlastic Forming (SPF); the bilayer was composed by: (i) a monolithic Titanium blank (cut by a rolled sheet); (b) a porous blank cut by a billet produced by Hot Isostatic Pressing (HIP) and subjected to a Solid State Foaming (SSF) heat treatment. Experimental bulge tests aimed to evaluate the effects of both the initial porosity level of the HIPed layer and the temperature (850 and 900 °C) of the SPF process. The bilayer sample was deformed by a constant argon gas pressure and the time evolution of the dome height was recorded during the test in order to be used as target in the IA to determine the constants of the rheological model of the porous layer, being the behavior of the monolithic layer already known. A 2D FE model of the free inflation test was integrated in an automatic optimization procedure with the aim to minimize the error between the calculated and the acquired dome height vs time curve. Finally, the obtained material constants determined were used to design the SPF process by means of the numerical simulation, identifying as a case study a zygomatic full custom prosthesis.

## Material and Methods

The experimental campaign aimed to evaluate the effectiveness of this approach to reduce the final component production time without disdaining the mechanical and functional performance of the final component. Unfortunately, as demonstrated in a previous work [8], it is not possible to characterize under superplastic conditions individually the HIPed layer because of the coalescence of porosities resulting in gas leakage during the test, which makes the blank deformation not possible. Based on this, experimental tests were performed directly on the bilayer by placing the monolithic blank in direct contact with Argon gas. Then, by implementing the data in the IA, it was possible to obtain the HIPed material's constants under superplastic conditions, considering also the constants referred to the BULK layer determined separately. Finally, the obtained values were implemented in a FE numerical model to design a hybrid full custom prosthesis.

## Material

The experimental activity was conducted using a 2 layers titanium blank obtained by rolling (named BULK) and by HIP process. Regarding the HIPed material, the manufacturing process was characterized by (i) Argon pressure of 0.2 MPa and (ii) HIP pressure equal to 80 MPa. Ti6Al4V-ELI powders with a diameter ranging from 50 to 100 µm were used. From a chemical point of view, the composition of the investigated materials is the same as listed in Table 1.
<table><caption>Table 1 Chemical composition of the investigated Ti alloys.</caption><thead><tr><th>Al%</th><th>V%</th><th>Fe%</th><th>C%</th><th>N%</th><th>H%</th><th>O%</th><th>Ti</th></tr></thead><tbody><tr><td>5.88</td><td>3.87</td><td>0.14</td><td>0.22</td><td>0.006</td><td>0.002</td><td>0.112</td><td>Bal.</td></tr></tbody></table>

Finally, for the experimental bulge tests, circular samples with a diameter of 75mm were used; the BULK and the HIPed material had a thickness of 1 mm and 1.5 mm, respectively.

## Methods

### Experimental tests

Experimental bulge tests were carried out by means of a proper equipped OMCN P30/W press machine. The overview of the experimental setup is provided in Fig. 1a. During bulge tests, the two different layers, placed as schematized in Fig. 1b and clamped by means of a blank-holder pressure of 10MPa, were simultaneously freely deformed into a cylindrical die (diameter and entry fillet radius equal respectively to 45mm and 3mm) heated by means of an induction system managed by a PID controller. The Dome Height (DH) and the temperature were continuously acquired using respectively an optoNCDT 1420 laser sensor and a K-type thermocouple (TC) directly welded on the periphery of the die (Fig. 1b).

<figure><img src="image_983.png" /><figcaption>Fig. 1 experimental setup: (a) overview and (b) scheme of the forming chamber.</figcaption></figure>

The effect of the initial porosity level was investigated in the first part of the experimental campaign. To do this, SSF tests were conducted on the HIPed material for 60 minutes at 850 °C using a furnace (Nabertherm N 61/H); the bulge tests were, thus, performed at 850 °C on both SSFed and HIPed samples.

In the second experimental step, instead of porosity, the effect of the temperature was evaluated by setting an additional temperature level (900 °C) to the previous one and focusing only on the condition after HIP process. Furthermore, as it is preparatory to the subsequent IA, in addition to the experimental curve related to the bilayer, it was necessary to determine the material constants; to do this, the analytical approach by Enikeev and Kruglov [6] was used to determine the BULK material constants. When the temperature was set to 850 °C, an extensive characterization was carried out in a previous work [9]; at 900 °C at least two tests were needed to calculate the material constants of the following power law model.

$$ \sigma = C\dot{\epsilon}^m \qquad (1) $$

where $\dot{\epsilon}$ is strain rate, C is a material constant and m represent the strain rate sensitivity index. This last parameter is of fundamental importance; in fact, the higher its value the more superplastic is the material. In Table 2 the experimental tests and the nomenclature adopted for each condition (repeated three times) are summarized.
<table><caption>Table 1 Chemical composition of the investigated Ti alloys.</caption><thead><tr><th>Al%</th><th>V%</th><th>Fe%</th><th>C%</th><th>N%</th><th>H%</th><th>O%</th><th>Ti</th></tr></thead><tbody><tr><td>5.88</td><td>3.87</td><td>0.14</td><td>0.22</td><td>0.006</td><td>0.002</td><td>0.112</td><td>Bal.</td></tr></tbody></table>

Finally, for the experimental bulge tests, circular samples with a diameter of 75mm were used; the BULK and the HIPed material had a thickness of 1 mm and 1.5 mm, respectively.

## Methods

### Experimental tests

Experimental bulge tests were carried out by means of a proper equipped OMCN P30/W press machine. The overview of the experimental setup is provided in Fig. 1a. During bulge tests, the two different layers, placed as schematized in Fig. 1b and clamped by means of a blank-holder pressure of 10MPa, were simultaneously freely deformed into a cylindrical die (diameter and entry fillet radius equal respectively to 45mm and 3mm) heated by means of an induction system managed by a PID controller. The Dome Height (DH) and the temperature were continuously acquired using respectively an optoNCDT 1420 laser sensor and a K-type thermocouple (TC) directly welded on the periphery of the die (Fig. 1b).

<figure><img src="image_990.png" /><figcaption>Fig. 1 experimental setup: (a) overview and (b) scheme of the forming chamber.</figcaption></figure>

The effect of the initial porosity level was investigated in the first part of the experimental campaign. To do this, SSF tests were conducted on the HIPed material for 60 minutes at 850 °C using a furnace (Nabertherm N 61/H); the bulge tests were, thus, performed at 850 °C on both SSFed and HIPed samples.

In the second experimental step, instead of porosity, the effect of the temperature was evaluated by setting an additional temperature level (900 °C) to the previous one and focusing only on the condition after HIP process. Furthermore, as it is preparatory to the subsequent IA, in addition to the experimental curve related to the bilayer, it was necessary to determine the material constants; to do this, the analytical approach by Enikeev and Kruglov [6] was used to determine the BULK material constants. When the temperature was set to 850 °C, an extensive characterization was carried out in a previous work [9]; at 900 °C at least two tests were needed to calculate the material constants of the following power law model.

$$ \sigma = C\dot{\epsilon}^m \quad (1) $$

where $\dot{\epsilon}$ is strain rate, C is a material constant and m represent the strain rate sensitivity index. This last parameter is of fundamental importance; in fact, the higher its value the more superplastic is the material. In Table 2 the experimental tests and the nomenclature adopted for each condition (repeated three times) are summarized.
<table><caption>Table 2 Experimental plan adopted.</caption><thead><tr><td>Temperature, °C</td><td>Pressure, MPa</td><td>Type of Sheet</td><td>SSF duration, min</td><td>Nomenclature adopted</td></tr></thead><tbody><tr><td>850</td><td>1.4</td><td>Bilayer</td><td>-</td><td>HIP_1.4_850</td></tr><tr><td>850</td><td>1.4</td><td>Bilayer</td><td>60</td><td>60SSF_1.4_850</td></tr><tr><td>900</td><td>1.4</td><td>Bilayer</td><td>-</td><td>HIP_1.4_900</td></tr><tr><td>900</td><td>0.56</td><td>Bulk</td><td>-</td><td>BULK_0.56_900</td></tr><tr><td>900</td><td>1.12</td><td>Bulk</td><td>-</td><td>BULK_1.12_900</td></tr></tbody></table>

As previously mentioned, the analyses related to these experimental tests were based on measurements of both DH over time and thickness after forming along the diametral direction. In this regard, for the measurements of the dome height data from laser sensor were considered; instead, the thickness distributions were obtained by means of a Mitutoyo digital caliper.

## Metallographic analyses

Diametral sections of each sample (both after SSF and bulge tests) were properly prepared by grinding and polishing to analyse their porosity. More in detail, the investigation aimed at determining both the average diameter and the percentage area of the porosities according to the different conditions reported in Table 2. For this purpose, the Nikon MA200 microscope and the ImageJ software for digital image processing were adopted respectively to acquire and process the micrographs.

## Inverse Analysis

The inverse determination of the material constants from the Backofen's constitutive equation was treated as an optimization problem in which the objective error function, calculated as the difference between the numerical and experimental dome height curve, had to be minimized. Therefore, a simple FE model was created according to a 2D axisymmetric approach (see Fig. 2) by modelling the two blanks as shell deformable bodies whereas keeping the die as a rigid body. Both the blanks were constrained in their outer nodes to simulate the action of the blankholder (not modelled to reduce the computational costs). A constant pressure of 1.4 MPa was applied on the top surface of the bulk blank (reproducing the loading condition of the experimental tests).

The FE model was integrated into an automatic optimization procedure managed by a multi-objective genetic algorithm (MOGA-II): basically, the genetic algorithm (GA) started from an initial population of designs (where the term “design” refers to a single numerical run) and iteratively changed the unknown parameters (the constants of the HIP material) with the final goal of minimizing the error function, expressed by Equation 2 (a more detailed description of the GA-based inverse approach can be found in [10]).

$$ERR = \sum_{i=0}^{N} (h_{i,num} - h_{i,exp})^2 \qquad (2)$$

Where $h_{i,num}$ refers to the dome height computed by the numerical simulation and $h_{i,exp}$ to the value measured during the test at the same i-th instant of time. The optimization procedure was carried out to determine the HIP material's constants at the two investigated levels of temperature according to the experimental plan proposed and discussed in section 2.2.1.
<table><caption>Table 2 Experimental plan adopted.</caption><thead><tr><td>Temperature, °C</td><td>Pressure, MPa</td><td>Type of Sheet</td><td>SSF duration, min</td><td>Nomenclature adopted</td></tr></thead><tbody><tr><td>850</td><td>1.4</td><td>Bilayer</td><td>-</td><td>HIP_1.4_850</td></tr><tr><td>850</td><td>1.4</td><td>Bilayer</td><td>60</td><td>60SSF_1.4_850</td></tr><tr><td>900</td><td>1.4</td><td>Bilayer</td><td>-</td><td>HIP_1.4_900</td></tr><tr><td>900</td><td>0.56</td><td>Bulk</td><td>-</td><td>BULK_0.56_900</td></tr><tr><td>900</td><td>1.12</td><td>Bulk</td><td>-</td><td>BULK_1.12_900</td></tr></tbody></table>

As previously mentioned, the analyses related to these experimental tests were based on measurements of both DH over time and thickness after forming along the diametral direction. In this regard, for the measurements of the dome height data from laser sensor were considered; instead, the thickness distributions were obtained by means of a Mitutoyo digital caliper.

## Metallographic analyses

Diametral sections of each sample (both after SSF and bulge tests) were properly prepared by grinding and polishing to analyse their porosity. More in detail, the investigation aimed at determining both the average diameter and the percentage area of the porosities according to the different conditions reported in Table 2. For this purpose, the Nikon MA200 microscope and the ImageJ software for digital image processing were adopted respectively to acquire and process the micrographs.

## Inverse Analysis

The inverse determination of the material constants from the Backofen's constitutive equation was treated as an optimization problem in which the objective error function, calculated as the difference between the numerical and experimental dome height curve, had to be minimized. Therefore, a simple FE model was created according to a 2D axisymmetric approach (see Fig. 2) by modelling the two blanks as shell deformable bodies whereas keeping the die as a rigid body. Both the blanks were constrained in their outer nodes to simulate the action of the blankholder (not modelled to reduce the computational costs). A constant pressure of 1.4 MPa was applied on the top surface of the bulk blank (reproducing the loading condition of the experimental tests).

The FE model was integrated into an automatic optimization procedure managed by a multi-objective genetic algorithm (MOGA-II): basically, the genetic algorithm (GA) started from an initial population of designs (where the term "design" refers to a single numerical run) and iteratively changed the unknown parameters (the constants of the HIP material) with the final goal of minimizing the error function, expressed by Equation 2 (a more detailed description of the GA-based inverse approach can be found in [10]).

$$ERR = \sum_{i=0}^{N} (h_{i,num} - h_{i,exp})^2 \qquad (2)$$

Where $h_{i,num}$ refers to the dome height computed by the numerical simulation and $h_{i,exp}$ to the value measured during the test at the same i-th instant of time. The optimization procedure was carried out to determine the HIP material's constants at the two investigated levels of temperature according to the experimental plan proposed and discussed in section 2.2.1.
<figure><img src="image_984.png" /><figcaption>Fig. 2 Scheme of the 2D axisymmetric FE model adopted for the inverse analysis (dimension are in mm).</figcaption></figure>

## Process design of the case study

The replacement of a partially damaged mandibular bone was chosen as a case study: once reconstructed the mandibular bone from the DICOM files, the region to be healed was isolated and the geometry of the prosthesis designed starting from mirroring the healthy bone from the opposite side. The final CAD geometry of the implant was used as the base for the definition of the die cavity for the manufacturing process. The sequence of steps is summarized in Fig. 3.

<figure><img src="image_985.png" /><figcaption>Fig. 3 Definition of the SPF die geometry starting from the DICOM acquisitions.</figcaption></figure>

The created SPF die geometry was implemented in a second FE model to design the SPF process of the case study. An overview of the modelled system is reported in Fig. 4. The two blanks were modelled as 3D shell deformable bodies (meshed by plane-stress elements with an average size of 0.5 mm), whereas the blank as a 3D discrete rigid body (discretized by 1 mm elements on average). Both the blanks periphery was pinned to simulate the action of the blankholder: a constant gas pressure of 1.4 MPa was applied on the top surface of the bulk blank.

<figure><img src="image_986.png" /><figcaption>Fig. 4 FE model for the process design of the case study.</figcaption></figure>
<figure><img src="image_991.png" /><figcaption>Fig. 2 Scheme of the 2D axisymmetric FE model adopted for the inverse analysis (dimension are in mm).</figcaption></figure>

## Process design of the case study

The replacement of a partially damaged mandibular bone was chosen as a case study: once reconstructed the mandibular bone from the DICOM files, the region to be healed was isolated and the geometry of the prosthesis designed starting from mirroring the healthy bone from the opposite side. The final CAD geometry of the implant was used as the base for the definition of the die cavity for the manufacturing process. The sequence of steps is summarized in Fig. 3.

<figure><img src="image_992.png" /><figcaption>Fig. 3 Definition of the SPF die geometry starting from the DICOM acquisitions.</figcaption></figure>

The created SPF die geometry was implemented in a second FE model to design the SPF process of the case study. An overview of the modelled system is reported in Fig. 4. The two blanks were modelled as 3D shell deformable bodies (meshed by plane-stress elements with an average size of 0.5 mm), whereas the blank as a 3D discrete rigid body (discretized by 1 mm elements on average). Both the blanks periphery was pinned to simulate the action of the blankholder: a constant gas pressure of 1.4 MPa was applied on the top surface of the bulk blank.

<figure><img src="image_993.png" /><figcaption>Fig. 4 FE model for the process design of the case study.</figcaption></figure>
# Results and Discussion

## Experimental tests

Results from bulge tests at a constant temperature of 850 °C, in terms of both the Dome Height profile and the thickness distribution, are reported in Fig. 5. These tests were stopped at a final DH of 14 mm.

<figure><img src="image_987.png" /><figcaption>Fig. 5 Experimental results obtained at 850 °C: (a) DH vs Time and (b) Thickness distribution after forming.</figcaption></figure>

It can be observed that a prior SSF of the HIPed material allows a faster increase of DH in the first part of the test; however, keeping the sample at a high temperature during the forming test activates foaming in the sample obtained by HIP. This motivates a gradually decreasing gap between the two conditions tested (Fig. 5a). In particular, the steady-state condition (characterized by a constant slope of the curve) appears to be about very similar for all conditions, suggesting a low effect of pre-foaming on the overall duration of the SPF process. At the same time, SSF allows for an increase in terms of thickness distribution, as demonstrated by the graph in Fig. 5b. This behavior, as better documented by the subsequent metallographic investigations (section 3.2), is due to a higher porosity attributable to the SSFed material.

Fig. 6a, on the other hand, shows the results for the tests conducted at 900 °C; as well as comparisons with what was obtained at 850 °C for the HIPed material are showed in Fig. 6b.

As mentioned in the methodological section, at 900 °C it was also necessary to conduct tests on the BULK material alone to determine in accordance with [6] the material constants (C and m). For this reason, the two tests on BULK were conducted at pressures such that a curve above (red) and below (green) that for the bilayer were obtained (Fig. 6a). The comparison between the two bilayers (Fig. 6b), in addition, shows a clear difference between the two investigated temperatures; despite the fact that the final heights of the two tests are not the same, a temperature increase of 50 °C allowed a DH value of 14 mm to be reached in 1300 sec rather than 9000 sec, recording a reduction in time of about 590 %. This result surely of remarkable importance for an industrial implementation. Furthermore, as concerns the BULK material, the obtained values of C (2396.65 MPa·s⁻⁶) and m-value (0.711) show a high propensity to superplasticity of the material in the above described pressure and temperature conditions.
# Results and Discussion

## Experimental tests

Results from bulge tests at a constant temperature of 850 °C, in terms of both the Dome Height profile and the thickness distribution, are reported in Fig. 5. These tests were stopped at a final DH of 14 mm.

<figure><img src="image_994.png" /><figcaption>Fig. 5 Experimental results obtained at 850 °C: (a) DH vs Time and (b) Thickness distribution after forming.</figcaption></figure>

It can be observed that a prior SSF of the HIPed material allows a faster increase of DH in the first part of the test; however, keeping the sample at a high temperature during the forming test activates foaming in the sample obtained by HIP. This motivates a gradually decreasing gap between the two conditions tested (Fig. 5a). In particular, the steady-state condition (characterized by a constant slope of the curve) appears to be about very similar for all conditions, suggesting a low effect of pre-foaming on the overall duration of the SPF process. At the same time, SSF allows for an increase in terms of thickness distribution, as demonstrated by the graph in Fig. 5b. This behavior, as better documented by the subsequent metallographic investigations (section 3.2), is due to a higher porosity attributable to the SSFed material.

Fig. 6a, on the other hand, shows the results for the tests conducted at 900 °C; as well as comparisons with what was obtained at 850 °C for the HIPed material are showed in Fig. 6b.

As mentioned in the methodological section, at 900 °C it was also necessary to conduct tests on the BULK material alone to determine in accordance with [6] the material constants (C and m). For this reason, the two tests on BULK were conducted at pressures such that a curve above (red) and below (green) that for the bilayer were obtained (Fig. 6a). The comparison between the two bilayers (Fig. 6b), in addition, shows a clear difference between the two investigated temperatures; despite the fact that the final heights of the two tests are not the same, a temperature increase of 50 °C allowed a DH value of 14 mm to be reached in 1300 sec rather than 9000 sec, recording a reduction in time of about 590 %. This result surely of remarkable importance for an industrial implementation. Furthermore, as concerns the BULK material, the obtained values of C (2396.65 MPa·s⁻⁴) and m-value (0.711) show a high propensity to superplasticity of the material in the above described pressure and temperature conditions.
<figure><img src="image_988.png" /><figcaption>Fig. 6 DH vs time referred to the HIPed and the BULK conditions at 900 °C (a); effect of the temperature (b)</figcaption></figure>

<figure><img src="image_989.png" /></figure>

## Metallographic analyses

SSF allowed to increase the porosity (in terms of both average diameter and percentage area) respect to the HIPed material. Marked differences can be found because of the SSF process. In fact, considering an initial average diameter (after HIP process) of only 0.91±0.05 µm, foaming allowed to increase this value up to 14.56±1.50 µm. At the same time, the percentage area changes from 0.65±0.10 % to 4.11±0.60 %. After bulge tests such differences are much smaller.

Beside this, considering the graphs shown in Fig. 7, it is also possible to observe how the level of porosity is strongly influenced by the deformation level (expressed in thinning) reached. The lowest level is referred to the flanged area; the highest is representative of the dome apex.

<figure><img src="image_990.png" /><figcaption>Fig. 7 Average diameter (a) and percentage area (b) of the porosity after bulge tests at 850 °C according to different thinning levels.</figcaption></figure>

<figure><img src="image_991.png" /></figure>

In the flanged area it is particularly evident that the effect of SSF before deformation is somewhat mitigated by the duration of the subsequent bulge test. All these considerations find support from the micrographs summarized in Fig. 8. The black areas are representative of the porosity.

<figure><img src="image_992.png" /><figcaption>Fig. 8 Porosity after bulge tests at 850 °C: (a) HIP_1.4_850 and (b) 60SSF_1.4_850.</figcaption></figure>

In light of the obtained experimental results, it is possible to say that the strategy based on SSF prior forming step appears to be not effective, since the effects of the SSF are drastically mitigated by the forming process itself.
<figure><img src="image_995.png" /><figcaption>Fig. 6 DH vs time referred to the HIPed and the BULK conditions at 900 °C (a); effect of the temperature (b)</figcaption></figure>

## Metallographic analyses

SSF allowed to increase the porosity (in terms of both average diameter and percentage area) respect to the HIPed material. Marked differences can be found because of the SSF process. In fact, considering an initial average diameter (after HIP process) of only 0.91±0.05 µm, foaming allowed to increase this value up to 14.56±1.50 µm. At the same time, the percentage area changes from 0.65±0.10 % to 4.11±0.60 %. After bulge tests such differences are much smaller.

Beside this, considering the graphs shown in Fig. 7, it is also possible to observe how the level of porosity is strongly influenced by the deformation level (expressed in thinning) reached. The lowest level is referred to the flanged area; the highest is representative of the dome apex.

<figure><img src="image_996.png" /><figcaption>Fig. 7 Average diameter (a) and percentage area (b) of the porosity after bulge tests at 850 °C according to different thinning levels.</figcaption></figure>

In the flanged area it is particularly evident that the effect of SSF before deformation is somewhat mitigated by the duration of the subsequent bulge test. All these considerations find support from the micrographs summarized in Fig. 8. The black areas are representative of the porosity.

<figure><img src="image_997.png" /><figcaption>Fig. 8 Porosity after bulge tests at 850 °C: (a) HIP_1.4_850 and (b) 60SSF_1.4_850.</figcaption></figure>

In light of the obtained experimental results, it is possible to say that the strategy based on SSF prior forming step appears to be not effective, since the effects of the SSF are drastically mitigated by the forming process itself.
On the contrary, regarding the increase in forming temperature (from 850 °C to 900 °C), the effect is also markedly different in terms of porosity. Once again, when considering the porosities as a function of the measured thinning and thus with reference to the two opposite zones (flange and dome apex), it is evident that the two investigated temperatures combined with markedly different forming times lead to varying levels of porosity.

This is supported by both the graphs in Fig. 9. and the micrographs in Fig. 10.

<figure><img src="image_993.png" /><figcaption>Fig. 9 Average diameter (a) and percentage area (b) of the porosity after bulge tests at different temperature according to different thinning levels.</figcaption></figure>

<figure><img src="image_994.png" /><figcaption>Fig. 10 Porosity after bulge tests at different temperature: (a) HIP_1.4_850 and (b) HIP_1.4_900.</figcaption></figure>

## Inverse Analysis

The optimization, for both the investigated levels of temperature, evolved throughout 20 generations (each of 50 individuals). Plotting the results by means of the history charts was an effective way to evaluate the convergence of the GA.

The history charts suggest that the inverse determination of the material constants at 850°C faced some convergence issues (see Fig. 11 a and b): in fact, after approximately 800 designs, the GA tended to stabilize around lower values of both the constants. Different results were recorded for the calibration of the constants at 900°C for which (as visible in Fig. 11 c and d), the convergence was more stable.

Eventually, at the end of each procedure, the best designs could be identified as those characterized by the optimal values of the two material constants able to minimize the error function (their values are resumed in Table 3).
On the contrary, regarding the increase in forming temperature (from 850 °C to 900 °C), the effect is also markedly different in terms of porosity. Once again, when considering the porosities as a function of the measured thinning and thus with reference to the two opposite zones (flange and dome apex), it is evident that the two investigated temperatures combined with markedly different forming times lead to varying levels of porosity.

This is supported by both the graphs in Fig. 9. and the micrographs in Fig. 10.

<figure><img src="image_998.png" /><figcaption>Fig. 9 Average diameter (a) and percentage area (b) of the porosity after bulge tests at different temperature according to different thinning levels.</figcaption></figure>

<figure><img src="image_999.png" /><figcaption>Fig. 10 Porosity after bulge tests at different temperature: (a) HIP_1.4_850 and (b) HIP_1.4_900.</figcaption></figure>

## Inverse Analysis

The optimization, for both the investigated levels of temperature, evolved throughout 20 generations (each of 50 individuals). Plotting the results by means of the history charts was an effective way to evaluate the convergence of the GA.

The history charts suggest that the inverse determination of the material constants at 850°C faced some convergence issues (see Fig. 11 a and b): in fact, after approximately 800 designs, the GA tended to stabilize around lower values of both the constants. Different results were recorded for the calibration of the constants at 900°C for which (as visible in Fig. 11 c and d), the convergence was more stable.

Eventually, at the end of each procedure, the best designs could be identified as those characterized by the optimal values of the two material constants able to minimize the error function (their values are resumed in Table 3).
<figure><img src="image_995.png" /><figcaption>Fig. 11 History charts plotting the evolution of the two parameters throughout the optimization: a) C and b) m at 850°C; c) C and d) m at 900°C.</figcaption></figure>

<table><caption>Table 3 Results from the inverse analysis.</caption><thead><tr><td>Nomenclature adopted</td><td>Temperature, °C</td><td>C, MPa·s<sup>m</sup></td><td>m</td></tr></thead><tbody><tr><td>HIP_1.4_850</td><td>850</td><td>374</td><td>0.3</td></tr><tr><td>HIP_1.4_900</td><td>900</td><td>2067.5</td><td>0.557</td></tr></tbody></table>

## FE simulation of the bilayer implant manufacturing

Results from FE simulations were primarily analysed in terms of die cavity filling with time. Fig. 12 shows the distribution of the COPEN variable which basically represents the clearance between surfaces in contact: more in details, regions of the HIP blank highlighted in blue are in contact with the die surface.

<figure><img src="image_996.png" /><figcaption>Fig. 12 Filling of the die cavity during the SPF process: a) T=850°C, b) T=900°C</figcaption></figure>

It could be then concluded that, at 850°C and even after 50000 s, the bilayer blank was not in contact with the bottom part of the die. On the other hand, when raising the temperature at 900°C, the blank got in contact with the bottom of the die cavity already after 25000 s. The different filling at the two investigated temperatures is coherent with the results from the material characterization (see Fig. 6b) and led to the conclusion that rising the temperature up to 900°C could be an effective way to obtain the final geometry without excessively prolonging the manufacturing time.
<figure><img src="image_1000.png" /><figcaption>Fig. 11 History charts plotting the evolution of the two parameters throughout the optimization: a) C and b) m at 850°C; c) C and d) m at 900°C.</figcaption></figure>

<table><caption>Table 3 Results from the inverse analysis.</caption><thead><tr><td>Nomenclature adopted</td><td>Temperature, °C</td><td>C, MPa·s<sup>m</sup></td><td>m</td></tr></thead><tbody><tr><td>HIP_1.4_850</td><td>850</td><td>374</td><td>0.3</td></tr><tr><td>HIP_1.4_900</td><td>900</td><td>2067.5</td><td>0.557</td></tr></tbody></table>

FE simulation of the bilayer implant manufacturing

Results from FE simulations were primarily analysed in terms of die cavity filling with time. Fig. 12 shows the distribution of the COPEN variable which basically represents the clearance between surfaces in contact: more in details, regions of the HIP blank highlighted in blue are in contact with the die surface.

<figure><img src="image_1001.png" /><figcaption>Fig. 12 Filling of the die cavity during the SPF process: a) T=850°C, b) T=900°C</figcaption></figure>

It could be then concluded that, at 850°C and even after 50000 s, the bilayer blank was not in contact with the bottom part of the die. On the other hand, when raising the temperature at 900°C, the blank got in contact with the bottom of the die cavity already after 25000 s. The different filling at the two investigated temperatures is coherent with the results from the material characterization (see Fig. 6b) and led to the conclusion that rising the temperature up to 900°C could be an effective way to obtain the final geometry without excessively prolonging the manufacturing time.
## Conclusions

In the present work, based on experimental tests, the effect of both initial level of the porosity and the temperature were investigated in order to numerically design a hybrid full custom prosthesis made from a Ti bilayer having one layer derived from HIP process and one monolithic layer. Main outcomes of the investigation are summarized below:

- The initial level of porosity seems to be less effective on the superplastic behavior of the HIPed layer, as the SPF conditions mitigated the effect of the previous SSF.

- Increasing the SPF temperature, on the other hand, makes it possible to significantly reduce the production time while still ensuring that the deformation behavior of the material is maintained, as supported by the high m-value which was determined.

- The IA showed a better capability to determine material constants at 900 °C; moreover, the obtained values of C and m suggest an emphasized propensity of the material to exhibit superplastic properties at 900 °C.

- The numerical design of the SPF process for the manufacturing of a full custom prosthesis has confirmed that the adoption of higher process temperature (900 °C) allows a drastic reduction in process time (while at 900 °C the prosthesis is formed after about 50000 seconds, at 850 °C, after the same time, the bilayer did not fully touch the die).

## References

[1] M.W. Kearns, Formation of Porous Bodies, 1987.

[2] S. Oppenheimer, D.C. Dunand, Solid-state foaming of Ti-6Al-4V by creep or superplastic expansion of argon-filled pores, Acta Mater 58 (2010) 4387-4397.
https://doi.org/10.1016/j.actamat.2010.04.034.

[3] P. Guglielmi, A. Piccininni, A. Cusanno, A.A. Kaya, G. Palumbo, Mechanical and microstructural evaluation of solid-state foamed Ti6Al4V-ELI alloy, Procedia CIRP 110 (2022) 105-110.
https://doi.org/10.1016/j.procir.2022.06.021.

[4] P. Guglielmi, A. Cusanno, I. Bagudanch, G. Centeno, I. Ferrer, M.L. Garcia-romeu, Experimental and numerical analysis of innovative processes for producing a resorbable cheekbone prosthesis, J Manuf Process 70 (2021) 1-14.
https://doi.org/10.1016/j.jmapro.2021.07.060.

[5] D. Sorgente, L. Tricarico, The role of the numerical simulation in superplastic forming process analysis and optimization, Key Eng Mater 433 (2010) 225-234.
https://doi.org/10.4028/www.scientific.net/KEM.433.225.

[6] F.U. Enikeev, A.A. Kruglov, An analysis of the superplastic forming of a thin circular diaphragm, Int J Mech Sci 37 (1995) 473-483.
https://doi.org/10.1016/0020-7403(94)00081-T.

[7] D. Sorgente, G. Palumbo, A. Piccininni, P. Guglielmi, L. Tricarico, Modelling the superplastic behaviour of the Ti6Al4V-ELI by means of a numerical/experimental approach, International Journal of Advanced Manufacturing Technology 90 (2017) 1-10.
https://doi.org/10.1007/s00170-016-9235-7.

[8] P. Guglielmi, A. Piccininni, A. Cusanno, G. Paumbo, Manufacturing of a hybrid component in Ti6Al4V-ELI alloy by combining diffusion bonding and superplastic forming, in: Materials Research Proceedings, Association of American Publishers, 2023: pp. 37-44.
https://doi.org/10.21741/9781644902714-5.

[9] D. Sorgente, G. Palumbo, A. Piccininni, P. Guglielmi, S. Aksenov, Investigation on the thickness distribution of highly customized titanium biomedical implants manufactured by superplastic forming, CIRP J Manuf Sci Technol Accepted, (2017).

[10] A. Piccininni, D. Sorgente, G. Palumbo, Genetic Algorithm based inverse analysis for the superplastic characterization of a Ti-6Al-4V biomedical grade, Finite Elements in Analysis and Design 148 (2018) 27-37.
## Conclusions

In the present work, based on experimental tests, the effect of both initial level of the porosity and the temperature were investigated in order to numerically design a hybrid full custom prosthesis made from a Ti bilayer having one layer derived from HIP process and one monolithic layer. Main outcomes of the investigation are summarized below:

- The initial level of porosity seems to be less effective on the superplastic behavior of the HIPed layer, as the SPF conditions mitigated the effect of the previous SSF.

- Increasing the SPF temperature, on the other hand, makes it possible to significantly reduce the production time while still ensuring that the deformation behavior of the material is maintained, as supported by the high m-value which was determined.

- The IA showed a better capability to determine material constants at 900 °C; moreover, the obtained values of C and m suggest an emphasized propensity of the material to exhibit superplastic properties at 900 °C.

- The numerical design of the SPF process for the manufacturing of a full custom prosthesis has confirmed that the adoption of higher process temperature (900 °C) allows a drastic reduction in process time (while at 900 °C the prosthesis is formed after about 50000 seconds, at 850 °C, after the same time, the bilayer did not fully touch the die).

## References

[1] M.W. Kearns, Formation of Porous Bodies, 1987.

[2] S. Oppenheimer, D.C. Dunand, Solid-state foaming of Ti-6Al-4V by creep or superplastic expansion of argon-filled pores, Acta Mater 58 (2010) 4387-4397.
https://doi.org/10.1016/j.actamat.2010.04.034.

[3] P. Guglielmi, A. Piccininni, A. Cusanno, A.A. Kaya, G. Palumbo, Mechanical and microstructural evaluation of solid-state foamed Ti6Al4V-ELI alloy, Procedia CIRP 110 (2022) 105-110.
https://doi.org/10.1016/j.procir.2022.06.021.

[4] P. Guglielmi, A. Cusanno, I. Bagudanch, G. Centeno, I. Ferrer, M.L. Garcia-romeu, Experimental and numerical analysis of innovative processes for producing a resorbable cheekbone prosthesis, J Manuf Process 70 (2021) 1-14.
https://doi.org/10.1016/j.jmapro.2021.07.060.

[5] D. Sorgente, L. Tricarico, The role of the numerical simulation in superplastic forming process analysis and optimization, Key Eng Mater 433 (2010) 225-234.
https://doi.org/10.4028/www.scientific.net/KEM.433.225.

[6] F.U. Enikeev, A.A. Kruglov, An analysis of the superplastic forming of a thin circular diaphragm, Int J Mech Sci 37 (1995) 473-483.
https://doi.org/10.1016/0020-7403(94)00081-T.

[7] D. Sorgente, G. Palumbo, A. Piccininni, P. Guglielmi, L. Tricarico, Modelling the superplastic behaviour of the Ti6Al4V-ELI by means of a numerical/experimental approach, International Journal of Advanced Manufacturing Technology 90 (2017) 1-10.
https://doi.org/10.1007/s00170-016-9235-7.

[8] P. Guglielmi, A. Piccininni, A. Cusanno, G. Paumbo, Manufacturing of a hybrid component in Ti6Al4V-ELI alloy by combining diffusion bonding and superplastic forming, in: Materials Research Proceedings, Association of American Publishers, 2023: pp. 37-44.
https://doi.org/10.21741/9781644902714-5.

[9] D. Sorgente, G. Palumbo, A. Piccininni, P. Guglielmi, S. Aksenov, Investigation on the thickness distribution of highly customized titanium biomedical implants manufactured by superplastic forming, CIRP J Manuf Sci Technol Accepted, (2017).

[10] A. Piccininni, D. Sorgente, G. Palumbo, Genetic Algorithm based inverse analysis for the superplastic characterization of a Ti-6Al-4V biomedical grade, Finite Elements in Analysis and Design 148 (2018) 27-37.
# Parametric failure limit detection for the sheet metal forming of a floating photovoltaic (FPV) aluminum alloy structure

TVEIT Sigbjørn¹,ᵃ*, REYES Aase¹,ᵇ and ERDURAN Emrah¹,ᶜ

¹Department of Built Environment, Oslo Metropolitan University, Postboks 4, St. Olavs plass, NO-0130 Oslo, Norway

ᵃsigbjorn.tveit@oslomet.no, ᵇaase.reyes@oslomet.no, ᶜemrah.erduran@oslomet.no

**Keywords:** Sheet Metal Forming, Aluminum Alloys, Parametric Analysis

**Abstract.** The sheet metal forming process of a floating photovoltaic (FPV) structure is simulated in LS-DYNA. An anisotropic yield criterion and a two-term Voce hardening law are used to model the plastic behavior of AA5083-H111 sheets. The numerical model incorporates thickness variations to trigger local necking and uses a critical thickness strain as a fracture criterion. To establish a methodology that can be expanded for further studies, the research explores the relationship between cup depth and drawbead distance by proposing an algorithm to distinguish between successful and unsuccessful sheet metal forming operations.

**Introduction**

In the competitive race of developing new methods for renewable energy production, the floating solar producer Sunlit Sea AS has come up with an innovative solution where formed aluminum alloy sheet structures are used as supports for floating photovoltaic (FPV) panels. In each FPV float, two formed components are joined together back-to-back, and sealed off along the rim to create a buoyant platform. The choice of material represents a key attribute of the design: As the power output of FPV modules decreases at elevated operational temperatures, the highly conductive aluminum alloy body, which acts as a thermal bridge between the FPV module and the cool sea water, allows the units to operate with increased efficiency.

A critical feature in the design is the repeated cup shape, which acts as a spacer to provide structural integrity to the platform. This allows for walking on top of the solar panels, which facilitates effortless installation, operation, and maintenance. The cup depth directly influences the buoyancy of the platform, and, as the cups transfer heat and forces, their geometry and constellation have a massive influence on the FPV's operational performance. Hence, for the technology to reach its potential with respect to cost per unit energy output, the optimization of the structural geometry with respect to several parameters, including the heat transfer, and the material costs, is vital.

A challenge emerges in the radical deep-drawing of marine-grade aluminum, as aggressively pushing the drawing limit can lead to increased failure rates in the production line. Sunlit Sea's collaborative partners have painstakingly developed a shape that is feasible to manufacture by deep-drawing 1.5 mm thick AA5083-H111 sheets close to their forming limit. However, in developing the next generation of floats, the company seeks a more flexible and efficient design process, that enables rapid testing of multiple design variations in a virtual environment, by combining explicit non-linear finite element (FE) simulations in LS-DYNA [1], with machine learning techniques.

As an initial part of this work, the current paper presents the precursory investigations of the comprehensive study on the influence of various geometric parameters on the sheet metal forming process of the Sunlit Sea's FPV structure. The overarching aim of the ongoing study is to establish a parametrized FE model that can be used to generate data for the training of an artificial neural network (ANN) that will be applied in the optimization process to develop the structural design

<figure><img src="image_997.png" /></figure>
# Parametric failure limit detection for the sheet metal forming of a floating photovoltaic (FPV) aluminum alloy structure

TVEIT Sigbjørn¹,ᵃ*, REYES Aase¹,ᵇ and ERDURAN Emrah¹,ᶜ

¹Department of Built Environment, Oslo Metropolitan University, Postboks 4, St. Olavs plass, NO-0130 Oslo, Norway

ᵃsigbjorn.tveit@oslomet.no, ᵇaase.reyes@oslomet.no, ᶜemrah.erduran@oslomet.no

**Keywords:** Sheet Metal Forming, Aluminum Alloys, Parametric Analysis

**Abstract.** The sheet metal forming process of a floating photovoltaic (FPV) structure is simulated in LS-DYNA. An anisotropic yield criterion and a two-term Voce hardening law are used to model the plastic behavior of AA5083-H111 sheets. The numerical model incorporates thickness variations to trigger local necking and uses a critical thickness strain as a fracture criterion. To establish a methodology that can be expanded for further studies, the research explores the relationship between cup depth and drawbead distance by proposing an algorithm to distinguish between successful and unsuccessful sheet metal forming operations.

**Introduction**

In the competitive race of developing new methods for renewable energy production, the floating solar producer Sunlit Sea AS has come up with an innovative solution where formed aluminum alloy sheet structures are used as supports for floating photovoltaic (FPV) panels. In each FPV float, two formed components are joined together back-to-back, and sealed off along the rim to create a buoyant platform. The choice of material represents a key attribute of the design: As the power output of FPV modules decreases at elevated operational temperatures, the highly conductive aluminum alloy body, which acts as a thermal bridge between the FPV module and the cool sea water, allows the units to operate with increased efficiency.

A critical feature in the design is the repeated cup shape, which acts as a spacer to provide structural integrity to the platform. This allows for walking on top of the solar panels, which facilitates effortless installation, operation, and maintenance. The cup depth directly influences the buoyancy of the platform, and, as the cups transfer heat and forces, their geometry and constellation have a massive influence on the FPV's operational performance. Hence, for the technology to reach its potential with respect to cost per unit energy output, the optimization of the structural geometry with respect to several parameters, including the heat transfer, and the material costs, is vital.

A challenge emerges in the radical deep-drawing of marine-grade aluminum, as aggressively pushing the drawing limit can lead to increased failure rates in the production line. Sunlit Sea's collaborative partners have painstakingly developed a shape that is feasible to manufacture by deep-drawing 1.5 mm thick AA5083-H111 sheets close to their forming limit. However, in developing the next generation of floats, the company seeks a more flexible and efficient design process, that enables rapid testing of multiple design variations in a virtual environment, by combining explicit non-linear finite element (FE) simulations in LS-DYNA [1], with machine learning techniques.

As an initial part of this work, the current paper presents the precursory investigations of the comprehensive study on the influence of various geometric parameters on the sheet metal forming process of the Sunlit Sea's FPV structure. The overarching aim of the ongoing study is to establish a parametrized FE model that can be used to generate data for the training of an artificial neural network (ANN) that will be applied in the optimization process to develop the structural design

<figure><img src="image_1002.png" /></figure>
for the FVP system. While the investigation presented in this paper is limited, its findings will be
used as a basis to develop the methodology for the further research.

In the realm of machine learning classification, decision boundaries are used to distinguish classes. In order to efficiently train the ANN to classify whether a sheet metal forming process will be successful or not for a given set of process parameters, it is crucial to assure the quality of the training data. Ideally, the database should be balanced, with roughly the same number of instances in the different classes, i.e. successful and unsuccessful forming operations [2]. Consequently, it is advantageous to get a sense of the mutual dependencies between important process parameters when generating the training data. Moreover, the predictive capability of the trained model relies on the accuracy of the training data, which calls for precise numerical models equipped with capable phenomenological models that accurately capture the material behavior.

As sheet metals are plastically deformed into complex shapes, a series of plastic instabilities may occur, as the material is subjected to large strains. The sequence of instability phenomena is normally initiated by diffuse necking, followed by localized necking and/or shear instability, until the coalescence of voids concludes in ductile failure [3]. While diffuse necking is often regarded as acceptable in industrial practices, the onset of a localized neck, or a shear localization, is commonly referred to as the material's forming limit.

Over the years, considerable research has been dedicated to the development of phenomenological models aiming to describe the occurrence of the various plastic instabilities that govern formability. Marciniak and Kuczynski (MK) [4] described how the presence of an imperfection, such as a small reduction in thickness, or an inhomogeneity in the material characteristics, causes the plastic strains to localize for sheet metals in biaxial tension. Since then, the concept has been further developed [5], and employed in conjunction with advanced anisotropic plasticity models, demonstrating the significance of the yield function and the plastic strain hardening law on the predicted forming limits. The MK approach has also been implemented in non-linear FE codes, where a non-local instability criterion (NLIC) is used to detect strain localization that are triggered by imperfections in the form of randomly distributed thickness variations [6-8]. A through-thickness shear instability criterion (TTSIC) was proposed by Bressan and Williams [9] for applications in isotropic sheet metals, a formulation that was later generalized by Hopperstad et al. [10] to comply with materials with orthotropic anisotropy. Reyes et al. [11] proved the robustness of NLIC and TTSIC when used in an FE model with random shell thickness variations, by demonstrating that the models could predict the decreased formability observed for certain aluminum alloy sheet metals subjected to biaxial pre-strain.

In this work, the deep-drawing of the indented cups in Sunlit Sea's FPV structure is simulated using a non-linear finite element model in LS-DYNA. A non-quadratic anisotropic yield criterion [12] with the associated flow rule, and a two-term Voce hardening law [13] are used to describe the plastic behavior of the sheet metal. Microscopic sheet thickness variations are implemented to trigger localized necking, and a simple fracture criterion based on a critical thickness strain is employed [14]. To establish a methodology that can be expanded upon in further studies, the model is used to explore the relationship between two geometric parameters: cup depth and drawbead distance. A simple algorithm is proposed to determine the curve that separates successful and unsuccessful sheet metal forming operations in the two-dimensional parameter space.

## Material models

The anisotropic Yld2003 [12], referred to in the LS-DYNA keyword as the strong texture model (STM), was employed as yield criterion, and as plastic potential by adopting the associated flow rule. The function is a generalization of the isotropic Hershey-Hosford yield function, which, for plane stress, can be written as
for the FVP system. While the investigation presented in this paper is limited, its findings will be used as a basis to develop the methodology for the further research.

In the realm of machine learning classification, decision boundaries are used to distinguish classes. In order to efficiently train the ANN to classify whether a sheet metal forming process will be successful or not for a given set of process parameters, it is crucial to assure the quality of the training data. Ideally, the database should be balanced, with roughly the same number of instances in the different classes, i.e. successful and unsuccessful forming operations [2]. Consequently, it is advantageous to get a sense of the mutual dependencies between important process parameters when generating the training data. Moreover, the predictive capability of the trained model relies on the accuracy of the training data, which calls for precise numerical models equipped with capable phenomenological models that accurately capture the material behavior.

As sheet metals are plastically deformed into complex shapes, a series of plastic instabilities may occur, as the material is subjected to large strains. The sequence of instability phenomena is normally initiated by diffuse necking, followed by localized necking and/or shear instability, until the coalescence of voids concludes in ductile failure [3]. While diffuse necking is often regarded as acceptable in industrial practices, the onset of a localized neck, or a shear localization, is commonly referred to as the material's forming limit.

Over the years, considerable research has been dedicated to the development of phenomenological models aiming to describe the occurrence of the various plastic instabilities that govern formability. Marciniak and Kuczynski (MK) [4] described how the presence of an imperfection, such as a small reduction in thickness, or an inhomogeneity in the material characteristics, causes the plastic strains to localize for sheet metals in biaxial tension. Since then, the concept has been further developed [5], and employed in conjunction with advanced anisotropic plasticity models, demonstrating the significance of the yield function and the plastic strain hardening law on the predicted forming limits. The MK approach has also been implemented in non-linear FE codes, where a non-local instability criterion (NLIC) is used to detect strain localization that are triggered by imperfections in the form of randomly distributed thickness variations [6-8]. A through-thickness shear instability criterion (TTSIC) was proposed by Bressan and Williams [9] for applications in isotropic sheet metals, a formulation that was later generalized by Hopperstad et al. [10] to comply with materials with orthotropic anisotropy. Reyes et al. [11] proved the robustness of NLIC and TTSIC when used in an FE model with random shell thickness variations, by demonstrating that the models could predict the decreased formability observed for certain aluminum alloy sheet metals subjected to biaxial pre-strain.

In this work, the deep-drawing of the indented cups in Sunlit Sea's FPV structure is simulated using a non-linear finite element model in LS-DYNA. A non-quadratic anisotropic yield criterion [12] with the associated flow rule, and a two-term Voce hardening law [13] are used to describe the plastic behavior of the sheet metal. Microscopic sheet thickness variations are implemented to trigger localized necking, and a simple fracture criterion based on a critical thickness strain is employed [14]. To establish a methodology that can be expanded upon in further studies, the model is used to explore the relationship between two geometric parameters: cup depth and drawbead distance. A simple algorithm is proposed to determine the curve that separates successful and unsuccessful sheet metal forming operations in the two-dimensional parameter space.

## Material models

The anisotropic Yld2003 [12], referred to in the LS-DYNA keyword as the strong texture model (STM), was employed as yield criterion, and as plastic potential by adopting the associated flow rule. The function is a generalization of the isotropic Hershey-Hosford yield function, which, for plane stress, can be written as
$$|\sigma_1|^m + |\sigma_2|^m + |\sigma_1 - \sigma_2|^m = 2\sigma_Y^m \quad (1)$$

where $\sigma_Y$ is the flow stress. $\sigma_1$ and $\sigma_2$ are the non-zero principal stress tensor components:

$$\left. \begin{array}{l} \sigma_1 \\ \sigma_2 \end{array} \right\} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2} \quad (2)$$

The Yld2003 yield function introduces anisotropy by dividing the function in Eq. (1) into two separate parts:

$$\phi = \phi' + \phi'' = 2\sigma_Y^m \quad (3)$$

where

$$\phi' = |\sigma'_1|^m + |\sigma'_2|^m \quad (4)$$

$$\phi'' = |\sigma''_1 - \sigma''_2|^m \quad (5)$$

The generalized principal stress terms $\sigma'_1$, $\sigma'_2$, $\sigma''_1$, and $\sigma''_2$, are constructed analogous to the principal stress formulae in Eq. (2) as

$$\left. \begin{array}{l} \sigma'_1 \\ \sigma'_2 \end{array} \right\} = \frac{a_8 \cdot \sigma_x + a_1 \cdot \sigma_y}{2} \pm \sqrt{\left(\frac{a_2 \cdot \sigma_x - a_3 \cdot \sigma_y}{2}\right)^2 + a_4^2 \cdot \tau_{xy}^2} \quad (6)$$

$$\left. \begin{array}{l} \sigma''_1 \\ \sigma''_2 \end{array} \right\} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{a_5 \cdot \sigma_x - a_6 \cdot \sigma_y}{2}\right)^2 + a_7^2 \cdot \tau_{xy}^2} \quad (7)$$

Here, $a_1$ to $a_8$ are the eight anisotropy coefficients that can be calibrated to experimental flow stress ratios and strain ratios obtained in different directions of uniaxial tensile stress, as well as in equibiaxial tensile stress, or a stress state that is equivalent with respect to the deviatoric stress.

For uniaxial tension at an angle $\theta$ to the sheet's rolling direction (RD), the strain ratio is defined as

$$R_{\theta} = \frac{d\epsilon_{w}}{d\epsilon_{t}} \quad (8)$$

where $d\epsilon_w$ and $d\epsilon_t$ are respectively the incremental strains in the width and thickness direction of the specimen. Moreover, the uniaxial flow stress ratio is defined as

$$r_{\theta} = \left. \frac{\sigma_{\theta}}{\sigma_{\theta=0}} \right|_{W_p} \quad (9)$$

where $\sigma_\theta$ is the directional flow stress at a certain level of plastic work $W_p$.

The disc compression test generates a deviatoric stress state that is equivalent to that of equibiaxial tensile stress ($\sigma_b = \sigma_1 = \sigma_2$). As the plastic process is independent of the hydrostatic pressure, this simple test is used to experimentally measure the in-plane plastic strains of the
$$|\sigma_1|^m + |\sigma_2|^m + |\sigma_1 - \sigma_2|^m = 2\sigma_Y^m \quad (1)$$

where $\sigma_Y$ is the flow stress. $\sigma_1$ and $\sigma_2$ are the non-zero principal stress tensor components:

$$\left. \begin{array}{l} \sigma_1 \\ \sigma_2 \end{array} \right\} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2} \quad (2)$$

The Yld2003 yield function introduces anisotropy by dividing the function in Eq. (1) into two separate parts:

$$\phi = \phi' + \phi'' = 2\sigma_Y^m \quad (3)$$

where

$$\phi' = |\sigma'_1|^m + |\sigma'_2|^m \quad (4)$$

$$\phi'' = |\sigma''_1 - \sigma''_2|^m \quad (5)$$

The generalized principal stress terms $\sigma'_1$, $\sigma'_2$, $\sigma''_1$, and $\sigma''_2$, are constructed analogous to the principal stress formulae in Eq. (2) as

$$\left. \begin{array}{l} \sigma'_1 \\ \sigma'_2 \end{array} \right\} = \frac{a_8 \cdot \sigma_x + a_1 \cdot \sigma_y}{2} \pm \sqrt{\left(\frac{a_2 \cdot \sigma_x - a_3 \cdot \sigma_y}{2}\right)^2 + a_4^2 \cdot \tau_{xy}^2} \quad (6)$$

$$\left. \begin{array}{l} \sigma''_1 \\ \sigma''_2 \end{array} \right\} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{a_5 \cdot \sigma_x - a_6 \cdot \sigma_y}{2}\right)^2 + a_7^2 \cdot \tau_{xy}^2} \quad (7)$$

Here, $a_1$ to $a_8$ are the eight anisotropy coefficients that can be calibrated to experimental flow stress ratios and strain ratios obtained in different directions of uniaxial tensile stress, as well as in equibiaxial tensile stress, or a stress state that is equivalent with respect to the deviatoric stress.

For uniaxial tension at an angle $\theta$ to the sheet's rolling direction (RD), the strain ratio is defined as

$$R_{\theta} = \frac{d\epsilon_{w}}{d\epsilon_{t}} \quad (8)$$

where $d\epsilon_w$ and $d\epsilon_t$ are respectively the incremental strains in the width and thickness direction of the specimen. Moreover, the uniaxial flow stress ratio is defined as

$$r_{\theta} = \left. \frac{\sigma_{\theta}}{\sigma_{\theta=0}} \right|_{W_p} \quad (9)$$

where $\sigma_\theta$ is the directional flow stress at a certain level of plastic work $W_p$.

The disc compression test generates a deviatoric stress state that is equivalent to that of equibiaxial tensile stress ($\sigma_b = \sigma_1 = \sigma_2$). As the plastic process is independent of the hydrostatic pressure, this simple test is used to experimentally measure the in-plane plastic strains of the
equibiaxial stress state [15]. By measuring the in-plane strains at different levels of compression, the incremental strains in TD and RD are obtained. The strain ratio in equibiaxial tensile stress is then taken as

$$R_b = \frac{d\epsilon_{TD}}{d\epsilon_{RD}} \qquad (10)$$

where $d\epsilon_{TD}$ and $d\epsilon_{RD}$ are the incremental strains in the transverse and the rolling directions, respectively. Moreover, the equibiaxial stress ratio is typically obtained by comparing the yield stress measured in bulge tests, to that measured in a uniaxial tensile test in RD ($\theta = 0$):

$$r_b = \frac{\sigma_b}{\sigma_0} \qquad (11)$$

Plastic strain hardening is modeled using the two-term Voce hardening rule [13], which reads

$$\sigma_Y(\epsilon_{pl}) = \sigma_0 + \sum_{i=1}^{2} Q_{Ri} [1 - \exp(-C_{Ri} \epsilon_{pl})] \qquad (12)$$

Here, $\epsilon_{pl}$ is the effective plastic strain, $\sigma_0$ is the stress at the onset of yielding, and $Q_{Ri}$ and $C_{Ri}$ are the Voce coefficients.

To trigger necking instability, which is quickly followed by failure, the material inhomogeneities were modeled as a field of random thickness variation, generated by spectral decomposition using Karhunen-Loève expansions [1, 16]. For this purpose, an isotropic Gaussian correlation function B was adopted:

$$B(t) = e^{-(at)^2} \qquad (13)$$

where a is the correlation parameter. In LS-DYNA, the user has the option to use a random seed, which generates a new Gaussian random thickness field for every run, or to provide a fixed seed number, to facilitate a deterministic analysis. A scale factor $f_{SCL}$ is employed to control the magnitude of the thickness perturbations.

A fracture criterion based on the local critical thickness strain was adopted [14]:

$$\epsilon_t \le \epsilon_{t,cr} \qquad (14)$$

where $\epsilon_{t,cr}$ is the critical thickness strain as fracture.

## Numerical model

Sunlit Sea's current design features 34 cups with center-to-center distance of 260 mm (see Fig. 1 (a)). The cups have a radius $r_{cup} = 80$ mm, and a depth $d_{cup} = 38.5$ mm. To avoid wrinkling in the drawing process, a circular drawbead is located at a distance $d_{db} = 5$ mm from the edge of the cup. A schematic diagram of the drawing operation of the cup is provided in Fig. 1 (b).

An FE model was established to simulate the sheet metal forming of the float component's cups in LS-DYNA R12.0. For simplicity, it was conservatively assumed that the drawbead, which circumferences the cup geometry, completely restricts any material flow. This assumption, which is conservative from a formability perspective, allows the sheet metal inside the drawbead to be modeled as a circular plate with a fixed constraint along its rim.
equibiaxial stress state [15]. By measuring the in-plane strains at different levels of compression, the incremental strains in TD and RD are obtained. The strain ratio in equibiaxial tensile stress is then taken as

$$R_b = \frac{d\epsilon_{TD}}{d\epsilon_{RD}} \qquad (10)$$

where $d\epsilon_{TD}$ and $d\epsilon_{RD}$ are the incremental strains in the transverse and the rolling directions, respectively. Moreover, the equibiaxial stress ratio is typically obtained by comparing the yield stress measured in bulge tests, to that measured in a uniaxial tensile test in RD ($\theta = 0$):

$$r_b = \frac{\sigma_b}{\sigma_0} \qquad (11)$$

Plastic strain hardening is modeled using the two-term Voce hardening rule [13], which reads

$$\sigma_Y(\epsilon_{pl}) = \sigma_0 + \sum_{i=1}^{2} Q_{Ri} [1 - \exp(-C_{Ri}\epsilon_{pl})] \qquad (12)$$

Here, $\epsilon_{pl}$ is the effective plastic strain, $\sigma_0$ is the stress at the onset of yielding, and $Q_{Ri}$ and $C_{Ri}$ are the Voce coefficients.

To trigger necking instability, which is quickly followed by failure, the material inhomogeneities were modeled as a field of random thickness variation, generated by spectral decomposition using Karhunen-Loève expansions [1, 16]. For this purpose, an isotropic Gaussian correlation function $B$ was adopted:

$$B(t) = e^{-(at)^2} \qquad (13)$$

where $a$ is the correlation parameter. In LS-DYNA, the user has the option to use a random seed, which generates a new Gaussian random thickness field for every run, or to provide a fixed seed number, to facilitate a deterministic analysis. A scale factor $f_{SCL}$ is employed to control the magnitude of the thickness perturbations.

A fracture criterion based on the local critical thickness strain was adopted [14]:

$$\epsilon_t \le \epsilon_{t,cr} \qquad (14)$$

where $\epsilon_{t,cr}$ is the critical thickness strain as fracture.

## Numerical model

Sunlit Sea's current design features 34 cups with center-to-center distance of 260 mm (see Fig. 1 (a)). The cups have a radius $r_{cup} = 80$ mm, and a depth $d_{cup} = 38.5$ mm. To avoid wrinkling in the drawing process, a circular drawbead is located at a distance $d_{db} = 5$ mm from the edge of the cup. A schematic diagram of the drawing operation of the cup is provided in Fig. 1 (b).

An FE model was established to simulate the sheet metal forming of the float component's cups in LS-DYNA R12.0. For simplicity, it was conservatively assumed that the drawbead, which circumferences the cup geometry, completely restricts any material flow. This assumption, which is conservative from a formability perspective, allows the sheet metal inside the drawbead to be modeled as a circular plate with a fixed constraint along its rim.
To allow the investigation of failure modes without restricting possible non-symmetric outcomes, the full circular symmetric cup system was modelled. The die and punch were modeled with rigid shell elements, based on the CAD geometry of the original forming press tool. The motion of the punch was prescribed according to a quadratic polynomial function, and the contact between the forming tools and the blank was modeled using a one-way surface-to surface rule, with a constant friction coefficient $\mu$. Selective mass scaling was employed to reduce computational costs by accelerating the explicit time-integration scheme without introducing significant dynamic effects upon impact between the forming tools and the blank.

The deformable blank was modeled using reduced integration Belytschko-Tsay elements with five through-thickness integration points, and approximate lengths of 1.2 mm. The part was equipped with the material models described in the previous section, through the keyword *MAT_WTM_STM. The material parameters used in the input are reported in Table 1.

The parameters of the two-term Voce plastic strain hardening law were calibrated to stress-strain data from five parallel uniaxial tensile tests in RD [17], by minimizing an error function based on the sum of squared residuals.

The anisotropy coefficients of Yld2003 were calibrated in LS-DYNA to experimental data from [17]. Data from uniaxial tensile tests at $\theta = 0°, 45°$, and $90°$ to RD provided the stress ratios $r_\theta$, and the strain ratios $R_\theta$. Moreover, the equibiaxial strain ratio $R_b$ was obtained from the disc compression test data. Due to lack of experimental bulge test data, the equibiaxial stress ratio was assumed to be isotropic, $r_\theta = 1$. As is common practice for face-centered cubic (FCC) materials, the yield function exponent was taken as $m = 8$. In Fig. 2, the yield function's consistency with measurements from uniaxial tensile tests at seven different angles $\theta$ is demonstrated.

Through the *PERTURBATION keyword, the random distribution was modelled, creating microscopic deviations in the shell thickness among the elements in the blank. Due to the absence of specific data on the surface topology of the sheet metal used in the current study, the correlation parameter $a$ was taken as unity, based on a qualitative assessment of the resulting thickness fields, while the perturbation scale factor was taken as $f_{SCL} = 0.005$ mm.

Experimental tests performed by Sunlit Sea's collaborative partners in the development of the design reportedly revealed that a cup depth of 40 mm resulted in failure, while a cup depth of 38.5 mm remained intact during forming, without showing any signs of neck formation. These reported results were used to estimate the constant friction coefficient $\mu$, and the critical thickness strain at fracture $\epsilon_{t,cr}$, in the absence of detailed experimental evidence. To do this, the forming of the original geometry ($d_{cup} = 38.5$ mm), and a model with $d_{cup} = 40$ mm, were simulated using different friction coefficients, and the original drawbead distance $d_{db} = 5$ mm. To alter the cup

<figure><img src="image_998.png" /><figcaption>Fig. 1 (a) FPV structural component with repeated cup geometry, approximately 2 by 2 m in size; (b) sheet metal forming scheme of the indented cups of the FPV structure.</figcaption></figure>
To allow the investigation of failure modes without restricting possible non-symmetric outcomes, the full circular symmetric cup system was modelled. The die and punch were modeled with rigid shell elements, based on the CAD geometry of the original forming press tool. The motion of the punch was prescribed according to a quadratic polynomial function, and the contact between the forming tools and the blank was modeled using a one-way surface-to surface rule, with a constant friction coefficient $\mu$. Selective mass scaling was employed to reduce computational costs by accelerating the explicit time-integration scheme without introducing significant dynamic effects upon impact between the forming tools and the blank.

The deformable blank was modeled using reduced integration Belytschko-Tsay elements with five through-thickness integration points, and approximate lengths of 1.2 mm. The part was equipped with the material models described in the previous section, through the keyword *MAT_WTM_STM. The material parameters used in the input are reported in Table 1.

The parameters of the two-term Voce plastic strain hardening law were calibrated to stress-strain data from five parallel uniaxial tensile tests in RD [17], by minimizing an error function based on the sum of squared residuals.

The anisotropy coefficients of Yld2003 were calibrated in LS-DYNA to experimental data from [17]. Data from uniaxial tensile tests at $\theta = 0°, 45°$, and $90°$ to RD provided the stress ratios $r_\theta$, and the strain ratios $R_\theta$. Moreover, the equibiaxial strain ratio $R_b$ was obtained from the disc compression test data. Due to lack of experimental bulge test data, the equibiaxial stress ratio was assumed to be isotropic, $r_\theta = 1$. As is common practice for face-centered cubic (FCC) materials, the yield function exponent was taken as $m = 8$. In Fig. 2, the yield function's consistency with measurements from uniaxial tensile tests at seven different angles $\theta$ is demonstrated.

Through the *PERTURBATION keyword, the random distribution was modelled, creating microscopic deviations in the shell thickness among the elements in the blank. Due to the absence of specific data on the surface topology of the sheet metal used in the current study, the correlation parameter $a$ was taken as unity, based on a qualitative assessment of the resulting thickness fields, while the perturbation scale factor was taken as $f_{SCL} = 0.005$ mm.

Experimental tests performed by Sunlit Sea's collaborative partners in the development of the design reportedly revealed that a cup depth of 40 mm resulted in failure, while a cup depth of 38.5 mm remained intact during forming, without showing any signs of neck formation. These reported results were used to estimate the constant friction coefficient $\mu$, and the critical thickness strain at fracture $\epsilon_{t,cr}$, in the absence of detailed experimental evidence. To do this, the forming of the original geometry ($d_{cup} = 38.5$ mm), and a model with $d_{cup} = 40$ mm, were simulated using different friction coefficients, and the original drawbead distance $d_{db} = 5$ mm. To alter the cup

<figure><img src="image_1003.png" /><figcaption>Fig. 1 (a) FPV structural component with repeated cup geometry, approximately 2 by 2 m in size; (b) sheet metal forming scheme of the indented cups of the FPV structure.</figcaption></figure>
depth while keeping the initial tool distances of the native model, the vertical nodal coordinates of the drawing tools were scaled and shifted. Distributions of strains in the space of $ε_1$ and $ε_2$ were manually evaluated in order to determine a friction coefficient that did not predict necking at $d_{cup} = 38.5$ mm, while predicting significant necking at $d_{cup} = 40$ mm. Fig. 3 (a) and (b) respectively display the element strains in the entire cup for $d_{cup} = 38.5$ and 40.0 mm, with $μ = 0.085$. It is apparent from the plot for $d_{cup} = 38.5$ mm that the strains in different elements are densely grouped in the strain space, while the plot for $d_{cup} = 40$ mm shows derivative strain levels in selected elements, indicating local neck formation. To comply with the reported observations, a critical thickness strain $ε_{t,cr} = -0.45$ was adopted as a fracture limit. The sensitivity of $ε_{t,cr}$ was analyzed, and it was concluded that the magnitude of $ε_{t,cr}$ has insignificant impact on the predicted forming limits as long as plastic instability occurs before the thickness strain exceeds $ε_{t,cr}$.

## Detection of parametric failure limits

To establish a methodology that can later be expanded to a multidimensional parameter space, an algorithmic approach was developed to locate the curve that separates safe from unsafe configurations of process parameters. In the current study, the relationship between the drawbead distance $d_{ab}$, and the cup depth $d_{cup}$, was investigated (cf. Fig. 1 (b)). MATLAB scripts were developed to parametrize the model, by manipulating the LS-DYNA input files. As described in the previous section, the cup depth was controlled by scaling and shifting the nodal coordinates of the forming tool parts. To vary the drawbead distance, the radius of the circular blank was scaled, while the element size (approximately 1.2 mm) was kept as consistent as possible.

<figure><img src="image_999.png" /><figcaption>Fig. 2 (a) Stress- and (b) strain ratios vs. load angle for the fitted Yld2003 yield criterion and uniaxial test data from [17]. The solid markers represent the input for the calibration in LS-DYNA.</figcaption></figure>
depth while keeping the initial tool distances of the native model, the vertical nodal coordinates of the drawing tools were scaled and shifted. Distributions of strains in the space of $ε_1$ and $ε_2$ were manually evaluated in order to determine a friction coefficient that did not predict necking at $d_{cup}$ = 38.5 mm, while predicting significant necking at $d_{cup}$ = 40 mm. Fig. 3 (a) and (b) respectively display the element strains in the entire cup for $d_{cup}$ = 38.5 and 40.0 mm, with $μ$ = 0.085. It is apparent from the plot for $d_{cup}$ = 38.5 mm that the strains in different elements are densely grouped in the strain space, while the plot for $d_{cup}$ = 40 mm shows derivative strain levels in selected elements, indicating local neck formation. To comply with the reported observations, a critical thickness strain $ε_{t,cr}$ = -0.45 was adopted as a fracture limit. The sensitivity of $ε_{t,cr}$ was analyzed, and it was concluded that the magnitude of $ε_{t,cr}$ has insignificant impact on the predicted forming limits as long as plastic instability occurs before the thickness strain exceeds $ε_{t,cr}$.

## Detection of parametric failure limits

To establish a methodology that can later be expanded to a multidimensional parameter space, an algorithmic approach was developed to locate the curve that separates safe from unsafe configurations of process parameters. In the current study, the relationship between the drawbead distance $d_{ab}$, and the cup depth $d_{cup}$, was investigated (cf. Fig. 1 (b)). MATLAB scripts were developed to parametrize the model, by manipulating the LS-DYNA input files. As described in the previous section, the cup depth was controlled by scaling and shifting the nodal coordinates of the forming tool parts. To vary the drawbead distance, the radius of the circular blank was scaled, while the element size (approximately 1.2 mm) was kept as consistent as possible.

<figure><img src="image_1004.png" /><figcaption>Fig. 2 (a) Stress- and (b) strain ratios vs. load angle for the fitted Yld2003 yield criterion and uniaxial test data from [17]. The solid markers represent the input for the calibration in LS-DYNA.</figcaption></figure>
<figure><img src="image_1000.png" /><figcaption>Fig. 3 Major and minor true strains from LS-DYNA used to estimate the friction parameter μ, and the critical thickness strain εt,cr: (a) Original design (d<sub>cup</sub> = 38.5 mm), and (b) with increased cup depth (d<sub>cup</sub> = 40 mm).</figcaption></figure>

<table><caption>Table 1 Model parameters</caption><thead><tr><th colspan="10">Elastic/mechanical properties</th></tr><tr><th>Density, ρ [g/cm³]</th><th colspan="4">Young's modulus, E [N/mm²]</th><th colspan="4">Poisson ratio, ν [-]</th></tr></thead><tbody><tr><td>2.650</td><td colspan="4">72000</td><td colspan="4">0.33</td></tr></tbody></table><table><thead><tr><th colspan="10">Yld2003 input [17]</th></tr><tr><th>r<sub>0</sub> [-]</th><th>r<sub>45</sub> [-]</th><th>r<sub>90</sub> [-]</th><th>r<sub>b</sub> [-]</th><th>R<sub>0</sub> [-]</th><th>R<sub>45</sub> [-]</th><th>R<sub>90</sub> [-]</th><th>R<sub>b</sub> [-]</th><th>m [-]</th></tr></thead><tbody><tr><td>1.000</td><td>0.9752</td><td>0.9908</td><td>1.000</td><td>0.7142</td><td>0.8357</td><td>0.6410</td><td>1.1263</td><td>8</td></tr></tbody></table><table><thead><tr><th colspan="10">Voce parameters</th></tr><tr><th>σ<sub>0</sub> [MPa]</th><th>Q<sub>R1</sub> [MPa]</th><th>C<sub>R1</sub> [-]</th><th>Q<sub>R2</sub> [MPa]</th><th>C<sub>R2</sub> [-]</th></tr></thead><tbody><tr><td>153.7</td><td>94.33</td><td>2.352</td><td>187.1</td><td>15.35</td></tr></tbody></table><table><thead><tr><th colspan="10">Perturbation parameters</th></tr><tr><th colspan="2">Friction</th><th colspan="2">Fracture</th></tr><tr><th>a [-]</th><th>f<sub>SCL</sub> [mm]</th><th>μ [-]</th><th>ε<sub>t,cr</sub> [-]</th></tr></thead><tbody><tr><td>1.0</td><td>0.005</td><td>0.085</td><td>-0.45</td></tr></tbody></table>

To search for the failure front in the two-dimensional parameter space, the following algorithm was developed. The drawbead distance $d_{db}$ was taken as an independent variable, and a step size $\Delta d_{db} = 5$ mm was used to cover the parameter interval from $d_{db,min} = 5$ mm, the original geometry, to $d_{db,max} = 45$ mm, the maximum drawbead distance that can fit between the cups in the original constellation.

Since the model was calibrated to experience failure at a cup depth of 40 mm for a drawbead distance of 5 mm, the initial value $d_{cup,0}$ was taken as 39.5 mm at this drawbead distance, which predicts a successful forming operation, where the fracture criterion in Eq. (14) is not satisfied. With a step size of $\Delta d_{cup} = 0.5$ mm, the cup size was increased, and a new simulation was run with the updated variables. Upon the occurrence of a failed forming operation, $d_{cup}$ was decreased
<figure><img src="image_1005.png" /><figcaption>Fig. 3 Major and minor true strains from LS-DYNA used to estimate the friction parameter μ, and the critical thickness strain εt,cr: (a) Original design (d<sub>cup</sub> = 38.5 mm), and (b) with increased cup depth (d<sub>cup</sub> = 40 mm).</figcaption></figure>

<table><caption>Table 1 Model parameters</caption><thead><tr><th colspan="10">Elastic/mechanical properties</th></tr><tr><th>Density, ρ [g/cm<sup>3</sup>]</th><th colspan="3">Young's modulus, E [N/mm<sup>2</sup>]</th><th colspan="4">Poisson ratio, ν [-]</th></tr></thead><tbody><tr><td>2.650</td><td colspan="3">72000</td><td colspan="4">0.33</td></tr></tbody></table><table><thead><tr><th colspan="10">Yld2003 input [17]</th></tr><tr><th>r<sub>0</sub> [-]</th><th>r<sub>45</sub> [-]</th><th>r<sub>90</sub> [-]</th><th>r<sub>b</sub> [-]</th><th>R<sub>0</sub> [-]</th><th>R<sub>45</sub> [-]</th><th>R<sub>90</sub> [-]</th><th>R<sub>b</sub> [-]</th><th>m [-]</th></tr></thead><tbody><tr><td>1.000</td><td>0.9752</td><td>0.9908</td><td>1.000</td><td>0.7142</td><td>0.8357</td><td>0.6410</td><td>1.1263</td><td>8</td></tr></tbody></table><table><thead><tr><th colspan="10">Voce parameters</th></tr><tr><th>σ<sub>0</sub> [MPa]</th><th>Q<sub>R1</sub> [MPa]</th><th>C<sub>R1</sub> [-]</th><th>Q<sub>R2</sub> [MPa]</th><th>C<sub>R2</sub> [-]</th></tr></thead><tbody><tr><td>153.7</td><td>94.33</td><td>2.352</td><td>187.1</td><td>15.35</td></tr></tbody></table><table><thead><tr><th colspan="2">Perturbation parameters</th><th>Friction</th><th>Fracture</th></tr><tr><th>a [-]</th><th>f<sub>SCL</sub> [mm]</th><th>μ [-]</th><th>ε<sub>t,cr</sub> [-]</th></tr></thead><tbody><tr><td>1.0</td><td>0.005</td><td>0.085</td><td>-0.45</td></tr></tbody></table>

To search for the failure front in the two-dimensional parameter space, the following algorithm was developed. The drawbead distance $d_{db}$ was taken as an independent variable, and a step size $\Delta d_{db} = 5$ mm was used to cover the parameter interval from $d_{db,min} = 5$ mm, the original geometry, to $d_{db,max} = 45$ mm, the maximum drawbead distance that can fit between the cups in the original constellation.

Since the model was calibrated to experience failure at a cup depth of 40 mm for a drawbead distance of 5 mm, the initial value $d_{cup,0}$ was taken as 39.5 mm at this drawbead distance, which predicts a successful forming operation, where the fracture criterion in Eq. (14) is not satisfied. With a step size of $\Delta d_{cup} = 0.5$ mm, the cup size was increased, and a new simulation was run with the updated variables. Upon the occurrence of a failed forming operation, $d_{cup}$ was decreased
by half a step size in the next run, refining the prediction of the failure limit to a precision of $0.5 \cdot \Delta d_{cup}$. The algorithm then proceeded by increasing the drawbead distance by $\Delta d_{db}$, adopting the largest $d_{cup}$ value that resulted in a successful forming operation, repeating the procedure, until $d_{db}$ exceeds $d_{ab,max}$. A flowchart of the search algorithm is provided in Fig. 4.

The results from the sensitivity analysis are displayed in Fig. 5. In Fig. 5 (a), the path taken by the algorithm is indicated with the dotted line connecting the datapoints, which represent the 32 different simulations that were run to complete the algorithmic search. The squares and crosses, respectively indicating the successful and unsuccessful forming processes, are separated by a solid line indicating the predicted parametric failure limit. It is evident from the plot that the FE model predicts a proportional relationship between the two parameters on the interval from $d_{db} = 5$ mm to 25 mm, after which a plateau is reached. When reviewing the presented results, it should be noted that the solid line in Fig. 5 (a) represents the predicted fracture, using the simple critical thickness strain criterion in Eq. (14), while the forming limit of sheet metals is typically taken as the onset of a localized neck, or a shear localization.

The predicted fracture patterns are presented in Fig. 5 (b). All instances display fracture perpendicular to the transverse direction, approximately 50 mm away from the cup center. This is according to expectations, as the minimum strain ratio $R_{\theta} = d\epsilon_{w}/d\epsilon_{t}$ was observed at $\theta = 90^{\circ}$ (cf. Fig 2 (b)), indicating that the material is mostly prone to thinning when subjected to strains in this direction.

<figure><img src="image_1001.png" /><figcaption>Fig. 4 Flowchart of algorithm for detection of the relationship between drawbead distance, and fracture cup depth.</figcaption></figure>
by half a step size in the next run, refining the prediction of the failure limit to a precision of $0.5 \cdot \Delta d_{cup}$. The algorithm then proceeded by increasing the drawbead distance by $\Delta d_{db}$, adopting the largest $d_{cup}$ value that resulted in a successful forming operation, repeating the procedure, until $d_{db}$ exceeds $d_{ab,max}$. A flowchart of the search algorithm is provided in Fig. 4.

The results from the sensitivity analysis are displayed in Fig. 5. In Fig. 5 (a), the path taken by the algorithm is indicated with the dotted line connecting the datapoints, which represent the 32 different simulations that were run to complete the algorithmic search. The squares and crosses, respectively indicating the successful and unsuccessful forming processes, are separated by a solid line indicating the predicted parametric failure limit. It is evident from the plot that the FE model predicts a proportional relationship between the two parameters on the interval from $d_{db} = 5$ mm to 25 mm, after which a plateau is reached. When reviewing the presented results, it should be noted that the solid line in Fig. 5 (a) represents the predicted fracture, using the simple critical thickness strain criterion in Eq. (14), while the forming limit of sheet metals is typically taken as the onset of a localized neck, or a shear localization.

The predicted fracture patterns are presented in Fig. 5 (b). All instances display fracture perpendicular to the transverse direction, approximately 50 mm away from the cup center. This is according to expectations, as the minimum strain ratio $R_{\theta} = d\epsilon_{w}/d\epsilon_{t}$ was observed at $\theta = 90^{\circ}$ (cf. Fig 2 (b)), indicating that the material is mostly prone to thinning when subjected to strains in this direction.

<figure><img src="image_1006.png" /><figcaption>Fig. 4 Flowchart of algorithm for detection of the relationship between drawbead distance, and fracture cup depth.</figcaption></figure>
<figure><img src="image_1002.png" /><figcaption>Fig. 5 (a) Fracture cup depth vs. drawbead distance. The error bars indicate the cup depth interval between the simulation that predicted failure, and the simulation that predicted an intact sheet. (b) Fracture patterns of the failed simulations.</figcaption></figure>

## Concluding remarks

In this preliminary study, the sheet metal forming process of a floating photovoltaic structure with a repeated cup feature was investigated through non-linear finite element analysis in LS-DYNA. The anisotropic Yld2003 yield criterion, and the isotropic two-term Voce hardening rule was calibrated to plasticity data from previous experimental tests of the 1.51 mm thick AA5083-H111 aluminum alloy sheets [17]. First, the cup geometry of the original design was assessed by varying the constant friction parameter, which proved to have a significant effect on the principal strain distribution and the critical cup depth. The results from these simulations were used along with the reported observations from preliminary experimental tests to estimate a constant friction parameter, and a critical thickness strain at fracture. Finally, the calibrated model was used in a parametric study, where a simple search algorithm was developed for the purpose of locating the failure limit in a two-dimensional parameter space.

The authors will continue with the work of developing a high-quality dataset for the training of a prediction model based on an artificial neural network. To this aim, the authors will focus on further developing the numerical simulation model, e.g. by implementing state-of-the-art criteria for the prediction of local necking, shear fracture, and ductile fracture, which governs the parametric forming limit. Since the case investigated in this study displayed a significant dependency on the constant friction parameter, a refined formulation that accounts for the static and dynamic contact friction would also be beneficial to ensure applicability to other geometric configurations. This calls for additional experimental tests to calibrate new model parameters, and to validate the FE model for various constellations of process parameters. Moreover, efforts will be dedicated to generalizing the simple search algorithm for extending its application to parameter spaces of higher dimensionality, starting with a review of existing approaches from the literature.

## References

[1] LIVERMORE SOFTWARE TECHNOLOGY (LSTC), LS-DYNA® Theory Manual 2019.
<figure><img src="image_1007.png" /><figcaption>Fig. 5 (a) Fracture cup depth vs. drawbead distance. The error bars indicate the cup depth interval between the simulation that predicted failure, and the simulation that predicted an intact sheet. (b) Fracture patterns of the failed simulations.</figcaption></figure>

## Concluding remarks

In this preliminary study, the sheet metal forming process of a floating photovoltaic structure with a repeated cup feature was investigated through non-linear finite element analysis in LS-DYNA. The anisotropic Yld2003 yield criterion, and the isotropic two-term Voce hardening rule was calibrated to plasticity data from previous experimental tests of the 1.51 mm thick AA5083-H111 aluminum alloy sheets [17]. First, the cup geometry of the original design was assessed by varying the constant friction parameter, which proved to have a significant effect on the principal strain distribution and the critical cup depth. The results from these simulations were used along with the reported observations from preliminary experimental tests to estimate a constant friction parameter, and a critical thickness strain at fracture. Finally, the calibrated model was used in a parametric study, where a simple search algorithm was developed for the purpose of locating the failure limit in a two-dimensional parameter space.

The authors will continue with the work of developing a high-quality dataset for the training of a prediction model based on an artificial neural network. To this aim, the authors will focus on further developing the numerical simulation model, e.g. by implementing state-of-the-art criteria for the prediction of local necking, shear fracture, and ductile fracture, which governs the parametric forming limit. Since the case investigated in this study displayed a significant dependency on the constant friction parameter, a refined formulation that accounts for the static and dynamic contact friction would also be beneficial to ensure applicability to other geometric configurations. This calls for additional experimental tests to calibrate new model parameters, and to validate the FE model for various constellations of process parameters. Moreover, efforts will be dedicated to generalizing the simple search algorithm for extending its application to parameter spaces of higher dimensionality, starting with a review of existing approaches from the literature.

## References

[1] LIVERMORE SOFTWARE TECHNOLOGY (LSTC), LS-DYNA® Theory Manual 2019.
[2] A. Dal Pozzolo, O. Caelen, R.A. Johnson, G. Bontempi, Calibrating probability with undersampling for unbalanced classification, in: 2015 IEEE symposium series on computational intelligence, IEEE, 2015, pp. 159-166. https://doi.org/10.1109/SSCI.2015.33

[3] D. Banabic, D.-S. Comsa, P. Eyckens, A. Kami, M. Gologanu, Advanced models for the prediction of forming limit curves, in: Multiscale Modelling in Sheet Metal Forming, 2016, pp. 205-300. https://doi.org/10.1007/978-3-319-44070-5_5

[4] Z. Marciniak, K. Kuczyński, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci., 9 (1967) 609-620. https://doi.org/10.1016/0020-7403(67)90066-5

[5] D. Banabic, A review on recent developments of Marciniak-Kuczynski model, Comput. Methods in Mater. Sci., 10 (2010) 225-237

[6] O. Lademo, T. Berstad, O. Hopperstad, K. Pedersen, A numerical tool for formability analysis of aluminium alloys. Part I: Theory, Steel Grips, 2 (2004) 427-431

[7] O. Lademo, T. Berstad, O. Hopperstad, K. Pedersen, A numerical tool for formability analysis of aluminium alloys. Part II: Experimental validation, Steel Grips, 2 (2004) 433-437.

[8] Ø. Fyllingen, O. Hopperstad, O.-G. Lademo, M. Langseth, Estimation of forming limit diagrams by the use of the finite element method and Monte Carlo simulation, Comput. Struct., 87 (2009) 128-139. https://doi.org/10.1016/j.compstruc.2008.07.002

[9] J.D. Bressan, J.A. Williams, The use of a shear instability criterion to predict local necking in sheet metal deformation, Int. J. Mech. Sci., 25 (1983) 155-168. https://doi.org/10.1016/0020-7403(83)90089-9

[10] O. Hopperstad, T. Berstad, O. Lademo, M. Langseth, Shear instability criterion for plastic anisotropy, SINTEF, Trondheim, 2006.

[11] A. Reyes, O.S. Hopperstad, T. Berstad, O.-G. Lademo, Prediction of necking for two aluminum alloys under non-proportional loading by using an FE-based approach, Int. J. Mater. Form., 1 (2008) 211-232. https://doi.org/10.1007/s12289-008-0384-6

[12] H. Aretz, A non-quadratic plane stress yield function for orthotropic sheet metals, J. Mater. Process. Technol., 168 (2005) 1-9. https://doi.org/10.1016/j.jmatprotec.2004.10.008

[13] E. Voce, The relationship between stress and strain for homogeneous deformation, J. Inst. Met., 74 (1948) 537-562

[14] Z. Marciniak, K. Kuczyński, T. Pokora, Influence of the plastic properties of a material on the forming limit diagram for sheet metal in tension, Int. J. Mech. Sci., 15 (1973) 789-800. https://doi.org/10.1016/0020-7403(73)90068-4

[15] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourgoghrat, S.H. Choi, E. Chu, Plane stress yield function for aluminum alloy sheets - part 1: theory, Int. J. Plast., 19 (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0

[16] K.K. Phoon, S.P. Huang, S.T. Quek, Implementation of Karhunen–Loeve expansion for simulation using a wavelet-Galerkin scheme, Probabilistic Eng. Mech., 17 (2002) 293-303. https://doi.org/10.1016/S0266-8920(02)00013-9

[17] S. Tveit, A. Reyes, A submodeling technique for incorporating sheet metal forming effects in an AA5083 FPV structure, submitted for publication, (2024).
[2] A. Dal Pozzolo, O. Caelen, R.A. Johnson, G. Bontempi, Calibrating probability with undersampling for unbalanced classification, in: 2015 IEEE symposium series on computational intelligence, IEEE, 2015, pp. 159-166. https://doi.org/10.1109/SSCI.2015.33

[3] D. Banabic, D.-S. Comsa, P. Eyckens, A. Kami, M. Gologanu, Advanced models for the prediction of forming limit curves, in: Multiscale Modelling in Sheet Metal Forming, 2016, pp. 205-300. https://doi.org/10.1007/978-3-319-44070-5_5

[4] Z. Marciniak, K. Kuczyński, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci., 9 (1967) 609-620. https://doi.org/10.1016/0020-7403(67)90066-5

[5] D. Banabic, A review on recent developments of Marciniak-Kuczynski model, Comput. Methods in Mater. Sci., 10 (2010) 225-237

[6] O. Lademo, T. Berstad, O. Hopperstad, K. Pedersen, A numerical tool for formability analysis of aluminium alloys. Part I: Theory, Steel Grips, 2 (2004) 427-431

[7] O. Lademo, T. Berstad, O. Hopperstad, K. Pedersen, A numerical tool for formability analysis of aluminium alloys. Part II: Experimental validation, Steel Grips, 2 (2004) 433-437.

[8] Ø. Fyllingen, O. Hopperstad, O.-G. Lademo, M. Langseth, Estimation of forming limit diagrams by the use of the finite element method and Monte Carlo simulation, Comput. Struct., 87 (2009) 128-139. https://doi.org/10.1016/j.compstruc.2008.07.002

[9] J.D. Bressan, J.A. Williams, The use of a shear instability criterion to predict local necking in sheet metal deformation, Int. J. Mech. Sci., 25 (1983) 155-168. https://doi.org/10.1016/0020-7403(83)90089-9

[10] O. Hopperstad, T. Berstad, O. Lademo, M. Langseth, Shear instability criterion for plastic anisotropy, SINTEF, Trondheim, 2006.

[11] A. Reyes, O.S. Hopperstad, T. Berstad, O.-G. Lademo, Prediction of necking for two aluminum alloys under non-proportional loading by using an FE-based approach, Int. J. Mater. Form., 1 (2008) 211-232. https://doi.org/10.1007/s12289-008-0384-6

[12] H. Aretz, A non-quadratic plane stress yield function for orthotropic sheet metals, J. Mater. Process. Technol., 168 (2005) 1-9. https://doi.org/10.1016/j.jmatprotec.2004.10.008

[13] E. Voce, The relationship between stress and strain for homogeneous deformation, J. Inst. Met., 74 (1948) 537-562

[14] Z. Marciniak, K. Kuczyński, T. Pokora, Influence of the plastic properties of a material on the forming limit diagram for sheet metal in tension, Int. J. Mech. Sci., 15 (1973) 789-800. https://doi.org/10.1016/0020-7403(73)90068-4

[15] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourgoghrat, S.H. Choi, E. Chu, Plane stress yield function for aluminum alloy sheets - part 1: theory, Int. J. Plast., 19 (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0

[16] K.K. Phoon, S.P. Huang, S.T. Quek, Implementation of Karhunen–Loeve expansion for simulation using a wavelet-Galerkin scheme, Probabilistic Eng. Mech., 17 (2002) 293-303. https://doi.org/10.1016/S0266-8920(02)00013-9

[17] S. Tveit, A. Reyes, A submodeling technique for incorporating sheet metal forming effects in an AA5083 FPV structure, submitted for publication, (2024).
# Investigation on diametral compression of an aluminium alloy AA5005 using EBSD and microhardness measurements

FRANCILLETTE Henri¹,ª* and GARAND Christian²,ᵇ

¹Univ Rennes, INSA Rennes, CNRS, ISCR UMR 6226, 35000 Rennes, France

²Univ Rennes, INSA Rennes, LGCGM, 35000 Rennes, France

ªhenri.francillette@insa-rennes.fr, ᵇchristian.garand@insa-rennes.fr

**Keywords:** Aluminium, Diametral, Compression, Microhardness, EBSD

**Abstract.** The study of the link between the mechanical behaviour of metallic materials and the inspection of their microstructure is a particularly important subject of investigation. The plastic deformation of a commercial aluminium alloy AA5005 is analyzed in this work using mainly diametral compression tests and different experimental techniques of observation and characterization. Uniaxial compression tests and microhardness measurements have also been carried out in the study. For the characterization of the specimens, Electron Back Scattering Diffraction (EBSD) has been used in order to ascertain the state of the microstructure of the material at different stages. Characterizations are performed with EBSD notably for the received state and after diametral compression followed by heat treatments at 400 °C.

## Introduction

In the transportation industry, the use of lightweight metals with high specific mechanical resistances is an important part of research. In this context, aluminum alloys and particularly the 5xxx Al-Mg series are of great interest [1-4]. The study of the mechanical behaviour of aluminium alloys in relation to their microstructure is necessary to understand the practical use of this type of metals [5,6]. Among the possible mechanical tests, diametral compression, the usual uniaxial compression and Vickers microhardness measurements are used in the study to characterize a commercial AA5005 aluminium bar. The diametral compression test is a less common mechanical test. Historically, this latter test has been developed few decades ago in order to determine the mechanical strength of concrete cylinders and pharmaceutical tablets [7,8]. It allows the measurement of the strength of a material in an indirect way when tensile tests are difficult because of the properties of the material [7]. It has been used more recently for metallic glass [9] and for sprayed coated and uncoated circular disc specimens [10]. Diametral compression has been used also in the characterization of the mechanical behaviour of metals notably for tubes and rings [11-13]. Interesting applications of this test (compared to the uniaxial one) can be found particularly for nuclear materials when determination of ductility in circumferential directions is investigated [12]. One can notice the recent use of this test (ring compression test) to study the influence of heterogeneities in an aluminium 6006 [13]. In the study presented here, the advantage of diametral compression test compared to uniaxial compression is that when diametral compression is considered one can expect more heterogeneities in the deformed cross-sections of the specimens. For instance, the geometry of the deformed specimens is expected to be less isotropic in the compression plane for diametral compression. When we use microhardness and EBSD measurements, the characterization of ductility of the metallic material can give important informations to find the links between the microstructure and the mechanical behaviour. Furthermore, diametral compression has the advantage or interest to provide experimental results (obtained with compressive stress and strain components) complementary to the usual

<figure><img src="image_1003.png" /></figure>
# Investigation on diametral compression of an aluminium alloy AA5005 using EBSD and microhardness measurements

FRANCILLETTE Henri¹,ª* and GARAND Christian²,ᵇ

¹Univ Rennes, INSA Rennes, CNRS, ISCR UMR 6226, 35000 Rennes, France

²Univ Rennes, INSA Rennes, LGCGM, 35000 Rennes, France

ªhenri.francillette@insa-rennes.fr, ᵇchristian.garand@insa-rennes.fr

**Keywords:** Aluminium, Diametral, Compression, Microhardness, EBSD

**Abstract.** The study of the link between the mechanical behaviour of metallic materials and the inspection of their microstructure is a particularly important subject of investigation. The plastic deformation of a commercial aluminium alloy AA5005 is analyzed in this work using mainly diametral compression tests and different experimental techniques of observation and characterization. Uniaxial compression tests and microhardness measurements have also been carried out in the study. For the characterization of the specimens, Electron Back Scattering Diffraction (EBSD) has been used in order to ascertain the state of the microstructure of the material at different stages. Characterizations are performed with EBSD notably for the received state and after diametral compression followed by heat treatments at 400 °C.

## Introduction

In the transportation industry, the use of lightweight metals with high specific mechanical resistances is an important part of research. In this context, aluminum alloys and particularly the 5xxx Al-Mg series are of great interest [1-4]. The study of the mechanical behaviour of aluminium alloys in relation to their microstructure is necessary to understand the practical use of this type of metals [5,6]. Among the possible mechanical tests, diametral compression, the usual uniaxial compression and Vickers microhardness measurements are used in the study to characterize a commercial AA5005 aluminium bar. The diametral compression test is a less common mechanical test. Historically, this latter test has been developed few decades ago in order to determine the mechanical strength of concrete cylinders and pharmaceutical tablets [7,8]. It allows the measurement of the strength of a material in an indirect way when tensile tests are difficult because of the properties of the material [7]. It has been used more recently for metallic glass [9] and for sprayed coated and uncoated circular disc specimens [10]. Diametral compression has been used also in the characterization of the mechanical behaviour of metals notably for tubes and rings [11-13]. Interesting applications of this test (compared to the uniaxial one) can be found particularly for nuclear materials when determination of ductility in circumferential directions is investigated [12]. One can notice the recent use of this test (ring compression test) to study the influence of heterogeneities in an aluminium 6006 [13]. In the study presented here, the advantage of diametral compression test compared to uniaxial compression is that when diametral compression is considered one can expect more heterogeneities in the deformed cross-sections of the specimens. For instance, the geometry of the deformed specimens is expected to be less isotropic in the compression plane for diametral compression. When we use microhardness and EBSD measurements, the characterization of ductility of the metallic material can give important informations to find the links between the microstructure and the mechanical behaviour. Furthermore, diametral compression has the advantage or interest to provide experimental results (obtained with compressive stress and strain components) complementary to the usual

<figure><img src="image_1008.png" /></figure>
compression test. In the next sections, after the presentation of the experimental procedure, we will present the experimental results with their analysis, followed by conclusion.

## Experimental procedure

A commercial bar of an AA5005 aluminium alloy is investigated by the consideration of two positions of cylindrical specimens which are determined by the compression direction of the mechanical device. These positions correspond to diametral compression and correspond to the usual uniaxial compression. The diametral compression test represents however the main part of the study. Vickers microhardness measurements are also performed in order to characterize the microstructures after the compression tests and heat treatments at 400 °C. Microhardness measurements have already been mentioned for AA5005 alloy [14]. Generally, heat treatments of aluminium alloys notably in 2xxx and 7xxx series may introduce important strain hardening effects when plastic deformation is applied, due to precipitates. Relatively low stress levels are expected due to heat treatments in the alloy investigated in this study AA5005 [19]. The new insight of the study presented here concerning heat treatments of an aluminium alloy is mainly their application after plastic deformation by diametral compression. Without these treatments, practically, it is particularly difficult to determine texture evolution by EBSD after plastic deformation.

Cylindrical specimens with a diameter of 8 mm were cut from a received commercial cylindrical bar with the same diameter. In the study the initial height of the samples was fixed at 10 mm. The mechanical tests were performed with a 200 kN capacity Instron press. All the compression tests were carried out at room temperature with a 1 mm/min cross-head speed. The total time for each mechanical test was fixed at 300 s. Concerning the Vickers microhardness measurements, a Mitutoyo device was used with a load of 0.3 kg. For each hardness measurement, the mean value of the two diagonals of the indent was determined. Optical microscopy has been used for the observation of the microstructure and a Dino-Lite optical microscope for the determination of macrographs. Electron Back Scattering Diffraction (EBSD) was also used to ascertain the microstructures of the analyzed samples, notably the Oxford HKL EBSD system in a Jeol scanning electron microscope (SEM). This technique is efficient to relate the crystallographic orientation of the grains of the material to its mechanical properties [15-19]. For the preparation of the specimens, after mechanical polishing, a solution with 0.5 % hydrofluoric acid was considered. For the Vickers hardness measurements and the EBSD characterizations the specimens were systematically wrapped in a phenolic resin.

In the experimental part of the study, a comparison was first performed between specimens of the received material in the initial state and specimens recovered 1 h at 300 °C before the mechanical tests. This temperature of 300 °C was chosen sufficiently high to provide potentially stress level differences in the measurements. All the compression tests were performed at room temperature. Next, for specimens in the received state, a particular attention was paid to microhardness measurements for diametral compression and for heat treatments performed at 400°C during 0.5 h and 3 h. The EBSD technique was used in order to ascertain the evolution of the microstructure.

## Results and analysis

The presentation of the microstructure of the received material in its initial state is given in Fig. 1 with sections corresponding to the inverse pole figures (IPF) of the horizontal X direction and the vertical Y direction, using the EBSD technique. The Y direction corresponds to the compression direction during diametral compression and the observation plane corresponds to a cross-section of the cylindrical bar before deformation. One can notice grains of the order of 100 µm or less. This cartography has been obtained with a step of the electron beam of 2 µm. The selected region corresponds to the center of the cylindrical surface. The crystallographic texture of the scanned zone of Fig.1 has been determined through the {100}, {110} and {111} pole figures plotted in
compression test. In the next sections, after the presentation of the experimental procedure, we will present the experimental results with their analysis, followed by conclusion.

## Experimental procedure

A commercial bar of an AA5005 aluminium alloy is investigated by the consideration of two positions of cylindrical specimens which are determined by the compression direction of the mechanical device. These positions correspond to diametral compression and correspond to the usual uniaxial compression. The diametral compression test represents however the main part of the study. Vickers microhardness measurements are also performed in order to characterize the microstructures after the compression tests and heat treatments at 400 °C. Microhardness measurements have already been mentioned for AA5005 alloy [14]. Generally, heat treatments of aluminium alloys notably in 2xxx and 7xxx series may introduce important strain hardening effects when plastic deformation is applied, due to precipitates. Relatively low stress levels are expected due to heat treatments in the alloy investigated in this study AA5005 [19]. The new insight of the study presented here concerning heat treatments of an aluminium alloy is mainly their application after plastic deformation by diametral compression. Without these treatments, practically, it is particularly difficult to determine texture evolution by EBSD after plastic deformation.

Cylindrical specimens with a diameter of 8 mm were cut from a received commercial cylindrical bar with the same diameter. In the study the initial height of the samples was fixed at 10 mm. The mechanical tests were performed with a 200 kN capacity Instron press. All the compression tests were carried out at room temperature with a 1 mm/min cross-head speed. The total time for each mechanical test was fixed at 300 s. Concerning the Vickers microhardness measurements, a Mitutoyo device was used with a load of 0.3 kg. For each hardness measurement, the mean value of the two diagonals of the indent was determined. Optical microscopy has been used for the observation of the microstructure and a Dino-Lite optical microscope for the determination of macrographs. Electron Back Scattering Diffraction (EBSD) was also used to ascertain the microstructures of the analyzed samples, notably the Oxford HKL EBSD system in a Jeol scanning electron microscope (SEM). This technique is efficient to relate the crystallographic orientation of the grains of the material to its mechanical properties [15-19]. For the preparation of the specimens, after mechanical polishing, a solution with 0.5 % hydrofluoric acid was considered. For the Vickers hardness measurements and the EBSD characterizations the specimens were systematically wrapped in a phenolic resin.

In the experimental part of the study, a comparison was first performed between specimens of the received material in the initial state and specimens recovered 1 h at 300 °C before the mechanical tests. This temperature of 300 °C was chosen sufficiently high to provide potentially stress level differences in the measurements. All the compression tests were performed at room temperature. Next, for specimens in the received state, a particular attention was paid to microhardness measurements for diametral compression and for heat treatments performed at 400°C during 0.5 h and 3 h. The EBSD technique was used in order to ascertain the evolution of the microstructure.

## Results and analysis

The presentation of the microstructure of the received material in its initial state is given in Fig. 1 with sections corresponding to the inverse pole figures (IPF) of the horizontal X direction and the vertical Y direction, using the EBSD technique. The Y direction corresponds to the compression direction during diametral compression and the observation plane corresponds to a cross-section of the cylindrical bar before deformation. One can notice grains of the order of 100 µm or less. This cartography has been obtained with a step of the electron beam of 2 µm. The selected region corresponds to the center of the cylindrical surface. The crystallographic texture of the scanned zone of Fig.1 has been determined through the {100}, {110} and {111} pole figures plotted in
Fig. 2. We can notice a significant concentration of {100} poles along the normal of the observation plane.

<figure><img src="image_1004.png" /><figcaption>Fig. 1: EBSD characterization of the received material. The observation plane corresponds to a cross-section of the cylindrical specimens.</figcaption></figure>

<figure><img src="image_1005.png" /><figcaption>Fig. 2: Texture characterization of the received material : {100}, {110} and {111} pole figures. z along the axis of the initial cylinders.</figcaption></figure>

The comparison of the mechanical tests for the received material and the material heat treated before the tests at 300 °C during 1 h are presented in the following figures. Fig. 3 and Fig. 4 correspond respectively to diametral compression and uniaxial compression at room temperature. For Fig. 3, the displacement u is normalized by the initial diameter d₀ of the specimen and the applied force F is normalized by the product of the initial diameter by the initial height h₀ of the specimen, as mentioned in the literature [9]. For uniaxial compression in Fig. 4, there is the presentation of the true stress-strain curves. We can notice greater values of the normalized force for the received specimen compared to the specimen recovered at 300 °C for a fixed value of the normalized displacement in Fig. 3. The same remark is true for the stress-strain curves of the uniaxial compression tests.
Fig. 2. We can notice a significant concentration of {100} poles along the normal of the observation plane.

<figure><img src="image_1009.png" /><figcaption>Fig. 1: EBSD characterization of the received material. The observation plane corresponds to a cross-section of the cylindrical specimens.</figcaption></figure>

<figure><img src="image_1010.png" /><figcaption>Fig. 2: Texture characterization of the received material : {100}, {110} and {111} pole figures. z along the axis of the initial cylinders.</figcaption></figure>

The comparison of the mechanical tests for the received material and the material heat treated before the tests at 300 °C during 1 h are presented in the following figures. Fig. 3 and Fig. 4 correspond respectively to diametral compression and uniaxial compression at room temperature. For Fig. 3, the displacement u is normalized by the initial diameter d₀ of the specimen and the applied force F is normalized by the product of the initial diameter by the initial height h₀ of the specimen, as mentioned in the literature [9]. For uniaxial compression in Fig. 4, there is the presentation of the true stress-strain curves. We can notice greater values of the normalized force for the received specimen compared to the specimen recovered at 300 °C for a fixed value of the normalized displacement in Fig. 3. The same remark is true for the stress-strain curves of the uniaxial compression tests.
<figure><img src="image_1006.png" /><figcaption>Fig. 3: Normalized force-displacement curves F/(d₀ × h₀) versus u/d₀ for diametral compression at room temperature for the received material and for the received material recovered 1 h at 300 °C before the test.</figcaption></figure>

<figure><img src="image_1007.png" /><figcaption>Fig. 4: True stress-strain curves of the uniaxial compression tests at room temperature for the received material and for the received material recovered 1 h at 300 °C before the test.</figcaption></figure>

Macrographs of the compression surface of the specimens are presented in Fig. 5 after the mechanical tests. An irregular shape for diametral compression is obtained in Fig. 5(a) for which the compression direction corresponds to the normal of the figure and the initial axis of the (initial) cylinder (IAC) corresponds to the horizontal direction. Other macrographs of cross-sections are also presented in Fig. 6.
<figure><img src="image_1011.png" /><figcaption>Fig. 3: Normalized force-displacement curves F/(d₀ × h₀) versus u/d₀ for diametral compression at room temperature for the received material and for the received material recovered 1 h at 300 °C before the test.</figcaption></figure>

<figure><img src="image_1012.png" /><figcaption>Fig. 4: True stress-strain curves of the uniaxial compression tests at room temperature for the received material and for the received material recovered 1 h at 300 °C before the test.</figcaption></figure>

Macrographs of the compression surface of the specimens are presented in Fig. 5 after the mechanical tests. An irregular shape for diametral compression is obtained in Fig. 5(a) for which the compression direction corresponds to the normal of the figure and the initial axis of the (initial) cylinder (IAC) corresponds to the horizontal direction. Other macrographs of cross-sections are also presented in Fig. 6.
<figure><img src="image_1008.png" /><figcaption>Fig. 5: Macrographs of the compression plane of the specimens after the mechanical tests: (a) diametral compression (red line: 13.586 mm) and (b) uniaxial compression (red line: 9.935 mm). The initial axis of the cylinder (IAC) is horizontal in (a) and along the normal of the figure in (b).</figcaption></figure>

For Fig. 6(a) initially, before deformation, the section corresponds to a disc and Fig. 6(b) initially contains the initial axis of the cylindrical specimen. For the two sections, the compression direction (CD) is along the vertical direction of the figures.

<figure><img src="image_1009.png" /><figcaption>Fig. 6: Macrographs of cross-sections after the mechanical tests: (a) diametral compression (red line: 12.497 mm) and (b) uniaxial compression (red line: 11.577 mm). The compression direction (CD) and the initial axis of the cylinder (IAC) are indicated in (a) and (b).</figcaption></figure>

The evolution of Vickers hardness along two perpendicular lines (with extremities fixed as origins) of deformed specimens is considered experimentally for uniaxial compression and diametral compression, as illustrated in Fig. 6 by the colored lines. The comparison of the hardness values is performed for the uniaxial compression tests by considering firstly a specimen in the starting state of the received bar (without heat treatment) and secondly for a specimen with a recovery of 1 h at 300 °C before the mechanical test. The plane surface for which the measurements are performed corresponds to a section cutting the sample through its center and perpendicular to the observation plane of Fig. 5(b). The intersection between the plane of the measurements and the plane of observation of Fig. 5 (b) may be for example the visible red trace: it corresponds to Fig. 6(b). The lines for the Vickers microhardness measurements were selected in the following
<figure><img src="image_1013.png" /><figcaption>Fig. 5: Macrographs of the compression plane of the specimens after the mechanical tests : (a) diametral compression (red line :13.586 mm) and (b) uniaxial compression (red line : 9.935 mm). The initial axis of the cylinder (IAC) is horizontal in (a) and along the normal of the figure in (b).</figcaption></figure>

For Fig. 6(a) initially, before deformation, the section corresponds to a disc and Fig. 6(b) initially contains the initial axis of the cylindrical specimen. For the two sections, the compression direction (CD) is along the vertical direction of the figures.

<figure><img src="image_1014.png" /><figcaption>Fig. 6: Macrographs of cross-sections after the mechanical tests: (a) diametral compression (red line : 12.497 mm) and (b) uniaxial compression (red line : 11.577 mm). The compression direction (CD) and the initial axis of the cylinder (IAC) are indicated in (a) and (b).</figcaption></figure>

The evolution of Vickers hardness along two perpendicular lines (with extremities fixed as origins) of deformed specimens is considered experimentally for uniaxial compression and diametral compression, as illustrated in Fig. 6 by the colored lines. The comparison of the hardness values is performed for the uniaxial compression tests by considering firstly a specimen in the starting state of the received bar (without heat treatment) and secondly for a specimen with a recovery of 1 h at 300 °C before the mechanical test. The plane surface for which the measurements are performed corresponds to a section cutting the sample through its center and perpendicular to the observation plane of Fig. 5(b). The intersection between the plane of the measurements and the plane of observation of Fig. 5 (b) may be for example the visible red trace : it corresponds to Fig. 6(b). The lines for the Vickers microhardness measurements were selected in the following
way: line HC (Horizontal centered, or Central horizontal line) and line VC (Vertical centered, or Central vertical line). A step size of 200 µm and a load of 0.3 kg were taken into account for the microhardness measurements on the surfaces. Fig. 7 presents an illustration of experimental indents along two perpendicular lines (HC and VC) which have been performed during the measurements.

<figure><img src="image_1010.png" /><figcaption>Fig. 7: Microhardness indents along horizontal and vertical centered lines (load of 0.3 kg) after diametral compression. The compression direction (CD) is vertical and the initial axis of the cylinder (IAC) is along the normal of the figure.</figcaption></figure>

The experimental values are presented in Fig. 8 for uniaxial compression and in Fig. 9 for diametral compression. The origin of the axis for the distance corresponds in these figures respectively to the (left) beginning of the red lines and to the (top) beginning of the green lines illustrated in Fig. 6. The measurements indicate that the values are great along the horizontal line for the untreated sample with microhardness values close to 90.0 HV. Most of the values are near or greater than 60.0 HV in the other case. In diametral compression, the values of the microhardness are relatively low at the extremities of the HC line, notably for the sample recovered at 300 °C during 1 h before the mechanical test. Values below 58.0 HV are obtained. In the next sections, only specimens in the received state before diametral compression are considered.

<figure><img src="image_1011.png" /><figcaption>Fig. 8: Microhardness evolution in the specimens deformed by uniaxial compression at room temperature for the received material and for the material recovered 1 h at 300 °C before the test. Measurements (with origins) taken along the HC and VC lines (as illustrated in Fig. 6(b)).</figcaption></figure>
way: line HC (Horizontal centered, or Central horizontal line) and line VC (Vertical centered, or Central vertical line). A step size of 200 µm and a load of 0.3 kg were taken into account for the microhardness measurements on the surfaces. Fig. 7 presents an illustration of experimental indents along two perpendicular lines (HC and VC) which have been performed during the measurements.

<figure><img src="image_1015.png" /><figcaption>Fig. 7: Microhardness indents along horizontal and vertical centered lines (load of 0.3 kg) after diametral compression. The compression direction (CD) is vertical and the initial axis of the cylinder (IAC) is along the normal of the figure.</figcaption></figure>

The experimental values are presented in Fig. 8 for uniaxial compression and in Fig. 9 for diametral compression. The origin of the axis for the distance corresponds in these figures respectively to the (left) beginning of the red lines and to the (top) beginning of the green lines illustrated in Fig. 6. The measurements indicate that the values are great along the horizontal line for the untreated sample with microhardness values close to 90.0 HV. Most of the values are near or greater than 60.0 HV in the other case. In diametral compression, the values of the microhardness are relatively low at the extremities of the HC line, notably for the sample recovered at 300 °C during 1 h before the mechanical test. Values below 58.0 HV are obtained. In the next sections, only specimens in the received state before diametral compression are considered.

<figure><img src="image_1016.png" /><figcaption>Fig. 8: Microhardness evolution in the specimens deformed by uniaxial compression at room temperature for the received material and for the material recovered 1 h at 300 °C before the test. Measurements (with origins) taken along the HC and VC lines (as illustrated in Fig. 6(b)).</figcaption></figure>
<figure><img src="image_1012.png" /><figcaption>Fig. 9: Microhardness evolution in the specimens deformed by diametral compression at room temperature for the received material and for the material recovered 1 h at 300 °C before the test. Measurements (with origins) taken along the HC and VC lines (as illustrated in Fig. 6(a)).</figcaption></figure>

In this section, the evolution of the microstructure is characterized after two heat treatments : in the first case, we consider 30 minutes at 400 °C and in the second case we consider 3 h at 400 °C. The corresponding microhardness measurements are presented in Fig. 10. We can observe lower values in the center of the central horizontal line for the curve corresponding to 30 minutes (values lower than 50.0 HV) compared to the extremities (values close to 55.0 HV - 60.0 HV). The values indicate a decrease of the hardness when the time of the heat treatment increases. On the one hand, Fig. 11 presents an EBSD analyzed zone at a short range for the specimen heat treated 0.5 h at 400 °C. A step size of 0.1 µm was chosen for the determination of this zone. The normal of the plane of observation is parallel to the initial axis of the initial cylinders and the compression direction (or normal direction ND) is in a vertical position. This also remains true for the next figure, notably Fig. 12. The microstructure in Fig. 11 is quite heterogeneous at this scale because of the previous plastic deformation of the specimen. For a greater heat treatment time at this temperature, notably 3 h at 400 °C, the microstructure is most regular at the grain scale. It was possible to determine a larger map for the orientation of the grains with EBSD. This corresponds to microstructure presented in Fig. 12. A step size of 2 µm was chosen for this map.

<figure><img src="image_1013.png" /><figcaption>Fig. 10: Microhardness measurements of the received material deformed by diametral compression and recovered 0.5 h at 400 °C and 3 h at 400 °C. Measurements in a plane as in Fig. 6(a).</figcaption></figure>
<figure><img src="image_1017.png" /><figcaption>Fig. 9: Microhardness evolution in the specimens deformed by diametral compression at room temperature for the received material and for the material recovered 1 h at 300 °C before the test. Measurements (with origins) taken along the HC and VC lines (as illustrated in Fig. 6(a)).</figcaption></figure>

In this section, the evolution of the microstructure is characterized after two heat treatments : in the first case, we consider 30 minutes at 400 °C and in the second case we consider 3 h at 400 °C. The corresponding microhardness measurements are presented in Fig. 10. We can observe lower values in the center of the central horizontal line for the curve corresponding to 30 minutes (values lower than 50.0 HV) compared to the extremities (values close to 55.0 HV - 60.0 HV). The values indicate a decrease of the hardness when the time of the heat treatment increases. On the one hand, Fig. 11 presents an EBSD analyzed zone at a short range for the specimen heat treated 0.5 h at 400 °C. A step size of 0.1 µm was chosen for the determination of this zone. The normal of the plane of observation is parallel to the initial axis of the initial cylinders and the compression direction (or normal direction ND) is in a vertical position. This also remains true for the next figure, notably Fig. 12. The microstructure in Fig. 11 is quite heterogeneous at this scale because of the previous plastic deformation of the specimen. For a greater heat treatment time at this temperature, notably 3 h at 400 °C, the microstructure is most regular at the grain scale. It was possible to determine a larger map for the orientation of the grains with EBSD. This corresponds to microstructure presented in Fig. 12. A step size of 2 µm was chosen for this map.

<figure><img src="image_1018.png" /><figcaption>Fig. 10: Microhardness measurements of the received material deformed by diametral compression and recovered 0.5 h at 400 °C and 3 h at 400 °C. Measurements in a plane as in Fig. 6(a).</figcaption></figure>
<figure><img src="image_1014.png" /><figcaption>Fig. 11: EBSD analysis of the received material deformed by diametral compression and recovered 0.5 h at 400 °C: Inverse pole figures (IPF).</figcaption></figure>

<figure><img src="image_1015.png" /><figcaption>Fig. 12: EBSD zone analysis for the received material deformed by diametral compression and recovered 3 h at 400 °C.</figcaption></figure>

To follow the crystallographic texture, the {100}, {110} and {111} pole figures associated with the analyzed zone of Fig. 12 were determined. The corresponding results are presented in Fig. 13. The vertical direction corresponds to normal direction (ND) during the mechanical test. We still observe an important concentration of {100} poles along the center of the pole figure. However, there is the appearance of poles in the vicinity of the compression direction.

<figure><img src="image_1016.png" /><figcaption>Fig. 13: Texture characterization of the received material deformed by diametral compression and recovered 3 h at 400 °C: {100}, {110} and {111} pole figures.</figcaption></figure>
<figure><img src="image_1019.png" /><figcaption>Fig. 11: EBSD analysis of the received material deformed by diametral compression and recovered 0.5 h at 400 °C: Inverse pole figures (IPF).</figcaption></figure>

<figure><img src="image_1020.png" /><figcaption>Fig. 12: EBSD zone analysis for the received material deformed by diametral compression and recovered 3 h at 400 °C.</figcaption></figure>

To follow the crystallographic texture, the {100}, {110} and {111} pole figures associated with the analyzed zone of Fig. 12 were determined. The corresponding results are presented in Fig. 13. The vertical direction corresponds to normal direction (ND) during the mechanical test. We still observe an important concentration of {100} poles along the center of the pole figure. However, there is the appearance of poles in the vicinity of the compression direction.

<figure><img src="image_1021.png" /><figcaption>Fig. 13: Texture characterization of the received material deformed by diametral compression and recovered 3 h at 400 °C: {100}, {110} and {111} pole figures.</figcaption></figure>
## Conclusion

Specimens of an aluminium alloy AA5005 have been investigated in this study using different mechanical tests and different experimental techniques. Diametral compression tests were carried out and completed with uniaxial compression tests and Vickers microhardness measurements. Samples with a height of 10 mm and with a diameter of 8 mm were cut from a received rod, deformed and heat treated. The EBSD technique in a SEM has been used to ascertain the microstructures by the determination of the orientation of the grains and crystallographic textures. Different conditions have been considered for the compression tests at room temperature. Normalized force-displacement curves were plotted for the diametral compression tests and indicated significant level differences. The results are consistent with the corresponding uniaxial compression tests and the associated microhardness measurements. For the two heat treatments performed at 400 °C during 0.5 h and 3 h at 400 °C after plastic deformation, the microhardness measurements indicate a significant decrease compared to the state of the material after diametral compression. The new insights in the study concerning the influence of heat treatments on an aluminium alloy concern their effects on texture evolution after plastic deformation performed by diametral compression, and the corresponding characterization thanks to the use of the EBSD technique. The heat treatments were practically compulsory in this study for performing EBSD measurements because of strain hardening introduced before by diametral compression. The EBSD analysis indicates local heterogeneities in the microstructure at the grain scale after 0.5 h compared to 3 h. An important concentration of the {100} poles along the initial direction of the axis of the initial cylinders is still measured after the heat treatment of 3 h at 400 °C.

## References

[1] A. Weck, P. Bisaillon, L. Nong, T. Meunier, H. Jin, M. Gallerneault, Mechanical properties of the aluminium roll-bond laminate AA5005-AA5083-AA5005. Mater. Sci. and Eng. A 528 (2011) 6186-6193. https://doi.org/10.1016/j.msea.2011.04.037

[2] J. Hirsch, T. Al-Samman, Superior light metals by texture engineering: Optimized aluminium and magnesium alloys for automotive applications. Acta Mater. 61 (2013) 818-843. https://doi.org/10.1016/j.actamat.2012.10.044

[3] H. Francillette, C. Garand, Formability and macroscopic shearing of a titanium alloy Ti-6Al-4V under channel die compression. AIP Conference Proceedings 1896, 020005 (2017). https://doi.org/10.1063/1.5007962

[4] D. Scotto D'Antuono, J. Gaies, W. Golumbfskie, M.L. Taheri, Direct measurement of the effect of cold rolling on β phase precipitation kinetics in 5xxx series aluminium alloys. Acta Mater. 123 (2017) 264-271. https://doi.org/10.1016/j.actamat.2016.10.060

[5] P.B. Trivedi, R.S. Yassar, D.P. Field, R. Alldredge, Microstructural evolution and observed stress response during hot deformation of 5005 and 6022 Al alloys. Mater. Sci. and Eng. A 425 (2006) 205-212. https://doi.org/10.1016/j.msea.2006.03.059

[6] F. Shen, D. Yi, B. Wang, H. Liu, Y. Jiang, C. Tang, B. Jiang, Semi-quantitative evaluation of texture components and anisotropy of the yield strength in 2524 T3 alloy sheets. Mater. Sci. and Eng. A 675 (2016) 386-395. http://dx.doi.org/10.1016/j.msea.2016.08.013

[7] V. Mazel, S. Guerard, B. Croquelois, J.B. Kopp, J. Girardot, H. Diarra, V. Busignies, Reevaluation of the diametral compression test for tablets using the flattened disc geometry. International journal of pharmaceutics 513 (2016) 669-677. http://dx.doi.org/10.1016/j.ijpharm.2016.09.088

[8] M. Mellor, I. Hawkes, Measurement of tensile strength by diametral compression of discs and annuli. Eng. Geol. 5 (3) (1971) 173-225. https://doi.org/10.1016/0013-7952(71)90001-9
## Conclusion

Specimens of an aluminium alloy AA5005 have been investigated in this study using different mechanical tests and different experimental techniques. Diametral compression tests were carried out and completed with uniaxial compression tests and Vickers microhardness measurements. Samples with a height of 10 mm and with a diameter of 8 mm were cut from a received rod, deformed and heat treated. The EBSD technique in a SEM has been used to ascertain the microstructures by the determination of the orientation of the grains and crystallographic textures. Different conditions have been considered for the compression tests at room temperature. Normalized force-displacement curves were plotted for the diametral compression tests and indicated significant level differences. The results are consistent with the corresponding uniaxial compression tests and the associated microhardness measurements. For the two heat treatments performed at 400 °C during 0.5 h and 3 h at 400 °C after plastic deformation, the microhardness measurements indicate a significant decrease compared to the state of the material after diametral compression. The new insights in the study concerning the influence of heat treatments on an aluminium alloy concern their effects on texture evolution after plastic deformation performed by diametral compression, and the corresponding characterization thanks to the use of the EBSD technique. The heat treatments were practically compulsory in this study for performing EBSD measurements because of strain hardening introduced before by diametral compression. The EBSD analysis indicates local heterogeneities in the microstructure at the grain scale after 0.5 h compared to 3 h. An important concentration of the {100} poles along the initial direction of the axis of the initial cylinders is still measured after the heat treatment of 3 h at 400 °C.

## References

[1] A. Weck, P. Bisaillon, L. Nong, T. Meunier, H. Jin, M. Gallerneault, Mechanical properties of the aluminium roll-bond laminate AA5005-AA5083-AA5005. Mater. Sci. and Eng. A 528 (2011) 6186-6193. https://doi.org/10.1016/j.msea.2011.04.037

[2] J. Hirsch, T. Al-Samman, Superior light metals by texture engineering: Optimized aluminium and magnesium alloys for automotive applications. Acta Mater. 61 (2013) 818-843. https://doi.org/10.1016/j.actamat.2012.10.044

[3] H. Francillette, C. Garand, Formability and macroscopic shearing of a titanium alloy Ti-6Al-4V under channel die compression. AIP Conference Proceedings 1896, 020005 (2017). https://doi.org/10.1063/1.5007962

[4] D. Scotto D'Antuono, J. Gaies, W. Golumbfskie, M.L. Taheri, Direct measurement of the effect of cold rolling on β phase precipitation kinetics in 5xxx series aluminium alloys. Acta Mater. 123 (2017) 264-271. https://doi.org/10.1016/j.actamat.2016.10.060

[5] P.B. Trivedi, R.S. Yassar, D.P. Field, R. Alldredge, Microstructural evolution and observed stress response during hot deformation of 5005 and 6022 Al alloys. Mater. Sci. and Eng. A 425 (2006) 205-212. https://doi.org/10.1016/j.msea.2006.03.059

[6] F. Shen, D. Yi, B. Wang, H. Liu, Y. Jiang, C. Tang, B. Jiang, Semi-quantitative evaluation of texture components and anisotropy of the yield strength in 2524 T3 alloy sheets. Mater. Sci. and Eng. A 675 (2016) 386-395. http://dx.doi.org/10.1016/j.msea.2016.08.013

[7] V. Mazel, S. Guerard, B. Croquelois, J.B. Kopp, J. Girardot, H. Diarra, V. Busignies, Reevaluation of the diametral compression test for tablets using the flattened disc geometry. International journal of pharmaceutics 513 (2016) 669-677. http://dx.doi.org/10.1016/j.ijpharm.2016.09.088

[8] M. Mellor, I. Hawkes, Measurement of tensile strength by diametral compression of discs and annuli. Eng. Geol. 5 (3) (1971) 173-225. https://doi.org/10.1016/0013-7952(71)90001-9
[9] C. Bernard, S. Hin, L. Charleux, E. Roux, Y. Yokoyama, A. Tanguy, V. Keryvin, Intense shear band plasticity in metallic glass as revealed by a diametral compression test. Mater. Sci. and Eng. A 864 (2023) 144533. https://doi.org/10.1016/j.msea.2022.144533

[10] N.H. Faisal, L. Mann, C. Duncan, E. Dunbar, M. Clayton, M. Frost, J. McConnachie, A. Fardan, R. Ahmed, Diametral compression test method to analyse relative surface stresses in thermally sprayed coated and uncoated circular disc specimens. Surface and Coat. Tech. 357 (2019) 497-514. https://doi.org/10.1016/j.surfcoat.2018.10.053

[11] N. K. Gupta, G.S. Sekhon, P.K. Gupta, Study of lateral compression of round metallic tubes. Thin-walled structures 43 (2005) 895-922. https://doi.org/10.1016/j.tws.2004.12.002

[12] Y. Coget, Y. Demarty, C. Czarnota, A. Bracq, J.S. Brest, A. Rusinek, Lateral ring compression test applied to a small calibre steel jacket : Identification of a constitutive model. Defence technology. In Press. https://doi.org/10.1016/j.dt.2023.11.001

[13] L. Bizet, L. Charleux, P. Balland, L. Tabourot, Influence of heterogeneities introduces into the modelling of a ring compression test. Arch. of Civ. and Mech Eng. 17 (2017) 365-374. http://dx.doi.org/10.1016/j.acme.2016.11.002

[14] R. Abrahams, J. Mikhail, P. Fasihi, Effect of friction stir process parameters on the mechanical properties of 5005-H34 and 7075-T651 aluminium alloy. Mater. Sci. and Eng. A 751 (2019) 363-373. https://doi.org/10.1016/j.msea.2019.02.065

[15] I. Carneiro, S. Simões, Recent advances in EBSD characterization of metals. Metals (2020) 10, 1097. https://doi.org/10.3390/met10081097

[16] H. Francillette, Effects of the interaction length between hollow shapes : Application to a HCP metal under mechanical loading. Procedia Manufacturing 47 (2020) 1203-1210. https://doi.org/10.1016/j.promfg.2020.04.351

[17] H. Francillette, B. Bacroix, M. Gaspérini, J.L. Béchade, Grain orientation effects in Zr702α polycristalline samples deformed in channel die compression at room temperature. Acta mater. 46 (1998), 4131-4142. https://doi.org/10.1016/S1359-6454(98)00121-9

[18] G. Zhao, M. Sun, J. Li, H. Li, L. Ma, Y. Li, Study on quasi-in-situ tensile microstructure evolution law of 5052-O aluminium alloy based on EBSD. Materials Today Communications 33 (2022) 104572. https://doi.org/10.1016/j.mtcomm.2022.104572

[19] J. Xie, Y. Liang, F. Yan, J. Zhou, P. Du, W. Liu. Effect of high-pressure heat treatment on the recrystallization and recrystallization texture of CC 5005 aluminium alloy. Journ. of Mater. Res. Tech. 26 (2023) 2017-2027. https://doi.org/10.1016/j.jmrt.2023.08.018
[9] C. Bernard, S. Hin, L. Charleux, E. Roux, Y. Yokoyama, A. Tanguy, V. Keryvin, Intense shear band plasticity in metallic glass as revealed by a diametral compression test. Mater. Sci. and Eng. A 864 (2023) 144533. https://doi.org/10.1016/j.msea.2022.144533

[10] N.H. Faisal, L. Mann, C. Duncan, E. Dunbar, M. Clayton, M. Frost, J. McConnachie, A. Fardan, R. Ahmed, Diametral compression test method to analyse relative surface stresses in thermally sprayed coated and uncoated circular disc specimens. Surface and Coat. Tech. 357 (2019) 497-514. https://doi.org/10.1016/j.surfcoat.2018.10.053

[11] N. K. Gupta, G.S. Sekhon, P.K. Gupta, Study of lateral compression of round metallic tubes. Thin-walled structures 43 (2005) 895-922. https://doi.org/10.1016/j.tws.2004.12.002

[12] Y. Coget, Y. Demarty, C. Czarnota, A. Bracq, J.S. Brest, A. Rusinek, Lateral ring compression test applied to a small calibre steel jacket : Identification of a constitutive model. Defence technology. In Press. https://doi.org/10.1016/j.dt.2023.11.001

[13] L. Bizet, L. Charleux, P. Balland, L. Tabourot, Influence of heterogeneities introduces into the modelling of a ring compression test. Arch. of Civ. and Mech Eng. 17 (2017) 365-374. http://dx.doi.org/10.1016/j.acme.2016.11.002

[14] R. Abrahams, J. Mikhail, P. Fasihi, Effect of friction stir process parameters on the mechanical properties of 5005-H34 and 7075-T651 aluminium alloy. Mater. Sci. and Eng. A 751 (2019) 363-373. https://doi.org/10.1016/j.msea.2019.02.065

[15] I. Carneiro, S. Simões, Recent advances in EBSD characterization of metals. Metals (2020) 10, 1097. https://doi.org/10.3390/met10081097

[16] H. Francillette, Effects of the interaction length between hollow shapes : Application to a HCP metal under mechanical loading. Procedia Manufacturing 47 (2020) 1203-1210. https://doi.org/10.1016/j.promfg.2020.04.351

[17] H. Francillette, B. Bacroix, M. Gaspérini, J.L. Béchade, Grain orientation effects in Zr702α polycristalline samples deformed in channel die compression at room temperature. Acta mater. 46 (1998), 4131-4142. https://doi.org/10.1016/S1359-6454(98)00121-9

[18] G. Zhao, M. Sun, J. Li, H. Li, L. Ma, Y. Li, Study on quasi-in-situ tensile microstructure evolution law of 5052-O aluminium alloy based on EBSD. Materials Today Communications 33 (2022) 104572. https://doi.org/10.1016/j.mtcomm.2022.104572

[19] J. Xie, Y. Liang, F. Yan, J. Zhou, P. Du, W. Liu. Effect of high-pressure heat treatment on the recrystallization and recrystallization texture of CC 5005 aluminium alloy. Journ. of Mater. Res. Tech. 26 (2023) 2017-2027. https://doi.org/10.1016/j.jmrt.2023.08.018
# Rope-driven biaxial tensile testing apparatus with specified load ratio

TAKIZAWA Hideo¹,ᵃ*, FUJIYA Kazuto²

¹Department of Mechanical Engineering, Nippon Institute of Technology, 4-1 Gakuendai, Miyashiro-machi, Saitama, 345-8501, Japan

²Graduate student, Graduate School, Nippon Institute of Technology, ibid., Japan

ᵃhtaki@nit.ac.jp

**Keywords:** Biaxial Tensile Test, Cruciform Specimen, Sheet Metal, Anisotropy, Pulley, Rope

**Abstract.** A novel biaxial tensile apparatus that can be installed on a universal testing machine is proposed. A tensile load was applied using a looped rope and pulley. The biaxial load can be changed by varying the angle of the rope hanging on the movable pulleys, and the ratio of the biaxial tensile force can be specified. A small cruciform specimen with a stress evaluation area of 12.5 mm square was used for preliminary tests. Biaxial tensile tests were conducted on mild steel sheets to measure the stress points on the contours of equal plastic work. A comparison with the yield surface measured using a standard biaxial tensile testing machine revealed that biaxial tensile tests using the proposed method can be performed appropriately.

## Introduction

To improve the accuracy of sheet metal forming analysis, an anisotropic yield surface can be modeled based on the results of material tests in a biaxial stress state. A biaxial tensile test using a cruciform specimen was standardized by ISO 16842:2021 [1]; however, a specialized and expensive testing machine was required to load a constant biaxial stress ratio [2,3,4].

To address this issue, Ferron et. al. [5], Nagayasu et. al. [6], and Merklein et. al. [7] developed an apparatus that could be installed in a general-purpose tension-compression testing machine. These apparatuses were designed to be loaded at a constant biaxial displacement ratio using a link mechanism. Using this type of apparatus, the ratio of the tensile displacement of the arms of the cruciform specimen can be modified by recombining the link mechanism. However, in the case of biaxial plastic deformation, the histories of the strain and stress paths generally differ depending on the cases of constant displacement, strain, and stress ratios. The yield surface, which can be measured using a biaxial tensile test with a constant displacement ratio, is limited to a stress state close to equibiaxial stress.

In this study, we present the concept of a novel biaxial tensile testing apparatus driven by pulleys and looped rope and report preliminary measurements of the yield surface (stress points on the contour of equal plastic work) of mild steel using the proposed apparatus. The measured stress points were compared with the yield surface modeled by a standard biaxial tensile testing machine.

<figure><img src="image_1017.png" /></figure>
# Rope-driven biaxial tensile testing apparatus with specified load ratio

TAKIZAWA Hideo¹,ᵃ*, FUJIYA Kazuto²

¹Department of Mechanical Engineering, Nippon Institute of Technology, 4-1 Gakuendai, Miyashiro-machi, Saitama, 345-8501, Japan

²Graduate student, Graduate School, Nippon Institute of Technology, ibid., Japan

ᵃhtaki@nit.ac.jp

**Keywords:** Biaxial Tensile Test, Cruciform Specimen, Sheet Metal, Anisotropy, Pulley, Rope

**Abstract.** A novel biaxial tensile apparatus that can be installed on a universal testing machine is proposed. A tensile load was applied using a looped rope and pulley. The biaxial load can be changed by varying the angle of the rope hanging on the movable pulleys, and the ratio of the biaxial tensile force can be specified. A small cruciform specimen with a stress evaluation area of 12.5 mm square was used for preliminary tests. Biaxial tensile tests were conducted on mild steel sheets to measure the stress points on the contours of equal plastic work. A comparison with the yield surface measured using a standard biaxial tensile testing machine revealed that biaxial tensile tests using the proposed method can be performed appropriately.

## Introduction

To improve the accuracy of sheet metal forming analysis, an anisotropic yield surface can be modeled based on the results of material tests in a biaxial stress state. A biaxial tensile test using a cruciform specimen was standardized by ISO 16842:2021 [1]; however, a specialized and expensive testing machine was required to load a constant biaxial stress ratio [2,3,4].

To address this issue, Ferron et. al. [5], Nagayasu et. al. [6], and Merklein et. al. [7] developed an apparatus that could be installed in a general-purpose tension-compression testing machine. These apparatuses were designed to be loaded at a constant biaxial displacement ratio using a link mechanism. Using this type of apparatus, the ratio of the tensile displacement of the arms of the cruciform specimen can be modified by recombining the link mechanism. However, in the case of biaxial plastic deformation, the histories of the strain and stress paths generally differ depending on the cases of constant displacement, strain, and stress ratios. The yield surface, which can be measured using a biaxial tensile test with a constant displacement ratio, is limited to a stress state close to equibiaxial stress.

In this study, we present the concept of a novel biaxial tensile testing apparatus driven by pulleys and looped rope and report preliminary measurements of the yield surface (stress points on the contour of equal plastic work) of mild steel using the proposed apparatus. The measured stress points were compared with the yield surface modeled by a standard biaxial tensile testing machine.

<figure><img src="image_1022.png" /></figure>
## Outline of proposed apparatus

<figure><img src="image_1018.png" /><figcaption>**Fig. 1** Schematic for proposed biaxial tensile testing apparatus using a looped rope and pulleys, installed in conventional uniaxial tensile testing machine.</figcaption></figure>

Fig. 1 shows the configuration of the proposed biaxial tensile apparatus. As shown in Fig. 1(a), a cruciform specimen was placed at the center of the apparatus, and a tensile load was applied from the top, bottom, left, and right of the figure. The first to fourth quadrants of the figure show the test configurations for different load ratios of $F_1: F_2 = 4:1, 2:1, 4:3$, and $1:1$.

On the central side of each tensile axis, the tensile grip can freely move along the linear rail on the axis, as shown in Fig. 1(b). A load cell and movable pulley are connected to the back end of the grip. In this apparatus, the rope was hung on a movable pulley on a linear rail, and the tensile load on each axis was applied by the rope tension. Because the rope was single and looped, the tensile force generated in the rope was constant at all positions on the rope. In the left and right axes in

<figure><img src="image_1019.png" /><figcaption>**Fig. 2** Photograph of prototype biaxial tensile apparatus installed in general purpose tensile testing machine.</figcaption></figure>
## Outline of proposed apparatus

<figure><img src="image_1023.png" /><figcaption>**Fig. 1** Schematic for proposed biaxial tensile testing apparatus using a looped rope and pulleys, installed in conventional uniaxial tensile testing machine.</figcaption></figure>

Fig. 1 shows the configuration of the proposed biaxial tensile apparatus. As shown in Fig. 1(a), a cruciform specimen was placed at the center of the apparatus, and a tensile load was applied from the top, bottom, left, and right of the figure. The first to fourth quadrants of the figure show the test configurations for different load ratios of $F_1: F_2 = 4:1, 2:1, 4:3$, and $1:1$.

On the central side of each tensile axis, the tensile grip can freely move along the linear rail on the axis, as shown in Fig. 1(b). A load cell and movable pulley are connected to the back end of the grip. In this apparatus, the rope was hung on a movable pulley on a linear rail, and the tensile load on each axis was applied by the rope tension. Because the rope was single and looped, the tensile force generated in the rope was constant at all positions on the rope. In the left and right axes in

<figure><img src="image_1024.png" /><figcaption>**Fig. 2** Photograph of prototype biaxial tensile apparatus installed in general purpose tensile testing machine.</figcaption></figure>
Fig. 1(a), the rope under tension is parallel to the tensile axis, and in the top and bottom axes, the rope is hung on the movable pulley with the angle $\theta$ specified by the positions of the fixed pulleys. Consequently, the ratios of the left and right loads $F_1$ and the top and bottom loads $F_2$ acting on the cruciform specimen are

$$F_1: F_2 = 1: \sin\theta. \qquad (1)$$

In this apparatus, the load ratio of the biaxial tension can be arbitrarily set by determining the path of the looped rope based on the positions of the fixed pulleys on the base plate. In addition, please note that in this explanation the change in angle $\theta$ due to slight deformation of the specimen and the friction of the pulleys are ignored.

A photograph of the prototype apparatus is shown in Fig. 2. As shown in Fig. 2(a), the base plate of the apparatus is fixed to the table of a general-purpose tensile-testing machine (Autograph AG-100kN, Shimadzu Corporation). The looped rope was hung on four pulleys fixed to the bottom face of the crosshead, and a tensile load was applied to the rope by raising the crosshead. The rope was made of high-strength fibers (Vectran SV-26), and the maximum test load of the biaxial tension was designed to be 5 kN, based on the strength of the tied rope. Fig. 2(b) shows an enlarged image of the test section. Wedge grips with 5kN capacity (MARK-10 G1061) were used in four directions.

## Test material and testing conditions

To evaluate the validity of the measurement using the proposed apparatus, specimens were cut from an identical lot of mild steel sheets, which yield surface (contour of equal plastic work) was measured with a standard biaxial tensile testing machine by Kuwabara et. al. [8]. The test material is an SPCD (equivalent to ISO 3574 CR2 steel) with a nominal thickness of 1.0 mm.

The dimensions of the cruciform specimen specified by ISO 16842 [1,9] were reduced to a small size with a stress evaluation area of 12.5 mm square and cut by fiber laser processing. However, the 0.15 mm width of the slits on the arms was relatively large for the small specimen, and the strength of the arm to apply a tensile load was insufficient. Strain was measured using the smallest biaxial strain gauge (TML, FCAB-1) attached to a specified position in the stress evaluation area of the specimen.

<figure><img src="image_1020.png" /><figcaption>Fig. 3 Dimensions of small cruciform specimen with 12.5mm square stress evaluation area.</figcaption></figure>
Fig. 1(a), the rope under tension is parallel to the tensile axis, and in the top and bottom axes, the rope is hung on the movable pulley with the angle $\theta$ specified by the positions of the fixed pulleys. Consequently, the ratios of the left and right loads $F_1$ and the top and bottom loads $F_2$ acting on the cruciform specimen are

$$F_1: F_2 = 1: \sin\theta. \qquad (1)$$

In this apparatus, the load ratio of the biaxial tension can be arbitrarily set by determining the path of the looped rope based on the positions of the fixed pulleys on the base plate. In addition, please note that in this explanation the change in angle $\theta$ due to slight deformation of the specimen and the friction of the pulleys are ignored.

A photograph of the prototype apparatus is shown in Fig. 2. As shown in Fig. 2(a), the base plate of the apparatus is fixed to the table of a general-purpose tensile-testing machine (Autograph AG-100kN, Shimadzu Corporation). The looped rope was hung on four pulleys fixed to the bottom face of the crosshead, and a tensile load was applied to the rope by raising the crosshead. The rope was made of high-strength fibers (Vectran SV-26), and the maximum test load of the biaxial tension was designed to be 5 kN, based on the strength of the tied rope. Fig. 2(b) shows an enlarged image of the test section. Wedge grips with 5kN capacity (MARK-10 G1061) were used in four directions.

## Test material and testing conditions

To evaluate the validity of the measurement using the proposed apparatus, specimens were cut from an identical lot of mild steel sheets, which yield surface (contour of equal plastic work) was measured with a standard biaxial tensile testing machine by Kuwabara et. al. [8]. The test material is an SPCD (equivalent to ISO 3574 CR2 steel) with a nominal thickness of 1.0 mm.

The dimensions of the cruciform specimen specified by ISO 16842 [1,9] were reduced to a small size with a stress evaluation area of 12.5 mm square and cut by fiber laser processing. However, the 0.15 mm width of the slits on the arms was relatively large for the small specimen, and the strength of the arm to apply a tensile load was insufficient. Strain was measured using the smallest biaxial strain gauge (TML, FCAB-1) attached to a specified position in the stress evaluation area of the specimen.

<figure><img src="image_1025.png" /><figcaption>Fig. 3 Dimensions of small cruciform specimen with 12.5mm square stress evaluation area.</figcaption></figure>
The stress ratio was set to $\sigma_x: \sigma_y=4:1, 2:1, 4:3, 1:1, 3:4, 1:2$, and $1:4$ as the stress in the rolling direction (RD) $\sigma_x$ and transvesal direction (TD) $\sigma_y$, respectively. For a stress ratio of $\sigma_x: \sigma_y=1:1$, experiments were conducted in which the RD and TD were swapped between the horizontal and vertical axes of the testing apparatus.

## Results and Discussions

Fig. 4 shows the true stress-true plastic strain curves obtained using the proposed biaxial tensile apparatus. The graphs in the upper row show the results of the experiment in which the RD of the sheet metal was parallel to the $F_1$ axis of the apparatus, and the lower row shows the results in which the TD was parallel to $F_1$.

The curves represent the results of three experiments under each condition. Because a small specimen was used, the variations in the measured stress-strain curves were significant even under the same conditions; however, the tendency of the experimental results was consistent.

Based on the specific plastic work obtained by integrating the true stress-true plastic strain curve, the stress points on the contour of the equal plastic work were determined. The magnitude of plastic work is shown as the reference plastic strain, which is the tensile plastic strain of the RD uniaxial tension.

Fig. 5(a) shows the validation of the linearity of the stress paths of the proposed apparatus. Broken lines indicate the intended stress paths. The experimental stress paths of the solid line run along the intended line. The biaxial tensile tests with almost constant stress ratios have been achieved.

Fig. 5(b) shows the experimentally measured stress points. The stress points were limited to a maximum reference plastic strain of 0.03 due to the early fracture of the arms. The curves in the figure show the yield loci measured with a standard biaxial tensile testing machine with hydraulic power and modeled by Kuwabara et. al. [9] using Yld2000-2d. The stress points measured using the proposed apparatus are close to the reference yield surface.

The stress points measured in this experiment are slightly higher than those of the reference yield loci. Because the size of the specimen was very small, variations in the dimensions of the

<figure><img src="image_1021.png" /><figcaption>Fig. 4 True stress-true plastic strain curves measured by prototype of proposed biaxial tensile testing machine (upper: RD is parallel to axis, lower: TD is parallel to axis)</figcaption></figure>
The stress ratio was set to $\sigma_x: \sigma_y=4:1, 2:1, 4:3, 1:1, 3:4, 1:2$, and $1:4$ as the stress in the rolling direction (RD) $\sigma_x$ and transvesal direction (TD) $\sigma_y$, respectively. For a stress ratio of $\sigma_x: \sigma_y=1:1$, experiments were conducted in which the RD and TD were swapped between the horizontal and vertical axes of the testing apparatus.

## Results and Discussions

Fig. 4 shows the true stress-true plastic strain curves obtained using the proposed biaxial tensile apparatus. The graphs in the upper row show the results of the experiment in which the RD of the sheet metal was parallel to the $F_1$ axis of the apparatus, and the lower row shows the results in which the TD was parallel to $F_1$.

The curves represent the results of three experiments under each condition. Because a small specimen was used, the variations in the measured stress-strain curves were significant even under the same conditions; however, the tendency of the experimental results was consistent.

Based on the specific plastic work obtained by integrating the true stress-true plastic strain curve, the stress points on the contour of the equal plastic work were determined. The magnitude of plastic work is shown as the reference plastic strain, which is the tensile plastic strain of the RD uniaxial tension.

Fig. 5(a) shows the validation of the linearity of the stress paths of the proposed apparatus. Broken lines indicate the intended stress paths. The experimental stress paths of the solid line run along the intended line. The biaxial tensile tests with almost constant stress ratios have been achieved.

Fig. 5(b) shows the experimentally measured stress points. The stress points were limited to a maximum reference plastic strain of 0.03 due to the early fracture of the arms. The curves in the figure show the yield loci measured with a standard biaxial tensile testing machine with hydraulic power and modeled by Kuwabara et. al. [9] using Yld2000-2d. The stress points measured using the proposed apparatus are close to the reference yield surface.

The stress points measured in this experiment are slightly higher than those of the reference yield loci. Because the size of the specimen was very small, variations in the dimensions of the

<figure><img src="image_1026.png" /><figcaption>Fig. 4 True stress-true plastic strain curves measured by prototype of proposed biaxial tensile testing machine (upper: RD is parallel to axis, lower: TD is parallel to axis)</figcaption></figure>