The fitting results are shown in Fig. 4. The identified constitutive parameters for the extended Voce hardening model are listed in Table 2. The PTR model parameters in Eq. (3) are identified as A=0.7 and B=1 for all the testing temperatures. It is worthwhile to note that the Voce hardening parameters for the tensile detwin are identified by fitting to the cyclic loading data obtained from the in-plane cyclic tests, which will appear later on.

<figure><img src="image_1166.png" /><figcaption>Fig. 4. Measured and simulated stress-strain curves under (a) uniaxial tension and (b) uniaxial compression, and (c) r-values under uniaxial tension</figcaption></figure>

<table><caption>Table 2 Identified constitutive parameters for the extended Voce hardening model</caption><thead><tr><th>Temperature</th><th>Slip/Twin Systems</th><th>τ<sub>0</sub></th><th>τ<sub>1</sub></th><th>θ<sub>0</sub></th><th>θ<sub>1</sub></th><th>q<sub>ij</sub></th></tr></thead><tbody><tr><td rowspan="5">25</td><td>Basal</td><td>25</td><td>1</td><td>90</td><td>0</td><td>2.5</td></tr><tr><td>Prismatic</td><td>105</td><td>21</td><td>390</td><td>10</td><td>2.5</td></tr><tr><td>Pyramidal &lt;a+c&gt;</td><td>108</td><td>29</td><td>440</td><td>15</td><td>4</td></tr><tr><td>Tensile twin</td><td>28</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td>Tensile detwin</td><td>21</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td rowspan="5">100</td><td>Basal</td><td>25</td><td>1</td><td>90</td><td>0</td><td>2.5</td></tr><tr><td>Prismatic</td><td>74</td><td>16</td><td>280</td><td>7.5</td><td>2.5</td></tr><tr><td>Pyramidal &lt;a+c&gt;</td><td>76</td><td>20.5</td><td>320</td><td>11</td><td>4</td></tr><tr><td>Tensile twin</td><td>28</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td>Tensile detwin</td><td>21</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td rowspan="5">150</td><td>Basal</td><td>25</td><td>1</td><td>90</td><td>0</td><td>2.5</td></tr><tr><td>Prismatic</td><td>50</td><td>12</td><td>185</td><td>6</td><td>2.5</td></tr><tr><td>Pyramidal &lt;a+c&gt;</td><td>51</td><td>15</td><td>210</td><td>8.5</td><td>4</td></tr><tr><td>Tensile twin</td><td>28</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td>Tensile detwin</td><td>21</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td rowspan="5">200</td><td>Basal</td><td>25</td><td>1</td><td>90</td><td>0</td><td>2.5</td></tr><tr><td>Prismatic</td><td>33</td><td>7</td><td>130</td><td>3</td><td>2.5</td></tr><tr><td>Pyramidal &lt;a+c&gt;</td><td>34</td><td>8</td><td>140</td><td>5</td><td>4</td></tr><tr><td>Tensile twin</td><td>28</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td>Tensile detwin</td><td>21</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr></tbody></table>
The scaling factor representing the temperature dependency is determined by fitting the identified Voce hardening parameters of the non-basal slip systems at different temperatures in Table 2. The hardening parameters are normalized to those at a reference temperature, i.e., RT. The normalized hardening parameters are fitted using a phenomenological equation, $k(T) = 1/(1 + ((T - T_R)/h_1)^{h_2})$ as $h_1=124$ and $h_2= 2.05$. The normalized hardening parameters and the fitted curve are shown in Fig. 5.

<figure><img src="image_1165.png" /><figcaption>Fig. 5. Normalized Voce hardening parameters for non-basal slip systems at different temperatures and best-fit</figcaption></figure>

The stress-strain curves under cyclic loadings, namely, compression-tension-compression (CTC) and tension-compression-tension (TCT), at RT, 100, 150, and 200°C are predicted using identified constitutive parameters and compared with the experimental data. The results are shown in Fig. 6.

During the first loading step in TCT, the tensile twin is hardly active. Therefore, the stress-strain curve is concave down shaped. During the second loading step, a stress plateau is observed at the beginning since the tensile twin is active. However, as the twin activation is terminated, non-basal slip systems are supposed to be active. Therefore, a concave down-shaped stress-strain curve is observed. During the third loading step, detwinning should be active in twinned areas during the second loading step, and a concave-up shaped stress-strain curve is observed with the stress plateau at the beginning. A similar phenomenon in terms of the stress-strain curve in consequence of twin and detwin activation in CTC. The CPFE well reproduce such abnormal stress-strain curves. The predictions at higher temperatures are also presented in Fig. 6, and the predictions are in good agreement with the measured data.
The scaling factor representing the temperature dependency is determined by fitting the identified Voce hardening parameters of the non-basal slip systems at different temperatures in Table 2. The hardening parameters are normalized to those at a reference temperature, i.e., RT. The normalized hardening parameters are fitted using a phenomenological equation, $k(T) = 1/(1 + ((T - T_R)/h_1)^{h_2})$ as $h_1=124$ and $h_2= 2.05$. The normalized hardening parameters and the fitted curve are shown in Fig. 5.

<figure><img src="image_1167.png" /><figcaption>Fig. 5. Normalized Voce hardening parameters for non-basal slip systems at different temperatures and best-fit</figcaption></figure>

The stress-strain curves under cyclic loadings, namely, compression-tension-compression (CTC) and tension-compression-tension (TCT), at RT, 100, 150, and 200°C are predicted using identified constitutive parameters and compared with the experimental data. The results are shown in Fig. 6.

During the first loading step in TCT, the tensile twin is hardly active. Therefore, the stress-strain curve is concave down shaped. During the second loading step, a stress plateau is observed at the beginning since the tensile twin is active. However, as the twin activation is terminated, non-basal slip systems are supposed to be active. Therefore, a concave down-shaped stress-strain curve is observed. During the third loading step, detwinning should be active in twinned areas during the second loading step, and a concave-up shaped stress-strain curve is observed with the stress plateau at the beginning. A similar phenomenon in terms of the stress-strain curve in consequence of twin and detwin activation in CTC. The CPFE well reproduce such abnormal stress-strain curves. The predictions at higher temperatures are also presented in Fig. 6, and the predictions are in good agreement with the measured data.
a

<figure><img src="image_1166.png" /><figcaption>Fig. 6. Measured and predicted stress-strain curves under cyclic loading conditions at (a) RT, (b) 100°C, (c) 150°C, and (d) 200°C</figcaption></figure>
a

<figure><img src="image_1168.png" /><figcaption>Fig. 6. Measured and predicted stress-strain curves under cyclic loading conditions at (a) RT, (b) 100°C, (c) 150°C, and (d) 200°C</figcaption></figure>
**Summary**

In the current work, a CPFE model to predict the mechanical behaviors of a magnesium alloy sheet under different temperature conditions and cyclic loading scenarios is developed. For such, the temperature dependency of the non-basal slip systems is modeled systematically by modifying the Voce hardening model, and the predominant twinning reorientation (PTR) scheme is further enhanced by incorporating the twinning-detwinning mechanism. The newly developed crystal plasticity model is applied in predicting mechanical responses under in-plane tensile loadings, compressive loadings, and cyclic loadings at different temperatures up to 200 °C. Finally, the predictions are compared with measured ones. The proposed model reproduces stress-strain responses under not only monotonic loadings but also cyclic loadings at various testing temperatures from room temperature to 200°C.

**Acknowledgments**

This work was supported by the Fundamental Research Program of the Korea Institute of Materials Science (KIMS, PNK9800) and the National Research Council of Science and Technology (NST) grant by the Korea government (MSIT) (CRC23011-210).

**References**

[1] S.A. Habib, A.S. Khan, T. Gnäupel-Herold, J.T. Lloyd, S.E. Schoenfeld, Anisotropy, tension-compression asymmetry and texture evolution of a rare-earth-containing magnesium alloy sheet, ZEK100, at different strain rates and temperatures: Experiments and modeling, Int. J. Plast. 95 (2017) 163–190. https://doi.org/10.1016/j.ijplas.2017.04.006

[2] H. Yoshinaga, R. Horiuchi, On the Nonbasal Slip in Magnesium Crystals, Trans. Japan Inst. Met. 5 (1964) 14–21. https://doi.org/10.2320/matertrans1960.5.14

[3] T. Obara, H. Yoshinga, S. Morozumi, {1122} (1123) Slip system in magnesium, Acta Metall. 21 (1973) 845–853. https://doi.org/10.1016/0001-6160(73)90141-7

[4] S.R. Agnew, Ö. Duygulu, Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B, Int. J. Plast. 21 (2005) 1161–1193. https://doi.org/10.1016/j.ijplas.2004.05.018

[5] H.J. Bong, X. Hu, X. Sun, Y. Ren, Mechanism-based constitutive modeling of ZEK100 magnesium alloy with crystal plasticity and in-situ HEXRD experiment, Int. J. Plast. 113 (2019) 35–51. https://doi.org/10.1016/j.ijplas.2018.09.005

[6] H.J. Bong, J. Lee, X. Hu, X. Sun, M.-G. Lee, Predicting forming limit diagrams for magnesium alloys using crystal plasticity finite elements, Int. J. Plast. 126 (2020) 102630. https://doi.org/10.1016/j.ijplas.2019.11.009

[7] A. Jain, S.R. Agnew, Modeling the temperature dependent effect of twinning on the behavior of magnesium alloy AZ31B sheet, Mater. Sci. Eng. A. 462 (2007) 29–36. https://doi.org/10.1016/j.msea.2006.03.160

[8] H. Wang, P.D. Wu, J. Wang, Modeling inelastic behavior of magnesium alloys during cyclic loading-unloading, Int. J. Plast. 47 (2013) 49–64. https://doi.org/10.1016/j.ijplas.2013.01.007

[9] H. Zhang, A. Jérusalem, E. Salvati, C. Papadaki, K.S. Fong, X. Song, A.M. Korsunsky, Multi-scale mechanisms of twinning-detwinning in magnesium alloy AZ31B simulated by crystal plasticity modeling and validated via in situ synchrotron XRD and in situ SEM-EBSD, Int. J. Plast. 119 (2019) 43–56. https://doi.org/10.1016/j.ijplas.2019.02.018
**Summary**

In the current work, a CPFE model to predict the mechanical behaviors of a magnesium alloy sheet under different temperature conditions and cyclic loading scenarios is developed. For such, the temperature dependency of the non-basal slip systems is modeled systematically by modifying the Voce hardening model, and the predominant twinning reorientation (PTR) scheme is further enhanced by incorporating the twinning-detwinning mechanism. The newly developed crystal plasticity model is applied in predicting mechanical responses under in-plane tensile loadings, compressive loadings, and cyclic loadings at different temperatures up to 200 °C. Finally, the predictions are compared with measured ones. The proposed model reproduces stress-strain responses under not only monotonic loadings but also cyclic loadings at various testing temperatures from room temperature to 200°C.

**Acknowledgments**

This work was supported by the Fundamental Research Program of the Korea Institute of Materials Science (KIMS, PNK9800) and the National Research Council of Science and Technology (NST) grant by the Korea government (MSIT) (CRC23011-210).

**References**

[1] S.A. Habib, A.S. Khan, T. Gnäupel-Herold, J.T. Lloyd, S.E. Schoenfeld, Anisotropy, tension-compression asymmetry and texture evolution of a rare-earth-containing magnesium alloy sheet, ZEK100, at different strain rates and temperatures: Experiments and modeling, Int. J. Plast. 95 (2017) 163–190. https://doi.org/10.1016/j.ijplas.2017.04.006

[2] H. Yoshinaga, R. Horiuchi, On the Nonbasal Slip in Magnesium Crystals, Trans. Japan Inst. Met. 5 (1964) 14–21. https://doi.org/10.2320/matertrans1960.5.14

[3] T. Obara, H. Yoshinga, S. Morozumi, {1122} (1123) Slip system in magnesium, Acta Metall. 21 (1973) 845–853. https://doi.org/10.1016/0001-6160(73)90141-7

[4] S.R. Agnew, Ö. Duygulu, Plastic anisotropy and the role of non-basal slip in magnesium alloy AZ31B, Int. J. Plast. 21 (2005) 1161–1193. https://doi.org/10.1016/j.ijplas.2004.05.018

[5] H.J. Bong, X. Hu, X. Sun, Y. Ren, Mechanism-based constitutive modeling of ZEK100 magnesium alloy with crystal plasticity and in-situ HEXRD experiment, Int. J. Plast. 113 (2019) 35–51. https://doi.org/10.1016/j.ijplas.2018.09.005

[6] H.J. Bong, J. Lee, X. Hu, X. Sun, M.-G. Lee, Predicting forming limit diagrams for magnesium alloys using crystal plasticity finite elements, Int. J. Plast. 126 (2020) 102630. https://doi.org/10.1016/j.ijplas.2019.11.009

[7] A. Jain, S.R. Agnew, Modeling the temperature dependent effect of twinning on the behavior of magnesium alloy AZ31B sheet, Mater. Sci. Eng. A. 462 (2007) 29–36. https://doi.org/10.1016/j.msea.2006.03.160

[8] H. Wang, P.D. Wu, J. Wang, Modeling inelastic behavior of magnesium alloys during cyclic loading-unloading, Int. J. Plast. 47 (2013) 49–64. https://doi.org/10.1016/j.ijplas.2013.01.007

[9] H. Zhang, A. Jérusalem, E. Salvati, C. Papadaki, K.S. Fong, X. Song, A.M. Korsunsky, Multi-scale mechanisms of twinning-detwinning in magnesium alloy AZ31B simulated by crystal plasticity modeling and validated via in situ synchrotron XRD and in situ SEM-EBSD, Int. J. Plast. 119 (2019) 43–56. https://doi.org/10.1016/j.ijplas.2019.02.018
[10] Q. Liu, A. Roy, V. V Silberschmidt, Temperature-dependent crystal-plasticity model for magnesium: A bottom-up approach, Mech. Mater. 113 (2017) 44-56.
https://doi.org/https://doi.org/10.1016/j.mechmat.2017.07.008

[11] M.G. Lee, J.H. Kim, D. Kim, O.S. Seo, N.T. Nguyen, H.Y. Kim, Anisotropic Hardening of Sheet Metals at Elevated Temperature: Tension-Compressions Test Development and Validation, Exp. Mech. 53 (2013) 1039-1055. https://doi.org/10.1007/s11340-012-9694-1

[12] R.J. Asaro, Geometrical effects in the inhomogeneous deformation of ductile single crystals, Acta Metall. 27 (1979) 445-453. https://doi.org/10.1016/0001-6160(79)90036-1

[13] S.R. Kalidindi, C.A. Bronkhorst, L. Anand, Crystallographic texture evolution in bulk deformation processing of FCC metals, J. Mech. Phys. Solids. 40 (1992) 537-569.
https://doi.org/10.1016/0022-5096(92)80003-9

[14] H.J. Bong, J. Lee, M.-G. Lee, Modeling crystal plasticity with an enhanced twinning-detwinning model to simulate cyclic behavior of AZ31B magnesium alloy at various temperatures, Int. J. Plast. 150 (2022) 103190. https://doi.org/10.1016/j.ijplas.2021.103190

[15] J. Singh, S. Mahesh, S. Roy, G. Kumar, D. Srivastava, G.K. Dey, N. Saibaba, I. Samajdar, Temperature dependence of work hardening in sparsely twinning zirconium, Acta Mater. 123 (2017) 337-349. https://doi.org/10.1016/j.actamat.2016.10.049

[16] H.J. Bong, J. Lee, Crystal plasticity finite element–Marciniak-Kuczynski approach with surface roughening effect in predicting formability of ultra-thin ferritic stainless steel sheets, Int. J. Mech. Sci. 191 (2021) 106066. https://doi.org/10.1016/j.ijmecsci.2020.106066

[17] H.J. Bong, J. Lee, M.-G. Lee, Study on Plastic Response Under Biaxial Tension and Its Correlation with Formability for Wrought Magnesium Alloys, JOM. 72 (2020) 2568-2577.
https://doi.org/10.1007/s11837-020-04084-w
[10] Q. Liu, A. Roy, V. V Silberschmidt, Temperature-dependent crystal-plasticity model for magnesium: A bottom-up approach, Mech. Mater. 113 (2017) 44-56.
https://doi.org/https://doi.org/10.1016/j.mechmat.2017.07.008

[11] M.G. Lee, J.H. Kim, D. Kim, O.S. Seo, N.T. Nguyen, H.Y. Kim, Anisotropic Hardening of Sheet Metals at Elevated Temperature: Tension-Compressions Test Development and Validation, Exp. Mech. 53 (2013) 1039-1055. https://doi.org/10.1007/s11340-012-9694-1

[12] R.J. Asaro, Geometrical effects in the inhomogeneous deformation of ductile single crystals, Acta Metall. 27 (1979) 445-453. https://doi.org/10.1016/0001-6160(79)90036-1

[13] S.R. Kalidindi, C.A. Bronkhorst, L. Anand, Crystallographic texture evolution in bulk deformation processing of FCC metals, J. Mech. Phys. Solids. 40 (1992) 537-569.
https://doi.org/10.1016/0022-5096(92)80003-9

[14] H.J. Bong, J. Lee, M.-G. Lee, Modeling crystal plasticity with an enhanced twinning-detwinning model to simulate cyclic behavior of AZ31B magnesium alloy at various temperatures, Int. J. Plast. 150 (2022) 103190. https://doi.org/10.1016/j.ijplas.2021.103190

[15] J. Singh, S. Mahesh, S. Roy, G. Kumar, D. Srivastava, G.K. Dey, N. Saibaba, I. Samajdar, Temperature dependence of work hardening in sparsely twinning zirconium, Acta Mater. 123 (2017) 337-349. https://doi.org/10.1016/j.actamat.2016.10.049

[16] H.J. Bong, J. Lee, Crystal plasticity finite element–Marciniak-Kuczynski approach with surface roughening effect in predicting formability of ultra-thin ferritic stainless steel sheets, Int. J. Mech. Sci. 191 (2021) 106066. https://doi.org/10.1016/j.ijmecsci.2020.106066

[17] H.J. Bong, J. Lee, M.-G. Lee, Study on Plastic Response Under Biaxial Tension and Its Correlation with Formability for Wrought Magnesium Alloys, JOM. 72 (2020) 2568-2577.
https://doi.org/10.1007/s11837-020-04084-w
# Enhanced formability in two-step forming for AA7075 sheet in -T6 and -W tempers

HA Jinjin<fnref n="1" />, a*, CHOI Yumi<fnref n="2" />, b, LEE Myoung-Gyu<fnref n="2" />, c, and KORKOLIS Yannis P.<fnref n="3" />, d

**Keywords:** Formability, Multi-Step Forming, Aa7075, Hole Expansion, Anisotropy

**Abstract.** In response to stringent environmental regulations, the automotive industry is increasingly prioritizing lightweighting, prompting a shift towards high-strength aluminum alloys while the low formability of these alloys remain a limiting factor. This study explores a solution through a two-step forming process applied to AA7075 sheets utilizing -T6 and -W tempers. Firstly, two-step uniaxial tension experiments are performed at two prestraining levels in the -T6 temper followed by subsequent tensions in the -W. Both cases exhibit significant plastic deformation before fracture, overcoming the thinning accumulated in the first step. Additionally, a two-step hole expansion experiment is conducted under the same tempering conditions. Results are compared with single operations in each temper, evaluating force-displacement curves and thickness strain distribution around the hole. The study highlights the substantial contribution to formability enhancement, demonstrating 80% higher cup height and twice greater thinning to fracture compared to conventional single-step operations.

## Introduction

The auto industry's focus on lightweighting, driven by environmental concerns, has escalated the demand for aluminum alloys, particularly the high-strength 6xxx and 7xxx series, as substitutes for steel components. However, their limited ductility requires technical interventions, such as forming at elevated temperatures or specific tempering conditions, to enhance formability. Tackling these formability challenges is crucial for advancing environmentally conscious and efficient automotive solutions, necessitating innovative approaches and tailored processes within the industry [1-3].

One metric to evaluate the stretch flange formability is expanding of a circular hole in a sheet sample [4,5]. In the hole expansion experiment, the free edge of the hole is stretched uniaxially along a circumferential direction but the compatibility with surrounding materials enables larger deformation near the edge than in a simple uniaxial tension test [4,6]. As a result, the simple uniaxial tension experiment always underestimates a formability limit, which represents an inappropriateness as an evaluation method for the stretch flange formability.

In this paper, the formability improvement in two-step forming is investigated by utilizing benefits of different tempering conditions in AA7075 aluminum sheet. Two experiments of each uniaxial tension and hole expansion in two-step operation are performed with combination of -T6 and -W tempers and the results are compared with single operation of individual tempers. The comparison shows significant improvement of formability in two-step forming with two tempers.

## Material characterization in uniaxial tension

Mechanical property of AA7075 in -T6 (UTT₆) and -W (UTW) is characterized by uniaxial tension experiment in 0°, 45°, and 90° from the rolling direction (RD). The material is received in -T6 and

<figure><img src="image_1167.png" /></figure>

<footnote n="1">¹University of New Hampshire, Mechanical Engineering, Durham, NH 03824, USA</footnote>

<footnote n="2">²Seoul National University, Materials Science and Engineering, Seoul 08826, Korea</footnote>

<footnote n="3">³TU Dortmund University, Institut für Umformtechnik und Leichtbau, Dortmund 44227, Germany</footnote>

<footnote>ajinjin.ha@unh.edu, byumichoi@snu.ac.kr, myounglee@snu.ac.kr,</footnote>

<footnote>dyannis.korkolis@iul.tu-dortmund.de</footnote>
# Enhanced formability in two-step forming for AA7075 sheet in -T6 and -W tempers

HA Jinjin<fnref n="1" />, a*, CHOI Yumi<fnref n="2" />, b, LEE Myoung-Gyu<fnref n="2" />, c, and KORKOLIS Yannis P.<fnref n="3" />, d

**Keywords:** Formability, Multi-Step Forming, Aa7075, Hole Expansion, Anisotropy

**Abstract.** In response to stringent environmental regulations, the automotive industry is increasingly prioritizing lightweighting, prompting a shift towards high-strength aluminum alloys while the low formability of these alloys remain a limiting factor. This study explores a solution through a two-step forming process applied to AA7075 sheets utilizing -T6 and -W tempers. Firstly, two-step uniaxial tension experiments are performed at two prestraining levels in the -T6 temper followed by subsequent tensions in the -W. Both cases exhibit significant plastic deformation before fracture, overcoming the thinning accumulated in the first step. Additionally, a two-step hole expansion experiment is conducted under the same tempering conditions. Results are compared with single operations in each temper, evaluating force-displacement curves and thickness strain distribution around the hole. The study highlights the substantial contribution to formability enhancement, demonstrating 80% higher cup height and twice greater thinning to fracture compared to conventional single-step operations.

## Introduction

The auto industry's focus on lightweighting, driven by environmental concerns, has escalated the demand for aluminum alloys, particularly the high-strength 6xxx and 7xxx series, as substitutes for steel components. However, their limited ductility requires technical interventions, such as forming at elevated temperatures or specific tempering conditions, to enhance formability. Tackling these formability challenges is crucial for advancing environmentally conscious and efficient automotive solutions, necessitating innovative approaches and tailored processes within the industry [1-3].

One metric to evaluate the stretch flange formability is expanding of a circular hole in a sheet sample [4,5]. In the hole expansion experiment, the free edge of the hole is stretched uniaxially along a circumferential direction but the compatibility with surrounding materials enables larger deformation near the edge than in a simple uniaxial tension test [4,6]. As a result, the simple uniaxial tension experiment always underestimates a formability limit, which represents an inappropriateness as an evaluation method for the stretch flange formability.

In this paper, the formability improvement in two-step forming is investigated by utilizing benefits of different tempering conditions in AA7075 aluminum sheet. Two experiments of each uniaxial tension and hole expansion in two-step operation are performed with combination of -T6 and -W tempers and the results are compared with single operation of individual tempers. The comparison shows significant improvement of formability in two-step forming with two tempers.

## Material characterization in uniaxial tension

Mechanical property of AA7075 in -T6 (UTT₆) and -W (UTW) is characterized by uniaxial tension experiment in 0°, 45°, and 90° from the rolling direction (RD). The material is received in -T6 and

<figure><img src="image_1169.png" /></figure>

<footnote n="1">¹University of New Hampshire, Mechanical Engineering, Durham, NH 03824, USA</footnote>

<footnote n="2">²Seoul National University, Materials Science and Engineering, Seoul 08826, Korea</footnote>

<footnote n="3">³TU Dortmund University, Institut für Umformtechnik und Leichtbau, Dortmund 44227, Germany</footnote>

<footnote>ajinjin.ha@unh.edu, byumichoi@snu.ac.kr, myounglee@snu.ac.kr,</footnote>

<footnote>dyannis.korkolis@iul.tu-dortmund.de</footnote>
then heat treated to alter the temper to -W as shown in Fig. 1 [7]. The timing is rigorously maintained for a duration of 15 minutes, as the -W temper condition is temporary and highly time sensitive, and significant changes in mechanical properties can occur after a certain time. A constant crosshead velocity of 3.6 mm/min is applied in the both monotonic and two-step uniaxial tension experiments, inducing the strain rate of ~10⁻³/s, assuming the deformation is uniform in the gauge area for both tempers.

The stress-strain curves from monotonic uniaxial tension under each temper are depicted in Fig. 2a [8]. UTₜ₆ in the RD exhibits the highest yield stress at approximately 530 MPa and ultimate tensile strength around 590 MPa. However, its uniform elongation is only 10%, presenting a challenge in terms of formability. In contrast, UT<sub>w</sub> in the RD displays significantly lower flow stress than UT<sub>t6</sub>, e.g., yield stress around 165 MPa, but demonstrates greater ductility until fracture, signifying a meaningful improvement in formability under uniaxial tension in the -W temper. It's noteworthy that the engineering stress and strain of UT<sub>w</sub> are calculated without considering deformation heterogeneity in the gauge region caused by Portevin-Le Chatelier (PLC) bands [9]. Additionally, plastic anisotropy in r-value, an indicator of thinning resistance during uniaxial tension deformation, is compared in Fig. 2b. The values are highest at 45° and lowest in the RD in both tempers, and the temper-based difference is minor. Thus, under the given conditions, plastic anisotropy in r-value is more pronounced than in flow stress for both tempers, highlighting the greater influence of temper changes on stress-strain behavior than on r-value.

<figure><img src="image_1168.png" /><figcaption>Figure 1. Heat treatment for monotonic uniaxial tension in -W temper</figcaption></figure>

<figure><img src="image_1169.png" /><figcaption>Figure 2. Mechanical properties of the AA7075 in -T6 (UT<sub>T6</sub>) and -W (UT<sub>w</sub>): (a) engineering stress-strain curves and (b) r-values in the three orientations</figcaption></figure>

## Two-step uniaxial tension

To evaluate formability improvement with -T6 and -W tempers, two-step uniaxial tension experiment is first performed with different prestraining and investigates the effect of prior deformation on the heat treatment and subsequent loading. ASTM E8 specimen in -T6 is stretched
then heat treated to alter the temper to -W as shown in Fig. 1 [7]. The timing is rigorously maintained for a duration of 15 minutes, as the -W temper condition is temporary and highly time sensitive, and significant changes in mechanical properties can occur after a certain time. A constant crosshead velocity of 3.6 mm/min is applied in the both monotonic and two-step uniaxial tension experiments, inducing the strain rate of ~10⁻³/s, assuming the deformation is uniform in the gauge area for both tempers.

The stress-strain curves from monotonic uniaxial tension under each temper are depicted in Fig. 2a [8]. UTₜ₆ in the RD exhibits the highest yield stress at approximately 530 MPa and ultimate tensile strength around 590 MPa. However, its uniform elongation is only 10%, presenting a challenge in terms of formability. In contrast, UT<sub>w</sub> in the RD displays significantly lower flow stress than UT<sub>t6</sub>, e.g., yield stress around 165 MPa, but demonstrates greater ductility until fracture, signifying a meaningful improvement in formability under uniaxial tension in the -W temper. It's noteworthy that the engineering stress and strain of UT<sub>w</sub> are calculated without considering deformation heterogeneity in the gauge region caused by Portevin-Le Chatelier (PLC) bands [9]. Additionally, plastic anisotropy in r-value, an indicator of thinning resistance during uniaxial tension deformation, is compared in Fig. 2b. The values are highest at 45° and lowest in the RD in both tempers, and the temper-based difference is minor. Thus, under the given conditions, plastic anisotropy in r-value is more pronounced than in flow stress for both tempers, highlighting the greater influence of temper changes on stress-strain behavior than on r-value.

<figure><img src="image_1170.png" /><figcaption>Figure 1. Heat treatment for monotonic uniaxial tension in -W temper</figcaption></figure>

<figure><img src="image_1171.png" /><figcaption>Figure 2. Mechanical properties of the AA7075 in -T6 (UT<sub>T6</sub>) and -W (UT<sub>w</sub>): (a) engineering stress-strain curves and (b) r-values in the three orientations</figcaption></figure>

## Two-step uniaxial tension

To evaluate formability improvement with -T6 and -W tempers, two-step uniaxial tension experiment is first performed with different prestraining and investigates the effect of prior deformation on the heat treatment and subsequent loading. ASTM E8 specimen in -T6 is stretched
with two grip displacements $δ_{UT_1}$=5 mm and $δ_{UT_1}$=8 mm in the first step (UT$_{1,T6}$), and then unloaded for the solution heat treatment to -W temper. After 15 minutes of holding in air, the subsequent uniaxial tension in -W (UT$_{2,W}$) is performed until the fracture. The schematic is shown in Fig. 3.

Figure 4 shows the engineering stress-strain curves of two-step uniaxial tension experiments. Two prestraining levels applied by $δ_{UT_1}$=5 mm and $δ_{UT_1}$=8 mm in the UT$_{1,T6}$ are led to different conditions: $δ_{UT_1}$=5 mm produces uniform deformation of 0.073 true strain in average while $δ_{UT_1}$=8 mm triggers the diffused necking with varying true strain range from 0.1 to 0.12 as shown in the strain field measured by stereo-type digital image correlation (DIC) system. Nevertheless, the both engineering stress-strain curves of UT$_{2,W}$ can reach significant deformation until the fracture and even UT$_{2,W}$ with $δ_{UT_1}$=5 mm shows the negligible ductility loss compared to the monotonic UT$_{W}$.

<figure><img src="image_1170.png" /><figcaption>Figure 3. Schematic for two-step uniaxial tension in -T6 (UT$_{1,T6}$) and -W (UT$_{2,W}$) tempers</figcaption></figure>

<figure><img src="image_1171.png" /><figcaption>Figure 4. Engineering stress-strain curves of monotonic (gray dashed) in -T6 and -W tempers and two-step uniaxial tensions (red and blue solid) of $δ_{UT_1}$=5 and 8 mm and the axial strain fields</figcaption></figure>

## Two-step hole expansion

In addition to the two-step uniaxial tension, two-step hole expansion experiment is performed with the same heat treatment and prestraining approach with the two-step uniaxial tension as in Fig. 5. A double-action hydraulic Greenerd press in Fig. 6 is used with the stereo-type DIC system to capture the surface strain field [10,11]. A square specimen with a circular hole of 17.5 mm diameter is prepared by end-milling for the best surface quality especially for the hole edge. The punch face at the contact with a blank is lubricated using diluted Drawsol. A constant punch velocity of ~21 mm/min is applied in the hole expansion experiments for both tempers, assuming
with two grip displacements $δ_{UT_1}$=5 mm and $δ_{UT_1}$=8 mm in the first step (UT$_{1,T6}$), and then unloaded for the solution heat treatment to -W temper. After 15 minutes of holding in air, the subsequent uniaxial tension in -W (UT$_{2,W}$) is performed until the fracture. The schematic is shown in Fig. 3.

Figure 4 shows the engineering stress-strain curves of two-step uniaxial tension experiments. Two prestraining levels applied by $δ_{UT_1}$=5 mm and $δ_{UT_1}$=8 mm in the UT$_{1,T6}$ are led to different conditions: $δ_{UT_1}$=5 mm produces uniform deformation of 0.073 true strain in average while $δ_{UT_1}$=8 mm triggers the diffused necking with varying true strain range from 0.1 to 0.12 as shown in the strain field measured by stereo-type digital image correlation (DIC) system. Nevertheless, the both engineering stress-strain curves of UT$_{2,W}$ can reach significant deformation until the fracture and even UT$_{2,W}$ with $δ_{UT_1}$=5 mm shows the negligible ductility loss compared to the monotonic UT$_{W}$.

<figure><img src="image_1172.png" /><figcaption>Figure 3. Schematic for two-step uniaxial tension in -T6 (UT$_{1,T6}$) and -W (UT$_{2,W}$) tempers</figcaption></figure>

<figure><img src="image_1173.png" /><figcaption>Figure 4. Engineering stress-strain curves of monotonic (gray dashed) in -T6 and -W tempers and two-step uniaxial tensions (red and blue solid) of $δ_{UT_1}$=5 and 8 mm and the axial strain fields</figcaption></figure>

## Two-step hole expansion

In addition to the two-step uniaxial tension, two-step hole expansion experiment is performed with the same heat treatment and prestraining approach with the two-step uniaxial tension as in Fig. 5. A double-action hydraulic Greenerd press in Fig. 6 is used with the stereo-type DIC system to capture the surface strain field [10,11]. A square specimen with a circular hole of 17.5 mm diameter is prepared by end-milling for the best surface quality especially for the hole edge. The punch face at the contact with a blank is lubricated using diluted Drawsol. A constant punch velocity of ~21 mm/min is applied in the hole expansion experiments for both tempers, assuming
the average strain rate in the hole periphery is ~10⁻³/s with ignoring the deformation heterogeneity due to anisotropy and PLC bands.

The initial hole expansion in -T6 temper (HE₁,Τ₆) involves a punch displacement of δ<sub>HE₁</sub>=13 mm, yielding a prestraining level similar to UT₁,Τ₆ with δ<sub>UT₁</sub>=8 mm in two-step uniaxial tension (see right figure in Fig. 7a for the circumferential strain field in HE₁,Τ₆). Following identical heat treatment and holding time, the subsequent hole expansion is conducted in -W temper (HE₂,W) until the fracture. The punch force-displacement curves in Fig. 7a shows significant increase of punch displacement at the onset of fracture in two-step hole expansion compared to the single operation at each temper (HE<sub>T6</sub> and HE<sub>W</sub>, respectively). Notably, the total enhancement in two-step hole expansion, in terms of punch displacement at fracture, is nearly 80% compared to the lowest case in HE<sub>T6</sub>, evidently surpassing HE<sub>W</sub> as well.

<figure><img src="image_1172.png" /><figcaption>Figure 5. Schematic for two-step hole expansion in -T6 (HE<sub>1,T6</sub>) and -W (HE<sub>2,W</sub>) tempers</figcaption></figure>

<figure><img src="image_1173.png" /><figcaption>Figure 6. Experimental setup for hole expansion: (a) machine and equipment and (b) tools</figcaption></figure>

Figure 8 shows the thickness strain variations around the hole in HE₁,Τ₆ after prestraining with δ<sub>HE₁</sub>=13 mm and in the HE₂,W at the onset of fracture in comparison of single operation HE<sub>T6</sub> and HE<sub>W</sub>. The strain is measured by the stereo-type DIC system along a hoop with initial radius of 20 mm and the extracted data is plotted with respect to the angle from the RD. As seen in Fig. 7, two-step hole expansion can reach to the highest level of thinning at the onset of fracture in the second step compared to the single operations HE<sub>T6</sub> and HE<sub>W</sub>. It should be noted that the prestraining HE<sub>1,T6</sub> is factored into the calculation of the thickness strain in HE<sub>2,W</sub>, which is crucial to account for the cumulative deformation experienced throughout the entire process. Interestingly, the maximum thinning locations remain consistent along the RD, i.e., at 180°, regardless the temper conditions. This represents the heat treatment to alter the temper from -T6 to -W does not influence
the average strain rate in the hole periphery is ~10⁻³/s with ignoring the deformation heterogeneity due to anisotropy and PLC bands.

The initial hole expansion in -T6 temper (HE₁,Τ₆) involves a punch displacement of δ<sub>HE₁</sub>=13 mm, yielding a prestraining level similar to UT₁,Τ₆ with δ<sub>UT₁</sub>=8 mm in two-step uniaxial tension (see right figure in Fig. 7a for the circumferential strain field in HE₁,Τ₆). Following identical heat treatment and holding time, the subsequent hole expansion is conducted in -W temper (HE₂,W) until the fracture. The punch force-displacement curves in Fig. 7a shows significant increase of punch displacement at the onset of fracture in two-step hole expansion compared to the single operation at each temper (HE<sub>T6</sub> and HE<sub>W</sub>, respectively). Notably, the total enhancement in two-step hole expansion, in terms of punch displacement at fracture, is nearly 80% compared to the lowest case in HE<sub>T6</sub>, evidently surpassing HE<sub>W</sub> as well.

<figure><img src="image_1174.png" /><figcaption>Figure 5. Schematic for two-step hole expansion in -T6 (HE<sub>1,T6</sub>) and -W (HE<sub>2,W</sub>) tempers</figcaption></figure>

<figure><img src="image_1175.png" /><figcaption>Figure 6. Experimental setup for hole expansion: (a) machine and equipment and (b) tools</figcaption></figure>

Figure 8 shows the thickness strain variations around the hole in HE₁,Τ₆ after prestraining with δ<sub>HE₁</sub>=13 mm and in the HE₂,W at the onset of fracture in comparison of single operation HE<sub>T6</sub> and HE<sub>W</sub>. The strain is measured by the stereo-type DIC system along a hoop with initial radius of 20 mm and the extracted data is plotted with respect to the angle from the RD. As seen in Fig. 7, two-step hole expansion can reach to the highest level of thinning at the onset of fracture in the second step compared to the single operations HE<sub>T6</sub> and HE<sub>W</sub>. It should be noted that the prestraining HE<sub>1,T6</sub> is factored into the calculation of the thickness strain in HE<sub>2,W</sub>, which is crucial to account for the cumulative deformation experienced throughout the entire process. Interestingly, the maximum thinning locations remain consistent along the RD, i.e., at 180°, regardless the temper conditions. This represents the heat treatment to alter the temper from -T6 to -W does not influence
the plastic anisotropy in the current experimental conditions while it can significantly lower the force-displacement curve. This is the same result seen in Fig. 2.

<figure><img src="image_1174.png" /><figcaption>Figure 7. Formability enhancement in two-step hole expansion: (a) punch force-displacement curves of two-step (HE₁,Τ₆ and HE₂,W) and single operation (HEΤ₆ and HEW) with the circumferential strain field after prestraining in HE₁,Τ₆ with δHE₁=13 mm and (b) specimens side view to compare the height at fracture in the lowest (HEΤ₆) and the highest (two-step: HE₁,Τ₆ followed by HE₂,W) cases</figcaption></figure>

<figure><img src="image_1175.png" /><figcaption>Figure 8. Thickness strain variation along the angle from the RD measured from a hoop with an initial radius of 20 mm</figcaption></figure>

**Summary and conclusion**

This study investigates the formability improvement through a two-step forming approach, leveraging the advantages of different tempering conditions in AA7075 aluminum sheets. Two-
the plastic anisotropy in the current experimental conditions while it can significantly lower the force-displacement curve. This is the same result seen in Fig. 2.

<figure><img src="image_1176.png" /><figcaption>Figure 7. Formability enhancement in two-step hole expansion: (a) punch force-displacement curves of two-step (HE<sub>1,T6</sub> and HE<sub>2,W</sub>) and single operation (HE<sub>T6</sub> and HE<sub>W</sub>) with the circumferential strain field after prestraining in HE<sub>1,T6</sub> with δ<sub>HE<sub>1</sub></sub>=13 mm and (b) specimens side view to compare the height at fracture in the lowest (HE<sub>T6</sub>) and the highest (two-step: HE<sub>1,T6</sub> followed by HE<sub>2,W</sub>) cases</figcaption></figure>

<figure><img src="image_1177.png" /><figcaption>Figure 8. Thickness strain variation along the angle from the RD measured from a hoop with an initial radius of 20 mm</figcaption></figure>

**Summary and conclusion**

This study investigates the formability improvement through a two-step forming approach, leveraging the advantages of different tempering conditions in AA7075 aluminum sheets. Two-
step uniaxial tension and two-step hole expansion experiments are conducted, combining -T6 and -W tempers, with results compared to single operations under each temper. The key findings are as follows:

* Formability is significantly improved in two-step forming with both -T6 and -W tempers. This underscores the efficacy of the two-step forming strategy to exploit the exiting formability in -T6 and to maximize the overall formability through the additional deformation in -W temper.

* Heat treatment during two-step forming alters the temper from -T6 and -W, reducing flow stress in subsequent deformation without affecting the plastic anisotropy in the current experiment. Consequently, the maximum thinning location in the hole expansion remains consistent in the RD leading to the initial rupture.

## Acknowledgement

This research was partially supported by the US National Science Foundation (NSF) under awards 1563216 and 1929873. JH appreciates the support from Alexander von Humboldt fellowship for experienced researcher and C. Dunn for the help to operate the hydraulic press machine. Start-up funds at The Ohio State University are acknowledged for contributing to the visit of YC. YC appreciates partial support from National Research Foundation of Korea (NRF-2012R1A5A1048294). The material is supplied by MS-Autotech.

## References

[1] O.N. Senkov, M.R. Shagiev, S. V. Senkova, D.B. Miracle, Precipitation of Al3(Sc,Zr) particles in an Al-Zn-Mg-Cu-Sc-Zr alloy during conventional solution heat treatment and its effect on tensile properties, Acta Materialia 56 (2008) 3723-3738.
https://doi.org/10.1016/j.actamat.2008.04.005

[2] E. Sáenz de Argandoña, L. Galdos, R. Ortubay, J. Mendiguren, X. Agirretxe, Room Temperature Forming of AA7075 Aluminum Alloys: W-Temper Process, Key Engineering Materials 651-653 (2015) 199-204. https://doi.org/10.4028/www.scientific.net/kem.651-653.199

[3] Y. Choi, J. Lee, S.S. Panicker, H.-K. Jin, S.K. Panda, M.-G. Lee, Mechanical properties, springback, and formability of W-temper and peak aged 7075 aluminum alloy sheets: Experiments and modeling, Int J Mech Sci 170 (2020) 105344.
https://doi.org/10.1016/J.IJMECSCI.2019.105344

[4] A. Parmar, P.B. Mellor, Plastic expansion of a circular hole in sheet metal subjected to biaxial tensile stress, Int J Mech Sci 20 (1978) 707-720. https://doi.org/10.1016/0020-7403(78)90057-7

[5] T. Kuwabara, K. Hashimoto, E. Iizuka, J.W. Yoon, Effect of anisotropic yield functions on the accuracy of hole expansion simulations, Journal of Materials Processing Technology 211 (2011) 475-481. https://doi.org/10.1016/j.jmatprotec.2010.10.025

[6] J.Y. Lee, K.J. Lee, M.G. Lee, T. Kuwabara, F. Barlat, Numerical modeling for accurate prediction of strain localization in hole expansion of a steel sheet, Int J Solids Struct 156-157 (2019) 107-118. https://doi.org/10.1016/j.ijsolstr.2018.08.005

[7] E. Sáenz de Argandoña, L. Galdos, R. Ortubay, J. Mendiguren, X. Agirretxe, Room Temperature Forming of AA7075 Aluminum Alloys: W-Temper Process, Key Engineering Materials (2015). https://doi.org/10.4028/www.scientific.net/kem.651-653.199

[8] Y. Choi, J. Ha, M.G. Lee, Y.P. Korkolis, Effect of plastic anisotropy and Portevin-Le Chatelier bands on hole-expansion in AA7075 sheets in -T6 and -W tempers, J Mater Process Technol 296 (2021) 117211. https://doi.org/10.1016/J.JMATPROTEC.2021.117211
step uniaxial tension and two-step hole expansion experiments are conducted, combining -T6 and -W tempers, with results compared to single operations under each temper. The key findings are as follows:

* Formability is significantly improved in two-step forming with both -T6 and -W tempers. This underscores the efficacy of the two-step forming strategy to exploit the exiting formability in -T6 and to maximize the overall formability through the additional deformation in -W temper.

* Heat treatment during two-step forming alters the temper from -T6 and -W, reducing flow stress in subsequent deformation without affecting the plastic anisotropy in the current experiment. Consequently, the maximum thinning location in the hole expansion remains consistent in the RD leading to the initial rupture.

## Acknowledgement

This research was partially supported by the US National Science Foundation (NSF) under awards 1563216 and 1929873. JH appreciates the support from Alexander von Humboldt fellowship for experienced researcher and C. Dunn for the help to operate the hydraulic press machine. Start-up funds at The Ohio State University are acknowledged for contributing to the visit of YC. YC appreciates partial support from National Research Foundation of Korea (NRF-2012R1A5A1048294). The material is supplied by MS-Autotech.

## References

[1] O.N. Senkov, M.R. Shagiev, S. V. Senkova, D.B. Miracle, Precipitation of Al3(Sc,Zr) particles in an Al-Zn-Mg-Cu-Sc-Zr alloy during conventional solution heat treatment and its effect on tensile properties, Acta Materialia 56 (2008) 3723-3738. https://doi.org/10.1016/j.actamat.2008.04.005

[2] E. Sáenz de Argandoña, L. Galdos, R. Ortubay, J. Mendiguren, X. Agirretxe, Room Temperature Forming of AA7075 Aluminum Alloys: W-Temper Process, Key Engineering Materials 651-653 (2015) 199-204. https://doi.org/10.4028/www.scientific.net/kem.651-653.199

[3] Y. Choi, J. Lee, S.S. Panicker, H.-K. Jin, S.K. Panda, M.-G. Lee, Mechanical properties, springback, and formability of W-temper and peak aged 7075 aluminum alloy sheets: Experiments and modeling, Int J Mech Sci 170 (2020) 105344. https://doi.org/10.1016/J.IJMECSCI.2019.105344

[4] A. Parmar, P.B. Mellor, Plastic expansion of a circular hole in sheet metal subjected to biaxial tensile stress, Int J Mech Sci 20 (1978) 707-720. https://doi.org/10.1016/0020-7403(78)90057-7

[5] T. Kuwabara, K. Hashimoto, E. Iizuka, J.W. Yoon, Effect of anisotropic yield functions on the accuracy of hole expansion simulations, Journal of Materials Processing Technology 211 (2011) 475-481. https://doi.org/10.1016/j.jmatprotec.2010.10.025

[6] J.Y. Lee, K.J. Lee, M.G. Lee, T. Kuwabara, F. Barlat, Numerical modeling for accurate prediction of strain localization in hole expansion of a steel sheet, Int J Solids Struct 156-157 (2019) 107-118. https://doi.org/10.1016/j.ijsolstr.2018.08.005

[7] E. Sáenz de Argandoña, L. Galdos, R. Ortubay, J. Mendiguren, X. Agirretxe, Room Temperature Forming of AA7075 Aluminum Alloys: W-Temper Process, Key Engineering Materials (2015). https://doi.org/10.4028/www.scientific.net/kem.651-653.199

[8] Y. Choi, J. Ha, M.G. Lee, Y.P. Korkolis, Effect of plastic anisotropy and Portevin-Le Chatelier bands on hole-expansion in AA7075 sheets in -T6 and -W tempers, J Mater Process Technol 296 (2021) 117211. https://doi.org/10.1016/J.JMATPROTEC.2021.117211
[9] Y. Choi, J. Ha, M.G. Lee, Y.P. Korkolis, Observation of Portevin-le Chatelier effect in aluminum alloy 7075-w under a heterogeneous stress field, Scr Mater 205 (2021) 114178.
https://doi.org/10.1016/J.SCRIPTAMAT.2021.114178

[10] J. Ha, S. Coppieters, Y.P. Korkolis, on the Expansion of a Circular Hole in an Orthotropic Elastoplastic Thin Sheet, International Journal of Mechanical Sciences (2020) 105706.
https://doi.org/10.1016/j.ijmecsci.2020.105706

[11] Y.P. Korkolis, B. Brownell, S. Coppieters, H. Tian, Modeling of hole-expansion of AA6022-T4 aluminum sheets with anisotropic non-quadratic yield functions, Journal of Physics: Conference Series 734 (2016). https://doi.org/10.1088/1742-6596/734/3/032083
[9] Y. Choi, J. Ha, M.G. Lee, Y.P. Korkolis, Observation of Portevin-le Chatelier effect in aluminum alloy 7075-w under a heterogeneous stress field, Scr Mater 205 (2021) 114178.
https://doi.org/10.1016/J.SCRIPTAMAT.2021.114178

[10] J. Ha, S. Coppieters, Y.P. Korkolis, on the Expansion of a Circular Hole in an Orthotropic Elastoplastic Thin Sheet, International Journal of Mechanical Sciences (2020) 105706.
https://doi.org/10.1016/j.ijmecsci.2020.105706

[11] Y.P. Korkolis, B. Brownell, S. Coppieters, H. Tian, Modeling of hole-expansion of AA6022-T4 aluminum sheets with anisotropic non-quadratic yield functions, Journal of Physics: Conference Series 734 (2016). https://doi.org/10.1088/1742-6596/734/3/032083
# Modelling ductile damage of a textured aluminum alloy based on a non-quadratic yield function

BRITO João P.<fnref n="1" />,ª, OLIVEIRA Marta C.<fnref n="1" />,ᵇ* and ALVES José Luís<fnref n="2" />,ᶜ

**Keywords:** Ductile damage, Porosity, Homogenization, Deep drawing, Orthotropy

**Abstract.** The development of more sophisticated constitutive models is essential for improving the reliability of metal forming process simulations. The main objective of this work is to employ a Gurson-type [1] porous criterion to assess the ductile damage distribution of a strongly textured AA5042-H2 sheet during a single-stage cup-drawing process. The anisotropy of the dense phase is described with the non-quadratic form of the CPB06ex2 [2] criterion using two linear transformations. In line with Gurson's homogenization theory, the plastic behavior of the porous solid is described by an approximate macroscopic strain-rate potential (SRP) using the classical Rice and Tracey trial fields. The particularity of this implementation is that the macroscopic potentials are not evaluated via analytical functions, but by numerical integration of the local fields [3]. It is shown that such approach is viable from the computational standpoint and opens the door for materials with intricate plastic behavior to be modeled within the framework of porous media.

**Introduction**

Traditionally, constitutive models are represented by closed-form expressions for two primary reasons: first, to minimize computational costs, and second, to facilitate their integration into numerical simulation schemes such as the Finite Element Method (FEM). Regrettably, due to inherent mathematical constraints, obtaining closed-form expressions in the context of porous media may require the incorporation of one or more simplifying hypotheses, often accompanied with detrimental modelling effects (e.g. the Cauchy-Schwartz inequality), or may even be unattainable. Opportunely, recent strides in the computational power-to-cost ratio have facilitated the adoption of a fully numerical approach to constitutive modelling. Such approach, by design, circumvents the challenges associated with obtaining closed-form expressions and presently allow for numerical simulations to be conducted within reasonable computational timeframes. Based on this notion, Brito [3] recently proposed a numerical integration scheme for evaluating the stress- and strain-rate- plastic potentials for ductile porous solids containing randomly distributed spherical voids whose matrix behaviour is described by the non-quadratic orthotropic stress potential of Plunkett et al. [2]. The goal of this paper is to apply the latter formulation to evaluate the ductile damage distribution of a strongly textured AA5042-H2 sheet during a single-stage cylindrical cup-drawing process. Indeed, given its industrial significance and geometric simplicity, the cylindrical cup-drawing process stands out as one of the most frequently employed benchmarks for evaluating the performance of constitutive criteria. The manifestation of plastic anisotropy gives rise to earning phenomena, characterized by the undulation of the top edge of the fully drawn cup. The extent of texture in the material can dictate the occurrence of four, six, or more ears resulting from the drawing process. The application example adopted in this work is based on the experimental results of Yoon et al. [4], which reported a cup with eight ears. Moreover, these

<figure><img src="image_1176.png" /></figure>

<footnote n="1">¹CEMMPRE, ARISE, Department of Mechanical Engineering, University of Coimbra, Pinhal de Marrocos, 3030-788 Coimbra, Portugal</footnote>

<footnote n="2">²Microelectromechanical Systems Research Unit, University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugal</footnote>

<footnote>ªjoao.brito@uc.pt, ᵇmarta.oliveira@dem.uc.pt, ᶜjlalves@dem.uminho.pt</footnote>
# Modelling ductile damage of a textured aluminum alloy based on a non-quadratic yield function

BRITO João P.¹,ª, OLIVEIRA Marta C.¹,ᵇ* and ALVES José Luís²,ᶜ

¹CEMMPRE, ARISE, Department of Mechanical Engineering, University of Coimbra, Pinhal de Marrocos, 3030-788 Coimbra, Portugal

²Microelectromechanical Systems Research Unit, University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugal

ªjoao.brito@uc.pt, ᵇmarta.oliveira@dem.uc.pt, ᶜjlalves@dem.uminho.pt

**Keywords:** Ductile damage, Porosity, Homogenization, Deep drawing, Orthotropy

**Abstract.** The development of more sophisticated constitutive models is essential for improving the reliability of metal forming process simulations. The main objective of this work is to employ a Gurson-type [1] porous criterion to assess the ductile damage distribution of a strongly textured AA5042-H2 sheet during a single-stage cup-drawing process. The anisotropy of the dense phase is described with the non-quadratic form of the CPB06ex2 [2] criterion using two linear transformations. In line with Gurson's homogenization theory, the plastic behavior of the porous solid is described by an approximate macroscopic strain-rate potential (SRP) using the classical Rice and Tracey trial fields. The particularity of this implementation is that the macroscopic potentials are not evaluated via analytical functions, but by numerical integration of the local fields [3]. It is shown that such approach is viable from the computational standpoint and opens the door for materials with intricate plastic behavior to be modeled within the framework of porous media.

**Introduction**

Traditionally, constitutive models are represented by closed-form expressions for two primary reasons: first, to minimize computational costs, and second, to facilitate their integration into numerical simulation schemes such as the Finite Element Method (FEM). Regrettably, due to inherent mathematical constraints, obtaining closed-form expressions in the context of porous media may require the incorporation of one or more simplifying hypotheses, often accompanied with detrimental modelling effects (e.g. the Cauchy-Schwartz inequality), or may even be unattainable. Opportunely, recent strides in the computational power-to-cost ratio have facilitated the adoption of a fully numerical approach to constitutive modelling. Such approach, by design, circumvents the challenges associated with obtaining closed-form expressions and presently allow for numerical simulations to be conducted within reasonable computational timeframes. Based on this notion, Brito [3] recently proposed a numerical integration scheme for evaluating the stress- and strain-rate- plastic potentials for ductile porous solids containing randomly distributed spherical voids whose matrix behaviour is described by the non-quadratic orthotropic stress potential of Plunkett et al. [2]. The goal of this paper is to apply the latter formulation to evaluate the ductile damage distribution of a strongly textured AA5042-H2 sheet during a single-stage cylindrical cup-drawing process. Indeed, given its industrial significance and geometric simplicity, the cylindrical cup-drawing process stands out as one of the most frequently employed benchmarks for evaluating the performance of constitutive criteria. The manifestation of plastic anisotropy gives rise to earning phenomena, characterized by the undulation of the top edge of the fully drawn cup. The extent of texture in the material can dictate the occurrence of four, six, or more ears resulting from the drawing process. The application example adopted in this work is based on the experimental results of Yoon et al. [4], which reported a cup with eight ears. Moreover, these

<figure><img src="image_1178.png" /></figure>
authors employed the Plunkett et al. [2] criterion with two transformations (CPB06ex2) to simulate the forming process and found that the model predicted a similar earing profile. While the geometry of the completely drawn cup can be assessed with such formulation (valid for dense, plastic isochoric, materials), no conclusion can be drawn regarding the damage distribution in the component. This problem is now readdressed within the scope of porous media, for which the porosity distribution quantifies the damage state. It is shown that the numerical homogenization scheme is, indeed, feasible from the computational perspective and is a promising tool for evaluating the damage state of anisotropic ductile metals and alloys.

The general scheme of notation is as follows. Scalars and scalar-valued functions are denoted with light-face letters, e.g. (A, b, ...); vectors with under bar italic Latin and Greek bold-face letters, e.g. (A, b, ...); second-order tensors or tensor-valued functions with italic Latin and Greek bold-face letters (A, b, ...); fourth-order tensors with calligraphic majuscules, e.g. (A, B, ...). The corresponding pseudo-vectorial and matrix Voigt notation are represented with upright Latin bold-face letters (A, b, ...). The double-contracted tensor products is defined as $A:B = A_{ij}B_{ij}$, and $(C: A)_{ij} = C_{ijkl}A_{kl}$. The summation convention over repeated indices is employed.

## Plasticity model

The employed macroscopic strain-rate potential results from the kinematic Hill-Mandel homogenization framework considering: (i) an hollow sphere of domain $\Omega$, with outer radius $B$, containing a concentric spherical void of radius $A$ occupying a domain $\omega$; (ii) the (kinematically admissible) Rice and Tracey [5] velocity fields; and (iii) the microscale (alias local) plasticity model governed by the yield criterion, $f(\sigma)$, of Plunkett et al. [2], i.e.:

$$ 
\begin{gathered}
f(\sigma, k^{(n)}, \mathbf{L}^{(n)}, a, \sigma_1^T) \Box \varphi(\sigma, k^{(n)}, \mathbf{L}^{(n)}, a) - \sigma_1^T = 0, \quad n=1, \dots, n_{\text{ex}}, \\
\varphi = m(F)^{\frac{1}{n_{\text{ex}}}}, \quad \text{with} \quad F = \sum_{i=1}^{n_{\text{ex}}} F^{(n)} \quad \text{and} \quad F^{(n)} = \sum_{i=1}^{3} (|\hat{s}_{p_i}^{(n)}| - k^{(n)} \hat{s}_{p_i}^{(n)})^a,
\end{gathered}
\quad (1) $$

where $n_{\text{ex}} \ge 1$ is the number of orthotropic transformations, $a$ is the the homogeneity degree, $\sigma_1^T$ is the yield stress in uniaxial tension along the 1-direction of the orthotropy axes, $\hat{s}_{p_i}^{(n)}$, $i=1,2,3$, are the eigenvalues of the $n$-th transformed stress tensor, $\hat{s}^{(n)} = \mathbf{L}^{(n)}:\mathbf{s}$, $\mathbf{L}^{(n)} = \mathbf{C}^{(n)}:\mathbf{K}$ is the $n$-th fourth-order deviatoric transformation tensor, and $\mathbf{C}^{(n)}$ is a major-symmetric orthotropic tensor whose Voigt 6x6-matrix notation, $C_{IJ}^{(n)} \leftarrow C_{ijkl}^{(n)}$, $I,J=1, \dots, 6$, $i,j,k,l=1,2,3$, is in the form:

$$ \mathbf{C}^{(n)} = \begin{bmatrix} C_{11}^{(n)} & C_{12}^{(n)} & C_{13}^{(n)} & 0 & 0 & 0 \\ C_{12}^{(n)} & C_{22}^{(n)} & C_{23}^{(n)} & 0 & 0 & 0 \\ C_{13}^{(n)} & C_{23}^{(n)} & C_{33}^{(n)} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44}^{(n)} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{55}^{(n)} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66}^{(n)} \end{bmatrix}, \quad (2) $$

$K_{ijkl} = 1/2(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) - 1/3(\delta_{ij}\delta_{kl})$, $i, j, k, l = 1, 2, 3$, is the isotropic fourth-order deviatoric unit tensor and $\delta_{ij}$ is the Kronecker delta. In Eq. 1, $k^{(n)}$ is the tension-compression asymmetry parameter associated with the transformation $n$, and $m$ is a material constant defined such that $\varphi(\sigma)$ reduces $\sigma_1^T$ for uniaxial tensile loadings and it is given in terms of $\mathbf{L}^{(n)}$ and $k^{(n)}$ as:
authors employed the Plunkett et al. [2] criterion with two transformations (CPB06ex2) to simulate the forming process and found that the model predicted a similar earing profile. While the geometry of the completely drawn cup can be assessed with such formulation (valid for dense, plastic isochoric, materials), no conclusion can be drawn regarding the damage distribution in the component. This problem is now readdressed within the scope of porous media, for which the porosity distribution quantifies the damage state. It is shown that the numerical homogenization scheme is, indeed, feasible from the computational perspective and is a promising tool for evaluating the damage state of anisotropic ductile metals and alloys.

The general scheme of notation is as follows. Scalars and scalar-valued functions are denoted with light-face letters, e.g. (A, b, ...); vectors with under bar italic Latin and Greek bold-face letters, e.g. (A, b, ...); second-order tensors or tensor-valued functions with italic Latin and Greek bold-face letters (A, b, ...); fourth-order tensors with calligraphic majuscules, e.g. (A, B, ...). The corresponding pseudo-vectorial and matrix Voigt notation are represented with upright Latin bold-face letters (A, b, ...). The double-contracted tensor products is defined as $A:B = A_{ij}B_{ij}$, and $(C: A)_{ij} = C_{ijkl}A_{kl}$. The summation convention over repeated indices is employed.

## Plasticity model

The employed macroscopic strain-rate potential results from the kinematic Hill-Mandel homogenization framework considering: (i) an hollow sphere of domain $\Omega$, with outer radius $B$, containing a concentric spherical void of radius $A$ occupying a domain $\omega$; (ii) the (kinematically admissible) Rice and Tracey [5] velocity fields; and (iii) the microscale (alias local) plasticity model governed by the yield criterion, $f(\sigma)$, of Plunkett et al. [2], i.e.:

$$f(\sigma, k^{(n)}, L^{(n)}, a, \sigma_1^T) \Box \varphi(\sigma, k^{(n)}, L^{(n)}, a) - \sigma_1^T = 0, \quad n=1, \dots, n_{\text{ex}}, \\ \varphi = m(F)^{\frac{1}{n_{\text{ex}}}}, \text{ with } F = \sum_{n=1}^{n_{\text{ex}}} F^{(n)} \text{ and } F^{(n)} = \sum_{i=1}^{3} (|\hat{s}_{p_i}^{(n)}| - k^{(n)} \hat{s}_{p_i}^{(n)})^a, \quad (1)$$

where $n_{\text{ex}} \ge 1$ is the number of orthotropic transformations, $a$ is the the homogeneity degree, $\sigma_1^T$ is the yield stress in uniaxial tension along the 1-direction of the orthotropy axes, $\hat{s}_{p_i}^{(n)}$, $i=1,2,3$, are the eigenvalues of the $n$-th transformed stress tensor, $\hat{s}^{(n)} = L^{(n)}:s$, $L^{(n)} = C^{(n)}:K$ is the $n$-th fourth-order deviatoric transformation tensor, and $C^{(n)}$ is a major-symmetric orthotropic tensor whose Voigt 6x6-matrix notation, $C_{IJ}^{(n)} \leftarrow C_{ijkl}^{(n)}$, $I, J = 1, \dots, 6$, $i, j, k, l = 1, 2, 3$, is in the form:

$$C^{(n)} = \begin{bmatrix} C_{11}^{(n)} & C_{12}^{(n)} & C_{13}^{(n)} & 0 & 0 & 0 \\ C_{12}^{(n)} & C_{22}^{(n)} & C_{23}^{(n)} & 0 & 0 & 0 \\ C_{13}^{(n)} & C_{23}^{(n)} & C_{33}^{(n)} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44}^{(n)} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{55}^{(n)} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66}^{(n)} \end{bmatrix}, \quad (2)$$

$K_{ijkl} = 1/2(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}) - 1/3(\delta_{ij}\delta_{kl})$, $i, j, k, l = 1, 2, 3$, is the isotropic fourth-order deviatoric unit tensor and $\delta_{ij}$ is the Kronecker delta. In Eq. 1, $k^{(n)}$ is the tension-compression asymmetry parameter associated with the transformation $n$, and $m$ is a material constant defined such that $\varphi(\sigma)$ reduces $\sigma_1^T$ for uniaxial tensile loadings and it is given in terms of $L^{(n)}$ and $k^{(n)}$ as:
$$m = \left( \sum_{i=1}^{n_{\text{ex}}} \bar{m}^{(n)} \right)^{-\frac{1}{a}}, \quad \text{with} \quad \bar{m}^{(n)} = \sum_{i=1}^{3} \left( |\Phi_i^{(n)}| - k^{(n)} \Phi_i^{(n)} \right)^a, \qquad (3)$$

where $\Phi_i^{(n)}$, $i=1,2,3$ are the components $L_{ii11}^{(n)}$, $i=1,2,3$ (no sum), respectively, (or, equivalently, the non-zero components of the first column of the 6x6-matrix form representation of $L^{(n)}$). Convexity is guaranteed for any integer $a \ge 1$ and for $k^{(n)} \in [-1,1]$ (for the proof refer to [6]). In Eq. 1, $\varphi(\boldsymbol{\sigma}) = \bar{\sigma}$, is the stress potential (i.e. equivalent stress measure). Under associated plasticity conditions, it is possible to show (see [3]) that it exists a dual (alias conjugated or polar reciprocate) potential of $\varphi(\boldsymbol{\sigma}) = \bar{\sigma}$, also convex, known as the strain-rate potential (SRP), $\psi(d^p) = \dot{\lambda}_1^T$, such that the dissipation function reads $D(d^p) = \boldsymbol{\sigma} : d^p = \varphi(\boldsymbol{\sigma})\psi(d^p) = \sigma_1^T \dot{\lambda}_1^T$, where the scalar $\dot{\lambda}_1^T$ is the equivalent plastic strain rate associated with the adopted equivalent stress measure, i.e. the uniaxial tensile stress, $\boldsymbol{\sigma}$ is the Cauchy stress tensor and $d^p$ its work-conjugate, i.e., the Eulerian plastic strain-rate tensor.

The approximate<fnref n="1" /> macroscopic SRP, $\Psi(D)$, associated with the imposed macroscopic strain rate tensor, $D = \langle d \rangle_{\Omega}$, and the Rice and Tracey [5] microscale trial strain-rate field, $d^{\text{RT}}(\mathbf{x}) = D' + D_m(b/r)^3(-2\underline{e}_r \otimes \underline{e}_r + \underline{e}_\theta \otimes \underline{e}_\theta + \underline{e}_\phi \otimes \underline{e}_\phi)$, (where $D'$ and $D_m$ represent the deviatoric and the hydrostatic part of $D$, and $(\underline{e}_r, \underline{e}_\theta, \underline{e}_\phi)$ is the orthonormal base of the spherical $(r, \theta, \phi)$ coordinate system), is defined as:

$$\Psi(D) = (1-f) \langle \psi(d^{\text{RT}}) \rangle_{\Omega \setminus \omega} = \frac{1}{V} \int_A^B A(r) \langle \psi(d^{\text{RT}}) \rangle_{S(r)} dr, \qquad (4)$$

where the notations $\langle \cdot \rangle_V$ and $\langle \cdot \rangle_{S(r)}$ stand for volume averaging over the volume $V$, and over the surface $S(r) = \{\mathbf{x} \in \mathbb{R}^3 : \|\mathbf{x}\|_2 = r\}$, respectively, with $S(r)$ denoting the spherical surface of radius $r$ and area $A(r) = 4\pi r^2$. The above integral is determined numerically by the cubature procedure proposed in Brito [3]. Moreover, one can define its dual, the macroscopic stress potential, $\Phi(\boldsymbol{\Sigma})$, by applying the normality principle at the macroscale:

$$D = \dot{\Lambda} \frac{\partial \Phi}{\partial \boldsymbol{\Sigma}}(\boldsymbol{\Sigma}) \Leftrightarrow \begin{cases} \frac{\boldsymbol{\Sigma}}{\sigma_1^T} = \frac{\partial \Psi(D)}{\partial D}, & \dot{\Lambda} \ge 0, \\ \Psi(D) = \dot{\Lambda} \end{cases} \qquad (5)$$

where scalar $\dot{\Lambda}$ is the macroscopic plastic multiplier rate and $\boldsymbol{\Sigma}$ is the macroscopic Cauchy stress. The criterion in Eq. 4 is implemented in an academic FE research code (DD3IMP, University of Coimbra, e.g. [7]–[9]) considering a large-strain elastoplastic constitutive framework grounded on an hyperelastic-based elastic-plastic multiplicative split formulation and based on the strain-rate potential flow rule (i.e. right member of Eq. 5) to describe the plastic dissipation (see also [3] for details). Given the adopted numerical approach to the homogenization problem, the model presented above is coined the Computational Homogenization Model (CHM).

<footnote n="1">¹ Approximate in the sense that a single trial velocity field, $d^{\text{tr}} = d^{\text{tr}}(\underline{x}, D)$, $\forall \underline{x} \in \Omega \setminus \omega$, compatible with uniform strain-rate boundary conditions of the type $\underline{v}(\underline{x} = B\underline{e}_r) = D\underline{x}$, is employed, where $\underline{x}$ is the position vector in the Representative Volume Element and $\Omega \setminus \omega$ denotes the domain of the matrix phase.</footnote>
$$m = \left( \sum_{i=1}^{n_{\text{ex}}} \bar{m}^{(n)} \right)^{-\frac{1}{a}}, \quad \text{with} \quad \bar{m}^{(n)} = \sum_{i=1}^{3} \left( |\Phi_{i}^{(n)}| - k^{(n)} \Phi_{i}^{(n)} \right)^{a}, \qquad (3)$$

where $\Phi_{i}^{(n)}$, $i=1,2,3$ are the components $L_{ii11}^{(n)}$, $i=1,2,3$ (no sum), respectively, (or, equivalently, the non-zero components of the first column of the 6x6-matrix form representation of $L^{(n)}$). Convexity is guaranteed for any integer $a \ge 1$ and for $k^{(n)} \in [-1,1]$ (for the proof refer to [6]). In Eq. 1, $\varphi(\boldsymbol{\sigma}) = \bar{\sigma}$, is the stress potential (i.e. equivalent stress measure). Under associated plasticity conditions, it is possible to show (see [3]) that it exists a dual (alias conjugated or polar reciprocate) potential of $\varphi(\boldsymbol{\sigma}) = \bar{\sigma}$, also convex, known as the strain-rate potential (SRP), $\psi(\boldsymbol{d}^p) = \dot{\lambda}_1^T$, such that the dissipation function reads $D(\boldsymbol{d}^p) = \boldsymbol{\sigma} : \boldsymbol{d}^p = \varphi(\boldsymbol{\sigma})\psi(\boldsymbol{d}^p) = \sigma_1^T \dot{\lambda}_1^T$, where the scalar $\dot{\lambda}_1^T$ is the equivalent plastic strain rate associated with the adopted equivalent stress measure, i.e. the uniaxial tensile stress, $\boldsymbol{\sigma}$ is the Cauchy stress tensor and $\boldsymbol{d}^p$ its work-conjugate, i.e., the Eulerian plastic strain-rate tensor.

The approximate<fnref n="1" /> macroscopic SRP, $\Psi(D)$, associated with the imposed macroscopic strain rate tensor, $D = \langle d \rangle_{\Omega}$, and the Rice and Tracey [5] microscale trial strain-rate field, $d^{\text{RT}}(\mathbf{x}) = D' + D_m(b/r)^3(-2\mathbf{e}_r \otimes \mathbf{e}_r + \mathbf{e}_\theta \otimes \mathbf{e}_\theta + \mathbf{e}_\phi \otimes \mathbf{e}_\phi)$, (where $D'$ and $D_m$ represent the deviatoric and the hydrostatic part of $D$, and $(\mathbf{e}_r, \mathbf{e}_\theta, \mathbf{e}_\phi)$ is the orthonormal base of the spherical $(r, \theta, \phi)$ coordinate system), is defined as:

$$\Psi(D) = (1-f) \langle \psi(d^{\text{RT}}) \rangle_{\Omega \setminus \omega} = \frac{1}{V} \int_A^B A(r) \langle \psi(d^{\text{RT}}) \rangle_{S(r)} dr, \qquad (4)$$

where the notations $\langle \cdot \rangle_V$ and $\langle \cdot \rangle_{S(r)}$ stand for volume averaging over the volume $V$, and over the surface $S(r) = \{\mathbf{x} \in \Omega^3 : \|\mathbf{x}\|_2 = r\}$, respectively, with $S(r)$ denoting the spherical surface of radius $r$ and area $A(r) = 4\pi r^2$. The above integral is determined numerically by the cubature procedure proposed in Brito [3]. Moreover, one can define its dual, the macroscopic stress potential, $\Phi(\boldsymbol{\Sigma})$, by applying the normality principle at the macroscale:

$$D = \dot{\Lambda} \frac{\partial \Phi}{\partial \boldsymbol{\Sigma}}(\boldsymbol{\Sigma}) \Leftrightarrow \left\{ \begin{array}{l} \frac{\boldsymbol{\Sigma}}{\sigma_1^T} = \frac{\partial \Psi(D)}{\partial D}, \quad \dot{\Lambda} \ge 0, \\ \Psi(D) = \dot{\Lambda} \end{array} \right. \qquad (5)$$

where scalar $\dot{\Lambda}$ is the macroscopic plastic multiplier rate and $\boldsymbol{\Sigma}$ is the macroscopic Cauchy stress. The criterion in Eq. 4 is implemented in an academic FE research code (DD3IMP, University of Coimbra, e.g. [7]–[9]) considering a large-strain elastoplastic constitutive framework grounded on an hyperelastic-based elastic-plastic multiplicative split formulation and based on the strain-rate potential flow rule (i.e. right member of Eq. 5) to describe the plastic dissipation (see also [3] for details). Given the adopted numerical approach to the homogenization problem, the model presented above is coined the Computational Homogenization Model (CHM).

<footnote n="1">¹ Approximate in the sense that a single trial velocity field, $d^{\text{tr}} = d^{\text{tr}}(\underline{x}, D)$, $\forall \underline{x} \in \Omega \setminus \omega$, compatible with uniform strain-rate boundary conditions of the type $\underline{v}(\underline{x} = B\underline{e}_r) = D\underline{x}$, is employed, where $\underline{x}$ is the position vector in the Representative Volume Element and $\Omega \setminus \omega$ denotes the domain of the matrix phase.</footnote>
Finite element model

The finite element (FE) simulation of the cup-drawing test is based on the work of Yoon et al. [4]. Both the process parameters and the constitutive identification of the yield behaviour of the AA5042-H2 matrix are extracted from the cited paper. A schematic view of the cylindrical cup drawing setup is shown in Fig. 1. The process parameters and the dimensions of the tools are given in Table 1. Given the geometric and orthotropic symmetry conditions of the cup drawing process, the analysis is confined to one quarter of the model. The adopted FE mesh is shown in Fig. 2. The structured zone is discretized with a 30×28 (radial × circumferential) partition in the plane of the sheet and two layers along the thickness direction are used. The FE mesh consists of a total of 1896 eight-node hexahedral FE with a selective reduced integration (SRI) technique. The tools are modelled as rigid bodies and the friction between the sheet and the tools is assumed constant and characterized by Coulomb's law with a friction coefficient $\mu = 0.008$ (after [4]).

The matrix plastic behaviour is characterized with the non-quadratic form of CPB06ex2 [2] yield criterion (i.e. using two ortotropic transformations); and its elastic behaviour is assumed isotropic and constant, as the degradation of the elastic modulus is neglected. The isotropic hardening of the matrix is described by a Voce-type law: $Y_{\text{Voce}}(\tilde{\alpha}_{M}^{p}) = Y_{0} + (Y_{\text{sat}} - Y_{0})(1 - e^{-C_{Y}\tilde{\alpha}_{M}^{p}})$, where $Y_{\text{Voce}}$ is the yield stress, $\tilde{\alpha}_{M}^{p}$ is the matrix equivalent plastic strain and $\{Y_{0}, Y_{\text{sat}}, C_{Y}\}$ are the Voce parameters. Strain-controlled nucleation is considered via Chu and Needleman's [10] normal distribution, yet stress-controlled nucleation is neglected. Table 2 summarizes the material model parameters.

Fig. 3 represents cross sections of the macroscopic SRP (Eq. 4) and the respective conjugate, the macroscopic yield surface, in the principal space with varying hydrostatic levels. Note that the shape of the sections changes drastically along the hydrostatic axis. Initially departing from a shape reminiscent to that of the dense phase, the sections progressively evolve towards a more smooth, rounded form (i.e. approaching isotropy) near the poles. Closed-form analytical criteria for porous solids are unable to capture such evolving shape behaviour as the underlying shape in the purely deviatoric plane is maintained throughout the hydrostatic axis. Moreover, it is worth noting that the positions of the extreme hydrostatic values are not on the hydrostatic axis. This is a result of the anisotropy alone (indeed, tension-compression asymmetry is assumed null for this matrix alloy) and renders the potentials remarkably asymmetrical with respect to the purely deviatoric plane (these are, however, symmetric with respect to the origin, since tension-compression asymmetry is neglected and thus the governing microscale criteria are even functions).

<figure><img src="image_1177.png" /><figcaption>Figure 1. Schematic representation of a cylindrical cup drawing process (adapted from [11]).</figcaption></figure>
Finite element model

The finite element (FE) simulation of the cup-drawing test is based on the work of Yoon et al. [4]. Both the process parameters and the constitutive identification of the yield behaviour of the AA5042-H2 matrix are extracted from the cited paper. A schematic view of the cylindrical cup drawing setup is shown in Fig. 1. The process parameters and the dimensions of the tools are given in Table 1. Given the geometric and orthotropic symmetry conditions of the cup drawing process, the analysis is confined to one quarter of the model. The adopted FE mesh is shown in Fig. 2. The structured zone is discretized with a 30×28 (radial × circumferential) partition in the plane of the sheet and two layers along the thickness direction are used. The FE mesh consists of a total of 1896 eight-node hexahedral FE with a selective reduced integration (SRI) technique. The tools are modelled as rigid bodies and the friction between the sheet and the tools is assumed constant and characterized by Coulomb's law with a friction coefficient $\mu = 0.008$ (after [4]).

The matrix plastic behaviour is characterized with the non-quadratic form of CPB06ex2 [2] yield criterion (i.e. using two ortotropic transformations); and its elastic behaviour is assumed isotropic and constant, as the degradation of the elastic modulus is neglected. The isotropic hardening of the matrix is described by a Voce-type law: $Y_{\text{Voce}}(\tilde{\alpha}_{M}^{p}) = Y_{0} + (Y_{\text{sat}} - Y_{0})(1 - e^{-C_{Y}\tilde{\alpha}_{M}^{p}})$, where $Y_{\text{Voce}}$ is the yield stress, $\tilde{\alpha}_{M}^{p}$ is the matrix equivalent plastic strain and $\{Y_{0}, Y_{\text{sat}}, C_{Y}\}$ are the Voce parameters. Strain-controlled nucleation is considered via Chu and Needleman's [10] normal distribution, yet stress-controlled nucleation is neglected. Table 2 summarizes the material model parameters.

Fig. 3 represents cross sections of the macroscopic SRP (Eq. 4) and the respective conjugate, the macroscopic yield surface, in the principal space with varying hydrostatic levels. Note that the shape of the sections changes drastically along the hydrostatic axis. Initially departing from a shape reminiscent to that of the dense phase, the sections progressively evolve towards a more smooth, rounded form (i.e. approaching isotropy) near the poles. Closed-form analytical criteria for porous solids are unable to capture such evolving shape behaviour as the underlying shape in the purely deviatoric plane is maintained throughout the hydrostatic axis. Moreover, it is worth noting that the positions of the extreme hydrostatic values are not on the hydrostatic axis. This is a result of the anisotropy alone (indeed, tension-compression asymmetry is assumed null for this matrix alloy) and renders the potentials remarkably asymmetrical with respect to the purely deviatoric plane (these are, however, symmetric with respect to the origin, since tension-compression asymmetry is neglected and thus the governing microscale criteria are even functions).

<figure><img src="image_1179.png" /><figcaption>Figure 1. Schematic representation of a cylindrical cup drawing process (adapted from [11]).</figcaption></figure>
<table><caption>Table 1. Process parameters and the dimensions of the tools (after [4]).</caption><thead><tr><th>Parameter</th><th></th><th>Value</th><th></th></tr></thead><tbody><tr><td>Blank diameter</td><td>D<sub>b</sub></td><td>76.07 [mm]</td><td></td></tr><tr><td>Die opening diameter</td><td>D<sub>d</sub></td><td>46.74 [mm]</td><td></td></tr><tr><td>Punch diameter</td><td>D<sub>p</sub></td><td>45.72 [mm]</td><td></td></tr><tr><td>Die profile radius</td><td>r<sub>d</sub></td><td>2.28 [mm]</td><td></td></tr><tr><td>Punch profile radius</td><td>r<sub>p</sub></td><td>2.28 [mm]</td><td></td></tr><tr><td>Initial blank thickness</td><td>t<sub>0</sub></td><td>0.274 [mm]</td><td></td></tr><tr><td>Punch stroke</td><td>u<sub>p</sub></td><td>22.75 [mm]</td><td></td></tr><tr><td>Blank-holder force (total)</td><td>f<sub>bh</sub></td><td>10.0 [kN]</td><td></td></tr></tbody></table>

<figure><img src="image_1178.png" /><figcaption>Figure 2. FE mesh of the blank in the reference configuration. Two through-thickness layers are used.</figcaption></figure>

<figure><img src="image_1179.png" /><figcaption>Figure 3. π-plane representation of macroscopic: (a) strain-rate potential (Eq. 4); and (b) stress potential; unitary isovalue surfaces in their respective principal space for a porous solid with porosity $f_0 = 0.001$, with a AA5042-H2 alloy matrix. Cross sections with varying hydrostatic fraction, $\bar{f}_{(\cdot)_m}$, with respect to the tensile and compressive hydrostatic extreme values are plotted.</figcaption></figure>
<table><caption>Table 1. Process parameters and the dimensions of the tools (after [4]).</caption><thead><tr><th>Parameter</th><th></th><th>Value</th><th></th></tr></thead><tbody><tr><td>Blank diameter</td><td>D<sub>b</sub></td><td>76.07 [mm]</td><td></td></tr><tr><td>Die opening diameter</td><td>D<sub>d</sub></td><td>46.74 [mm]</td><td></td></tr><tr><td>Punch diameter</td><td>D<sub>p</sub></td><td>45.72 [mm]</td><td></td></tr><tr><td>Die profile radius</td><td>r<sub>d</sub></td><td>2.28 [mm]</td><td></td></tr><tr><td>Punch profile radius</td><td>r<sub>p</sub></td><td>2.28 [mm]</td><td></td></tr><tr><td>Initial blank thickness</td><td>t<sub>0</sub></td><td>0.274 [mm]</td><td></td></tr><tr><td>Punch stroke</td><td>u<sub>p</sub></td><td>22.75 [mm]</td><td></td></tr><tr><td>Blank-holder force (total)</td><td>f<sub>bh</sub></td><td>10.0 [kN]</td><td></td></tr></tbody></table>

<figure><img src="image_1180.png" /><figcaption>Figure 2. FE mesh of the blank in the reference configuration. Two through-thickness layers are used.</figcaption></figure>

<figure><img src="image_1181.png" /><figcaption>Figure 3. π-plane representation of macroscopic: (a) strain-rate potential (Eq. 4); and (b) stress potential; unitary isovalue surfaces in their respective principal space for a porous solid with porosity $f_0 = 0.001$, with a AA5042-H2 alloy matrix. Cross sections with varying hydrostatic fraction, $\bar{f}_{(\cdot)_m}$, with respect to the tensile and compressive hydrostatic extreme values are plotted.</figcaption></figure>
<table><caption>Table 2. Material coefficients for the AA5042-H2 alloy (partially after [4]).</caption><thead><tr><th colspan="2">Elasticity (isotropic)</th></tr><tr><th>Young modulus [GPa]<br>E</th><th>Poisson ratio [-]<br>υ</th></tr></thead><tbody><tr><td>68.9</td><td>0.33</td></tr></tbody></table><table><thead><tr><th colspan="10">Plasticity: matrix behaviour</th></tr><tr><th colspan="10">CPB06ex2 parameters (a = 10)</th></tr><tr><th>k<sup>(1)</sup></th><th>C<sub>11</sub><sup>(1)</sup></th><th>C<sub>12</sub><sup>(1)</sup></th><th>C<sub>13</sub><sup>(1)</sup></th><th>C<sub>22</sub><sup>(1)</sup></th><th>C<sub>23</sub><sup>(1)</sup></th><th>C<sub>33</sub><sup>(1)</sup></th><th>C<sub>44</sub><sup>(1)</sup></th><th>C<sub>55</sub><sup>(1)</sup></th><th>C<sub>66</sub><sup>(1)</sup></th></tr></thead><tbody><tr><td>0.0</td><td>1.0</td><td>-0.0272</td><td>-0.6011</td><td>1.2870</td><td>0.6864</td><td>-0.2736</td><td>1.0</td><td>1.0</td><td>1.1514</td></tr><tr><th>k<sup>(2)</sup></th><th>C<sub>11</sub><sup>(2)</sup></th><th>C<sub>12</sub><sup>(2)</sup></th><th>C<sub>13</sub><sup>(2)</sup></th><th>C<sub>22</sub><sup>(2)</sup></th><th>C<sub>23</sub><sup>(2)</sup></th><th>C<sub>33</sub><sup>(2)</sup></th><th>C<sub>44</sub><sup>(2)</sup></th><th>C<sub>55</sub><sup>(2)</sup></th><th>C<sub>66</sub><sup>(2)</sup></th></tr><tbody><tr><td>0.0</td><td>1.0</td><td>-0.0897</td><td>0.0112</td><td>1.1322</td><td>-0.1092</td><td>-1.2009</td><td>1.0</td><td>1.0</td><td>1.3093</td></tr></tbody></table><table><thead><tr><th colspan="4">Isotropic hardening</th></tr><tr><th colspan="4">Voce law</th></tr><tr><th>Y<sub>0</sub> [MPa]</th><th>Y<sub>sat</sub> [MPa]</th><th>C<sub>Y</sub> [-]</th><th></th></tr></thead><tbody><tr><td>267.80</td><td>375.08</td><td>17.859</td><td></td></tr></tbody></table><table><thead><tr><th colspan="5">Porosity</th></tr><tr><th>Initial porosity</th><th>Critical porosity</th><th colspan="3">Strain-controlled nucleation</th></tr><tr><th>f<sub>0</sub></th><th>f<sub>C</sub></th><th>f<sub>N</sub></th><th>f&#773;<sub>N</sub></th><th>s<sub>N</sub></th></tr></thead><tbody><tr><td>0.001</td><td>0.1</td><td>0.01</td><td>0.3</td><td>0.1</td></tr></tbody></table>

## Results and discussion

The simulation of the cup drawing test is performed for three SRP plasticity models: (i) the dense model (the dual of the CPB06ex2 criterion in Eq. 1); (ii) the Gurson [1] (isotropic) porous criterion; and (iii) the CHM described in Eq. 4. All simulations were executed on a computer equipped with an Intel® Core™ i7-8700K (6C/12T) CPU on a 64-bit Windows 10 Pro for Workstations operating system. Table 3 summarizes the computational performance of the numerical simulations for each material model. Fig. 4 depicts the predicted geometry of the completely drawn cup using the CHM. This model predicts the occurrence of eight ears of varying height, which agrees with experimental data, as shown in Fig 5. Fig. 5 also shows that a comparable earing height trend is captured using the CPB06ex2 (dense) criterion, yet, quantitatively, the latter model predicts higher earing profile. This is thought to result from the restrictive condition of plastic incompressibility in the CPB06ex2 model, which facilitates the elongation of the sheet in the blank holder zone.

<table><caption>Table 3. Computational performance measurements of the cup drawing test simulations for the three plasticity models: the dense model (CPB06ex2, Eq. 1); the Gurson (1977) model; and the computational homogenization model (CHM, Eq. 4)</caption><thead><tr><th></th><th></th><th>Dense</th><th>Gurson</th><th>CHM</th></tr></thead><tbody><tr><td>No. of increments</td><td></td><td>129</td><td>121</td><td>124</td></tr><tr><td rowspan="2">Computational time</td><td>Absolute [h]</td><td>2.73</td><td>0.18</td><td>46.97</td></tr><tr><td>Relative</td><td>1.0</td><td>0.066</td><td>20.42</td></tr></tbody></table>
<table><caption>Table 2. Material coefficients for the AA5042-H2 alloy (partially after [4]).</caption><thead><tr><th colspan="2">Elasticity (isotropic)</th></tr><tr><th>Young modulus [GPa]<br>E</th><th>Poisson ratio [-]<br>υ</th></tr></thead><tbody><tr><td>68.9</td><td>0.33</td></tr></tbody></table><table><thead><tr><th colspan="10">Plasticity: matrix behaviour</th></tr><tr><th colspan="10">CPB06ex2 parameters (a = 10)</th></tr><tr><th>k<sup>(1)</sup></th><th>C<sub>11</sub><sup>(1)</sup></th><th>C<sub>12</sub><sup>(1)</sup></th><th>C<sub>13</sub><sup>(1)</sup></th><th>C<sub>22</sub><sup>(1)</sup></th><th>C<sub>23</sub><sup>(1)</sup></th><th>C<sub>33</sub><sup>(1)</sup></th><th>C<sub>44</sub><sup>(1)</sup></th><th>C<sub>55</sub><sup>(1)</sup></th><th>C<sub>66</sub><sup>(1)</sup></th></tr></thead><tbody><tr><td>0.0</td><td>1.0</td><td>-0.0272</td><td>-0.6011</td><td>1.2870</td><td>0.6864</td><td>-0.2736</td><td>1.0</td><td>1.0</td><td>1.1514</td></tr><tr><th>k<sup>(2)</sup></th><th>C<sub>11</sub><sup>(2)</sup></th><th>C<sub>12</sub><sup>(2)</sup></th><th>C<sub>13</sub><sup>(2)</sup></th><th>C<sub>22</sub><sup>(2)</sup></th><th>C<sub>23</sub><sup>(2)</sup></th><th>C<sub>33</sub><sup>(2)</sup></th><th>C<sub>44</sub><sup>(2)</sup></th><th>C<sub>55</sub><sup>(2)</sup></th><th>C<sub>66</sub><sup>(2)</sup></th></tr><tbody><tr><td>0.0</td><td>1.0</td><td>-0.0897</td><td>0.0112</td><td>1.1322</td><td>-0.1092</td><td>-1.2009</td><td>1.0</td><td>1.0</td><td>1.3093</td></tr></tbody></table><table><thead><tr><th colspan="4">Isotropic hardening</th></tr><tr><th colspan="4">Voce law</th></tr><tr><th>Y<sub>0</sub> [MPa]</th><th>Y<sub>sat</sub> [MPa]</th><th>C<sub>Y</sub> [-]</th><th></th></tr></thead><tbody><tr><td>267.80</td><td>375.08</td><td>17.859</td><td></td></tr></tbody></table><table><thead><tr><th colspan="5">Porosity</th></tr><tr><th>Initial porosity</th><th>Critical porosity</th><th colspan="3">Strain-controlled nucleation</th></tr><tr><th>f<sub>0</sub></th><th>f<sub>C</sub></th><th>f<sub>N</sub></th><th>f&#773;<sub>N</sub></th><th>s<sub>N</sub></th></tr></thead><tbody><tr><td>0.001</td><td>0.1</td><td>0.01</td><td>0.3</td><td>0.1</td></tr></tbody></table>

## Results and discussion

The simulation of the cup drawing test is performed for three SRP plasticity models: (i) the dense model (the dual of the CPB06ex2 criterion in Eq. 1); (ii) the Gurson [1] (isotropic) porous criterion; and (iii) the CHM described in Eq. 4. All simulations were executed on a computer equipped with an Intel® Core™ i7-8700K (6C/12T) CPU on a 64-bit Windows 10 Pro for Workstations operating system. Table 3 summarizes the computational performance of the numerical simulations for each material model. Fig. 4 depicts the predicted geometry of the completely drawn cup using the CHM. This model predicts the occurrence of eight ears of varying height, which agrees with experimental data, as shown in Fig 5. Fig. 5 also shows that a comparable earing height trend is captured using the CPB06ex2 (dense) criterion, yet, quantitatively, the latter model predicts higher earing profile. This is thought to result from the restrictive condition of plastic incompressibility in the CPB06ex2 model, which facilitates the elongation of the sheet in the blank holder zone.

<table><caption>Table 3. Computational performance measurements of the cup drawing test simulations for the three plasticity models: the dense model (CPB06ex2, Eq. 1); the Gurson (1977) model; and the computational homogenization model (CHM, Eq. 4)</caption><thead><tr><th></th><th></th><th>Dense</th><th>Gurson</th><th>CHM</th></tr></thead><tbody><tr><td>No. of increments</td><td></td><td>129</td><td>121</td><td>124</td></tr><tr><td rowspan="2">Computational time</td><td>Absolute [h]</td><td>2.73</td><td>0.18</td><td>46.97</td></tr><tr><td>Relative</td><td>1.0</td><td>0.066</td><td>20.42</td></tr></tbody></table>
<figure><img src="image_1180.png" /><figcaption>Figure 4. Completely drawn cup. Isocontours of the: (a) accumulated equivalent plastic strain, $\Lambda = \int \dot{\Lambda} dt$; and (b) the porosity, $f$, using the computational homogenization model (CHM, Eq. 4).</figcaption></figure>

<figure><img src="image_1181.png" /><figcaption>Figure 5. FE results of the earing profile of the completely drawn cup using: the dense model (CPB06ex2, Eq. 1); the Gurson (1977) model; and the computational homogenization model (CHM, Eq. 4). Comparison with experimental data (after [4]).</figcaption></figure>

<figure><img src="image_1182.png" /><figcaption>Figure 6. FE results of the punch force-displacement curves using: the dense model (CPB06ex2, Eq. 1); the Gurson (1977) model; and the computational homogenization model (CHM, Eq. 4).</figcaption></figure>

The punch force-displacement evolution as predicted by each model is represented in Fig. 6, including the detail for the instance the blank lost contact with the blank-holder. The criteria that consider matrix orthotropy (CPB06ex2 and the CHM) predict virtually the same result, whereas
<figure><img src="image_1182.png" /><figcaption>Figure 4. Completely drawn cup. Isocontours of the: (a) accumulated equivalent plastic strain, $\Lambda = \int \dot{\Lambda} dt$; and (b) the porosity, $f$, using the computational homogenization model (CHM, Eq. 4).</figcaption></figure>

<figure><img src="image_1183.png" /><figcaption>Figure 5. FE results of the earing profile of the completely drawn cup using: the dense model (CPB06ex2, Eq. 1); the Gurson (1977) model; and the computational homogenization model (CHM, Eq. 4). Comparison with experimental data (after [4]).</figcaption></figure>

<figure><img src="image_1184.png" /><figcaption>Figure 6. FE results of the punch force-displacement curves using: the dense model (CPB06ex2, Eq. 1); the Gurson (1977) model; and the computational homogenization model (CHM, Eq. 4).</figcaption></figure>

The punch force-displacement evolution as predicted by each model is represented in Fig. 6, including the detail for the instance the blank lost contact with the blank-holder. The criteria that consider matrix orthotropy (CPB06ex2 and the CHM) predict virtually the same result, whereas
the curve associated with Gurson's isotropic model appears to be a lower bound estimate throughout the entire drawing process. Unfortunately, no experimental data is available to assess the quality of these predictions.

## Conclusions

In this work, a numerical approach to the Gurson-type homogenization problem was employed to describe the damage state of a highly textured alloy whose orthotropy is modelled based on a non-quadratic yield function with multiple linear transformations. In contrast with existing closed-form *ad hoc* criteria, the employed formulation accounts for the micro-macro plasticity coupling as accurately as allowed by the scale transition operations (Hill-Mandel lemma and limit analysis theory). The primary drawback lies in the higher computational cost incurred with the utilization of cubature methods for the assessment of volume integrals. Nonetheless, it is shown that this cost is not prohibitively expensive. Moreover, a surge in the number of cores of new chips is due in next few years driven by the demands of emerging technologies such as artificial intelligence (AI) and virtual reality (VR), hence the heightened computational effort associated with the employed numerical homogenization scheme is expected to be mitigated by the rapid advancements in microprocessor performance and parallelism within a short timeframe.

## References

[1] A. Gurson, "Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I-Yield Criteria and Flow Rules for Porous Ductile Media," *J Eng Mater Technol*, vol. 99, no. 1, pp. 2-15, Jan. 1977. https://doi.org/10.1115/1.3443401

[2] B. Plunkett, O. Cazacu, and F. Barlat, "Orthotropic yield criteria for description of the anisotropy in tension and compression of sheet metals," *Int J Plast*, vol. 24, no. 5, pp. 847-866, 2008. https://doi.org/10.1016/j.ijplas.2007.07.013

[3] J. P. Brito, "Numerical-based plastic potentials for anisotropic porous metallic materials: development and implementation," PhD Thesis, University of Coimbra, 2023.

[4] J.-H. Yoon, O. Cazacu, J. W. Yoon, and R. E. Dick, "Earing predictions for strongly textured aluminum sheets," *Int J Mech Sci*, vol. 52, no. 12, pp. 1563-1578, 2010. https://doi.org/10.1016/j.ijmecsci.2010.07.005

[5] J. R. Rice and D. M. Tracey, "On the ductile enlargement of voids in triaxial stress fields," *J Mech Phys Solids*, vol. 17, no. 3, pp. 201-217, 1969. https://doi.org/10.1016/0022-5096(69)90033-7

[6] O. Cazacu, B. Plunkett, and F. Barlat, "Orthotropic yield criterion for hexagonal closed packed metals," *Int J Plast*, vol. 22, no. 7, pp. 1171-1194, 2006. https://doi.org/10.1016/j.ijplas.2005.06.001

[7] M. C. Oliveira, J. L. Alves, and L. F. Menezes, "Algorithms and strategies for treatment of large deformation frictional contact in the numerical simulation of deep drawing process," *Archives of Computational Methods in Engineering*, vol. 15, no. 2, pp. 113-162, 2008. https://doi.org/10.1007/s11831-008-9018-x

[8] L. F. Menezes and C. Teodosiu, "Three-dimensional numerical simulation of the deep-drawing process using solid finite elements," *J Mater Process Technol*, vol. 97, no. 1-3, pp. 100-106, 2000. https://doi.org/10.1016/S0924-0136(99)00345-3

[9] D. M. Neto, M. C. Oliveira, L. F. Menezes, and J. L. Alves, "Applying Nagata patches to smooth discretized surfaces used in 3D frictional contact problems," *Comput Methods Appl Mech Eng*, vol. 271, pp. 296-320, 2014. https://doi.org/10.1016/j.cma.2013.12.008
the curve associated with Gurson's isotropic model appears to be a lower bound estimate throughout the entire drawing process. Unfortunately, no experimental data is available to assess the quality of these predictions.

## Conclusions

In this work, a numerical approach to the Gurson-type homogenization problem was employed to describe the damage state of a highly textured alloy whose orthotropy is modelled based on a non-quadratic yield function with multiple linear transformations. In contrast with existing closed-form *ad hoc* criteria, the employed formulation accounts for the micro-macro plasticity coupling as accurately as allowed by the scale transition operations (Hill-Mandel lemma and limit analysis theory). The primary drawback lies in the higher computational cost incurred with the utilization of cubature methods for the assessment of volume integrals. Nonetheless, it is shown that this cost is not prohibitively expensive. Moreover, a surge in the number of cores of new chips is due in next few years driven by the demands of emerging technologies such as artificial intelligence (AI) and virtual reality (VR), hence the heightened computational effort associated with the employed numerical homogenization scheme is expected to be mitigated by the rapid advancements in microprocessor performance and parallelism within a short timeframe.

## References

[1] A. Gurson, “Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I-Yield Criteria and Flow Rules for Porous Ductile Media,” *J Eng Mater Technol*, vol. 99, no. 1, pp. 2–15, Jan. 1977. https://doi.org/10.1115/1.3443401

[2] B. Plunkett, O. Cazacu, and F. Barlat, “Orthotropic yield criteria for description of the anisotropy in tension and compression of sheet metals,” *Int J Plast*, vol. 24, no. 5, pp. 847–866, 2008. https://doi.org/10.1016/j.ijplas.2007.07.013

[3] J. P. Brito, “Numerical-based plastic potentials for anisotropic porous metallic materials: development and implementation,” PhD Thesis, University of Coimbra, 2023.

[4] J.-H. Yoon, O. Cazacu, J. W. Yoon, and R. E. Dick, “Earing predictions for strongly textured aluminum sheets,” *Int J Mech Sci*, vol. 52, no. 12, pp. 1563–1578, 2010. https://doi.org/10.1016/j.ijmecsci.2010.07.005

[5] J. R. Rice and D. M. Tracey, “On the ductile enlargement of voids in triaxial stress fields,” *J Mech Phys Solids*, vol. 17, no. 3, pp. 201–217, 1969. https://doi.org/10.1016/0022-5096(69)90033-7

[6] O. Cazacu, B. Plunkett, and F. Barlat, “Orthotropic yield criterion for hexagonal closed packed metals,” *Int J Plast*, vol. 22, no. 7, pp. 1171–1194, 2006. https://doi.org/10.1016/j.ijplas.2005.06.001

[7] M. C. Oliveira, J. L. Alves, and L. F. Menezes, “Algorithms and strategies for treatment of large deformation frictional contact in the numerical simulation of deep drawing process,” *Archives of Computational Methods in Engineering*, vol. 15, no. 2, pp. 113–162, 2008. https://doi.org/10.1007/s11831-008-9018-x

[8] L. F. Menezes and C. Teodosiu, “Three-dimensional numerical simulation of the deep-drawing process using solid finite elements,” *J Mater Process Technol*, vol. 97, no. 1–3, pp. 100–106, 2000. https://doi.org/10.1016/S0924-0136(99)00345-3

[9] D. M. Neto, M. C. Oliveira, L. F. Menezes, and J. L. Alves, “Applying Nagata patches to smooth discretized surfaces used in 3D frictional contact problems,” *Comput Methods Appl Mech Eng*, vol. 271, pp. 296–320, 2014. https://doi.org/10.1016/j.cma.2013.12.008
[10] C. C. Chu and A. Needleman, “Void Nucleation Effects in Biaxially Stretched Sheets,” *J Eng Mater Technol*, vol. 102, no. 3, pp. 249–256, Jul. 1980. https://doi.org/10.1115/1.3224807

[11] J.-W. Yoon, F. Barlat, R. E. Dick, K. Chung, and T. J. Kang, “Plane stress yield function for aluminum alloy sheets-part II: FE formulation and its implementation,” *Int J Plast*, vol. 20, no. 3, pp. 495–522, 2004. https://doi.org/10.1016/S0749-6419(03)00099-8
[10] C. C. Chu and A. Needleman, “Void Nucleation Effects in Biaxially Stretched Sheets,” *J Eng Mater Technol*, vol. 102, no. 3, pp. 249–256, Jul. 1980. https://doi.org/10.1115/1.3224807

[11] J.-W. Yoon, F. Barlat, R. E. Dick, K. Chung, and T. J. Kang, “Plane stress yield function for aluminum alloy sheets-part II: FE formulation and its implementation,” *Int J Plast*, vol. 20, no. 3, pp. 495–522, 2004. https://doi.org/10.1016/S0749-6419(03)00099-8
# A parameter-free strain estimation method to prevent occurrence of splits in sheet forming of complex CAD designs

EYCKENS Philip<fnref n="1" />,ᵃ*, SALAETS Rob<fnref n="2" />,ᵇ and NELISSEN Gert<fnref n="3" />,ᶜ

**Keywords:** Strain Estimate, Conformal Projection, Hydroforming

**Abstract.** Early detection of manufacturability issues in the design stage is key to avoid long lead times in product development. For various sheet forming processes a critical issue is the occurrence of splits, due to forming beyond the intrinsic formability properties represented by the Forming Limit Curve (FLC). Nevertheless, within industrial engineering practice a detailed manufacturing process simulation including forming limit prediction is not obvious. While commercial Finite Element analysis tools for sheet metal forming are readily available on the market, because of current limitations in availability of accurate data, companies may be rather inclined to choose experimental verification over process simulation analysis. For formability analysis, the required accurate data comprises both material data (hardening data, plastic anisotropy data, FLC data) as well as process data (e.g. non-linear friction data in lubricated system; static or dynamic tool elasticity properties). Therefore, we propose a first-order approximation of the forming deformation, without any parameter (e.g. for material, friction), based solely on conformal flattening of the CAD sheet surface. In this work, we show the usefulness of this method in hydroforming of complex-shaped, industrial parts (bipolar plates). It is shown that from the strain estimation, we can identify the most critically strained regions for individual categories of strain, as formability is known to be greatly dependent on strain mode. The tooling assists to compare design alternatives in terms of formability in the absence of FLC or even basic material data.

**Challenges in Hydroforming of bipolar plates for fuel cells**

Polymer Electrolyte Membrane (PEM) fuel cells require an intricate network of fine-mazed channels to guide gasses and coolant in-between sheets of MEA (Membrane Electrode Assembly). As illustrated in Fig. 1, such channels can be realized by bipolar plates, which are manufactured by stamping foils of materials such as 316L and Ti alloys. Common foil thicknesses are (75µm-500µm. The bipolar plate design determines the flow efficiency of gasses and coolant, and are therefore crucial for the fuel cell performance.

One of the commercial production technologies for bipolar plates is Hydrogate™, a continuous hydroforming process that delivers pressures up to 2000 bar, and that has been developed and patented by Borit NV. As all manufacturing processes, hydroforming has specific limits that may be translated in design practice by so-called manufacturing rules; a critical one of these is the occurrence of splits due to local deformation beyond the Forming Limit. However, experimental Forming Limit Curves are typically not available for commercial foils grades, and even basic material data to derive FLCs theoretically (see e.g. review article [1]), is usually lacking. In this industrial context, a strain estimation method for complex parts such as bipolar plates, offers a valuable tool to assess risk of splits.

<figure><img src="image_1183.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.

<footnote n="1">¹Flanders Make, corelab CodesignS, Gaston Geeslaan 8, 3001 Leuven, Belgium</footnote>

<footnote n="2">²Flanders Make, corelab MotionS, Gaston Geeslaan 8, 3001 Leuven, Belgium</footnote>

<footnote n="3">³Borit NV, Lammerdries 18ᵉ, 2440 Geel, Belgium</footnote>

<footnote>ᵃphilip.eyckens@flandersmake.be, ᵇrob.salaets@flandersmake.be, ᶜgert.nelissen@borit.be</footnote>
# A parameter-free strain estimation method to prevent occurrence of splits in sheet forming of complex CAD designs

EYCKENS Philip<fnref n="1" />,ᵃ*, SALAETS Rob<fnref n="2" />,ᵇ and NELISSEN Gert<fnref n="3" />,ᶜ

**Keywords:** Strain Estimate, Conformal Projection, Hydroforming

**Abstract.** Early detection of manufacturability issues in the design stage is key to avoid long lead times in product development. For various sheet forming processes a critical issue is the occurrence of splits, due to forming beyond the intrinsic formability properties represented by the Forming Limit Curve (FLC). Nevertheless, within industrial engineering practice a detailed manufacturing process simulation including forming limit prediction is not obvious. While commercial Finite Element analysis tools for sheet metal forming are readily available on the market, because of current limitations in availability of accurate data, companies may be rather inclined to choose experimental verification over process simulation analysis. For formability analysis, the required accurate data comprises both material data (hardening data, plastic anisotropy data, FLC data) as well as process data (e.g. non-linear friction data in lubricated system; static or dynamic tool elasticity properties). Therefore, we propose a first-order approximation of the forming deformation, without any parameter (e.g. for material, friction), based solely on conformal flattening of the CAD sheet surface. In this work, we show the usefulness of this method in hydroforming of complex-shaped, industrial parts (bipolar plates). It is shown that from the strain estimation, we can identify the most critically strained regions for individual categories of strain, as formability is known to be greatly dependent on strain mode. The tooling assists to compare design alternatives in terms of formability in the absence of FLC or even basic material data.

**Challenges in Hydroforming of bipolar plates for fuel cells**

Polymer Electrolyte Membrane (PEM) fuel cells require an intricate network of fine-mazed channels to guide gasses and coolant in-between sheets of MEA (Membrane Electrode Assembly). As illustrated in Fig. 1, such channels can be realized by bipolar plates, which are manufactured by stamping foils of materials such as 316L and Ti alloys. Common foil thicknesses are (75µm-500µm. The bipolar plate design determines the flow efficiency of gasses and coolant, and are therefore crucial for the fuel cell performance.

One of the commercial production technologies for bipolar plates is Hydrogate™, a continuous hydroforming process that delivers pressures up to 2000 bar, and that has been developed and patented by Borit NV. As all manufacturing processes, hydroforming has specific limits that may be translated in design practice by so-called manufacturing rules; a critical one of these is the occurrence of splits due to local deformation beyond the Forming Limit. However, experimental Forming Limit Curves are typically not available for commercial foils grades, and even basic material data to derive FLCs theoretically (see e.g. review article [1]), is usually lacking. In this industrial context, a strain estimation method for complex parts such as bipolar plates, offers a valuable tool to assess risk of splits.

<figure><img src="image_1185.png" /></figure>

<footnote n="1">¹Flanders Make, corelab CodesignS, Gaston Geeslaan 8, 3001 Leuven, Belgium</footnote>

<footnote n="2">²Flanders Make, corelab MotionS, Gaston Geeslaan 8, 3001 Leuven, Belgium</footnote>

<footnote n="3">³Borit NV, Lammerdries 18ᵉ, 2440 Geel, Belgium</footnote>

<footnote>ᵃphilip.eyckens@flandersmake.be, ᵇrob.salaets@flandersmake.be, ᶜgert.nelissen@borit.be</footnote>
<figure><img src="image_1184.png" /><figcaption>Fig. 1 (a) A fuel cell consists of stacks of (b) metal bipolar plates with intricate designs of channel-like features. Typical dimension: 80mm width. (c) shows a cross-section of these channels. (d) A hydroformed test plate showing the Borit logo and a series of channels.</figcaption></figure>

<figure><img src="image_1185.png" /><figcaption>Fig. 2. The Borit HydrogateTM process: a continuous sheet and foil hydroforming process by applying water pressure against a single-sided die.</figcaption></figure>
<figure><img src="image_1186.png" /><figcaption>Fig. 1 (a) A fuel cell consists of stacks of (b) metal bipolar plates with intricate designs of channel-like features. Typical dimension: 80mm width. (c) shows a cross-section of these channels. (d) A hydroformed test plate showing the Borit logo and a series of channels.</figcaption></figure>

<figure><img src="image_1187.png" /><figcaption>Fig. 2. The Borit HydrogateTM process: a continuous sheet and foil hydroforming process by applying water pressure against a single-sided die.</figcaption></figure>
Strain Estimation by Conformal Projection

Methodology.

Referring to Fig. 3, we use a conformal flattening method [2] to recover the initial position (undeformed state) of discrete points onto the design (final positions). As points, we choose the corner points of a discretized format of the CAD design by triangulation. A conformal map minimizes the angular distortion. The resulting projection has most distortion as areal distortion (scaling). This type of mapping is similar to the deformation in sheet forming processes, which tends to “smear out” the deformation in a homogeneous manner (at least until localization of deformation occurs). The physical link with elastic deformation of sheets is discussed extensively in [3].

Originating from cartography, conformal projection has been originally applied to map the spherical glob onto a (flat) surface. The mathematical principles can be generalized to project points and shapes from any non-flat surface onto a flat surface. In this application within sheet metal plasticity, the undeformed (flat) reference state forms the (a priori unknown) map. It can be shown that conformal projection or flattening is in fact equivalent to minimizing the Dirichlet energy or membrane energy of map $f: \mathcal{M} \to \mathbb{R}^{[0,1]\times[0,1]}$.

$$E_{memb} = \int_{\mathcal{M}} |\nabla f|^2 d\mathcal{M} \qquad (1)$$

This problem is typically discretized with the Laplace-Beltrami operator, with cotangent weights:

$$\Delta_{\mathcal{M}} f(v_i) = w_i \sum_{v_j \in N(v_i)} w_{ij} (f(v_j) - f(v_i)) \quad \text{with} \quad w_i = \frac{1}{A_i}, w_{ij} = \frac{1}{2} (\cot \alpha_{ij} + \cot \beta_{ij}) \qquad (2)$$

with $v_i$ the vertices of $\mathcal{M}$, $N(v_i)$ the 1-ring neighbourhood of a vertex $v_i$. The vertex area $A_i$ is one third the area of all triangles incident on the vertex and $\alpha_{ij}$, $\beta_{ij}$ are the opposing angles of an edge ij. A minimizer of the Dirichlet energy is given by solving the following Laplace problem, as a sparse linear system:

$$\Delta_{\mathcal{M}} f(v_i) = 0, \quad \begin{bmatrix} \Delta_{\mathcal{M}} \\ \mathbf{1} \end{bmatrix} \begin{bmatrix} u & v \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ b_u & b_v \end{bmatrix} \qquad (3)$$

with $b_u$ and $b_v$ the coordinates of the boundary vertices.

In hydroforming, it is assumed that the (flat) area of the foil that stays in contact with the (flat area of) the die throughout the process does not undergo any deformation, due to the “clamping” action of the water pressure. This flat area can be readily identified numerically and effectively provides boundary conditions for the solution of (1)-(3).

In a next step, from the initial and final positions of the 3 corners of any discrete triangular surface element, the deformation gradient tensor $\mathbf{F}$ is derived, in a similar way as is common for linear, 3-node shell elements in large-strain continuum mechanics. We assume triangles can be considered small enough so that we can find $\mathbf{F}$ by solving

$$\mathbf{F}[(u_2 - u_1) \quad (u_3 - u_1)] = [(v_2 - v_1) \quad (v_3 - v_1)] \qquad (4)$$

with $u_i, v_i$ vertices of undeformed and deformed state respectively. We can choose the reference frames aligned the triangle normal and the first edge, so that $u_i, v_i \in \mathbb{R}^2$ and $\mathbf{F} \in \mathbb{R}^{2 \times 2}$. A polar decomposition of $\mathbf{F} = \mathbf{R}U$ (with $\mathbf{R}$ unitary and $\mathbf{U}$ positive semi-definite symmetric) reveals the right stretch tensor. We can find the principal stretches, i.e. the eigenvalues of $\mathbf{U}$, by computing the eigenvalue decomposition of $\mathbf{F}^T \mathbf{F}$ since, $\mathbf{F}^T \mathbf{F} = \mathbf{U}^T \mathbf{U}$

This in turn defines the (large-strain) strain tensor $\boldsymbol{\epsilon} = \mathbf{U} - \mathbf{I}$ with $\mathbf{I}$ the identity tensor. The tensor $\boldsymbol{\epsilon}$ is characterized by its principal values, namely the minor and major in-plane strain and thickness strain, together with their respective principal directions (Fig. 3).
Strain Estimation by Conformal Projection

Methodology.

Referring to Fig. 3, we use a conformal flattening method [2] to recover the initial position (undeformed state) of discrete points onto the design (final positions). As points, we choose the corner points of a discretized format of the CAD design by triangulation. A conformal map minimizes the angular distortion. The resulting projection has most distortion as areal distortion (scaling). This type of mapping is similar to the deformation in sheet forming processes, which tends to “smear out” the deformation in a homogeneous manner (at least until localization of deformation occurs). The physical link with elastic deformation of sheets is discussed extensively in [3].

Originating from cartography, conformal projection has been originally applied to map the spherical glob onto a (flat) surface. The mathematical principles can be generalized to project points and shapes from any non-flat surface onto a flat surface. In this application within sheet metal plasticity, the undeformed (flat) reference state forms the (a priori unknown) map. It can be shown that conformal projection or flattening is in fact equivalent to minimizing the Dirichlet energy or membrane energy of map $f: \mathcal{M} \to \mathbb{R}^{[0,1]\times[0,1]}$.

$$E_{memb} = \int_{\mathcal{M}} |\nabla f|^2 d\mathcal{M} \qquad (1)$$

This problem is typically discretized with the Laplace-Beltrami operator, with cotangent weights:

$$\Delta_{\mathcal{M}} f(v_i) = w_i \sum_{v_j \in N(v_i)} w_{ij} (f(v_j) - f(v_i)) \quad \text{with } w_i = \frac{1}{A_i}, w_{ij} = \frac{1}{2} (\cot \alpha_{ij} + \cot \beta_{ij}) \qquad (2)$$

with $v_i$ the vertices of $\mathcal{M}$, $N(v_i)$ the 1-ring neighbourhood of a vertex $v_i$. The vertex area $A_i$ is one third the area of all triangles incident on the vertex and $\alpha_{ij}$, $\beta_{ij}$ are the opposing angles of an edge ij. A minimizer of the Dirichlet energy is given by solving the following Laplace problem, as a sparse linear system:

$$\Delta_{\mathcal{M}} f(v_i) = 0, \quad \begin{bmatrix} \Delta_{\mathcal{M}} \\ \mathbf{1} \end{bmatrix} \begin{bmatrix} u & v \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ b_u & b_v \end{bmatrix} \qquad (3)$$

with $b_u$ and $b_v$ the coordinates of the boundary vertices.

In hydroforming, it is assumed that the (flat) area of the foil that stays in contact with the (flat area of) the die throughout the process does not undergo any deformation, due to the “clamping” action of the water pressure. This flat area can be readily identified numerically and effectively provides boundary conditions for the solution of (1)-(3).

In a next step, from the initial and final positions of the 3 corners of any discrete triangular surface element, the deformation gradient tensor $\mathbf{F}$ is derived, in a similar way as is common for linear, 3-node shell elements in large-strain continuum mechanics. We assume triangles can be considered small enough so that we can find $\mathbf{F}$ by solving

$$\mathbf{F}[(u_2 - u_1) \quad (u_3 - u_1)] = [(v_2 - v_1) \quad (v_3 - v_1)] \qquad (4)$$

with $u_i, v_i$ vertices of undeformed and deformed state respectively. We can choose the reference frames aligned the triangle normal and the first edge, so that $u_i, v_i \in \mathbb{R}^2$ and $\mathbf{F} \in \mathbb{R}^{2 \times 2}$. A polar decomposition of $\mathbf{F} = \mathbf{R}U$ (with $\mathbf{R}$ unitary and $\mathbf{U}$ positive semi-definite symmetric) reveals the right stretch tensor. We can find the principal stretches, i.e. the eigenvalues of $\mathbf{U}$, by computing the eigenvalue decomposition of $\mathbf{F}^T \mathbf{F}$ since, $\mathbf{F}^T \mathbf{F} = \mathbf{U}^T \mathbf{U}$

This in turn defines the (large-strain) strain tensor $\boldsymbol{\epsilon} = \mathbf{U} - \mathbf{I}$ with $\mathbf{I}$ the identity tensor. The tensor $\boldsymbol{\epsilon}$ is characterized by its principal values, namely the minor and major in-plane strain and thickness strain, together with their respective principal directions (Fig. 3).
<figure><img src="image_1186.png" /><figcaption>Fig. 3 (left) Example of the projection of a discrete triangular element from undeformed (flat) state to the side of a channel in the deformed state. (right) This transformation is mathematically described by the deformation gradient tensor **F**, which can be constructed from the initial and final coordinates of the 3 triangle corners. From **F**, the strain of the triangular element can be derived, characterized by the major and minor strains with their respective principal directions.</figcaption></figure>

Implementation.

The methodology described above has been implemented as base functions within CameCAD, a
prototype manufacturability assessment software tool for CAD designs. As outlined in [4],
CameCAD interpretes intricate manufacturing knowledge that has been formalized by the
Knowledge Formalization Editor. The KFE is conceived to describe manufacturing knowledge as
geometry-based manufacturing rules within a generic formalism, meaning it is conceived to be
applicable to any manufacturing process for which geometric manufacturing knowledge may be
formulated; the strain estimation method as outlined in this paper just being one example among
many [5].

<figure><img src="image_1187.png" /><figcaption>Fig. 4 Example of strain estimation by conformal projection, showing (left) the (dimensionless) von Mises equivalent strain and (right) the (dimensionless) strain mode, defined as the ratio of major over minor strain. The flat, undeformed area is shown here with label "Not Calculated".
Plots generated by CameCAD UI [4-5].</figcaption></figure>

The CameCAD User Interface produces, among others, color plots of the described manufacturing knowledge as evaluated over the surface of CAD designs. Fig. 4 shows a result of strain estimate by conformal projection.
<figure><img src="image_1188.png" /><figcaption>Fig. 3 (left) Example of the projection of a discrete triangular element from undeformed (flat) state to the side of a channel in the deformed state. (right) This transformation is mathematically described by the deformation gradient tensor **F**, which can be constructed from the initial and final coordinates of the 3 triangle corners. From **F**, the strain of the triangular element can be derived, characterized by the major and minor strains with their respective principal directions.</figcaption></figure>

Implementation.

The methodology described above has been implemented as base functions within CameCAD, a
prototype manufacturability assessment software tool for CAD designs. As outlined in [4],
CameCAD interpretes intricate manufacturing knowledge that has been formalized by the
Knowledge Formalization Editor. The KFE is conceived to describe manufacturing knowledge as
geometry-based manufacturing rules within a generic formalism, meaning it is conceived to be
applicable to any manufacturing process for which geometric manufacturing knowledge may be
formulated; the strain estimation method as outlined in this paper just being one example among
many [5].

<figure><img src="image_1189.png" /><figcaption>Fig. 4 Example of strain estimation by conformal projection, showing (left) the (dimensionless) von Mises equivalent strain and (right) the (dimensionless) strain mode, defined as the ratio of major over minor strain. The flat, undeformed area is shown here with label "Not Calculated".
Plots generated by CameCAD UI [4-5].</figcaption></figure>

The CameCAD User Interface produces, among others, color plots of the described manufacturing knowledge as evaluated over the surface of CAD designs. Fig. 4 shows a result of strain estimate by conformal projection.
# Use Case: Hydroforming Test Die

## Experiments

Before ordering a production die suitable for mass-production of a new bipolar plate design, it is safe engineering practice to have try-out tests of the design using a smaller (and cheaper) ‘test die’, cf. Fig. 5. It is crucial in this practice that the test die features the critical regions of the overall bipolar plate, as missing such feature could result in splits that are noticed only at the start of full-scale production using the production die. It is in this respect that our strain estimation tool provides a practical and insightful engineering check, as is illustrated next with a real-life example of industrial practice.

<figure><img src="image_1188.png" /><figcaption>Fig. 5 Out of (a) a bipolar plate, a smaller region is selected for testing in a test die: (b) shows the corresponding CAD and (c) a forming test result with 2 splits.</figcaption></figure>

## Thinning estimation to improve the CAD design process

An existing customer to Borit wishes to design a next-generation fuel cell product, in which the bipolar plates are thinner and of a new alloy, in comparison to previous product version. However, preliminary testing with the available test die onto the new material, results in systematic splitting at the locations depicted in Fig. 5(c). This triggers an engineering discussion between Borit and its customer, resulting in 3 alternative design proposals. With the absence of formability data from material supplier, or even basic material data a detailed formability analysis is not an option.
# Use Case: Hydroforming Test Die

## Experiments

Before ordering a production die suitable for mass-production of a new bipolar plate design, it is safe engineering practice to have try-out tests of the design using a smaller (and cheaper) ‘test die’, cf. Fig. 5. It is crucial in this practice that the test die features the critical regions of the overall bipolar plate, as missing such feature could result in splits that are noticed only at the start of full-scale production using the production die. It is in this respect that our strain estimation tool provides a practical and insightful engineering check, as is illustrated next with a real-life example of industrial practice.

<figure><img src="image_1190.png" /><figcaption>Fig. 5 Out of (a) a bipolar plate, a smaller region is selected for testing in a test die: (b) shows the corresponding CAD and (c) a forming test result with 2 splits.</figcaption></figure>

## Thinning estimation to improve the CAD design process

An existing customer to Borit wishes to design a next-generation fuel cell product, in which the bipolar plates are thinner and of a new alloy, in comparison to previous product version. However, preliminary testing with the available test die onto the new material, results in systematic splitting at the locations depicted in Fig. 5(c). This triggers an engineering discussion between Borit and its customer, resulting in 3 alternative design proposals. With the absence of formability data from material supplier, or even basic material data a detailed formability analysis is not an option.
<figure><img src="image_1189.png" /><figcaption>Fig. 6 Simultaneous analysis for the largest-strained regions for original design (labeled A) and 3 alternative test die designs (labeled B, C and D), each within 4 “categories of strain mode”. The Forming Limit Diagrams shown indicate the respective strain mode category with colored pie-section (a strain mode category is defined here as a specific range of strain mode between a lower and upper limit). Per strain mode category, the areas that are less than 60% strained of the maximum value are colored green, more than 60% but less than 80% are colored orange, and more than 80% are colored red and are additionally encircled with an indication of the area size, in mm². Plots generated by CameCAD UI [4-5].</figcaption></figure>
<figure><img src="image_1191.png" /><figcaption>Fig. 6 Simultaneous analysis for the largest-strained regions for original design (labeled A) and 3 alternative test die designs (labeled B, C and D), each within 4 “categories of strain mode”. The Forming Limit Diagrams shown indicate the respective strain mode category with colored pie-section (a strain mode category is defined here as a specific range of strain mode between a lower and upper limit). Per strain mode category, the areas that are less than 60% strained of the maximum value are colored green, more than 60% but less than 80% are colored orange, and more than 80% are colored red and are additionally encircled with an indication of the area size, in mm². Plots generated by CameCAD UI [4-5].</figcaption></figure>
In this context, the CameCAD tool provides valuable insights to steer the design selection in terms of manufacturability. Firstly, as Seen in Fig. 6, it confirms that the split location is a highly-strained region under (near-)equibiaxial deformation conditions. Secondly, referring to Table 1, it confirms that the strain level are significantly lower with the alternative designs, which explains why there are no splits observed for these designs. In the end, the most performant design that is manufacturable, is selected.

<table><caption>Table 1: Summary of test die designs, with strain data obtained by CameCAD [4].</caption><thead><tr><th>Alternative Test Die Designs</th><th>Max. equivalent strain (equibiaxial category)</th><th>Experimental outcome</th><th>Conclusion</th></tr></thead><tbody><tr><td>A (original)</td><td>1.04</td><td>Systematic splits (cf. Fig. 5(c))</td><td>Rejected: manufacturing issues</td></tr><tr><td>B</td><td>0.84</td><td>Manufacturable</td><td><strong>Selected design</strong></td></tr><tr><td>C</td><td>0.75</td><td>Manufacturable</td><td>Rejected (poor functional performance)</td></tr><tr><td>D</td><td>0.73</td><td>Manufacturable</td><td>Rejected (poor functional performance)</td></tr></tbody></table>

## Summary

For intricate 3D designs to be manufactured by sheet forming, we propose a strain estimation method on the basis of conformal flattening. Given the limitations in accuracy of predictions, which is to date not yet sufficiently quantified, this methodology nevertheless forms a useful engineering tool in common industrial practice, notably since no material data is required and no process data is needed apart from the CAD design.

## Acknowledgement

The authors like to acknowledge the support towards SMEs from the AMULET project - The Advanced Materials & Manufacturing United for LightwEighT (https://amulet-h2020.eu/). AMULET is funded from the European Union's Horizon 2020 INNOSUP Programme under grant agreement No 101005435.

## References

[1] D. Banabic, A. Kami, D.-S. Comsa, & P. Eyckens, Philip. (2019). Developments of the Marciniak-Kuczynski Model for Sheet Metal Formability: a Review. Journal of Materials Processing Technology. 287. 10.1016/j.jmatprotec.2019.116446

[2] M. Botsch and O. Sorkine, "On Linear Variational Surface Deformation Methods," in IEEE Transactions on Visualization and Computer Graphics, vol. 14, no. 1, pp. 213-230, Jan.-Feb. 2008. https://doi.org/10.1109/TVCG.2007.1054

[3] Rumpf, M. and Wardetzky, M. (2014), Geometry processing from an elastic perspective. GAMM-Mitteilungen, 37: 184-216. https://doi.org/10.1002/gamm.201410009

[4] R. Salaets, B. Decraemer, P. Eyckens. A flexible approach for design rule formalization and evaluation. https://doi.org/10.1016/j.procir.2022.05.294

[5] P. Eyckens & B. Decraemer, webinar "Save time and money by manufacturability analysis of CAD designs", 6 Nov 2023, recording: https://youtu.be/VSoZ36GfxSo
In this context, the CameCAD tool provides valuable insights to steer the design selection in terms of manufacturability. Firstly, as Seen in Fig. 6, it confirms that the split location is a highly-strained region under (near-)equibiaxial deformation conditions. Secondly, referring to Table 1, it confirms that the strain level are significantly lower with the alternative designs, which explains why there are no splits observed for these designs. In the end, the most performant design that is manufacturable, is selected.

<table><caption>Table 1: Summary of test die designs, with strain data obtained by CameCAD [4].</caption><thead><tr><th>Alternative Test Die Designs</th><th>Max. equivalent strain (equibiaxial category)</th><th>Experimental outcome</th><th>Conclusion</th></tr></thead><tbody><tr><td>A (original)</td><td>1.04</td><td>Systematic splits (cf. Fig. 5(c))</td><td>Rejected: manufacturing issues</td></tr><tr><td>B</td><td>0.84</td><td>Manufacturable</td><td><strong>Selected design</strong></td></tr><tr><td>C</td><td>0.75</td><td>Manufacturable</td><td>Rejected (poor functional performance)</td></tr><tr><td>D</td><td>0.73</td><td>Manufacturable</td><td>Rejected (poor functional performance)</td></tr></tbody></table>

## Summary

For intricate 3D designs to be manufactured by sheet forming, we propose a strain estimation method on the basis of conformal flattening. Given the limitations in accuracy of predictions, which is to date not yet sufficiently quantified, this methodology nevertheless forms a useful engineering tool in common industrial practice, notably since no material data is required and no process data is needed apart from the CAD design.

## Acknowledgement

The authors like to acknowledge the support towards SMEs from the AMULET project - The Advanced Materials & Manufacturing United for LightwEighT (https://amulet-h2020.eu/). AMULET is funded from the European Union's Horizon 2020 INNOSUP Programme under grant agreement No 101005435.

## References

[1] D. Banabic, A. Kami, D.-S. Comsa, & P. Eyckens, Philip. (2019). Developments of the Marciniak-Kuczynski Model for Sheet Metal Formability: a Review. Journal of Materials Processing Technology. 287. 10.1016/j.jmatprotec.2019.116446

[2] M. Botsch and O. Sorkine, "On Linear Variational Surface Deformation Methods," in IEEE Transactions on Visualization and Computer Graphics, vol. 14, no. 1, pp. 213-230, Jan.-Feb. 2008. https://doi.org/10.1109/TVCG.2007.1054

[3] Rumpf, M. and Wardetzky, M. (2014), Geometry processing from an elastic perspective. GAMM-Mitteilungen, 37: 184-216. https://doi.org/10.1002/gamm.201410009

[4] R. Salaets, B. Decraemer, P. Eyckens. A flexible approach for design rule formalization and evaluation. https://doi.org/10.1016/j.procir.2022.05.294

[5] P. Eyckens & B. Decraemer, webinar "Save time and money by manufacturability analysis of CAD designs", 6 Nov 2023, recording: https://youtu.be/VSoZ36GfxSo
Friction and wear in metal forming

<figure><img src="image_1190.png" /></figure>
Friction and wear in metal forming

<figure><img src="image_1192.png" /></figure>
# Massive nitrogen super-saturation into CoCrMo Alloys for improvement of tribological performance

AIZAWA Tatsuhiko¹,ᵃ*, FUNAZUKA Tatsuya²,ᵇ, SHIRATORI Tomomi²,ᶜ and SUZUKI Yohei³,ᵈ

¹³-15-10 Minami-Rokugo, Ota-City, Tokyo 144-0045, Japan

²³190 Gofuku, Toyama-City, Toyama 930-8555, Japan

³⁹⁴²-2 Shiga-Kuwahara, Suwa-City, Nagano 392-0012, Japan

ᵃtaizawa@sic.shibaura-it.ac.jp, ᵇfunazuka@eng.u-toyama.ac.jp, ᶜshira@eng.u-toyama.ac.jp, ᵈy-suzuki@komatsuseiki.co.jp

**Keywords:** Plasma Nitriding, CoCrMo Alloys, Massive Nitrogen Interstitials, Low Friction and Wear

**Abstract.** The cobalt-chromium-molybdenum (CoCrMo) superalloys have been utilized in the dental, knee, and hip arthroplasty applications because of their unique combination of high strength, high wear and corrosion resistance. A massive nitrogen supersaturation (MNS) process was proposed as a surface treatment to improve these strength, hardness and ductility, further needed for hip replacement and dental prosthesis. CoCrMo disc specimens were prepared for plasma immersion nitriding at 673 K for 21.6 ks by 50 Pa. The nitrogen solute uniformly distributed in the MNS-layer with the average content of 5 mass%. Higher surface hardness than 1550 HV was attained under application of 10 N. The maximum friction coefficient was halved before and after MNS and this low frictional state with μ< 0.45 was preserved although the sliding condition by 0.1 m/s.

**Introduction**

Cobalt-chromium (CoCr) and cobalt-chromium-molybdenum (CoCrMo) superalloys have been utilized in the dental, knee and hip arthroplasty applications because of their unique combination of high strength, high wear and corrosion resistance, and excellent fatigue strength [1]. Various studies were reported in literature to improve the hardness by multi-phase structuring [2], to attain higher tensile and fatigue strength by heat treatment [3], to overcome their intrinsic low ductility by enrichment of chromium and nitrogen [4], and to much improve the mechanical balancing among strength, hardness and ductility by surface treatment [5, 6]. In these approaches, the plasma nitriding processes were high-lighted to increase the surface hardness without loss of high corrosion toughness [7-9]. Among them, the low temperature plasma nitriding provided a way to make nitrogen supersaturation to CoCr and CoCrMo alloys with much higher nitrogen solute content than their intrinsic solubility limit [10].

Owing to this massive nitrogen supersaturation (MNS), the original surface hardness of 450 HV<sub>1N</sub> was increased two to three times after plasma nitriding at 673 K for 14.4 ks (or 4 h). The average nitrogen content reached 5 mass% in the nitrided surface layer. To be noticed, this nitrogen supersaturated layer had an intensely textured microstructure with nearly the same orientation to (111) crystallographic orientation.

In the present study, this microstructure control effect on the mechanical properties of MNS-CoCrMo, is investigated by the hardness testing and tribotesting. First, the low temperature plasma nitriding conditions are described with the comments on the holding temperature and the mixture gas pressure. XRD, SEM-EDX, and EBSD are utilized to make microstructure analysis on the surface and cross-section of MNS-CoCrMo specimen. The hardness measurement and tribological

<figure><img src="image_1191.png" /></figure>
# Massive nitrogen super-saturation into CoCrMo Alloys for improvement of tribological performance

AIZAWA Tatsuhiko¹,ᵃ*, FUNAZUKA Tatsuya²,ᵇ, SHIRATORI Tomomi²,ᶜ and SUZUKI Yohei³,ᵈ

¹³-15-10 Minami-Rokugo, Ota-City, Tokyo 144-0045, Japan

²³¹⁹⁰ Gofuku, Toyama-City, Toyama 930-8555, Japan

³⁹⁴²-2 Shiga-Kuwahara, Suwa-City, Nagano 392-0012, Japan

ᵃtaizawa@sic.shibaura-it.ac.jp, ᵇfunazuka@eng.u-toyama.ac.jp, ᶜshira@eng.u-toyama.ac.jp, ᵈy-suzuki@komatsuseiki.co.jp

**Keywords:** Plasma Nitriding, CoCrMo Alloys, Massive Nitrogen Interstitials, Low Friction and Wear

**Abstract.** The cobalt-chromium-molybdenum (CoCrMo) superalloys have been utilized in the dental, knee, and hip arthroplasty applications because of their unique combination of high strength, high wear and corrosion resistance. A massive nitrogen supersaturation (MNS) process was proposed as a surface treatment to improve these strength, hardness and ductility, further needed for hip replacement and dental prosthesis. CoCrMo disc specimens were prepared for plasma immersion nitriding at 673 K for 21.6 ks by 50 Pa. The nitrogen solute uniformly distributed in the MNS-layer with the average content of 5 mass%. Higher surface hardness than 1550 HV was attained under application of 10 N. The maximum friction coefficient was halved before and after MNS and this low frictional state with μ< 0.45 was preserved although the sliding condition by 0.1 m/s.

**Introduction**

Cobalt-chromium (CoCr) and cobalt-chromium-molybdenum (CoCrMo) superalloys have been utilized in the dental, knee and hip arthroplasty applications because of their unique combination of high strength, high wear and corrosion resistance, and excellent fatigue strength [1]. Various studies were reported in literature to improve the hardness by multi-phase structuring [2], to attain higher tensile and fatigue strength by heat treatment [3], to overcome their intrinsic low ductility by enrichment of chromium and nitrogen [4], and to much improve the mechanical balancing among strength, hardness and ductility by surface treatment [5, 6]. In these approaches, the plasma nitriding processes were high-lighted to increase the surface hardness without loss of high corrosion toughness [7-9]. Among them, the low temperature plasma nitriding provided a way to make nitrogen supersaturation to CoCr and CoCrMo alloys with much higher nitrogen solute content than their intrinsic solubility limit [10].

Owing to this massive nitrogen supersaturation (MNS), the original surface hardness of 450 HV₁N was increased two to three times after plasma nitriding at 673 K for 14.4 ks (or 4 h). The average nitrogen content reached 5 mass% in the nitrided surface layer. To be noticed, this nitrogen supersaturated layer had an intensely textured microstructure with nearly the same orientation to (111) crystallographic orientation.

In the present study, this microstructure control effect on the mechanical properties of MNS-CoCrMo, is investigated by the hardness testing and tribotesting. First, the low temperature plasma nitriding conditions are described with the comments on the holding temperature and the mixture gas pressure. XRD, SEM-EDX, and EBSD are utilized to make microstructure analysis on the surface and cross-section of MNS-CoCrMo specimen. The hardness measurement and tribological

<figure><img src="image_1193.png" /></figure>
testing are employed to describe the mechanical and tribological behavior of MNS-CoCrMo and to discuss its possibility in working as a die for precise forging process.

## Methods and Materials

The low temperature plasma nitriding system is explained for massive nitrogen supersaturation to CoCrMo specimen. Microstructure analysis and mechanical characterization is performed to investigate the relationship between the MNS-induced nanostructure and hardness as well as tribological performance,

**Massive nitrogen supersaturation.** The plasma carburizing system is schematically illustrated in Fig. 1a. With reference to the previous studies in [11-12], the hollow cathode device was employed to increase the nitrogen-ion and NH-radical densities more than 3 x 10¹⁷ ions/m³ and to deepen the nitrogen supersaturated layer thickness. This system consisted of the RF (Radio Frequency) - and DC (Direct Current) - generators to independently ignite the nitrogen – hydrogen plasmas.

<figure><img src="image_1192.png" /><figcaption>Fig. 1. Plasma immersion nitriding for massive nitrogen supersaturation. a) A schematic view on the experimental setup, and b) overview on the nitriding system.</figcaption></figure>

This system in Fig. 1a consisted of the vacuum chamber, the evacuation unit, the RF-DC generators, the mixture gas supply of N₂ and H₂, the heating unit, and, the hollow cathode device. As shown in Fig. 1b, the specimen was placed in this hollow cathode setup for high densification of nitrogen ions, NH radicals and electrons. Both the hollow and the specimen were electrically connected to the same DC-bias. The ionization from the neutral molecules of N₂ and H₂ to the activated species {N⁺, N*, NH} was enhanced in this hollow cathode setup. This enhancement of plasma density was controlled by the ratio of gas flow rate in N₂ + H₂. The yield of NH radicals and nitrogen ions (N⁺) in the plasma sheath increases with decreasing H₂ flow rate after the quantitative plasma diagnosis in [11-12].

After setting the specimen into the hollow cathode setup, the chamber was evacuated down to 0.1 Pa, and, filled with the nitrogen gas until 100 Pa. Under this constant pressure, the specimen was presputtered at 673 K for 1.8 ks (or 0.5 h) with the DC-bias of -500 V. The plasma nitriding followed this pre-sputtering stage continuously. The specimen was plasma nitrided at 673 K for 21.6 ks (or 6 h) at 50 Pa. After the plasma diagnosis as stated in above, the plasma nitriding process parameters were controlled to have the RF voltage of 250V, the DC-bias of -500 V and the gas flow rate ratio of 160 ml/min for N₂ and 30 ml/min for H₂, respectively. After this nitriding process, the specimen was cooled down to RT in the chamber under the N₂ atmosphere
testing are employed to describe the mechanical and tribological behavior of MNS-CoCrMo and to discuss its possibility in working as a die for precise forging process.

## Methods and Materials

The low temperature plasma nitriding system is explained for massive nitrogen supersaturation to CoCrMo specimen. Microstructure analysis and mechanical characterization is performed to investigate the relationship between the MNS-induced nanostructure and hardness as well as tribological performance,

**Massive nitrogen supersaturation.** The plasma carburizing system is schematically illustrated in Fig. 1a. With reference to the previous studies in [11-12], the hollow cathode device was employed to increase the nitrogen-ion and NH-radical densities more than 3 x 10¹⁷ ions/m³ and to deepen the nitrogen supersaturated layer thickness. This system consisted of the RF (Radio Frequency) - and DC (Direct Current) - generators to independently ignite the nitrogen – hydrogen plasmas.

<figure><img src="image_1194.png" /><figcaption>Fig. 1. Plasma immersion nitriding for massive nitrogen supersaturation. a) A schematic view on the experimental setup, and b) overview on the nitriding system.</figcaption></figure>

This system in Fig. 1a consisted of the vacuum chamber, the evacuation unit, the RF-DC generators, the mixture gas supply of N₂ and H₂, the heating unit, and, the hollow cathode device. As shown in Fig. 1b, the specimen was placed in this hollow cathode setup for high densification of nitrogen ions, NH radicals and electrons. Both the hollow and the specimen were electrically connected to the same DC-bias. The ionization from the neutral molecules of N₂ and H₂ to the activated species {N⁺, N*, NH} was enhanced in this hollow cathode setup. This enhancement of plasma density was controlled by the ratio of gas flow rate in N₂ + H₂. The yield of NH radicals and nitrogen ions (N⁺) in the plasma sheath increases with decreasing H₂ flow rate after the quantitative plasma diagnosis in [11-12].

After setting the specimen into the hollow cathode setup, the chamber was evacuated down to 0.1 Pa, and, filled with the nitrogen gas until 100 Pa. Under this constant pressure, the specimen was presputtered at 673 K for 1.8 ks (or 0.5 h) with the DC-bias of -500 V. The plasma nitriding followed this pre-sputtering stage continuously. The specimen was plasma nitrided at 673 K for 21.6 ks (or 6 h) at 50 Pa. After the plasma diagnosis as stated in above, the plasma nitriding process parameters were controlled to have the RF voltage of 250V, the DC-bias of -500 V and the gas flow rate ratio of 160 ml/min for N₂ and 30 ml/min for H₂, respectively. After this nitriding process, the specimen was cooled down to RT in the chamber under the N₂ atmosphere
**CoCrMo disc.** The CoCrMo alloy rod (Cobarion; Eiwa, Co., Ltd., Iwate, Japan) was cut and sliced to the disc specimens with the diameter of 20 mm and the thickness of 5 mm. Each disc specimen was adjusted in dimension and mirror polished on its either surface. Its chemical composition was listed in Table 1. Its normal grain size in average was 20 µm.

<table><caption>Table 1. Chemical composition of CoCrMo disc specimen for nitrogen supersaturation.</caption><thead><tr><th>Element</th><th>Cr</th><th>Mo</th><th>Si</th><th>Mn</th><th>Fe</th><th>Ni</th><th>N</th><th>C</th><th>Co</th></tr></thead><tbody><tr><th>Content</th><td>27.14</td><td>5.99</td><td>0.59</td><td>0.61</td><td>0.06</td><td>&lt; 0.01</td><td>0.11</td><td>0.052</td><td>Balance</td></tr></tbody></table>

*mass%

**Microstructural analysis.** XRD (X-Ray Diffraction) was first utilized to describe the phase change before and after MNS. SEM (Scanning Electron Microscopy), EDX (Energy Dispersive X-ray spectroscopy) and EBSD (Electron Back-Scattering Diffraction) were used for microstructure analysis and element mapping on the surface and cross-section of MNS-CoCrMo disc specimens.

The cross-section of sample was mechanically polished by SiC sand paper with increasing the mesh-grades and by the alumina powder solution before finishing. The mirror-like surface condition was adjusted and finished below the mask by the ion milling (HITACH-IM400) with argon etching.

In EBSD analysis, IPF (Inverse Pole Figure) was mapped in the ND (Normal Direction), RD (Rolling Direction) and TD (Tangential Direction), respectively. The crystallographic structure was analyzed by this IPF. KAM (Kernel Angle Misorientation) was employed to describe the local plastic strain distribution in the MNS-specimen. Phase mapping was also utilized to analyze the effect of plastic straining to the transformation from γ-phase. The surface profile of nitrided specimen was observed by the optical microscopy and SEM.

**Mechanical characterization.** Micro-Vickers testing (Mitsutoyo, Co., Ltd.; Kanagawa, Japan) was employed to measure the surface hardness by varying the applied load. The hardness depth profile was also analyzed under the constant applied load by 10 N. Tribotesting (CVS; Switzerland) was utilized to measure the variation of friction coefficient with increasing the sliding distance up to L = 500 m. As depicted in Fig. 2a, the friction coefficient (f) was defined as follows; f = F / W for the measured shear force (F), and the applied normal load (W). As shown in Fig. 2b, the sliding speed (V) was kept constant by 0.1 m/s.

<figure><img src="image_1193.png" /><figcaption>Fig. 2. Ball-on-disc tribological testing system. a) Schematic view on the rotational sliding test, and b) overview of testing system.</figcaption></figure>
**CoCrMo disc.** The CoCrMo alloy rod (Cobarion; Eiwa, Co., Ltd., Iwate, Japan) was cut and sliced to the disc specimens with the diameter of 20 mm and the thickness of 5 mm. Each disc specimen was adjusted in dimension and mirror polished on its either surface. Its chemical composition was listed in Table 1. Its normal grain size in average was 20 µm.

<table><caption>Table 1. Chemical composition of CoCrMo disc specimen for nitrogen supersaturation.</caption><thead><tr><th>Element</th><th>Cr</th><th>Mo</th><th>Si</th><th>Mn</th><th>Fe</th><th>Ni</th><th>N</th><th>C</th><th>Co</th></tr></thead><tbody><tr><th>Content</th><td>27.14</td><td>5.99</td><td>0.59</td><td>0.61</td><td>0.06</td><td>&lt; 0.01</td><td>0.11</td><td>0.052</td><td>Balance</td></tr></tbody></table>

*mass%

**Microstructural analysis.** XRD (X-Ray Diffraction) was first utilized to describe the phase change before and after MNS. SEM (Scanning Electron Microscopy), EDX (Energy Dispersive X-ray spectroscopy) and EBSD (Electron Back-Scattering Diffraction) were used for microstructure analysis and element mapping on the surface and cross-section of MNS-CoCrMo disc specimens.

The cross-section of sample was mechanically polished by SiC sand paper with increasing the mesh-grades and by the alumina powder solution before finishing. The mirror-like surface condition was adjusted and finished below the mask by the ion milling (HITACH-IM400) with argon etching.

In EBSD analysis, IPF (Inverse Pole Figure) was mapped in the ND (Normal Direction), RD (Rolling Direction) and TD (Tangential Direction), respectively. The crystallographic structure was analyzed by this IPF. KAM (Kernel Angle Misorientation) was employed to describe the local plastic strain distribution in the MNS-specimen. Phase mapping was also utilized to analyze the effect of plastic straining to the transformation from γ-phase. The surface profile of nitrided specimen was observed by the optical microscopy and SEM.

**Mechanical characterization.** Micro-Vickers testing (Mitsutoyo, Co., Ltd.; Kanagawa, Japan) was employed to measure the surface hardness by varying the applied load. The hardness depth profile was also analyzed under the constant applied load by 10 N. Tribotesting (CVS; Switzerland) was utilized to measure the variation of friction coefficient with increasing the sliding distance up to L = 500 m. As depicted in Fig. 2a, the friction coefficient (f) was defined as follows; f = F / W for the measured shear force (F), and the applied normal load (W). As shown in Fig. 2b, the sliding speed (V) was kept constant by 0.1 m/s.

<figure><img src="image_1195.png" /><figcaption>Fig. 2. Ball-on-disc tribological testing system. a) Schematic view on the rotational sliding test, and b) overview of testing system.</figcaption></figure>
## Results

The microstructure of MNS-CoCrMo discs at 673 K was analyzed on their surface and cross-section. Their crystallographic textures were investigated to describe the microstructural evolution through MNS. Micro-hardness testing was used to analyze the hardening behavior at the MNS-layer. Tribotesting was employed to prove that lower frictional state was preserved at the MNS-specimen than the bare CoCrMo alloys.

**Microstructure analysis on the surface of NS-CoCrMo.** High resolution optical microscopy was utilized to observe the MNS-layer surface with comparison to a bare CoCrMo disc. Fig. 3 depicts the overview of the MNS-CoCrMo specimen as well we the crystallographic structure. As explained before, the bare CoCrMo had a uniform microstructure with the average grain size of 20 µm. On the other hand, as pointed by the white arrows in Fig. 3b, the most of grains with fine grain boundaries have a lot of fine skewed scratches on their surface with small-sized spouts. The original crystallographic structure with the average grain size of 20 µm was significantly modified to have plastically deformed grains with fine slip-lines in each inside.

<figure><img src="image_1194.png" /><figcaption>Fig. 3. Optical-microscopic image of CoCrMo disc after plasm immersion nitriding. a) Overview of MNS-CoCrMo disc, and b) microscopic image on its surface.</figcaption></figure>

<figure><img src="image_1195.png" /><figcaption>Fig. 4. Comparison of the XRD diagrams for CoCrMo discs before and after plasma immersion nitriding. a) XRD diagram before MNS, and b) XRD diagram after MNS.</figcaption></figure>
## Results

The microstructure of MNS-CoCrMo discs at 673 K was analyzed on their surface and cross-section. Their crystallographic textures were investigated to describe the microstructural evolution through MNS. Micro-hardness testing was used to analyze the hardening behavior at the MNS-layer. Tribotesting was employed to prove that lower frictional state was preserved at the MNS-specimen than the bare CoCrMo alloys.

**Microstructure analysis on the surface of NS-CoCrMo.** High resolution optical microscopy was utilized to observe the MNS-layer surface with comparison to a bare CoCrMo disc. Fig. 3 depicts the overview of the MNS-CoCrMo specimen as well we the crystallographic structure. As explained before, the bare CoCrMo had a uniform microstructure with the average grain size of 20 µm. On the other hand, as pointed by the white arrows in Fig. 3b, the most of grains with fine grain boundaries have a lot of fine skewed scratches on their surface with small-sized spouts. The original crystallographic structure with the average grain size of 20 µm was significantly modified to have plastically deformed grains with fine slip-lines in each inside.

<figure><img src="image_1196.png" /><figcaption>Fig. 3. Optical-microscopic image of CoCrMo disc after plasm immersion nitriding. a) Overview of MNS-CoCrMo disc, and b) microscopic image on its surface.</figcaption></figure>

<figure><img src="image_1197.png" /><figcaption>Fig. 4. Comparison of the XRD diagrams for CoCrMo discs before and after plasma immersion nitriding. a) XRD diagram before MNS, and b) XRD diagram after MNS.</figcaption></figure>
XRD was utilized to describe the nitrogen supersaturation and the chromium nitride (CrN) formation via the plasma nitriding at 673 K for 21.6 ks. Fig. 4 compares the XRD diagrams between the bare and nitrided specimens. The XRD diagram for the bare CoCrMo alloy is characterized by two ◎-phase peaks at 2θ = 44° and 2θ = 51°, respectively. Under the alloying design in Table 1, the CoCrMo alloy disc specimen had fully austenitic phase without ε-phases. The MNS-CoCrMo alloy has an expanded γ-phase with CrN in the trace level. After [13], the grain size of CrN was estimated from the measured full width half maximum of this CrN broad peak to be 10 nm.

<figure><img src="image_1196.png" /><figcaption>Fig. 5. The pole Fig. in ND (Normal Direction) on the MNS-CoCrMo by the texture analysis using EBSD.</figcaption></figure>

EBSD was utilized to characterize the crystallographic surface structure on the MNS-CoCrMo disc. The pole figure in ND was shown in Fig. 5. MNS-CoCrMo is intensely textured to have a lot of nano-grains with the orientation to (111). Since the as-sliced CoCrMo disc from the rolled bar feedstock had no specific crystallographic orientations, this texturing must be induced by massive nitrogen supersaturation process.

**Microstructure analysis on the cross-section of NS-CoCrMo.** MNS-CoCrMo disc specimen was precisely cut to halves and polished for SEM-EDX analysis on the cross-sectional microstructure.

<figure><img src="image_1197.png" /><figcaption>Fig. 6. SEM-EDX analysis on the cross-section of the MNS-CoCrMo disc. a) SEM image on the cross-section of MNS-layer across the nitriding front end, and b) nitrogen mapping from the surface to the depth.</figcaption></figure>
XRD was utilized to describe the nitrogen supersaturation and the chromium nitride (CrN) formation via the plasma nitriding at 673 K for 21.6 ks. Fig. 4 compares the XRD diagrams between the bare and nitrided specimens. The XRD diagram for the bare CoCrMo alloy is characterized by two ◦-phase peaks at 2θ = 44° and 2θ = 51°, respectively. Under the alloying design in Table 1, the CoCrMo alloy disc specimen had fully austenitic phase without ε-phases. The MNS-CoCrMo alloy has an expanded γ-phase with CrN in the trace level. After [13], the grain size of CrN was estimated from the measured full width half maximum of this CrN broad peak to be 10 nm.

<figure><img src="image_1198.png" /><figcaption>Fig. 5. The pole Fig. in ND (Normal Direction) on the MNS-CoCrMo by the texture analysis using EBSD.</figcaption></figure>

EBSD was utilized to characterize the crystallographic surface structure on the MNS-CoCrMo disc. The pole figure in ND was shown in Fig. 5. MNS-CoCrMo is intensely textured to have a lot of nano-grains with the orientation to (111). Since the as-sliced CoCrMo disc from the rolled bar feedstock had no specific crystallographic orientations, this texturing must be induced by massive nitrogen supersaturation process.

**Microstructure analysis on the cross-section of NS-CoCrMo.** MNS-CoCrMo disc specimen was precisely cut to halves and polished for SEM-EDX analysis on the cross-sectional microstructure.

<figure><img src="image_1199.png" /><figcaption>Fig. 6. SEM-EDX analysis on the cross-section of the MNS-CoCrMo disc. a) SEM image on the cross-section of MNS-layer across the nitriding front end, and b) nitrogen mapping from the surface to the depth.</figcaption></figure>
Fig. 6a depicts the SEM image on the cross-section of MNS-CoCrMo. The MNS-layer was formed with the thickness of 12 µm. As shown in Fig. 6b, the nitrogen solute homogeneously distributes from the surface to the depth of 12 µm. The pointwise chemical composition analysis was performed at several positions from the surface to the depth; e.g., N₁ located at the vicinity of surface, N2 located just above the nitriding front end, and N3 located far from this nitriding front end. At these positions, the contents of Co, Cr, Mo, Si and N were measured to be nearly constant at every position on the cross-section; e.g., [Co] ~ 60 mass%, [Cr] ~ 25 mass%, [Mo] ~ 6.5 mass%, and [Si] = 0.6 mass%. The nitrogen content [N] distributes from the surface to the depth as depicted in Fig. 7.

<figure><img src="image_1198.png" /><figcaption>Fig. 7. Nitrogen solute content [N] from the surface to the depth across the nitriding front end.</figcaption></figure>

In the MNS-layer, the nitrogen content is more than 5 mass%. This [N] gradually decreases with the depth (d) across the nitriding front end. No nitrogen solute was detected when d > 15 µm. This high nitrogen content in the surface layer proves that nitrogen solute is massively saturated in the depth of CoCrMo specimen.

<figure><img src="image_1199.png" /><figcaption>Fig. 8. SEM image, phase mapping, plastic strain distribution and granular structure analyzed respectively by EBSD on the cross-section. a) SEM image, b) inverse pole figure, c) plastic strain distribution, and d) phase mapping.</figcaption></figure>

SEM image on the cross-section is depicted in Fig. 8a. The MNS-layer is formed as a mono-grain; no fine grain boundaries are detected in this layer. This implies that the impinged nitrogen solutes by MNS uniformly distribute in the MNS-layer and super-saturate the specimen irrespective to their chemical affinity to Cr, Co, and Mo. MNS below this layer, larger size grains than CoCrMo before MNS are aligned in the specified crystallographic orientation. Inverse pole
Fig. 6a depicts the SEM image on the cross-section of MNS-CoCrMo. The MNS-layer was formed with the thickness of 12 µm. As shown in Fig. 6b, the nitrogen solute homogeneously distributes from the surface to the depth of 12 µm. The pointwise chemical composition analysis was performed at several positions from the surface to the depth; e.g., N₁ located at the vicinity of surface, N2 located just above the nitriding front end, and N3 located far from this nitriding front end. At these positions, the contents of Co, Cr, Mo, Si and N were measured to be nearly constant at every position on the cross-section; e.g., [Co] ~ 60 mass%, [Cr] ~ 25 mass%, [Mo] ~ 6.5 mass%, and [Si] = 0.6 mass%. The nitrogen content [N] distributes from the surface to the depth as depicted in Fig. 7.

<figure><img src="image_1200.png" /><figcaption>Fig. 7. Nitrogen solute content [N] from the surface to the depth across the nitriding front end.</figcaption></figure>

In the MNS-layer, the nitrogen content is more than 5 mass%. This [N] gradually decreases with the depth (d) across the nitriding front end. No nitrogen solute was detected when d > 15 µm. This high nitrogen content in the surface layer proves that nitrogen solute is massively saturated in the depth of CoCrMo specimen.

<figure><img src="image_1201.png" /><figcaption>Fig. 8. SEM image, phase mapping, plastic strain distribution and granular structure analyzed respectively by EBSD on the cross-section. a) SEM image, b) inverse pole figure, c) plastic strain distribution, and d) phase mapping.</figcaption></figure>

SEM image on the cross-section is depicted in Fig. 8a. The MNS-layer is formed as a mono-grain; no fine grain boundaries are detected in this layer. This implies that the impinged nitrogen solutes by MNS uniformly distribute in the MNS-layer and super-saturate the specimen irrespective to their chemical affinity to Cr, Co, and Mo. MNS below this layer, larger size grains than CoCrMo before MNS are aligned in the specified crystallographic orientation. Inverse pole
figure in ND is shown in Fig. 8b. The MNS-layer is intensely textured to have an orientation in (111). In the literature [9], the crystallographic texturing by the plasma nitriding was reported to vary itself with dependence on the bias voltage. Its intensity for crystallographic orientation was much weaker as seen in Fig. 8.

Let us investigate the effect of plastic straining on this change. As depicted in Fig. 8c, formation of MNS-layer accompanies with high plastic straining. Since the easy-to-straining orientation is (111), Fig.s 8b and 8c reveal that intense crystallographic texturing to (111) in MNS-layer is directly induced by the plastic straining during MNS. The original CoCrMo grains are also affected by plastic straining to change their crystallographic orientation. Fig. 8d shows the phase mapping in cross-section. Most domains in each grain preserve the same ◎-phase as before MNS; phase transformation takes place at the nitriding front end. This phase change suggests that the original ◎-grains are modified by MNS process to drive the crystallographic structure change.

**Mechanical characterization.** Micro-Vickers hardness and BOD testing procedures were utilized to describe the change of hardness and tribological behavior before and after MNS. The surface hardness change was first analyzed by varying the applied load up 20 N (or 2 kgf).

<figure><img src="image_1200.png" /><figcaption>Fig. 9. Surface hardness change by varying the applied load for the NS-CoCrMo specimen at 673 K for 21.6 ks by 50 Pa.</figcaption></figure>

As shown in Fig. 9, the surface hardness is higher than 1500 HV under application of 10 N. Since the hardness of bare CoCrMo is 450 HV₀.₁N, the surface hardness increases by MNS three times or more. Let us consider the effect of penetration depth by indentation of diamond cone on the measured surface hardness in hardness testing. After [14], the surface hardness is affected by the substrate hardness when the penetration depth (D) is more than the surface layer thickness (h) in five to six times. When applying 1 N, D ~ 2 µm; since MNS layer thickness is 12 µm, h ~ 6 x D. Then, the measured hardness is equivalent to the hardness of MNS layer. With increasing W, the soft substrate hardness gradually affects onto the measured hardness. Monotonous decrease of the measured hardness with W in Fig. 9, proves this influence of substrate hardness to the measured hardness. In fact, when W = 20 N, the measured hardness becomes nearly equal to the substrate hardness. MNS-CoCrMo is characterized by this high hardness up to 2400 HV when applying 1N.

In literature [4, 7, 9], the high surface hardness was attained by nitrogen addition and plasma treatment. In most of those studies, the applied load was limited by 1 – 2 N; their maximum hardness at the vicinity of surface was also upper bound by 1100 HV. Higher hardness than 2000
figure in ND is shown in Fig. 8b. The MNS-layer is intensely textured to have an orientation in (111). In the literature [9], the crystallographic texturing by the plasma nitriding was reported to vary itself with dependence on the bias voltage. Its intensity for crystallographic orientation was much weaker as seen in Fig. 8.

Let us investigate the effect of plastic straining on this change. As depicted in Fig. 8c, formation of MNS-layer accompanies with high plastic straining. Since the easy-to-straining orientation is (111), Fig.s 8b and 8c reveal that intense crystallographic texturing to (111) in MNS-layer is directly induced by the plastic straining during MNS. The original CoCrMo grains are also affected by plastic straining to change their crystallographic orientation. Fig. 8d shows the phase mapping in cross-section. Most domains in each grain preserve the same ◎-phase as before MNS; phase transformation takes place at the nitriding front end. This phase change suggests that the original ◎-grains are modified by MNS process to drive the crystallographic structure change.

**Mechanical characterization.** Micro-Vickers hardness and BOD testing procedures were utilized to describe the change of hardness and tribological behavior before and after MNS. The surface hardness change was first analyzed by varying the applied load up 20 N (or 2 kgf).

<figure><img src="image_1202.png" /><figcaption>Fig. 9. Surface hardness change by varying the applied load for the NS-CoCrMo specimen at 673 K for 21.6 ks by 50 Pa.</figcaption></figure>

As shown in Fig. 9, the surface hardness is higher than 1500 HV under application of 10 N. Since the hardness of bare CoCrMo is 450 HV₀.₁N, the surface hardness increases by MNS three times or more. Let us consider the effect of penetration depth by indentation of diamond cone on the measured surface hardness in hardness testing. After [14], the surface hardness is affected by the substrate hardness when the penetration depth (D) is more than the surface layer thickness (h) in five to six times. When applying 1 N, D ~ 2 µm; since MNS layer thickness is 12 µm, h ~ 6 x D. Then, the measured hardness is equivalent to the hardness of MNS layer. With increasing W, the soft substrate hardness gradually affects onto the measured hardness. Monotonous decrease of the measured hardness with W in Fig. 9, proves this influence of substrate hardness to the measured hardness. In fact, when W = 20 N, the measured hardness becomes nearly equal to the substrate hardness. MNS-CoCrMo is characterized by this high hardness up to 2400 HV when applying 1N.

In literature [4, 7, 9], the high surface hardness was attained by nitrogen addition and plasma treatment. In most of those studies, the applied load was limited by 1 – 2 N; their maximum hardness at the vicinity of surface was also upper bound by 1100 HV. Higher hardness than 2000
$HV_{IN}$ in Fig. 9 implies that MNS-layer with the specific crystallographic orientation to (111) has a unique mechanical property due to its nanostructure.

<figure><img src="image_1201.png" /><figcaption>Fig. 10. Variation of the friction coefficient with increasing the sliding distance (L) up to L = 500 m for the bare CoCrMo.</figcaption></figure>

BOD testing was employed to describe the change of tribological performance before and after MNS. Fig. 10 shows the variation of friction coefficient with increasing the sliding distance (L) for the bare CoCrMo disc. Even except for the initial high frictional behavior, the average friction coefficient is 0.45 to 0.5 with a few metal-sticking or a spiky increase of $\int$. Large worn-out volume is experienced in this sliding mode testing in hard-to-hard materials. As suggested in [15], the friction coefficient as well as the specific wear volume are expected to decrease with hardening the die against the same counter materials. In this case, the normal hardness of SUJ2 is 700 HV. Then, this friction and wear behavior of bare CoCrMo alloys can be improved by MNS.

Fig. 11 shows the variation of friction coefficient with the sliding distance for MNS-CoCrMo. The maximum friction coefficient is reduced from 0.7 to 0.4. The low frictional state with $\int = 0.45$ is preserved in the whole sliding distance without any spiky peaks. This assures that MNS-layer is useful to significantly lower the friction and wear.

To be noticed in Fig. 11, the small spiky increase of $\int$ might come from the surface roughness. This might be induced by the slip-lines as seen in Fig. 3b.

<figure><img src="image_1202.png" /><figcaption>Fig. 11. Variation of the friction coefficient with increasing the sliding distance (L) up to L = 500 m for the NS-CoCrMo at 673 K for 21.6 ks by 50 Pa.</figcaption></figure>

## Discussion

The massive nitrogen supersaturation process is characterized by uniform distribution of nitrogen solute with high contents into CoCrMo substrate. Together with this supersaturation, CrN is slightly synthesized as a nano-sized precipitate in the MNS-CoCrMo. To be noticed in the microstructure change by MNS, the plastically strained nano-grains in Fig. 3 have intense textures
$HV_{IN}$ in Fig. 9 implies that MNS-layer with the specific crystallographic orientation to (111) has a unique mechanical property due to its nanostructure.

<figure><img src="image_1203.png" /><figcaption>Fig. 10. Variation of the friction coefficient with increasing the sliding distance (L) up to L = 500 m for the bare CoCrMo.</figcaption></figure>

BOD testing was employed to describe the change of tribological performance before and after MNS. Fig. 10 shows the variation of friction coefficient with increasing the sliding distance (L) for the bare CoCrMo disc. Even except for the initial high frictional behavior, the average friction coefficient is 0.45 to 0.5 with a few metal-sticking or a spiky increase of $\int = 0.45$ Large worn-out volume is experienced in this sliding mode testing in hard-to-hard materials. As suggested in [15], the friction coefficient as well as the specific wear volume are expected to decrease with hardening the die against the same counter materials. In this case, the normal hardness of SUJ2 is 700 HV. Then, this friction and wear behavior of bare CoCrMo alloys can be improved by MNS.

Fig. 11 shows the variation of friction coefficient with the sliding distance for MNS-CoCrMo. The maximum friction coefficient is reduced from 0.7 to 0.4. The low frictional state with $\int = 0.45$ is preserved in the whole sliding distance without any spiky peaks. This assures that MNS-layer is useful to significantly lower the friction and wear.

To be noticed in Fig. 11, the small spiky increase of $\int$ might come from the surface roughness. This might be induced by the slip-lines as seen in Fig. 3b.

<figure><img src="image_1204.png" /><figcaption>Fig. 11. Variation of the friction coefficient with increasing the sliding distance (L) up to L = 500 m for the NS-CoCrMo at 673 K for 21.6 ks by 50 Pa.</figcaption></figure>

## Discussion

The massive nitrogen supersaturation process is characterized by uniform distribution of nitrogen solute with high contents into CoCrMo substrate. Together with this supersaturation, CrN is slightly synthesized as a nano-sized precipitate in the MNS-CoCrMo. To be noticed in the microstructure change by MNS, the plastically strained nano-grains in Fig. 3 have intense textures
with nearly the same crystallographic orientation to (111). After EBSD analysis, the MNS-layer looks a single crystal with the common orientation to (111). KAM distribution in Fig. 8c proves that high plastic straining forms a dense slip-line field network into MNS-layer so that the modified nano-grains are forced to orient to the easy-to-plastically-deformation axis (111). As partially discussed in [9, 10], this specific crystallographic orientation by MNS must be dependent on the holding temperature and gas pressure in MNS.

Let us compare this MNS processes between the CoCrMo alloy and the AISI316. As studied in [16-17], the nitrogen-rich and nitrogen-poor clusters are microscopically formed in the MNS-AISI316 together with microscopic decomposition among the constituent elements of AISI316 or Fe, Cr and Ni with dependence on each different chemical affinity to nitrogen solute. This nanostructure change significantly influences on the mesoscopic analysis by EBSD; the original uniform austenitic structure changes itself to two-phase structure. On the other hand, almost all grains of MNS-CoCrMo have austenitic phase in Fig. 8d; no chemical decomposition occurs in MNS-layer of CoCrMo alloy. More precise description is expected by STEM analysis on MNS-CoCrMo to investigate the microscopic interaction among the constituent atoms of Co, Cr and Mo and the massively supersaturated nitrogen solutes.

Higher surface hardness than 2000 HV<sub>1N</sub> reveals that hardening mechanism must be different from the normal one working in MNS-AISI316/304 and MNS-tool-steels. In addition to fine CrN precipitation in MNS-CoCrMo, the intensely textured ©N-grains work as a blockage of dislocation movements. Further studies on the effect of strongly textured ©N-nanograins on the hardness and strength are needed by using the nano-indentation and STEM analysis.

Higher hardness than 1550 HV<sub>10N</sub> is also promising to apply this MNS-CoCrMo alloys to die-technology. As reported in [18], the bare CoCrMo was utilized as a die for hot extrusion of aluminum alloy with better performance. This highly hardened superalloy by MNS with preferable balancing among the strength, the ductility and the toughness, is useful for hot stamping and forging instead of the WC (Co) with coating.

The tribological data on the tool steel and high-speed steel dies against the SUJ2 were summarized in [19]. In every hard-to-hard material friction, much higher friction coefficients than 1.0 were referred in that data base. The low frictional behavior with μ < 0.45 by the sliding movement by 0.1 m/s under 10 N or 1 kgf, proves that MNS-CoCrMo works as an artificial knee and hip part in medical applications.

## Summary

Massive nitrogen supersaturation to CoCrMo alloys forms a uniform hardened layer up to 2400 HV with the high average nitrogen solute content of 5 mass% and the intensely textured crystallographic structure. Every nano-grain in this layer has common crystallographic orientation to (111) due to the slip-line field by the plastic straining during MNS. Hardness decreases gradually with increasing the penetration depth by indenter. This implies that nitrogen solute diffuses deeper than the MNS-layer thickness to modify the microstructure, neighboring to the nitriding front end. The tribological property is also improved by MNS. The friction coefficient is preserved to be less than 0.45 without any metallic sticking against the hard SUJ2 materials.

This significant improvement in mechanical property and performance by MNS assures that MNS-CoCrMo is useful as a new artificial part in the dental and knee-/hip-arthroplasty and as a die material for hot stamping and forging processes. In particular, MNS-CoCrMo is suitable to mold-stamping the oxide glass works into optical elements at the elevated holding temperature instead of WC (Co) and AISI316 steel molds.

## Acknowledgment

The authors would like to express their gratitude to S-I. Kurozumi (Nano-Film Coat, llc.) and K. Kubota (Graduate School of Engineering, Toyama University) for their help in experiments.
with nearly the same crystallographic orientation to (111). After EBSD analysis, the MNS-layer looks a single crystal with the common orientation to (111). KAM distribution in Fig. 8c proves that high plastic straining forms a dense slip-line field network into MNS-layer so that the modified nano-grains are forced to orient to the easy-to-plastically-deformation axis (111). As partially discussed in [9, 10], this specific crystallographic orientation by MNS must be dependent on the holding temperature and gas pressure in MNS.

Let us compare this MNS processes between the CoCrMo alloy and the AISI316. As studied in [16-17], the nitrogen-rich and nitrogen-poor clusters are microscopically formed in the MNS-AISI316 together with microscopic decomposition among the constituent elements of AISI316 or Fe, Cr and Ni with dependence on each different chemical affinity to nitrogen solute. This nanostructure change significantly influences on the mesoscopic analysis by EBSD; the original uniform austenitic structure changes itself to two-phase structure. On the other hand, almost all grains of MNS-CoCrMo have austenitic phase in Fig. 8d; no chemical decomposition occurs in MNS-layer of CoCrMo alloy. More precise description is expected by STEM analysis on MNS-CoCrMo to investigate the microscopic interaction among the constituent atoms of Co, Cr and Mo and the massively supersaturated nitrogen solutes.

Higher surface hardness than 2000 HV<sub>1N</sub> reveals that hardening mechanism must be different from the normal one working in MNS-AISI316/304 and MNS-tool-steels. In addition to fine CrN precipitation in MNS-CoCrMo, the intensely textured ©N-grains work as a blockage of dislocation movements. Further studies on the effect of strongly textured ©N-nanograins on the hardness and strength are needed by using the nano-indentation and STEM analysis.

Higher hardness than 1550 HV<sub>10N</sub> is also promising to apply this MNS-CoCrMo alloys to die-technology. As reported in [18], the bare CoCrMo was utilized as a die for hot extrusion of aluminum alloy with better performance. This highly hardened superalloy by MNS with preferable balancing among the strength, the ductility and the toughness, is useful for hot stamping and forging instead of the WC (Co) with coating.

The tribological data on the tool steel and high-speed steel dies against the SUJ2 were summarized in [19]. In every hard-to-hard material friction, much higher friction coefficients than 1.0 were referred in that data base. The low frictional behavior with μ < 0.45 by the sliding movement by 0.1 m/s under 10 N or 1 kgf, proves that MNS-CoCrMo works as an artificial knee and hip part in medical applications.

**Summary**

Massive nitrogen supersaturation to CoCrMo alloys forms a uniform hardened layer up to 2400 HV with the high average nitrogen solute content of 5 mass% and the intensely textured crystallographic structure. Every nano-grain in this layer has common crystallographic orientation to (111) due to the slip-line field by the plastic straining during MNS. Hardness decreases gradually with increasing the penetration depth by indenter. This implies that nitrogen solute diffuses deeper than the MNS-layer thickness to modify the microstructure, neighboring to the nitriding front end. The tribological property is also improved by MNS. The friction coefficient is preserved to be less than 0.45 without any metallic sticking against the hard SUJ2 materials.

This significant improvement in mechanical property and performance by MNS assures that MNS-CoCrMo is useful as a new artificial part in the dental and knee-/hip-arthroplasty and as a die material for hot stamping and forging processes. In particular, MNS-CoCrMo is suitable to mold-stamping the oxide glass works into optical elements at the elevated holding temperature instead of WC (Co) and AISI316 steel molds.

**Acknowledgment**

The authors would like to express their gratitude to S-I. Kurozumi (Nano-Film Coat, llc.) and K. Kubota (Graduate School of Engineering, Toyama University) for their help in experiments.
## References

[1] Bandyopaghyay, A., Traxel, K. D., Avila, J. D., Mitra, I., Bose. S., CoCr alloys. Biomaterials Science, 4th Ed. Academic Press 1.3.3C (2020) pp. 257-269. https://doi.org/10.1016/B978-0-12-816137-1.00020-9

[2] Carek, A.; Babic, J. Z., Schauperl, Z., Tomislav, B., Mechanical properties of Co-Cr alloys for metal base framework. Int. J. Prosthodont. Restor. Dent. 1 (2011) pp. 13-19. https://doi.org/10.5005/jp-journals-10019-1003

[3] Devine, T. M., Wulff, J., Cast vs. wrought cobalt-chromium surgical implant alloys. J. Biomed. Mater. Res. 9 (1975) pp. 151-167. https://doi.org/10.1002/jbm.820090205

[4] Lee, S., Nomura, N., Chiba, A., Significant improvement in mechanical properties of biomedical Co-Cr-Mo alloys with combination of N addition and Cr-enrichment. Materials Transactions. 2 (2008) pp.260-264. https://doi.org/10.2320/matertrans.MRA2007220

[5] Longquan, S.; Northwood, D., Cao, Z., The properties of a wrought biomedical Cobalt-Chromium alloy. J. Mat. Sci. 18 (1994) pp. 1233-1238. https://doi.org/10.1007/BF00975070

[6] Cheng, H.; Xu, M., Zhang, H., Wu, W., Zheng, M., Li, X., Cyclic fatigue properties of Cobalt-Chromium alloy clasps for partial removable dental protheses. J. Prosthetic Dent. 104 (2010) pp. 389-396. https://doi.org/10.1016/S0022-3913(10)60173-4

[7] Wang Q.; Zhang L., Shen H., Microstructure analysis of plasma nitrided cast/forged CoCrMo alloys. Surf. Coat. Technol. 205 (7) (2010) pp. 2654-2660. https://doi.org/10.1016/j.surfcoat.2010.10.031

[8] Dong H.; S-phase surface engineering of Fe-Cr, Co-Cr and Ni-Cr alloys. Int. Mater. Rev. 55 (2) (2011) pp. 65-98. https://doi.org/10.1179/095066009X12572530170589

[9] Shukla K.; Sugumaran A. A., Khan I., Ehiasarian A. P., Hovsepian P. E., Low pressure plasma nitrided CoCrMo alloy utilizing HIPIMS discharge for biomedical applications. J. Mech. Behavior of Biomedical Materials. 111 (2020) 104110. https://doi.org/10.1016/j.jmbbm.2020.104004

[10] Aizawa T.; Funazuka T., Shiratori T., Suzuki Y., Massive nitrogen supersaturation to CoCrMo alloys for surface microstructure control. Academia Mater. Sci. 1 (2023) pp. 1-9. https://doi.org/10.20935/AcadMatSci6139

[11] Farghali A., Aizawa T., Yoshino T., Microstructure/mechanical characterization of plasma nitrided fine-grain austenitic stainless steels in low temperature. Nitrogen 2 (2021) pp. 244-258. https://doi.org/10.3390/nitrogen2020016

[12] Aizawa T., Rsadi I., Yunata E. E., High density RF-DC plasma nitriding under optimized conditions by plasma diagnosis. Appl. Sci. 12, 3706 (2022) pp. 1-12. https://doi.org/10.3390/app12083706

[13] He K., Chen N., Wang C., Wei L., Chen J., Method for determining crystal grain size by X-ray diffraction. Crystal Res. Technol. 53 (1) (2018) 157. https://doi.org/10.1002/crat.201700157

[14] Frueh P., Heine A., Weber K. E., Wickert M., Effective depth-of-penetration range due to hardness variation for different lots of nominally identical target material. Defence Technol. 12 (2) (2016) pp. 171-176. https://doi.org/10.1016/j.dt.2015.10.002

[15] Fu J., Lyu D., Zhao J., Tribological characteristics of hard-to-hard matching materials of cylinder block/valve plate interface in electro-hydrostatic actuator pumps. J. Mech. Eng. Sci. 27 (2023) 61-68.
## References

[1] Bandyopaghyay, A., Traxel, K. D., Avila, J. D., Mitra, I., Bose. S., CoCr alloys. Biomaterials Science, 4th Ed. Academic Press 1.3.3C (2020) pp. 257-269. https://doi.org/10.1016/B978-0-12-816137-1.00020-9

[2] Carek, A.; Babic, J. Z., Schauperl, Z., Tomislav, B., Mechanical properties of Co-Cr alloys for metal base framework. Int. J. Prosthodont. Restor. Dent. 1 (2011) pp. 13-19. https://doi.org/10.5005/jp-journals-10019-1003

[3] Devine, T. M., Wulff, J., Cast vs. wrought cobalt-chromium surgical implant alloys. J. Biomed. Mater. Res. 9 (1975) pp. 151-167. https://doi.org/10.1002/jbm.820090205

[4] Lee, S., Nomura, N., Chiba, A., Significant improvement in mechanical properties of biomedical Co-Cr-Mo alloys with combination of N addition and Cr-enrichment. Materials Transactions. 2 (2008) pp.260-264. https://doi.org/10.2320/matertrans.MRA2007220

[5] Longquan, S.; Northwood, D., Cao, Z., The properties of a wrought biomedical Cobalt-Chromium alloy. J. Mat. Sci. 18 (1994) pp. 1233-1238. https://doi.org/10.1007/BF00975070

[6] Cheng, H.; Xu, M., Zhang, H., Wu, W., Zheng, M., Li, X., Cyclic fatigue properties of Cobalt-Chromium alloy clasps for partial removable dental protheses. J. Prosthetic Dent. 104 (2010) pp. 389-396. https://doi.org/10.1016/S0022-3913(10)60173-4

[7] Wang Q.; Zhang L., Shen H., Microstructure analysis of plasma nitrided cast/forged CoCrMo alloys. Surf. Coat. Technol. 205 (7) (2010) pp. 2654-2660. https://doi.org/10.1016/j.surfcoat.2010.10.031

[8] Dong H.; S-phase surface engineering of Fe-Cr, Co-Cr and Ni-Cr alloys. Int. Mater. Rev. 55 (2) (2011) pp. 65-98. https://doi.org/10.1179/095066009X12572530170589

[9] Shukla K.; Sugumaran A. A., Khan I., Ehiasarian A. P., Hovsepian P. E., Low pressure plasma nitrided CoCrMo alloy utilizing HIPIMS discharge for biomedical applications. J. Mech. Behavior of Biomedical Materials. 111 (2020) 104110. https://doi.org/10.1016/j.jmbbm.2020.104004

[10] Aizawa T.; Funazuka T., Shiratori T., Suzuki Y., Massive nitrogen supersaturation to CoCrMo alloys for surface microstructure control. Academia Mater. Sci. 1 (2023) pp. 1-9. https://doi.org/10.20935/AcadMatSci6139

[11] Farghali A., Aizawa T., Yoshino T., Microstructure/mechanical characterization of plasma nitrided fine-grain austenitic stainless steels in low temperature. Nitrogen 2 (2021) pp. 244-258. https://doi.org/10.3390/nitrogen2020016

[12] Aizawa T., Rsadi I., Yunata E. E., High density RF-DC plasma nitriding under optimized conditions by plasma diagnosis. Appl. Sci. 12, 3706 (2022) pp. 1-12. https://doi.org/10.3390/app12083706

[13] He K., Chen N., Wang C., Wei L., Chen J., Method for determining crystal grain size by X-ray diffraction. Crystal Res. Technol. 53 (1) (2018) 157. https://doi.org/10.1002/crat.201700157

[14] Frueh P., Heine A., Weber K. E., Wickert M., Effective depth-of-penetration range due to hardness variation for different lots of nominally identical target material. Defence Technol. 12 (2) (2016) pp. 171-176. https://doi.org/10.1016/j.dt.2015.10.002

[15] Fu J., Lyu D., Zhao J., Tribological characteristics of hard-to-hard matching materials of cylinder block/valve plate interface in electro-hydrostatic actuator pumps. J. Mech. Eng. Sci. 27 (2023) 61-68.
[16] Aizawa T., Shiratori T., Komatsu T., Integrated manufacturing of fine-grained stainless steels for industries and medicals. Chapter 1 in Engineering Steels and High Entropy-Alloys. IntechOpen, London, UK (2020) pp. 3-26. https://doi.org/10.5772/intechopen.89754

[17] Aizawa T., Micro-/meso-structure control of multi-hostmetal alloys by massive nitrogen supersaturation. J. Materials (2024) (in press). https://doi.org/10.3390/ma17061294

[18] Funazuka T., Horiuchi S., Dohda K., Shiratori T., Effect of CoCrMo die with nanotexture applied on micro-extrudability of micro backward extrusion of AA6063. Proc. WCMNM2023 (19th September, 2023; Evanston, Chicago, USA) 2023; 1-4.

[19] Silicolloy, https://www.silicolloy.co.jp/tribology/friction_wear Afion-suj2/. (Retrieved at 2023/12/15).
[16] Aizawa T., Shiratori T., Komatsu T., Integrated manufacturing of fine-grained stainless steels for industries and medicals. Chapter 1 in Engineering Steels and High Entropy-Alloys. IntechOpen, London, UK (2020) pp. 3-26. https://doi.org/10.5772/intechopen.89754

[17] Aizawa T., Micro-/meso-structure control of multi-hostmetal alloys by massive nitrogen supersaturation. J. Materials (2024) (in press). https://doi.org/10.3390/ma17061294

[18] Funazuka T., Horiuchi S., Dohda K., Shiratori T., Effect of CoCrMo die with nanotexture applied on micro-extrudability of micro backward extrusion of AA6063. Proc. WCMNM2023 (19th September, 2023; Evanston, Chicago, USA) 2023; 1-4.

[19] Silicolloy, https://www.silicolloy.co.jp/tribology/friction_wear ballon-suj2/. (Retrieved at 2023/12/15).
# Analysis of oxidized hot working tool steel tribological behavior in tube piercing

MURILLO-MARRODÁN Alberto¹, a*, MERESSE Damien², b, GARCÍA Eduardo¹, c,
MOREAU Philippe², d, LA BARBERA-SOSA Jose Gregorio², e and
DUBAR Laurent², f

¹Department of Mechanics, Design and Industrial Management, University of Deusto, 48007 Bilbao, Spain

²LAMIH UMR CNRS 8201, Université Polytechnique Hauts-de-France, F-59313, Valenciennes Cedex 9, France

aalberto.murillo@deusto.es, bdamien.meresse@uphf.fr, c'e.garcia@deusto.es,
dphilippe.moreau@uphf.fr, ejosegregorio.labarberasosa@uphf.fr, f'laurent.dubar@uphf.fr

**Keywords:** Friction, Hot Working Steel, Oxide Scale, Severe Contact Conditions

**Abstract.** In this paper the tribological behavior of the oxide layer present on the mandrel used in rotary piercing tube manufacturing operations has been analyzed. A hot upsetting sliding test campaign has been carried out in order to analyze the hot working steel oxide layer under different contact conditions. Optical microscopy, profilometry and XRD have been used to establish the type of contact and oxides generated, respectively. Results show abrasive sliding contact between mandrel and tube materials. Oxide transfer from the tube material to the mandrel material occurs in all the tested conditions. Regarding friction, the coefficient of friction COF is reduced by the presence of the oxide layer, confirming its lubricity. It has been observed how COF decreases with the increase of contact pressure and relative velocity. The value of COF is reduced from 0.42 to 0.18 if the contact pressure is increased from 165 to 200 MPa and the relative velocity is increased from 200 to 450 mm/s.

**Introduction**

In the rotary seamless tube piercing process, the tools are subjected to very demanding working conditions during their lifespan. The process is carried out at high temperature (the billet is heated, depending on the alloy, to values above 1200°C), high contact pressure and fast relative sliding velocities at billet-tool interfaces [1]. Of all the elements that make up the rolling-piercing mill, the mandrel, also known as the plug, is responsible for the generation of the hollow cavity of the pierced tube, starting from a solid billet. This is a critical element for the efficiency of the process and the final quality of the tube and, in addition, it is subjected to the most severe contact conditions. Therefore, special attention must be paid to the choice of the type of material to be used to design this tooling. In general, hot work tool steel, which contains Nickel, Chromium and Molybdenum as main alloying elements to improve its resistance to high temperatures and wear [2], is used when the material to be pierced is low carbon steel.

In order to improve the efficiency of the process and reduce its carbon footprint, one approach is improving the resistance of the tooling at high temperatures, reducing wear and friction. In the bulk metal forming field, lubricants are well extended, however, they have clear disadvantages from an environmental perspective as after forming, parts need to be cleaned and dried before going to subsequent stages [3]. Another alternative is the use of texturing, which can influence the friction coefficient and modify the material flow stress during the forming operation [4], but it does not improve substantially the wear performance. Furthermore, the use of coatings has been actively promoted in reducing wear in tooling systems for sheet metal forming operations as it can

<figure><img src="image_1203.png" /></figure>
# Analysis of oxidized hot working tool steel tribological behavior in tube piercing

MURILLO-MARRODÁN Alberto¹, a*, MERESSE Damien², b, GARCÍA Eduardo¹, c,
MOREAU Philippe², d, LA BARBERA-SOSA Jose Gregorio², e and
DUBAR Laurent², f

¹Department of Mechanics, Design and Industrial Management, University of Deusto, 48007 Bilbao, Spain

²LAMIH UMR CNRS 8201, Université Polytechnique Hauts-de-France, F-59313, Valenciennes Cedex 9, France

aalberto.murillo@deusto.es, bdamien.meresse@uphf.fr, c'e.garcia@deusto.es,
dphilippe.moreau@uphf.fr, ejosegregorio.labarberasosa@uphf.fr, f'laurent.dubar@uphf.fr

**Keywords:** Friction, Hot Working Steel, Oxide Scale, Severe Contact Conditions

**Abstract.** In this paper the tribological behavior of the oxide layer present on the mandrel used in rotary piercing tube manufacturing operations has been analyzed. A hot upsetting sliding test campaign has been carried out in order to analyze the hot working steel oxide layer under different contact conditions. Optical microscopy, profilometry and XRD have been used to establish the type of contact and oxides generated, respectively. Results show abrasive sliding contact between mandrel and tube materials. Oxide transfer from the tube material to the mandrel material occurs in all the tested conditions. Regarding friction, the coefficient of friction COF is reduced by the presence of the oxide layer, confirming its lubricity. It has been observed how COF decreases with the increase of contact pressure and relative velocity. The value of COF is reduced from 0.42 to 0.18 if the contact pressure is increased from 165 to 200 MPa and the relative velocity is increased from 200 to 450 mm/s.

**Introduction**

In the rotary seamless tube piercing process, the tools are subjected to very demanding working conditions during their lifespan. The process is carried out at high temperature (the billet is heated, depending on the alloy, to values above 1200°C), high contact pressure and fast relative sliding velocities at billet-tool interfaces [1]. Of all the elements that make up the rolling-piercing mill, the mandrel, also known as the plug, is responsible for the generation of the hollow cavity of the pierced tube, starting from a solid billet. This is a critical element for the efficiency of the process and the final quality of the tube and, in addition, it is subjected to the most severe contact conditions. Therefore, special attention must be paid to the choice of the type of material to be used to design this tooling. In general, hot work tool steel, which contains Nickel, Chromium and Molybdenum as main alloying elements to improve its resistance to high temperatures and wear [2], is used when the material to be pierced is low carbon steel.

In order to improve the efficiency of the process and reduce its carbon footprint, one approach is improving the resistance of the tooling at high temperatures, reducing wear and friction. In the bulk metal forming field, lubricants are well extended, however, they have clear disadvantages from an environmental perspective as after forming, parts need to be cleaned and dried before going to subsequent stages [3]. Another alternative is the use of texturing, which can influence the friction coefficient and modify the material flow stress during the forming operation [4], but it does not improve substantially the wear performance. Furthermore, the use of coatings has been actively promoted in reducing wear in tooling systems for sheet metal forming operations as it can

<figure><img src="image_1205.png" /></figure>
prevent adhesive-related problems such as galling [5]. In the case of hot working tool steels, in contact with oxygen, metal surfaces can form oxide layers. When working in air at temperatures higher than 400 °C, the wear of the steels is typical oxidation wear [6]. In certain cases, these oxides can also serve as a protective barrier, preventing the steels from undergoing further oxidation. In addition, oxide layers can reduce metal-to-metal contact during the sliding motion of metal components, thus protecting them from wear [7].

The tribological behavior of the oxide layer resulting from the oxidation of hot work steels depends on many variables. A fundamental aspect is the type of oxide that makes up the tribo-layer. In the case of low carbon steels treated in air at high temperature, Zambrano et. al. [8] distinguished up to three types of iron oxide, namely Hematite, Magnetite and Wüstite. At room temperature, Hematite exhibits a hardness of approximately 1000 HV, leading to abrasive wear, while Magnetite, with a hardness of 500 HV in similar conditions, demonstrates a comparatively lower resistance. In contrast, Wüstite is renowned for its diminished hardness of 300 HV and stands out for possessing the highest lubricity among these oxides. As the temperature is increased to 900°C, some authors have reported hardness values for FeO, Fe₃O₄, and Fe₂O₃ of 105, 366, and 516 HV, respectively [9]. In the case of Magnetite and Wüsite, these are the iron oxides that exhibit the highest ductility and resistance to wear at high temperature [10]. Typical morphology ratio reported at temperatures up to 1200°C is 95:4:1 (FeO:Fe₃O₄:Fe₂O₃) [11]. In addition, the thickness of the layer has been identified as a factor that severely affects the coefficient of friction, along with the temperature of the sample. A thick oxide scale produces a lower coefficient of friction, providing a certain degree of lubricity [12].

The presence of alloying elements in the steel also affects the oxide formation. Elements such as chromium, silicon or aluminum can enrich those oxide layers generated at the scale/substrate interface, leading to different mechanical properties and contact behavior. In addition, elements such as Ni or Cu, could increase the adherence of the oxides to the substrate, which increases the friction coefficient as oxides are not separating from the metallic interface, resulting in oxide-oxide contact [13].

In the case of hot rolling processes, several studies have focused on evaluating the effect of the oxide tribo-layer on the process. Pellizzari et al. [14] analysed the behaviour of high-speed steels and high chromium iron with regard to friction and wear at high temperature during hot rolling. Abrasion was identified as being caused by the oxide scale that develops on the hot carbon steel counterpart. Tieu et al. [15] determined that the rolling force and friction of pre-oxidised rolls are superior to those of virgin surface rolls for all thickness reductions and temperatures considered in their study. Utsunomiya et al. [16] identified that the relative sliding of the scale is the mechanism responsible for its apparent lubricating effect, and that the amount of the relative sliding increased with an increase in scale thickness. Nevertheless, one of the main challenges in examining the tribological behavior of a specific process is the lack of experimental procedure that effectively reproduces the forming process.

In this study, the tribological behavior of the oxide layer present on the mandrel used in rotary piercing tube manufacturing operations has been analyzed. A hot upsetting sliding test campaign has been carried out, which is able to reproduce industrial process contact conditions, including deformation during sliding contact. The contact of hot working tool steel with and without oxidation treatment has been evaluated and the type of contact and friction between tool-material assessed.

## Materials and methods

The materials considered in this study are a high-strength low-alloy (HSLA) steel grade for the tube and a hot working tool steel for the plug. Both the tube and tool materials were provided by an industrial partner, and its chemical composition was evaluated in the laboratory. The chemical composition is listed in Table 1.