forming forces. Overall, the forces for each forming parameter setup were highest for the ground and sintered variants. This can possibly be caused by the stronger particle cohesion after pressing and sintering and therefore stronger overall material strength which must be overcome in forming.

<table><caption>Table 2. Results of the maximum force measurements</caption><thead><tr><th rowspan="2"></th><th colspan="2">Reference</th><th colspan="2">Ball milled</th></tr><tr><th>Hot Forming</th><th>Sintering and Hot Forming</th><th>Hot Forming</th><th>Sintering and Hot Forming</th></tr></thead><tbody><tr><td>φ = -0.76, v = 30 mm/s, U<sub>T</sub> = 1050 °C</td><td>329 kN</td><td>317 kN</td><td>322 kN</td><td>331 kN</td></tr><tr><td>φ = -0.76, v = 30 mm/s, U<sub>T</sub> = 1150 °C</td><td>247 kN</td><td>252 kN</td><td>259 kN</td><td>275 kN</td></tr><tr><td>φ = -0.76, v = 30 mm/s, U<sub>T</sub> = 1250 °C</td><td>200 kN</td><td>208 kN</td><td>195 kN</td><td>185 kN</td></tr><tr><td>φ = -0.47, v = 30 mm/s, U<sub>T</sub> = 1250 °C</td><td>108 kN</td><td>111 kN</td><td>122 kN</td><td>118 kN</td></tr><tr><td>φ = -1.16, v = 30 mm/s, U<sub>T</sub> = 1250 °C</td><td>381 kN</td><td>368 kN</td><td>363 kN</td><td>384 kN</td></tr><tr><td>φ = -0.76, v = 10 mm/s, U<sub>T</sub> = 1250 °C</td><td>155 kN</td><td>151 kN</td><td>164 kN</td><td>176 kN</td></tr><tr><td>φ = -0.76, v = 50 mm/s, U<sub>T</sub> = 1250 °C</td><td>265 kN</td><td>282 kN</td><td>259 kN</td><td>260 kN</td></tr></tbody></table>

Porosity

Figure 2 shows the optical microscope images of representative samples. In the partial images a) to d), the titanium aluminide (grey) can be seen in the centre and the steel capsule (light grey) at the edge. As in the study by Jiang et al., the light micrographs show that the porosity is inhomogeneous and increases towards the edge of the sample [16]. The increase in black dots towards the edge shows this. This can be led back to local differences in plastic strain, which generally reach a maximum in the centre for cylinder compression tests. The unground variants have a higher porosity over the entire cross-section than the ground variants, see Figure 2 c) and d). It can therefore be concluded that higher green strength can reduce porosity of the finished part. In addition, the oxide layers act as a diffusion barrier during heat treatment, thus favouring porosity [19]. The optical microscope images showed that most of the samples contained cracks. A tendency depending on the process route or the forming parameters cannot be recognised. It can be assumed that the selected pressing speeds are too high. In the investigations by Bolz et. Al., pressing speeds of less than 0.05 mm/s were used to prevent cracking [20]. However, the longer contact time due to the low pressing speed led to an increase in the diffusion layer between the die and the component, which had to be removed afterwards. A diffusion layer between the steel capsule and the titanium aluminide can also be seen in Figure 2 a) and c). This can be seen in the colour transition from the grey titanium aluminide to the light grey steel capsule. As diffusion is temperature-dependent, the diffusion layer is less pronounced at the lower forming temperature, Figure 2 b).
<figure><img src="image_911.png" /><figcaption>Figure 2. Light micrographs: a) ball milled and sintered, φ= -0.76, v= 30 mm/s, U<sub>T</sub> = 1250 °C; b) ball milled and sintered, φ= -0.76, v= 30 mm/s, U<sub>T</sub> = 1050 °C; c) ball milled and not sintered, φ= -0.76, v= 30 mm/s, U<sub>T</sub> = 1250 °C; d) Reference and not sintered, φ= -0.76, v= 30 mm/s, U<sub>T</sub> = 1250 °C</figcaption></figure>

By forming, it is possible to achieve a porosity of less than 0.05 % and thus almost completely seal the material, see Figure 3. However, the results show that different porosities are achieved depending on the selected parameters. When varying the forming temperature, it can be seen that the porosity is highest at 1150 °C. This temperature is close to the eutectic temperature. This temperature is close to the eutectoid temperature (1125 °C) of TiAl, at which a phase transformation takes place [21]. It is assumed that this transformation causes a higher porosity. In addition, the variation of the pressing speed shows that the lowest pressing speed has the highest porosity. It is assumed that the higher heat loss during forming due to the lower speed means that the diffusion processes are not fully completed, so that the porosity is not further reduced. A higher pressing speed is therefore recommended for lower porosity, but this can lead to greater cracking. When varying the degree of deformation, it was found that a degree of deformation of -0.47 leads to high porosity and decreases at higher degrees of deformation. It can be assumed that the higher forming force due to the higher degree of deformation leads to further consolidation of the material and influences the mechanical properties.
<figure><img src="image_918.png" /><figcaption>Figure 2. Light micrographs: a) ball milled and sintered, φ= -0.76, v= 30 mm/s, U<sub>T</sub> = 1250 °C; b) ball milled and sintered, φ= -0.76, v= 30 mm/s, U<sub>T</sub> = 1050 °C; c) ball milled and not sintered, φ= -0.76, v= 30 mm/s, U<sub>T</sub> = 1250 °C; d) Reference and not sintered, φ= -0.76, v= 30 mm/s, U<sub>T</sub> = 1250 °C</figcaption></figure>

By forming, it is possible to achieve a porosity of less than 0.05 % and thus almost completely seal the material, see Figure 3. However, the results show that different porosities are achieved depending on the selected parameters. When varying the forming temperature, it can be seen that the porosity is highest at 1150 °C. This temperature is close to the eutectic temperature. This temperature is close to the eutectoid temperature (1125 °C) of TiAl, at which a phase transformation takes place [21]. It is assumed that this transformation causes a higher porosity. In addition, the variation of the pressing speed shows that the lowest pressing speed has the highest porosity. It is assumed that the higher heat loss during forming due to the lower speed means that the diffusion processes are not fully completed, so that the porosity is not further reduced. A higher pressing speed is therefore recommended for lower porosity, but this can lead to greater cracking. When varying the degree of deformation, it was found that a degree of deformation of -0.47 leads to high porosity and decreases at higher degrees of deformation. It can be assumed that the higher forming force due to the higher degree of deformation leads to further consolidation of the material and influences the mechanical properties.
<figure><img src="image_912.png" /><figcaption>Figure 3. Results of the porosity measurements for ball-milled, sintered specimens after forming</figcaption></figure>

## Hardness

The hardness results show that hot forming leads to an increase in hardness. All samples exhibited a significantly higher hardness than the sintered specimens (152 HV1) in the prior study [8]. During the forming process, the pores are closed by the wrought process, which improved the mechanical properties of the TiAl sample [9]. Figure 4 shows an example of the hardness values from a series of tests. It can be seen that the ground variants have a higher hardness than the unground variants. It can be assumed that the oxide-free surfaces created as a result of ball milling in a low-oxygen atmosphere lead to an increase in green strength during pressing due to the improved cold welding. These juvenile surfaces and the increased cold welding favour the diffusion process during sintering, which is why the hardness of the titanium aluminide increases again in the sintered variants. The hardness is further increased with the subsequent forming process. The sintering process before hot forming also leads to an increase in hardness in the untreated samples. In the untreated TiAl samples, the oxide layers hinder diffusion [19].

<figure><img src="image_913.png" /><figcaption>Figure 4. Comparison of hardness values within a test series: φ= -0.76, v= 30 mm/s, U<sub>T</sub>= 1150 °C</figcaption></figure>

The results of the milled and sintered variants from the study are shown in Figure 5. When varying the forming parameters, this study shows that the forming temperature has not an effect on the hardness of the samples after forming; the differences are within the standard deviation and therefore not significant. The standard deviation represents the material homogeneity. As the selected forming temperatures are above the recrystallisation temperature, the temperature
<figure><img src="image_919.png" /><figcaption>Figure 3. Results of the porosity measurements for ball-milled, sintered specimens after forming</figcaption></figure>

## Hardness

The hardness results show that hot forming leads to an increase in hardness. All samples exhibited a significantly higher hardness than the sintered specimens (152 HV1) in the prior study [8]. During the forming process, the pores are closed by the wrought process, which improved the mechanical properties of the TiAl sample [9]. Figure 4 shows an example of the hardness values from a series of tests. It can be seen that the ground variants have a higher hardness than the unground variants. It can be assumed that the oxide-free surfaces created as a result of ball milling in a low-oxygen atmosphere lead to an increase in green strength during pressing due to the improved cold welding. These juvenile surfaces and the increased cold welding favour the diffusion process during sintering, which is why the hardness of the titanium aluminide increases again in the sintered variants. The hardness is further increased with the subsequent forming process. The sintering process before hot forming also leads to an increase in hardness in the untreated samples. In the untreated TiAl samples, the oxide layers hinder diffusion [19].

<figure><img src="image_920.png" /><figcaption>Figure 4. Comparison of hardness values within a test series: φ= -0.76, v= 30 mm/s, U<sub>T</sub>= 1150 °C</figcaption></figure>

The results of the milled and sintered variants from the study are shown in Figure 5. When varying the forming parameters, this study shows that the forming temperature has not an effect on the hardness of the samples after forming; the differences are within the standard deviation and therefore not significant. The standard deviation represents the material homogeneity. As the selected forming temperatures are above the recrystallisation temperature, the temperature
differences have no significant influence on the global hardness [22]. However, it can be seen that the standard deviation is highest at the forming temperature of 1150 °C. Thus, as with the porosity measurements, an influence of the phase transformation near the eutectoid temperature (1125 °C) on the homogeneity of the hardness curve can be assumed [21]. When varying the pressing speed, hardness measurements also showed no significant differences; the deviations are within the standard deviation. However, it was found that the standard deviation decreases with increasing forming speed. It is assumed that the lower heat loss due to the higher pressing speed leads to a higher homogeneity of the hardness, as the diffusion processes can take longer. When varying the degree of deformation, it was found that the hardness increases with increasing degree of deformation. By increasing the degree of deformation, the porosity in the TiAl samples decreases as a result of the forging process, which increases the hardness [9]. The hardness of the variant with the highest degree of deformation was 412 HV1, an increase in hardness of over 170 % compared to sintering [8]. Compared to the cast material (285 HV10), the hardness of the moulded sample is also over 40 % higher. It can therefore be concluded that increased pre-consolidation and a higher degree of forming lead to an increase in hardness

<figure><img src="image_914.png" /><figcaption>Figure 5. Comparison of hardness values with variation of the forming parameters for ball-milled, sintered specimens after forming</figcaption></figure>

## Summary

In this study it is analysed how the properties of titanium aluminide produced by pressing and sintering can be enhanced through a hot forming process in an oxygen-reduced atmosphere. The influence of milling in an oxygen-reduced atmosphere before pressing and sintering and hot forming was analysed. Both the milling of the TiAl powder and the sintering before forming had no significant influence on the maximum forces during forming. The correlation between low porosity and high hardness was also confirmed. Ball milling and sintering as well as a higher degree of forming lead to a higher hardness and a lower porosity over the entire cross-section. Compared to the sintered sample, an increase in hardness of over 170% and compared to the cast material, an increase in hardness of over 40% could be achieved. In addition, the pores could be almost completely closed. However, it was shown that forming close to the eutectoid temperature, a low pressing speed and a low degree of forming lead to a high porosity. In subsequent work, the results should be validated through further volumetric density measurements. The influence of the eutectoid temperature on the density of TiAl will be examined in more detail. Further work is also required to homogenise the density globally throughout the specimens, as the high density
differences have no significant influence on the global hardness [22]. However, it can be seen that the standard deviation is highest at the forming temperature of 1150 °C. Thus, as with the porosity measurements, an influence of the phase transformation near the eutectoid temperature (1125 °C) on the homogeneity of the hardness curve can be assumed [21]. When varying the pressing speed, hardness measurements also showed no significant differences; the deviations are within the standard deviation. However, it was found that the standard deviation decreases with increasing forming speed. It is assumed that the lower heat loss due to the higher pressing speed leads to a higher homogeneity of the hardness, as the diffusion processes can take longer. When varying the degree of deformation, it was found that the hardness increases with increasing degree of deformation. By increasing the degree of deformation, the porosity in the TiAl samples decreases as a result of the forging process, which increases the hardness [9]. The hardness of the variant with the highest degree of deformation was 412 HV1, an increase in hardness of over 170 % compared to sintering [8]. Compared to the cast material (285 HV10), the hardness of the moulded sample is also over 40 % higher. It can therefore be concluded that increased pre-consolidation and a higher degree of forming lead to an increase in hardness

<figure><img src="image_921.png" /><figcaption>Figure 5. Comparison of hardness values with variation of the forming parameters for ball-milled, sintered specimens after forming</figcaption></figure>

## Summary

In this study it is analysed how the properties of titanium aluminide produced by pressing and sintering can be enhanced through a hot forming process in an oxygen-reduced atmosphere. The influence of milling in an oxygen-reduced atmosphere before pressing and sintering and hot forming was analysed. Both the milling of the TiAl powder and the sintering before forming had no significant influence on the maximum forces during forming. The correlation between low porosity and high hardness was also confirmed. Ball milling and sintering as well as a higher degree of forming lead to a higher hardness and a lower porosity over the entire cross-section. Compared to the sintered sample, an increase in hardness of over 170% and compared to the cast material, an increase in hardness of over 40% could be achieved. In addition, the pores could be almost completely closed. However, it was shown that forming close to the eutectoid temperature, a low pressing speed and a low degree of forming lead to a high porosity. In subsequent work, the results should be validated through further volumetric density measurements. The influence of the eutectoid temperature on the density of TiAl will be examined in more detail. Further work is also required to homogenise the density globally throughout the specimens, as the high density
presented here is limited to the centre with the highest plastic strain. The extent to which hardness can be increased and porosity reduced by increasing the degree of forming must be determined in subsequent studies.

## Acknowledgements

This research and development project is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 394563137 - SFB 1368 TP A03. The authors thank the DFG for the financial support of this project.

## References

[1] Mphahlele M. R.; Olubambi, P. A.; Olevsky, E.A. Advances in Sintering of Titanium Aluminide: A Review. JOM 2023, 75, 2877-2896. https://doi.org/10.1007/s11837-022-05664-8

[2] B. P. Bewlay, S. Nag, A. Suzuki, M. J. Weimer, TiAl alloys in commercial aircraft engines, 2016. https://doi.org/10.1080/09603409.2016.1183068

[3] Clemens, H.; Mayer, S. Design, Processing, Microstructure, Properties, and Applications of Advanced Intermetallic TiAl Alloys. Adv. Eng. Mater. 2013, 15, 191-215. https://doi.org/10.1002/adem.201200231

[4] Clemens, H.; Mayer, S.; Heilmaier, M. Pulvermetallurgische Herstellung von innovativen Hochtemperaturwerkstoffen. BHM Berg-Und Hüttenmännische Monatshefte 2021, 166, 1-7. https://doi.org/10.1007/s00501-018-0704-1

[5] L. Jiao, Q. Feng, S. He, B. Duan, Z. Dou, C. Li, X. Lu, Direct oxygen removal from titanium aluminide scraps by yttrium reduction. 2021, Nonferrous Metals Society of China 32, pp. 2428-2437. https://doi.org/10.1016/S1003-6326(22)65958-2

[6] Groza, J.R.; Zavaliangos, A. Sintering activation by external electrical field. Mater. Sci. Eng. A 2000, 287, 171-177. https://doi.org/10.1016/S0921-5093(00)00771-1

[7] Kariya, S.; Issariyapat, A.; Bahador, A.; Umeda, J.; Shen, J.; Kondoh, K.: Ductility improvement of high-strength Ti-O material upon heteromicrostructure formation. 2022. https://doi.org/10.1016/j.msea.2022.143041

[8] B.-A. Behrens, K. Brunotte, J. Peddinghaus, J. Ursinus, S. Döring, W. Maus-Friedrichs, R. Gustus, M. Szafarska, Pressing and sintering of titanium aluminide powder after ball milling in silane-doped atmosphere (2023), Journal of Manufacturing and Materials Processing. https://doi.org/10.3390/jmmp7050171

[9] E. Doege, B.-A. Behrens, Handbuch Umformtechnik 2010, Springer-Verlag. https://doi.org/10.1007/978-3-642-04249-2

[10] P. Zhang, S.X. Li, Z.F. Zhang, General relationship between strength and hardness, 2011. https://doi.org/10.1016/j.msea.2011.08.061

[11] G. H. Meier, F. S. Pettit; S. Hu, Oxidation behavior of titanium aluminides. 1993, Journal de Physique, pp. 395-402. https://doi.org/10.1051/jp4:1993941

[12] N. F. Mogale, W. R. Matizamhuka, Spark Plasma Sintering of Titanium Aluminides: A Progress Review on Processing, Structure-Property Relations, Alloy Development and Challenges. 2020, metals. https://doi.org/10.3390/met10081080

[13] L. Wang, Y. Liu, W Zhang, H. Wang, Q. Li, Optimization of pack parameters for hot deformation of TiAl alloys, 2011, Intermetallics 19 (1), S. 68-74. https://doi.org/10.1016/j.intermet.2010.09.011
presented here is limited to the centre with the highest plastic strain. The extent to which hardness can be increased and porosity reduced by increasing the degree of forming must be determined in subsequent studies.

## Acknowledgements

This research and development project is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 394563137 - SFB 1368 TP A03. The authors thank the DFG for the financial support of this project.

## References

[1] Mphahlele M. R.; Olubambi, P. A.; Olevsky, E.A. Advances in Sintering of Titanium Aluminide: A Review. JOM 2023, 75, 2877-2896. https://doi.org/10.1007/s11837-022-05664-8

[2] B. P. Bewlay, S. Nag, A. Suzuki, M. J. Weimer, TiAl alloys in commercial aircraft engines, 2016. https://doi.org/10.1080/09603409.2016.1183068

[3] Clemens, H.; Mayer, S. Design, Processing, Microstructure, Properties, and Applications of Advanced Intermetallic TiAl Alloys. Adv. Eng. Mater. 2013, 15, 191-215. https://doi.org/10.1002/adem.201200231

[4] Clemens, H.; Mayer, S.; Heilmaier, M. Pulvermetallurgische Herstellung von innovativen Hochtemperaturwerkstoffen. BHM Berg-Und Hüttenmännische Monatshefte 2021, 166, 1-7. https://doi.org/10.1007/s00501-018-0704-1

[5] L. Jiao, Q. Feng, S. He, B. Duan, Z. Dou, C. Li, X. Lu, Direct oxygen removal from titanium aluminide scraps by yttrium reduction. 2021, Nonferrous Metals Society of China 32, pp. 2428-2437. https://doi.org/10.1016/S1003-6326(22)65958-2

[6] Groza, J.R.; Zavaliangos, A. Sintering activation by external electrical field. Mater. Sci. Eng. A 2000, 287, 171-177. https://doi.org/10.1016/S0921-5093(00)00771-1

[7] Kariya, S.; Issariyapat, A.; Bahador, A.; Umeda, J.; Shen, J.; Kondoh, K.: Ductility improvement of high-strength Ti-O material upon heteromicrostructure formation. 2022. https://doi.org/10.1016/j.msea.2022.143041

[8] B.-A. Behrens, K. Brunotte, J. Peddinghaus, J. Ursinus, S. Döring, W. Maus-Friedrichs, R. Gustus, M. Szafarska, Pressing and sintering of titanium aluminide powder after ball milling in silane-doped atmosphere (2023), Journal of Manufacturing and Materials Processing. https://doi.org/10.3390/jmmp7050171

[9] E. Doege, B.-A. Behrens, Handbuch Umformtechnik 2010, Springer-Verlag. https://doi.org/10.1007/978-3-642-04249-2

[10] P. Zhang, S.X. Li, Z.F. Zhang, General relationship between strength and hardness, 2011. https://doi.org/10.1016/j.msea.2011.08.061

[11] G. H. Meier, F. S. Pettit; S. Hu, Oxidation behavior of titanium aluminides. 1993, Journal de Physique, pp. 395-402. https://doi.org/10.1051/jp4:1993941

[12] N. F. Mogale, W. R. Matizamhuka, Spark Plasma Sintering of Titanium Aluminides: A Progress Review on Processing, Structure-Property Relations, Alloy Development and Challenges. 2020, metals. https://doi.org/10.3390/met10081080

[13] L. Wang, Y. Liu, W Zhang, H. Wang, Q. Li, Optimization of pack parameters for hot deformation of TiAl alloys, 2011, Intermetallics 19 (1), S. 68-74. https://doi.org/10.1016/j.intermet.2010.09.011
[14] J. Li, Y. Liu, B. Liu, Y. Wang, X. Liang, Y. He, Microstructure characterization and mechanical behaviors of a hot forged high Nb containing PM-TiAl alloy, 2014, Materials Characterization 95, S. 148-156. https://doi.org/10.1016/j.matchar.2014.06.015

[15] B.-A. Behrens, K. Brunotte, J. Peddinghaus, A. Heymann, Influence of dwell time and pressure on SPS process with titanium aluminides, 2021, metals. https://doi.org/10.3390/met12010083

[16] Q. Bai, J. Lin, J. Jiang, T.A. Dean, J. Zou, G. Tian, A study of direct forging process for powder superalloys, 2014. https://doi.org/10.1016/j.msea.2014.10.039

[17] S.L. Semiatin, V. Seetharaman, V.K. Jain, Microstructure Development during Conventional and Isothermal Hot Forging of a Near-Gamma Titanium Aluminide, 1994, Metallurgical and Materials Transactions A. https://doi.org/10.1007/BF02649227

[18] Schaeffler Technologies GmbH & Co. KG, Schaeffler Technisches Taschenbuch, 2014.

[19] A. Gökce, F. Findik, Mechanical and physical properties of sintered aluminum powders. J. Achievements Mater. Manuf. Eng. 2008, 30, 157-164.

[20] S. Bolz, Eigenschaftsoptimierung und Prozessfensterbestimmung der γ-(TiAl)-Schmiedelegierung TNB-V4, 2015

[21] V. Ventzke, Reibschweißen der γ-TiAl-Feingusslegierung Ti-47Al-3.5(Mn+Cr+Nb)-0.8(B+Si) und der Titanlegierung Ti6Al4V – Prozessentwicklung und Charakterisierung der mikrostrukturellen und mechanischen Eigenschaften, 2014.

[22] M. J. Blackburn, The Science, Technology and Applications of Titanium Alloys, Plenum Press, New York, 1970, p. 633. https://doi.org/10.1016/C2013-0-01574-5
[14] J. Li, Y. Liu, B. Liu, Y. Wang, X. Liang, Y. He, Microstructure characterization and mechanical behaviors of a hot forged high Nb containing PM-TiAl alloy, 2014, Materials Characterization 95, S. 148-156. https://doi.org/10.1016/j.matchar.2014.06.015

[15] B.-A. Behrens, K. Brunotte, J. Peddinghaus, A. Heymann, Influence of dwell time and pressure on SPS process with titanium aluminides, 2021, metals. https://doi.org/10.3390/met12010083

[16] Q. Bai, J. Lin, J. Jiang, T.A. Dean, J. Zou, G. Tian, A study of direct forging process for powder superalloys, 2014. https://doi.org/10.1016/j.msea.2014.10.039

[17] S.L. Semiatin, V. Seetharaman, V.K. Jain, Microstructure Development during Conventional and Isothermal Hot Forging of a Near-Gamma Titanium Aluminide, 1994, Metallurgical and Materials Transactions A. https://doi.org/10.1007/BF02649227

[18] Schaeffler Technologies GmbH & Co. KG, Schaeffler Technisches Taschenbuch, 2014.

[19] A. Gökce, F. Findik, Mechanical and physical properties of sintered aluminum powders. J. Achievements Mater. Manuf. Eng. 2008, 30, 157-164.

[20] S. Bolz, Eigenschaftsoptimierung und Prozessfensterbestimmung der γ-(TiAl)-Schmiedelegierung TNB-V4, 2015

[21] V. Ventzke, Reibschweißen der γ-TiAl-Feingusslegierung Ti-47Al-3.5(Mn+Cr+Nb)-0.8(B+Si) und der Titanlegierung Ti6Al4V – Prozessentwicklung und Charakterisierung der mikrostrukturellen und mechanischen Eigenschaften, 2014.

[22] M. J. Blackburn, The Science, Technology and Applications of Titanium Alloys, Plenum Press, New York, 1970, p. 633. https://doi.org/10.1016/C2013-0-01574-5
# Influence of the forming pre-strain on the fatigue performance of upset bulge formed tubes

BECHLER Jan¹,ª*, WEBER Florian¹,ᵇ, BULZAKOVSKI Tose²,ᶜ,
VÖLLMECKE Jörg²,ᵈ, SULAIMAN Hosen²,ᵉ and TEKKAYA A. Erman¹,*f

¹Institute of Forming Technology and Lightweight Components, TU Dortmund University, Baroper Str. 303, Dortmund 44227, Germany

²Faurecia Autositze GmbH, Garbsener Landstraße 7, Hannover 30419, Germany

ªJan.Bechler@iul.tu-dortmund.de, ᵇFlorian.Weber@iul.tu-dortmund.de,

ᶜTose.Buzalkovski@forvia.com, ᵈJoerg.Voellmecke@forvia.com,

ᵉHosen.Sulaiman-dr@forvia.com, ṼErman.Tekkaya@iul.tu-dortmund.de

**Keywords:** Fatigue, Forming Pre-Strain, Upset Bulging

**Abstract.** Safety-relevant components in the automotive industry must fulfill stringent requirements for performance, particularly in endurance. The height adjustment system in a car seat is one example. The height adjustment system is made up of sheet metal parts which are joined with steel tubes through axial compression – also referred to as upset bulge forming. As a result of the forming process, initial cracks occur at the inner bending radius of the tubes which, along with the forming history, decrease the fatigue life by 450%. The aim of this study is to identify whether the initial cracks occur due to shear bending generated by the process or elastic springback. The influence of the elastic springback on the crack initiation is investigated through bending experiments of DP600, Al99.5, and 1.4301. To prevent elastic springback, the specimens are secured with screws post-bending and are removed afterwards. The bending radius is investigated with respect to the equivalent plastic strain to determine a minimum without failure. Cyclic tests of bulges with two different forming histories are tested to investigate the influence of the forming pre-strain on the fatigue behavior. In all bent materials, initial cracks occur at the inner bending radius where the elastic springback increases the initial cracks and Scanning Electron Microscopy (SEM) analysis confirms shearing as the cause. It was also found the grain shape and orientation have a significant influence on the crack propagation in the forming zone of the tube through the upset bulging process.

**Introduction**

The automotive industry is under constant pressure to increase performance, while decreasing the weight and cost of components. One of the most applied methods to achieve the objective is to use high-strength steels. In order to reduce the cost and to maintain the performance of the product, forming instead of welding processes are used in the assembly. Forming processes avoid the degradation of the assembly performance, which is often the case in welding due to the process temperatures when joining dissimilar materials. For safety-relevant components, it is especially important to ensure the performance over the entire lifetime of the vehicle. The height adjustment system is one example of a formed, safety-relevant component where it is made up of sheet metal parts which are joined with a tube. A bulge is formed in a first step by axial compression. The part is then placed onto the bulge and a second bulge is formed, finally joining both tube and sheet metal part (Fig. 1 a)).

<figure><img src="image_915.png" /></figure>
# Influence of the forming pre-strain on the fatigue performance of upset bulge formed tubes

BECHLER Jan¹,ª*, WEBER Florian¹,ᵇ, BULZAKOVSKI Tose²,ᶜ,
VÖLLMECKE Jörg²,ᵈ, SULAIMAN Hosen²,ᵉ and TEKKAYA A. Erman¹,*f

¹Institute of Forming Technology and Lightweight Components, TU Dortmund University, Baroper Str. 303, Dortmund 44227, Germany

²Faurecia Autositze GmbH, Garbsener Landstraße 7, Hannover 30419, Germany

ªJan.Bechler@iul.tu-dortmund.de, ᵇFlorian.Weber@iul.tu-dortmund.de,

ᶜTose.Buzalkovski@forvia.com, ᵈJoerg.Voellmecke@forvia.com,

ᵉHosen.Sulaiman-dr@forvia.com, ṼErman.Tekkaya@iul.tu-dortmund.de

**Keywords:** Fatigue, Forming Pre-Strain, Upset Bulging

**Abstract.** Safety-relevant components in the automotive industry must fulfill stringent requirements for performance, particularly in endurance. The height adjustment system in a car seat is one example. The height adjustment system is made up of sheet metal parts which are joined with steel tubes through axial compression – also referred to as upset bulge forming. As a result of the forming process, initial cracks occur at the inner bending radius of the tubes which, along with the forming history, decrease the fatigue life by 450%. The aim of this study is to identify whether the initial cracks occur due to shear bending generated by the process or elastic springback. The influence of the elastic springback on the crack initiation is investigated through bending experiments of DP600, A199.5, and 1.4301. To prevent elastic springback, the specimens are secured with screws post-bending and are removed afterwards. The bending radius is investigated with respect to the equivalent plastic strain to determine a minimum without failure. Cyclic tests of bulges with two different forming histories are tested to investigate the influence of the forming pre-strain on the fatigue behavior. In all bent materials, initial cracks occur at the inner bending radius where the elastic springback increases the initial cracks and Scanning Electron Microscopy (SEM) analysis confirms shearing as the cause. It was also found the grain shape and orientation have a significant influence on the crack propagation in the forming zone of the tube through the upset bulging process.

**Introduction**

The automotive industry is under constant pressure to increase performance, while decreasing the weight and cost of components. One of the most applied methods to achieve the objective is to use high-strength steels. In order to reduce the cost and to maintain the performance of the product, forming instead of welding processes are used in the assembly. Forming processes avoid the degradation of the assembly performance, which is often the case in welding due to the process temperatures when joining dissimilar materials. For safety-relevant components, it is especially important to ensure the performance over the entire lifetime of the vehicle. The height adjustment system is one example of a formed, safety-relevant component where it is made up of sheet metal parts which are joined with a tube. A bulge is formed in a first step by axial compression. The part is then placed onto the bulge and a second bulge is formed, finally joining both tube and sheet metal part (Fig. 1 a)).

<figure><img src="image_922.png" /></figure>
<figure><img src="image_916.png" /><figcaption>Fig. 1 a) Height adjustment system and b) The resulting initial cracks due to forming.</figcaption></figure>

Investigations of the process limits showed that initial cracks occurred at the inner bending radius [1] (Fig. 1 b)). Furthermore, the initial crack length increased with further axial compression of the tube. It was presumed the elastic springback was the root cause of the initial cracks. The analysis of the microstructure showed that the initial cracks occurred mostly at the grain boundaries. Therefore, [2] concluded that the cracks initiated by shear bands and that the elastic springback supported the crack initiation. The development of shear bands for sheets was numerically analyzed for pure bending [3]. For this, periodic imperfections were implemented at the surface of the outer and inner bending radius. The surface imperfections led to undulations both at the tensile and compression sides of the sheet. The shear bands developed at the point of strain concentrations due to the undulations. Triantafyllidis et al. [3] suggested that notch-like protuberances on the compressive side of the sheet served as a failure initiation. In impact compression tests of aluminum tubes, similar initial cracks occurred compared to the upset bulge-forming process [4]. Microscopic analysis revealed that the bending led to undulations, which led into folds and sharp discontinuities with further bending. The bending load reversal led to failure due to the sharp clefts and creases. Haley and Kyriakides [4] suggested that even unloading due to elastic springback caused the local fracture. Furthermore, the numerical implementation of the surface roughness in the compression test caused undulations and finally high equivalent plastic strains occurred at the inner bending radius in the area of initial cracks.

The forming at elevated temperatures of 700 °C prevented the formation of initial cracks at the inner bending radius in upset bulging [5]. For this, the tube was inductively heated and subsequently formed. Microscopy analysis of the bulge cross-section showed that for 500 °C little and for 600 °C and 700 °C no initial cracks occurred. The fatigue lifetime of the hot formed bulge was 450% higher compared to the formed bulge at room temperature in a subsequent cyclic tensile test in axial direction.

From the literature, a possible root cause may exists where local instabilities cause undulations and can lead to cracks where the elastic springback may support the crack initiation. However, until now it is not clear whether the elastic springback or the shear bands cause the initial cracks. Furthermore, the initial cracks and the forming history have an influence on the fatigue life, but their interaction is unknown. Therefore, the main objectives of this paper are to analyze whether the cracks occur due to shear fracture or due to the elastic springback and how the microstructure affects the fatigue life. At first, cyclic tests were interrupted to analyze the crack propagation. The fracture surface was investigated by SEM to link the crack propagation to the microstructure. Secondly, bending experiments and microscopic analysis with DP600, Al99.5, and 1.4301 sheet materials were performed to investigate the influence of elastic springback on the initiation of cracks. To prevent elastic springback, the specimens were secured with fasteners post-bending.
<figure><img src="image_923.png" /><figcaption>Fig. 1 a) Height adjustment system and b) The resulting initial cracks due to forming.</figcaption></figure>

Investigations of the process limits showed that initial cracks occurred at the inner bending radius [1] (Fig. 1 b)). Furthermore, the initial crack length increased with further axial compression of the tube. It was presumed the elastic springback was the root cause of the initial cracks. The analysis of the microstructure showed that the initial cracks occurred mostly at the grain boundaries. Therefore, [2] concluded that the cracks initiated by shear bands and that the elastic springback supported the crack initiation. The development of shear bands for sheets was numerically analyzed for pure bending [3]. For this, periodic imperfections were implemented at the surface of the outer and inner bending radius. The surface imperfections led to undulations both at the tensile and compression sides of the sheet. The shear bands developed at the point of strain concentrations due to the undulations. Triantafyllidis et al. [3] suggested that notch-like protuberances on the compressive side of the sheet served as a failure initiation. In impact compression tests of aluminum tubes, similar initial cracks occurred compared to the upset bulge-forming process [4]. Microscopic analysis revealed that the bending led to undulations, which led into folds and sharp discontinuities with further bending. The bending load reversal led to failure due to the sharp clefts and creases. Haley and Kyriakides [4] suggested that even unloading due to elastic springback caused the local fracture. Furthermore, the numerical implementation of the surface roughness in the compression test caused undulations and finally high equivalent plastic strains occurred at the inner bending radius in the area of initial cracks.

The forming at elevated temperatures of 700 °C prevented the formation of initial cracks at the inner bending radius in upset bulging [5]. For this, the tube was inductively heated and subsequently formed. Microscopy analysis of the bulge cross-section showed that for 500 °C little and for 600 °C and 700 °C no initial cracks occurred. The fatigue lifetime of the hot formed bulge was 450% higher compared to the formed bulge at room temperature in a subsequent cyclic tensile test in axial direction.

From the literature, a possible root cause may exists where local instabilities cause undulations and can lead to cracks where the elastic springback may support the crack initiation. However, until now it is not clear whether the elastic springback or the shear bands cause the initial cracks. Furthermore, the initial cracks and the forming history have an influence on the fatigue life, but their interaction is unknown. Therefore, the main objectives of this paper are to analyze whether the cracks occur due to shear fracture or due to the elastic springback and how the microstructure affects the fatigue life. At first, cyclic tests were interrupted to analyze the crack propagation. The fracture surface was investigated by SEM to link the crack propagation to the microstructure. Secondly, bending experiments and microscopic analysis with DP600, Al99.5, and 1.4301 sheet materials were performed to investigate the influence of elastic springback on the initiation of cracks. To prevent elastic springback, the specimens were secured with fasteners post-bending.
## Influence of the forming pre-strain on the fatigue behavior

The height adjustment system comprises of a tube material of DP600 and two sheet metal parts with comparable material properties, where one of the sheets has a gearing to adjust the height. The interlocking of the bulge into notches of the sheet allows the transmission of torque (Fig. 1 a)). The tubes were manufactured by cold rolling of sheet material. The yield strength of the tube was determined by a tensile test with an A80 specimen, lasered from the tube [6]. The outer diameter of the tube was d_a = 36 mm, the total length was l = 460 mm and the tube was thickness s_0 = 1.2 mm. The flow curve of the sheet material was derived from an A80 tensile specimen of the material DP600 from a different batch with an initial sheet thickness of s_0 = 1.5 mm for the comparison of the flow curves. Since the manufacturing of the tube was a bending operation the equivalent plastic strain was calculated by Eq. 2. Here, y is the half of the initial sheet thickness and the bending radius r_m is the half of the initial diameter of the tube. Shifting the tube's flow curve by the equivalent plastic strain of ε̄ = 0.039, caused by the bending operation, shows a sufficient agreement with the flow curve of the sheet material (Fig. 2). Therefore, for future analysis the forming history of the tube must be taken into account.

<figure><img src="image_917.png" /><figcaption>Fig. 2 Flow curve of the DP600 tube and sheet material.</figcaption></figure>

The microscopic analysis of a representative bulge of the height adjustment system reveals initial cracks with a length of 45 µm (Fig. 1 b)). Due to the complexity of the bulging process of the height adjustment system, single formed bulges were analyzed microscopically and cyclically. Two different formed bulges were examined to investigate the influence of forming pre-strain. One bulge was opened and therefore had lower forming pre-strain at the inner bending radius of ε̄<sub>Exp.</sub> = 1.19 and the other one was closed and therefore had higher pre-strain ε̄<sub>Exp.</sub> = 1.25 (see Eq. 1). The high pre-strain at the inner bending radius occurs due to the compressive stress state during upset bulge forming, which postpones the onset of failure. Cracks at the inner bending radius indicate the surpassing of the materials compressive fracture limit in this area rather than its respective tensile limit at the outer radius. The crack lengths were measured by analyzing three formed bulges at three positions 120° shifted around the circumference and averaged. The sheet thickness was examined to verify an orthogonal cut and no oblique measure of the crack length. The opened bulge has a crack length of 24 µm (Fig. 3). This length increases by a factor of 2.8 up to 67 µm for the closed bulge. Higher axial compression of the tubes leads to an increase in crack length, confirming the results from [1]. Furthermore, the initial crack length of the height adjustment system lies in the range of the single formed bulges, which justifies the simplification to a single formed bulge.

The cyclic experiments were performed in a servo-hydraulic testing machine from Walter + Bai AG LFV-100-HH. The clamping jaws were designed to hold the tube material with mandrels. The
## Influence of the forming pre-strain on the fatigue behavior

The height adjustment system comprises of a tube material of DP600 and two sheet metal parts with comparable material properties, where one of the sheets has a gearing to adjust the height. The interlocking of the bulge into notches of the sheet allows the transmission of torque (Fig. 1 a)). The tubes were manufactured by cold rolling of sheet material. The yield strength of the tube was determined by a tensile test with an A80 specimen, lasered from the tube [6]. The outer diameter of the tube was d_a = 36 mm, the total length was l = 460 mm and the tube was thickness s_0 = 1.2 mm. The flow curve of the sheet material was derived from an A80 tensile specimen of the material DP600 from a different batch with an initial sheet thickness of s_0 = 1.5 mm for the comparison of the flow curves. Since the manufacturing of the tube was a bending operation the equivalent plastic strain was calculated by Eq. 2. Here, y is the half of the initial sheet thickness and the bending radius r_m is the half of the initial diameter of the tube. Shifting the tube's flow curve by the equivalent plastic strain of ε̄ = 0.039, caused by the bending operation, shows a sufficient agreement with the flow curve of the sheet material (Fig. 2). Therefore, for future analysis the forming history of the tube must be taken into account.

<figure><img src="image_924.png" /><figcaption>Fig. 2 Flow curve of the DP600 tube and sheet material.</figcaption></figure>

The microscopic analysis of a representative bulge of the height adjustment system reveals initial cracks with a length of 45 µm (Fig. 1 b)). Due to the complexity of the bulging process of the height adjustment system, single formed bulges were analyzed microscopically and cyclically. Two different formed bulges were examined to investigate the influence of forming pre-strain. One bulge was opened and therefore had lower forming pre-strain at the inner bending radius of ε̄<sub>Exp.</sub> = 1.19 and the other one was closed and therefore had higher pre-strain ε̄<sub>Exp.</sub> = 1.25 (see Eq. 1). The high pre-strain at the inner bending radius occurs due to the compressive stress state during upset bulge forming, which postpones the onset of failure. Cracks at the inner bending radius indicate the surpassing of the materials compressive fracture limit in this area rather than its respective tensile limit at the outer radius. The crack lengths were measured by analyzing three formed bulges at three positions 120° shifted around the circumference and averaged. The sheet thickness was examined to verify an orthogonal cut and no oblique measure of the crack length. The opened bulge has a crack length of 24 µm (Fig. 3). This length increases by a factor of 2.8 up to 67 µm for the closed bulge. Higher axial compression of the tubes leads to an increase in crack length, confirming the results from [1]. Furthermore, the initial crack length of the height adjustment system lies in the range of the single formed bulges, which justifies the simplification to a single formed bulge.

The cyclic experiments were performed in a servo-hydraulic testing machine from Walter + Bai AG LFV-100-HH. The clamping jaws were designed to hold the tube material with mandrels. The
fatigue tests were conducted with a load of 10 kN and a frequency of 10 Hz. The fatigue lifetime was averaged over three samples.

<figure><img src="image_918.png" /><figcaption>Fig. 3 Influence of the pre-strain in the simplified bulge geometry for the a) Opened and b) Closed configuration.</figcaption></figure>

The fatigue life of the closed bulge is 742 cycles and for the opened bulge is 4092 cycles (Fig. 4 a)). The forming pre-strain decreases the fatigue life for the different bulges by a factor of 5.52. This confirms the influence of the forming history [1]. Because of the difference in the initial crack length, which differs by a factor of 2.8, the question arises if differences in crack length cause the discrepancy. Therefore, the development of the crack length was analyzed by interrupted cyclic tests. The tests were interrupted at 2.5%, 7.5%, 20%, 50%, 80%, and 95% of the fatigue life for each single formed bulge. For the closed bulge, the initial crack length was 67 µm. Under the cyclic load, this length increases rapidly after 18 cycles to 278 µm (Fig. 4 b)). After the increase in length, the crack propagation stabilizes until the remaining cross section is too small to bear the load and forces rupture. In contrast, in the opened bulge, the crack propagation is stable until the forced rupture appears. Furthermore, the slope in the region of the stable crack propagation is higher for the closed than for the opened bulge which indicates that besides the instantaneous increase in crack length, the crack propagates faster in the closed bulge.

<figure><img src="image_919.png" /><figcaption>Fig. 4 a) Fatigue lifetime of the opened and closed bulge and b) The difference in crack propagation</figcaption></figure>

The difference in the crack propagation was investigated by SEM of the fracture surface after failure. At the inner bending radius, both bulges show shear fracture (Fig. 5). The results show that the initial crack initiate due to shear fracture at the inner bending radii. In radial direction from the
fatigue tests were conducted with a load of 10 kN and a frequency of 10 Hz. The fatigue lifetime was averaged over three samples.

<figure><img src="image_925.png" /><figcaption>Fig. 3 Influence of the pre-strain in the simplified bulge geometry for the a) Opened and b) Closed configuration.</figcaption></figure>

The fatigue life of the closed bulge is 742 cycles and for the opened bulge is 4092 cycles (Fig. 4 a)). The forming pre-strain decreases the fatigue life for the different bulges by a factor of 5.52. This confirms the influence of the forming history [1]. Because of the difference in the initial crack length, which differs by a factor of 2.8, the question arises if differences in crack length cause the discrepancy. Therefore, the development of the crack length was analyzed by interrupted cyclic tests. The tests were interrupted at 2.5%, 7.5%, 20%, 50%, 80%, and 95% of the fatigue life for each single formed bulge. For the closed bulge, the initial crack length was 67 µm. Under the cyclic load, this length increases rapidly after 18 cycles to 278 µm (Fig. 4 b)). After the increase in length, the crack propagation stabilizes until the remaining cross section is too small to bear the load and forces rupture. In contrast, in the opened bulge, the crack propagation is stable until the forced rupture appears. Furthermore, the slope in the region of the stable crack propagation is higher for the closed than for the opened bulge which indicates that besides the instantaneous increase in crack length, the crack propagates faster in the closed bulge.

<figure><img src="image_926.png" /><figcaption>Fig. 4 a) Fatigue lifetime of the opened and closed bulge and b) The difference in crack propagation</figcaption></figure>

The difference in the crack propagation was investigated by SEM of the fracture surface after failure. At the inner bending radius, both bulges show shear fracture (Fig. 5). The results show that the initial crack initiate due to shear fracture at the inner bending radii. In radial direction from the
inner radius, the closed bulge shows shear fracture directly followed by trans- and intergranular fracture. At the outer bending radius a ductile fracture surface with cup and cone occurs. According to Atkins [7] the lack of remaining ductility caused the brittle failure in terms of trans- and intergranular failure. In contrast to this, the opened bulge shows the shear and the ductile fracture surfaces in the same order without brittle failure. Therefore, the difference in the fatigue lifetime was caused by the different fracture modes which is influenced by the forming pre-strain.

<figure><img src="image_920.png" /><figcaption>Fig. 5 Fracture surface analysis of the opened and closed bulge after the fatigue test</figcaption></figure>

The influence of the microstructure on the fracture mode was further analyzed by microscopic investigations and the results are schematically depicted in Fig. 6 a). The results of the microscopic investigations can be seen in Fig. 6 b). The grains are elongated at the inner bending radius orthogonal to the surface due to the compressive state. The grains become more globular towards the inner of the bulge and elongated at the outer bending radius due to the tensile stress state. At the inner bending radius, the grains are more elongated in the closed bulge compared to the open bulge because of the higher compressive stress state. Higher grain elongation is equivalent to higher forming pre-strains. Therefore, the remaining ductility of the material of the closed bulge is lower than of the open bulge. Microscopic analysis of the grain size distribution of the closed bulge interrupted at 95% (704 cycles) of the overall fatigue life for the closed bulge, show a less tortuous fracture path at the inner bending radius and the path becomes more tortuous towards the inner of the bulge (Fig. 6 b)). At the inner bending radius, the crack grows uniformly in the radial direction and becomes more tortuous in the area where the grains are more globular.
inner radius, the closed bulge shows shear fracture directly followed by trans- and intergranular fracture. At the outer bending radius a ductile fracture surface with cup and cone occurs. According to Atkins [7] the lack of remaining ductility caused the brittle failure in terms of trans- and intergranular failure. In contrast to this, the opened bulge shows the shear and the ductile fracture surfaces in the same order without brittle failure. Therefore, the difference in the fatigue lifetime was caused by the different fracture modes which is influenced by the forming pre-strain.

<figure><img src="image_927.png" /><figcaption>Fig. 5 Fracture surface analysis of the opened and closed bulge after the fatigue test</figcaption></figure>

The influence of the microstructure on the fracture mode was further analyzed by microscopic investigations and the results are schematically depicted in Fig. 6 a). The results of the microscopic investigations can be seen in Fig. 6 b). The grains are elongated at the inner bending radius orthogonal to the surface due to the compressive state. The grains become more globular towards the inner of the bulge and elongated at the outer bending radius due to the tensile stress state. At the inner bending radius, the grains are more elongated in the closed bulge compared to the open bulge because of the higher compressive stress state. Higher grain elongation is equivalent to higher forming pre-strains. Therefore, the remaining ductility of the material of the closed bulge is lower than of the open bulge. Microscopic analysis of the grain size distribution of the closed bulge interrupted at 95% (704 cycles) of the overall fatigue life for the closed bulge, show a less tortuous fracture path at the inner bending radius and the path becomes more tortuous towards the inner of the bulge (Fig. 6 b)). At the inner bending radius, the crack grows uniformly in the radial direction and becomes more tortuous in the area where the grains are more globular.
<figure><img src="image_921.png" /><figcaption>Fig. 6 a) Basic principle of the fracture path in upset bulge formed tubes and b) Microscopic image of the fracture path.</figcaption></figure>

## Influence of the springback onto the crack initiation

To investigate the influence of elastic springback on the initiation of cracks, sheets were bent and fastened by a screw as shown in Fig. 7. The sheet specimens had the length $l = 100$ mm, width $b = 40$ mm and thickness $s_0 = 1.5$ mm. The materials were high strength steels DP600, pure AL99.5 and austenitic stainless steel 1.4301. Subsequently, the specimens were cut open near the screw and analyzed microscopically. This methodology allowed the comparison of the same crack in the fastened and unfastened configuration.

All materials show initial cracks in the fastened configuration at the inner bending radius (Fig. 7 b)). The aluminum has a crack with 131 µm in length, the DP600 with 97 µm and the 1.4301 with 45 µm. SEM analysis of the fracture surfaces confirm shear fracture as the cause for the initial cracks (Fig. 7 c)).

<figure><img src="image_922.png" /><figcaption>Fig. 7 a) Bending of the specimen, b) The initial crack in the fastened configuration without elastic springback, c) The SEM analysis of the resulting shear fracture surface and d) The increase in gap height after the elastic springback.</figcaption></figure>

The influence of the elastic springback on the crack length was analyzed. For this, five bending specimens of the material DP600 with different forming pre-strain were examined with respect to
<figure><img src="image_928.png" /><figcaption>Fig. 6 a) Basic principle of the fracture path in upset bulge formed tubes and b) Microscopic image of the fracture path.</figcaption></figure>

## Influence of the springback onto the crack initiation

To investigate the influence of elastic springback on the initiation of cracks, sheets were bent and fastened by a screw as shown in Fig. 7. The sheet specimens had the length $l = 100$ mm, width $b = 40$ mm and thickness $s_0 = 1.5$ mm. The materials were high strength steels DP600, pure AL99.5 and austenitic stainless steel 1.4301. Subsequently, the specimens were cut open near the screw and analyzed microscopically. This methodology allowed the comparison of the same crack in the fastened and unfastened configuration.

All materials show initial cracks in the fastened configuration at the inner bending radius (Fig. 7 b)). The aluminum has a crack with 131 µm in length, the DP600 with 97 µm and the 1.4301 with 45 µm. SEM analysis of the fracture surfaces confirm shear fracture as the cause for the initial cracks (Fig. 7 c)).

<figure><img src="image_929.png" /><figcaption>Fig. 7 a) Bending of the specimen, b) The initial crack in the fastened configuration without elastic springback, c) The SEM analysis of the resulting shear fracture surface and d) The increase in gap height after the elastic springback.</figcaption></figure>

The influence of the elastic springback on the crack length was analyzed. For this, five bending specimens of the material DP600 with different forming pre-strain were examined with respect to
initial cracks. The different forming pre-strains were determined by different bending radii. The bending radii ranged from completely closed $r_m = 0.75$ mm to opened $r_m = 2.14$ mm. The elastic springback was quantified by measuring the height at the end of the specimen (Fig.7 d)). The height of the bent specimen increases by 1% - 4%, which is a change of a thousandth millimeter, from the fastened to the unfastened configuration due to the elastic springback (Fig. 8a)). Furthermore, the crack length increases for the majority of the specimens from the fastened to the unfastened configuration. The growth in crack length varies from 10% to over 100% increase, which is a change of a hundredth of a millimeter (Fig. 8b)). Only one specimen does not show an increase in crack length. One explanation could be the increase of a crack further into the material, which was not visible. However, even small elastic springback can cause a severe increase in crack length.

<figure><img src="image_923.png" /><figcaption>Fig. 8 Influence of the elastic springback a) On the gap height and b) On the crack length.</figcaption></figure>

## Prediction of initial crack for bending specimens

The highest bending radius of $r_m = 2.14$ mm showed no initial cracks whereby the next smallest bending radius of $r_m = 1.37$ mm had initial cracks. Therefore, a limit of the smallest bending radius between $r_m = 2.14$ mm and $r_m = 1.37$ mm must exist where no initial crack occur. For this, the experimental equivalent plastic strain $\bar{\epsilon}$ was calculated from the bending samples by the change of grain size

$$ \bar{\epsilon}_{\text{Exp.}} = \sqrt{\frac{2}{3} \left( \ln \left( \frac{d_1}{d_0} \right)^2 + \ln \left( \frac{d_2}{d_0} \right)^2 + \ln \left( \frac{w_1}{w_0} \right)^2 \right)}, \quad (1) $$

whereby $d_0$ is the initial diameter, $d_1$ is the changed length and $d_2$ the changed width of the grain. The change in width $w$ direction of the sheet is calculated by the volume constancy.

Additional bending specimens were analyzed and the equivalent plastic strain $\bar{\epsilon}$ was plotted over the bending radius $r_m$ normalized by the initial sheet thickness $s_0$ (Fig. 9 b)). The equivalent plastic strain $\bar{\epsilon}$ increases for smaller and decreases for higher bending radii $r_m$. Specimens with initial cracks are marked with a red cross. The bending experiments do not show a clear limit at which strain initial cracks occur. Therefore, the shear strain $\gamma$ from shear test is evaluated by the so-called time-dependent method to determine the equivalent plastic strain $\bar{\epsilon}$ - normally used to determine the forming limit curve of sheet metal components [8] (Fig. 9 a)). The shear strain $\gamma$ and the equivalent plastic strain $\bar{\epsilon}$ were evaluated over time. The shear strain $\gamma$ value increases over time and at the onset of failure $\gamma$ accelerates. The onset was determined by fitting a first linear line
initial cracks. The different forming pre-strains were determined by different bending radii. The bending radii ranged from completely closed $r_m = 0.75$ mm to opened $r_m = 2.14$ mm. The elastic springback was quantified by measuring the height at the end of the specimen (Fig.7 d)). The height of the bent specimen increases by 1% – 4%, which is a change of a thousandth millimeter, from the fastened to the unfastened configuration due to the elastic springback (Fig. 8a)). Furthermore, the crack length increases for the majority of the specimens from the fastened to the unfastened configuration. The growth in crack length varies from 10% to over 100% increase, which is a change of a hundredth of a millimeter (Fig. 8b)). Only one specimen does not show an increase in crack length. One explanation could be the increase of a crack further into the material, which was not visible. However, even small elastic springback can cause a severe increase in crack length.

<figure><img src="image_930.png" /><figcaption>Fig. 8 Influence of the elastic springback a) On the gap height and b) On the crack length.</figcaption></figure>

### Prediction of initial crack for bending specimens

The highest bending radius of $r_m = 2.14$ mm showed no initial cracks whereby the next smallest bending radius of $r_m = 1.37$ mm had initial cracks. Therefore, a limit of the smallest bending radius between $r_m = 2.14$ mm and $r_m = 1.37$ mm must exist where no initial crack occur. For this, the experimental equivalent plastic strain $\bar{\epsilon}$ was calculated from the bending samples by the change of grain size

$$ \bar{\epsilon}_{\text{Exp.}} = \sqrt{\frac{2}{3} \left( \ln \left( \frac{d_1}{d_0} \right)^2 + \ln \left( \frac{d_2}{d_0} \right)^2 + \ln \left( \frac{w_1}{w_0} \right)^2 \right)}, \quad (1) $$

whereby $d_0$ is the initial diameter, $d_1$ is the changed length and $d_2$ the changed width of the grain. The change in width $w$ direction of the sheet is calculated by the volume constancy.

Additional bending specimens were analyzed and the equivalent plastic strain $\bar{\epsilon}$ was plotted over the bending radius $r_m$ normalized by the initial sheet thickness $s_0$ (Fig. 9 b)). The equivalent plastic strain $\bar{\epsilon}$ increases for smaller and decreases for higher bending radii $r_m$. Specimens with initial cracks are marked with a red cross. The bending experiments do not show a clear limit at which strain initial cracks occur. Therefore, the shear strain $\gamma$ from shear test is evaluated by the so-called time-dependent method to determine the equivalent plastic strain $\bar{\epsilon}$ – normally used to determine the forming limit curve of sheet metal components [8] (Fig. 9 a)). The shear strain $\gamma$ and the equivalent plastic strain $\bar{\epsilon}$ were evaluated over time. The shear strain $\gamma$ value increases over time and at the onset of failure $\gamma$ accelerates. The onset was determined by fitting a first linear line
$L_1$ in the region of homogeneous deformation. The second linear line $L_2$ was fitted from the last two recorded points [8]. The intersection of the $L_1$ and $L_2$ determined the onset of shear failure and the equivalent plastic strain was evaluated by the onset of shear failure. The shear test determines a failure limit of $\bar{\varepsilon}_{SL} = 0.78$. The $\bar{\varepsilon}_{SL}$ determines the failure strain between the specimen of $r_m = 2.14$ mm ($\bar{\varepsilon}_{Exp.} = 0.65$) and $r_m = 1.37$ mm ($\bar{\varepsilon}_{Exp.} = 0.84$) and is therefore in good agreement. Furthermore, the closed $\bar{\varepsilon}_{Exp.} = 1.25$ and opened $\bar{\varepsilon}_{Exp.} = 1.19$ bulge lie above the determined shear limit.

However, the determination of the equivalent plastic strain $\bar{\varepsilon}_{Exp.}$ from the change of grain size for many bending experiments is time and cost consuming. The equivalent plastic strain $\bar{\varepsilon}$ for bending experiments can also be analytically determined at the inner bending radius

$$ \bar{\varepsilon}_{A.} = \frac{2}{\sqrt{3}} \left| \ln \left( 1 + \frac{y}{r_m} \right) \right|, \qquad (2) $$

whereby $y$ is the half of the initial sheet thickness starting from the neutral fiber of the bending radius. It is assumed that the neutral fiber does not shift. The analytical method is valid as long as pure bending moments are applied and the sheet thickness does not change. However, for small bending radii, shearing causes initial cracks and which alters the sheet thickness.

The equivalent plastic strain $\bar{\varepsilon}$ was evaluated at the inner bending radius since the initial cracks occur there. The $\bar{\varepsilon}_{A.}$ overestimates the experiments for small bending radii due to the violation of the previously named assumptions. For larger bending radii the analytically and experimentally determined $\bar{\varepsilon}_{A.}$ are in good agreement with $\bar{\varepsilon}_{Exp.}$. The intersection of the analytical determined $\bar{\varepsilon}_{A.}$ and the shear limit $\bar{\varepsilon}_{SL}$ determine a conservative lower limit of $r_m$ to which specimen can be bend without initial cracks at the inner bending radius.

<figure><img src="image_924.png" /><figcaption>Fig. 9 a) Evaluation of the shear strains to determine the equivalent strain and b) Equivalent strain over bending radius evaluated at the inner bending radius</figcaption></figure>

## Summary and Conclusions

The aim of this study was to investigate the root cause of initial cracks at the inner bending radius and the influences of the forming history on the fatigue life of upset bulged tubes. For this, a closed bulge with higher forming pre-strain and an open bulge with lower forming pre-strain were investigated. The closed bulge had higher elongated grains than the opened bulge. The closed bulge had a decreased fatigue life by the factor of 5.52 compared to the opened bulge. From the
$L_1$ in the region of homogeneous deformation. The second linear line $L_2$ was fitted from the last two recorded points [8]. The intersection of the $L_1$ and $L_2$ determined the onset of shear failure and the equivalent plastic strain was evaluated by the onset of shear failure. The shear test determines a failure limit of $\bar{\varepsilon}_{SL} = 0.78$. The $\bar{\varepsilon}_{SL}$ determines the failure strain between the specimen of $r_m = 2.14$ mm ($\bar{\varepsilon}_{Exp.} = 0.65$) and $r_m = 1.37$ mm ($\bar{\varepsilon}_{Exp.} = 0.84$) and is therefore in good agreement. Furthermore, the closed $\bar{\varepsilon}_{Exp.} = 1.25$ and opened $\bar{\varepsilon}_{Exp.} = 1.19$ bulge lie above the determined shear limit.

However, the determination of the equivalent plastic strain $\bar{\varepsilon}_{Exp.}$ from the change of grain size for many bending experiments is time and cost consuming. The equivalent plastic strain $\bar{\varepsilon}$ for bending experiments can also be analytically determined at the inner bending radius

$$ \bar{\varepsilon}_{A.} = \frac{2}{\sqrt{3}} \left| \ln \left( 1 + \frac{y}{r_m} \right) \right|, \qquad (2) $$

whereby $y$ is the half of the initial sheet thickness starting from the neutral fiber of the bending radius. It is assumed that the neutral fiber does not shift. The analytical method is valid as long as pure bending moments are applied and the sheet thickness does not change. However, for small bending radii, shearing causes initial cracks and which alters the sheet thickness.

The equivalent plastic strain $\bar{\varepsilon}$ was evaluated at the inner bending radius since the initial cracks occur there. The $\bar{\varepsilon}_{A.}$ overestimates the experiments for small bending radii due to the violation of the previously named assumptions. For larger bending radii the analytically and experimentally determined $\bar{\varepsilon}_{A.}$ are in good agreement with $\bar{\varepsilon}_{Exp.}$. The intersection of the analytical determined $\bar{\varepsilon}_{A.}$ and the shear limit $\bar{\varepsilon}_{SL}$ determine a conservative lower limit of $r_m$ to which specimen can be bend without initial cracks at the inner bending radius.

<figure><img src="image_931.png" /><figcaption>Fig. 9 a) Evaluation of the shear strains to determine the equivalent strain and b) Equivalent strain over bending radius evaluated at the inner bending radius</figcaption></figure>

## Summary and Conclusions

The aim of this study was to investigate the root cause of initial cracks at the inner bending radius and the influences of the forming history on the fatigue life of upset bulged tubes. For this, a closed bulge with higher forming pre-strain and an open bulge with lower forming pre-strain were investigated. The closed bulge had higher elongated grains than the opened bulge. The closed bulge had a decreased fatigue life by the factor of 5.52 compared to the opened bulge. From the
results of the interrupted cyclic tests, in combination with the microscopic analysis, it can be said the grain size distribution in the form of highly elongated grains, in combination with a decreased ductility due to the forming pre-strain both enhance the susceptibility of brittle failure. Therefore, the bulge with the higher forming pre-strain is more susceptible to brittle failure, which decreases the fatigue lifetime.

The fastened bending experiments in combination with SEM analysis of the fracture surface proved shearing as the cause for the initial cracks at the inner bending radius. Furthermore, the elastic springback did not cause, but supported the initial crack length. Additionally, a conservative lower boundary of bending radius was determined by the comparison of analytically bending strain and the limit strain of the shear tests.

Future analysis will focus on the numerical prediction of the initial cracks in upset bulge forming or bending processes. Furthermore, attempts could be made to predict the complex fatigue behavior of upset bulge formed tubes.

## Acknowledgments

We would like to thank Mr. Hanl for the accurate execution of the interrupted cyclic tests and Mr. Dombrowski for the execution and measurement of the bending specimens. Furthermore, we thank Ms. Trask for the linguistic correction of the paper.

## References

[1] P. Grützner, Prozessentwicklung für das Fügen durch Knickbauchen, Winter Industries, 2014.

[2] I. Sizova, A. Sviridov, and M. Bambach, Avoiding crack nucleation and propagation during upset bulging of tubes, International Journal of Material Forming 10 (2017) 443–451. https://doi.org/10.1007/s12289-016-1292-9

[3] N. Triantafyllidis, A. Needleman, and V. Tvergaard, On the development of shear bands in pure bending, International Journal of Solids and Structures 18 (1982) 121–138. https://doi.org/10.1016/0020-7683(82)90021-X

[4] J. A. Haley and S. Kyriakides, Bending induced wrinkling and creasing in axially crushed aluminum tubes, International Journal of Solids and Structures 202 (2020) 368–383. https://doi.org/10.1016/j.ijsolstr.2020.06.003

[5] A. Sviridov, P. Grützner, M. Rusch, and M. Bambach, Joining by upset bulging-tooling design and new concepts for online process control using servo presses and local heating, Journal of Machine Engineering 17 (2017) 78–87.

[6] 6892-1: 2017-02, Metallische Werkstoffe-Zugversuch-Teil 1: Prüfverfahren bei Raumtemperatur (ISO 6892-1: 2016), 6892-1, DIN 6892-1 (2017).
results of the interrupted cyclic tests, in combination with the microscopic analysis, it can be said the grain size distribution in the form of highly elongated grains, in combination with a decreased ductility due to the forming pre-strain both enhance the susceptibility of brittle failure. Therefore, the bulge with the higher forming pre-strain is more susceptible to brittle failure, which decreases the fatigue lifetime.

The fastened bending experiments in combination with SEM analysis of the fracture surface proved shearing as the cause for the initial cracks at the inner bending radius. Furthermore, the elastic springback did not cause, but supported the initial crack length. Additionally, a conservative lower boundary of bending radius was determined by the comparison of analytically bending strain and the limit strain of the shear tests.

Future analysis will focus on the numerical prediction of the initial cracks in upset bulge forming or bending processes. Furthermore, attempts could be made to predict the complex fatigue behavior of upset bulge formed tubes.

## Acknowledgments

We would like to thank Mr. Hanl for the accurate execution of the interrupted cyclic tests and Mr. Dombrowski for the execution and measurement of the bending specimens. Furthermore, we thank Ms. Trask for the linguistic correction of the paper.

## References

[1] P. Grützner, Prozessentwicklung für das Fügen durch Knickbauchen, Winter Industries, 2014.

[2] I. Sizova, A. Sviridov, and M. Bambach, Avoiding crack nucleation and propagation during upset bulging of tubes, International Journal of Material Forming 10 (2017) 443–451. https://doi.org/10.1007/s12289-016-1292-9

[3] N. Triantafyllidis, A. Needleman, and V. Tvergaard, On the development of shear bands in pure bending, International Journal of Solids and Structures 18 (1982) 121–138. https://doi.org/10.1016/0020-7683(82)90021-X

[4] J. A. Haley and S. Kyriakides, Bending induced wrinkling and creasing in axially crushed aluminum tubes, International Journal of Solids and Structures 202 (2020) 368–383. https://doi.org/10.1016/j.ijsolstr.2020.06.003

[5] A. Sviridov, P. Grützner, M. Rusch, and M. Bambach, Joining by upset bulging–tooling design and new concepts for online process control using servo presses and local heating, Journal of Machine Engineering 17 (2017) 78–87.

[6] 6892-1: 2017-02, Metallische Werkstoffe-Zugversuch-Teil 1: Prüfverfahren bei Raumtemperatur (ISO 6892-1: 2016), 6892-1, DIN 6892-1 (2017).
# The non-proportional loading of mild steel

HAZRA Sumit¹,ª*, DHARA Sisir¹,ᵇ, TAYLOR Scott¹,ᶜ and FIGIEL Łukasz¹,ᵈ

¹ WMG, University of Warwick, Coventry, CV4 7AL, UK

ªsumit.hazra@warwick.ac.uk, ᵇSisir.Dhara.2@warwick.ac.uk, ᶜScott.Taylor.1@warwick.ac.uk,

ᵈL.W.Figiel@warwick.ac.uk

**Keywords:** Strain Path Change, In-Situ Microstructure Evaluations, Formability

**Abstract.** The effect of proportional loading on formability has been extensively studied. However, non-proportional loading occurs frequently in a stamping operation and its impact on formability has been studied less. Non-proportional loading occurs in two areas of stamping: continuously during the draw stage of a complex geometry and discontinuously through stamping stages. Studies on non-proportionality (eg. [1]) show that it can change the formability of a material compared to when it is loaded proportionally. This paper introduces an experimental method that allows a comparison of the deformation in mild steel subject to continuous and discontinuous loading mechanisms. A notable innovation of the method is that it can be carried out inside the chamber of a scanning electron microscope, allowing in-situ measurements of plastic deformation. The experiments show that the plastic behavior of the material under the continuous mode is distinct to when it is deformed discontinuously

**Introduction**

Automotive stamping is a multi-stage process. In the first, drawing stage, the material is drawn over a complex tool to undergo the majority of its final deformation. In follow-on stages, the drawn component is allowed to springback and then may be processed in follow-on steps to produce the final geometry. This process results in continuous and discontinuous strain path change mechanisms. The continuous mechanism takes place during the draw stage when the material flows over complex geometry of the draw tool, which forces the strain path of the material to change when the material encounters transitions in geometry. The discontinuous mechanism occurs when the component is removed from the draw tooling, allowing it to springback and is subsequently deformed along a different strain path when it is re-drawn or flanged in a following stage.

The current experimental state-of-the-art has focused on the effects of the discontinuous mechanism on formability. Dhara et al. [1] simulated the discontinuous mechanism in an AA5754-O aluminum alloy by first pre-straining samples uniaxially in a tensile machine and subsequently re-loading along the biaxial strain path using a miniature Nakajima test. They found considerable increase in the subsequent forming limits of the material. More recently, Collins et al. [2] observed a similar phenomenon in low carbon ferritic steel samples. Using a purpose-built tester, cruciform samples were first pre-strained along a uniaxial strain path, unloaded and then loaded biaxially. They found a significant increase in the limit strains.

The effect of the continuous mechanism has not been studied as thoroughly as the discontinuous mechanism. Song et al. [3] and Leotoing and Guines [4] developed a complex cruciform specimen to change its strain path continuously from uniaxial to biaxial tension using a sophisticated servo-hydraulic testing machine. They found that continuously mechanism caused changes to the forming limit strains in steel (DP600) and aluminum (AA5086) but they did not contrast the changes to that produced by the discontinuous mechanism. Hence, an important contribution was made by Yoshida et al. [5] and Yoshida and Kuwabara [6], who compared the continuous and discontinuous mechanisms in stress space. They made three findings. First, the stress-based

<figure><img src="image_925.png" /></figure>
# The non-proportional loading of mild steel

HAZRA Sumit¹,ª*, DHARA Sisir¹,ᵇ, TAYLOR Scott¹,ᶜ and FIGIEL Łukasz¹,ᵈ

¹ WMG, University of Warwick, Coventry, CV4 7AL, UK

ªsumit.hazra@warwick.ac.uk, ᵇSisir.Dhara.2@warwick.ac.uk, ᶜScott.Taylor.1@warwick.ac.uk,

ᵈL.W.Figiel@warwick.ac.uk

**Keywords:** Strain Path Change, In-Situ Microstructure Evaluations, Formability

**Abstract.** The effect of proportional loading on formability has been extensively studied. However, non-proportional loading occurs frequently in a stamping operation and its impact on formability has been studied less. Non-proportional loading occurs in two areas of stamping: continuously during the draw stage of a complex geometry and discontinuously through stamping stages. Studies on non-proportionality (eg. [1]) show that it can change the formability of a material compared to when it is loaded proportionally. This paper introduces an experimental method that allows a comparison of the deformation in mild steel subject to continuous and discontinuous loading mechanisms. A notable innovation of the method is that it can be carried out inside the chamber of a scanning electron microscope, allowing in-situ measurements of plastic deformation. The experiments show that the plastic behavior of the material under the continuous mode is distinct to when it is deformed discontinuously

**Introduction**

Automotive stamping is a multi-stage process. In the first, drawing stage, the material is drawn over a complex tool to undergo the majority of its final deformation. In follow-on stages, the drawn component is allowed to springback and then may be processed in follow-on steps to produce the final geometry. This process results in continuous and discontinuous strain path change mechanisms. The continuous mechanism takes place during the draw stage when the material flows over complex geometry of the draw tool, which forces the strain path of the material to change when the material encounters transitions in geometry. The discontinuous mechanism occurs when the component is removed from the draw tooling, allowing it to springback and is subsequently deformed along a different strain path when it is re-drawn or flanged in a following stage.

The current experimental state-of-the-art has focused on the effects of the discontinuous mechanism on formability. Dhara et al. [1] simulated the discontinuous mechanism in an AA5754-O aluminum alloy by first pre-straining samples uniaxially in a tensile machine and subsequently re-loading along the biaxial strain path using a miniature Nakajima test. They found considerable increase in the subsequent forming limits of the material. More recently, Collins et al. [2] observed a similar phenomenon in low carbon ferritic steel samples. Using a purpose-built tester, cruciform samples were first pre-strained along a uniaxial strain path, unloaded and then loaded biaxially. They found a significant increase in the limit strains.

The effect of the continuous mechanism has not been studied as thoroughly as the discontinuous mechanism. Song et al. [3] and Leotoing and Guines [4] developed a complex cruciform specimen to change its strain path continuously from uniaxial to biaxial tension using a sophisticated servo-hydraulic testing machine. They found that continuously mechanism caused changes to the forming limit strains in steel (DP600) and aluminum (AA5086) but they did not contrast the changes to that produced by the discontinuous mechanism. Hence, an important contribution was made by Yoshida et al. [5] and Yoshida and Kuwabara [6], who compared the continuous and discontinuous mechanisms in stress space. They made three findings. First, the stress-based

<figure><img src="image_932.png" /></figure>
characterization of forming limits was path independent when the strain path change was discontinuous. Second, the limit stress was path dependent when it was continuous. Third, path-dependent nature was directly affected by the strain hardening behavior of the material. Erinosho et al. [7, 8] showed that strain hardening behavior is related to the orientation of grains in the microstructure and the evolution of dislocation density during the discontinuous mechanism.

To investigate the relationship between the externally applied loading and microstructural deformation, in-situ deformation techniques are increasingly used. During an in-situ test, a material is elongated using a micromechanical test rig that is placed in a scanning electron microscope (SEM) chamber. The deformation of the material can be evaluated using secondary electron or electron back-scatter diffraction detectors to image and measure textural changes respectively. For example, Ghadbeigi et al. [9] and Celotto et al. [10] deformed dual-phase 1000 (DP1000) steel and transformation induced plasticity 800 (TRIP800) steel samples uniaxially using a miniaturized loading rig. They captured SEM images of the sample surface and post-processed the images using the digital image correlation (DIC) technique to analyze the evolution of strain distribution in the ferritic and martensitic phases of their specimens.

In this work, the in-situ technique was adapted to investigate the effect of continuous and discontinuous mechanisms on its microstructural deformation. Tests were performed on DX54 in the uniaxial and then biaxial strain paths. A novel sample geometry was designed to enable the material to be deformed continuously and discontinuously. Using DIC, strains were measured at the macroscale and the microscale. The results showed that the material deformed differently at the macroscale and this was reflected in the microstructural deformation of the material.

## Method

A Micromecha Proxima test rig was used to perform the strain path change experiment (Fig.1). The rig was placed in a Zeiss Sigma SEM and was controlled with Micromecha's POROS 2 software. The 'inline' and 'transverse' grips of the rig are connected through a 45° wedge. The rig's motor pulls inline grip 1 and its motion is translated into a perpendicular motion to load the transverse grips via the wedge connection. The peak capacity of the rig is 3.5kN. Load and extension were captured by sensors on the rig and recorded to the POROS 2 software in real time. Displacement was applied to the inline grip 1 at a rate of 5µm/sec and the test was interrupted every 0.2mm to image the microstructure. Images were captured using a 10keV voltage source and a secondary electron detector.
characterization of forming limits was path independent when the strain path change was discontinuous. Second, the limit stress was path dependent when it was continuous. Third, path-dependent nature was directly affected by the strain hardening behavior of the material. Erinosho et al. [7, 8] showed that strain hardening behavior is related to the orientation of grains in the microstructure and the evolution of dislocation density during the discontinuous mechanism.

To investigate the relationship between the externally applied loading and microstructural deformation, in-situ deformation techniques are increasingly used. During an in-situ test, a material is elongated using a micromechanical test rig that is placed in a scanning electron microscope (SEM) chamber. The deformation of the material can be evaluated using secondary electron or electron back-scatter diffraction detectors to image and measure textural changes respectively. For example, Ghadbeigi et al. [9] and Celotto et al. [10] deformed dual-phase 1000 (DP1000) steel and transformation induced plasticity 800 (TRIP800) steel samples uniaxially using a miniaturized loading rig. They captured SEM images of the sample surface and post-processed the images using the digital image correlation (DIC) technique to analyze the evolution of strain distribution in the ferritic and martensitic phases of their specimens.

In this work, the in-situ technique was adapted to investigate the effect of continuous and discontinuous mechanisms on its microstructural deformation. Tests were performed on DX54 in the uniaxial and then biaxial strain paths. A novel sample geometry was designed to enable the material to be deformed continuously and discontinuously. Using DIC, strains were measured at the macroscale and the microscale. The results showed that the material deformed differently at the macroscale and this was reflected in the microstructural deformation of the material.

## Method

A Micromecha Proxima test rig was used to perform the strain path change experiment (Fig.1). The rig was placed in a Zeiss Sigma SEM and was controlled with Micromecha's POROS 2 software. The 'inline' and 'transverse' grips of the rig are connected through a 45° wedge. The rig's motor pulls inline grip 1 and its motion is translated into a perpendicular motion to load the transverse grips via the wedge connection. The peak capacity of the rig is 3.5kN. Load and extension were captured by sensors on the rig and recorded to the POROS 2 software in real time. Displacement was applied to the inline grip 1 at a rate of 5µm/sec and the test was interrupted every 0.2mm to image the microstructure. Images were captured using a 10keV voltage source and a secondary electron detector.
<figure><img src="image_926.png" /><figcaption>Fig. 1 Top view of the Micromecha Proxima rig. 1: Lead screw, 2: Motor, 3: In-line grip 1, 4: Specimen, 5: Transverse grip 1, 6: Connecting bolt, 7: In-line grip 2, 8: 45° wedge, 9: Transverse grip 2, 10: Load cell</figcaption></figure>

The 30 x 30mm cruciform sample design proposed by Caër and Pesci [11] was used for the tests. To perform the continuous tests, 30 x 30mm sample was modified to include 0.2mm length slots in the transverse arms of the sample (Fig.1, component 6). The slots were added to allow the connecting bolts of the transverse grips (Fig.1, component 1 and 9) to translate without loading the specimen at the start of the test. When the connecting bolts reached the end of the slots, it loaded the transverse arms of the specimen to load the sample biaxially. In this way, the material was deformed uniaxially until 0.108 strain, then loaded biaxially until failure. The discontinuous tests were carried out by loading the cruciform sample uniaxially until 0.108 strain by simply disconnecting the transverse grips. The sample was then unloaded to fix the transverse grips, then reloaded biaxially until failure. The key difference between the discontinuous and continuous modes is the unloading step in the discontinuous mode.

The deformation of the material was measured at the macroscopic and microscopic scales. At the macroscopic scale, the strain evolution was measured with a GOM 12M DIC system. The DIC system consisted of two 12MP cameras fitted with 75mm lenses. A paint speckled paint pattern was applied to the sample surface before the test and GOM ARAMIS version 6.1 software was used to acquire and process the captured images to calculate strain contour [12]. Length resolution was 0.36mm and measurement uncertainty of the system was measured to be ±0.0006 [13].

To measure microstrains, the deformation of the sample was captured in a Zeiss Sigma SEM and its images post-processed with LaVision's DIC software (DaVis 10) [13]. The samples were polished and etched with 2% Nital solution for 10s to reveal the microstructural features within the microstructure. The image of the microstructure was captured every 200µm of deformation. Using the manufacturer recommended subset size of 25pixels and step size of 8pixels (to give length resolution of 13µm), the features in the evolving microstructure was tracked by the DIC software to estimate the strain within the microstructure. Displacement accuracy of the system was 0.01 pixels, which corresponded to a strain resolution of about ±0.001 strains [13].

The mechanical properties of the DX54 steel used in these experiments were measured with tensile tests and are shown in Table 1.
<figure><img src="image_933.png" /><figcaption>Fig. 1 Top view of the Micromecha Proxima rig. 1: Lead screw, 2: Motor, 3: In-line grip 1, 4: Specimen, 5: Transverse grip 1, 6: Connecting bolt, 7: In-line grip 2, 8: 45° wedge, 9: Transverse grip 2, 10: Load cell</figcaption></figure>

The 30 x 30mm cruciform sample design proposed by Caër and Pesci [11] was used for the tests. To perform the continuous tests, 30 x 30mm sample was modified to include 0.2mm length slots in the transverse arms of the sample (Fig.1, component 6). The slots were added to allow the connecting bolts of the transverse grips (Fig.1, component 1 and 9) to translate without loading the specimen at the start of the test. When the connecting bolts reached the end of the slots, it loaded the transverse arms of the specimen to load the sample biaxially. In this way, the material was deformed uniaxially until 0.108 strain, then loaded biaxially until failure. The discontinuous tests were carried out by loading the cruciform sample uniaxially until 0.108 strain by simply disconnecting the transverse grips. The sample was then unloaded to fix the transverse grips, then reloaded biaxially until failure. The key difference between the discontinuous and continuous modes is the unloading step in the discontinuous mode.

The deformation of the material was measured at the macroscopic and microscopic scales. At the macroscopic scale, the strain evolution was measured with a GOM 12M DIC system. The DIC system consisted of two 12MP cameras fitted with 75mm lenses. A paint speckled paint pattern was applied to the sample surface before the test and GOM ARAMIS version 6.1 software was used to acquire and process the captured images to calculate strain contour [12]. Length resolution was 0.36mm and measurement uncertainty of the system was measured to be ±0.0006 [13].

To measure microstrains, the deformation of the sample was captured in a Zeiss Sigma SEM and its images post-processed with LaVision's DIC software (DaVis 10) [13]. The samples were polished and etched with 2% Nital solution for 10s to reveal the microstructural features within the microstructure. The image of the microstructure was captured every 200µm of deformation. Using the manufacturer recommended subset size of 25pixels and step size of 8pixels (to give length resolution of 13µm), the features in the evolving microstructure was tracked by the DIC software to estimate the strain within the microstructure. Displacement accuracy of the system was 0.01 pixels, which corresponded to a strain resolution of about ±0.001 strains [13].

The mechanical properties of the DX54 steel used in these experiments were measured with tensile tests and are shown in Table 1.
<table><caption>Table1 Mechanical properties of DX54</caption><tr><td>Material</td><td></td><td>DX54</td></tr><tr><td>Gauge [mm]</td><td></td><td>1</td></tr><tr><td>Thickness of central thinned region [mm]</td><td></td><td>0.3</td></tr><tr><td>Young modulus [GPa]</td><td></td><td>213</td></tr><tr><td>Yield strength [MPa]</td><td></td><td>225</td></tr><tr><td>Ultimate tensile strength [MPa]</td><td></td><td>329</td></tr><tr><td>Elongation [strain]</td><td></td><td>0.43</td></tr><tr><td rowspan="3">Anisotropy parameters</td><td>r<sub>0</sub></td><td>1.53</td></tr><tr><td>r<sub>45</sub></td><td>1.08</td></tr><tr><td>r<sub>90</sub></td><td>2.16</td></tr><tr><td rowspan="2">Hardening parameters according to Hollomon law</td><td>Strength coefficient, K [MPa]</td><td>555</td></tr><tr><td>Hardening exponent, n</td><td>0.20</td></tr></table>

## Results and discussion

The macroscopic deformation in strain space is shown in Fig.2, which compares the deformation of material undergoing discontinuous (red line) and continuous loading (blue line). Point A is the start of the deformation, point B is the abrupt transition from uniaxial to biaxial deformation and points C and C' are the failure points for the continuous and discontinuous tests respectively. As expected, the initial uniaxial strain path was identical for both loading conditions and the point of the strain path change occurred within 0.002 major strain. There are two key differences in the subsequent biaxial stage. First, the strain ratios ($\epsilon_{xx}/\epsilon_{yy}$) were 0.78 ± 0.04 for continuous loading and 0.97 ± 0.03 for the discontinuous loading, an approximately 0.19 difference in strain ratio. Secondly, the elongation of the discontinuous strain path change was found to be consistently higher (by about 0.07 major strain) than the continuous tests.

<figure><img src="image_927.png" /><figcaption>Fig 2 Deformation of continuous and discontinuous strain path change in strain space</figcaption></figure>

At the microscale, the applied deformation on the sample coincided with a heterogeneous of strain in the microstructure (Fig.3). Fig.3a shows strain distribution for the continuous loading and Fig.3b shows strain distribution for discontinuous loading. The strain path change occurred at 1000µm displacement and this corresponded to point B in Fig.2. For the discontinuous tests, the strain path change involved unloading the sample at 1000µm and then re-loading along the biaxial path (Fig.3b). Two observations can be made. First, the heterogeneous strain distribution had a banded
<table><caption>Table1 Mechanical properties of DX54</caption><tr><td>Material</td><td></td><td>DX54</td></tr><tr><td>Gauge [mm]</td><td></td><td>1</td></tr><tr><td>Thickness of central thinned region [mm]</td><td></td><td>0.3</td></tr><tr><td>Young modulus [GPa]</td><td></td><td>213</td></tr><tr><td>Yield strength [MPa]</td><td></td><td>225</td></tr><tr><td>Ultimate tensile strength [MPa]</td><td></td><td>329</td></tr><tr><td>Elongation [strain]</td><td></td><td>0.43</td></tr><tr><td rowspan="3">Anisotropy parameters</td><td>r<sub>0</sub></td><td>1.53</td></tr><tr><td>r<sub>45</sub></td><td>1.08</td></tr><tr><td>r<sub>90</sub></td><td>2.16</td></tr><tr><td rowspan="2">Hardening parameters according to Hollomon law</td><td>Strength coefficient, K [MPa]</td><td>555</td></tr><tr><td>Hardening exponent, n</td><td>0.20</td></tr></table>

## Results and discussion

The macroscopic deformation in strain space is shown in Fig.2, which compares the deformation of material undergoing discontinuous (red line) and continuous loading (blue line). Point A is the start of the deformation, point B is the abrupt transition from uniaxial to biaxial deformation and points C and C' are the failure points for the continuous and discontinuous tests respectively. As expected, the initial uniaxial strain path was identical for both loading conditions and the point of the strain path change occurred within 0.002 major strain. There are two key differences in the subsequent biaxial stage. First, the strain ratios ($\epsilon_{xx}/\epsilon_{yy}$) were 0.78 ± 0.04 for continuous loading and 0.97 ± 0.03 for the discontinuous loading, an approximately 0.19 difference in strain ratio. Secondly, the elongation of the discontinuous strain path change was found to be consistently higher (by about 0.07 major strain) than the continuous tests.

<figure><img src="image_934.png" /><figcaption>Fig 2 Deformation of continuous and discontinuous strain path change in strain space</figcaption></figure>

At the microscale, the applied deformation on the sample coincided with a heterogeneous of strain in the microstructure (Fig.3). Fig.3a shows strain distribution for the continuous loading and Fig.3b shows strain distribution for discontinuous loading. The strain path change occurred at 1000µm displacement and this corresponded to point B in Fig.2. For the discontinuous tests, the strain path change involved unloading the sample at 1000µm and then re-loading along the biaxial path (Fig.3b). Two observations can be made. First, the heterogeneous strain distribution had a banded
structure and these bands tended to remain consistent during each deformation path. Second, the continuous and discontinuous loadings can be distinguished by the change to this banded structure after the strain path change. During the uniaxial loading, the banded structure started vertically and gradually inclined about 35° to the x-axis for both the continuous and discontinuous loadings. After the strain path change, the banded structure in the continuous samples increased slightly to 36° until failure. In the discontinuous strain path change, the 35° banded structure that was observed during uniaxial deformation was replaced by a banded structure that was vertical and remained vertical until failure.

<figure><img src="image_928.png" /><figcaption>Fig. 3 x-axis micro-strain ($\epsilon_{xx}$) distribution. Left column: 200µm displacement, middle column: 1000µm displacement, right column: before failure, (a) row: continuous loading, (b) row: discontinuous loading. Strain path change occurs at 1000µm</figcaption></figure>

## Summary

Stamping is a complex process that involves continuous and discontinuous non-proportional loading of material. To address the gap in understanding of material under non-proportional loadings, an experiment was developed and carried out to evaluate deformation behavior of DX54 under continuous and discontinuous loading from the uniaxial to biaxial strain paths.

In-situ experiments were carried out on cruciform samples that were loaded by a mechanical rig that was small enough to fit in the chamber of a scanning electron microscope. This allowed the deformation of the material to be studied under macro and micro-scales. A comparison of results of the macro and micro scales showed that differences in macroscale behavior are reflected in the microstructural behavior of the material.

## References

[1] S Dhara, S Basak, SK Panda, S Hazra, B Shollock, R Dashwood, Formability analysis of pre-strained AA5754-O sheet metal using Yld96 plasticity theory: Role of amount and direction of uni-axial pre-strain, J Manuf Process 24 (2016) 270-82.
https://doi.org/10.1016/j.jmapro.2016.09.014
structure and these bands tended to remain consistent during each deformation path. Second, the continuous and discontinuous loadings can be distinguished by the change to this banded structure after the strain path change. During the uniaxial loading, the banded structure started vertically and gradually inclined about 35° to the x-axis for both the continuous and discontinuous loadings. After the strain path change, the banded structure in the continuous samples increased slightly to 36° until failure. In the discontinuous strain path change, the 35° banded structure that was observed during uniaxial deformation was replaced by a banded structure that was vertical and remained vertical until failure.

<figure><img src="image_935.png" /><figcaption>Fig. 3 x-axis micro-strain (εxx) distribution. Left column: 200μm displacement, middle column: 1000μm displacement, right column: before failure, (a) row: continuous loading, (b) row: discontinuous loading. Strain path change occurs at 1000μm</figcaption></figure>

## Summary

Stamping is a complex process that involves continuous and discontinuous non-proportional loading of material. To address the gap in understanding of material under non-proportional loadings, an experiment was developed and carried out to evaluate deformation behavior of DX54 under continuous and discontinuous loading from the uniaxial to biaxial strain paths.

In-situ experiments were carried out on cruciform samples that were loaded by a mechanical rig that was small enough to fit in the chamber of a scanning electron microscope. This allowed the deformation of the material to be studied under macro and micro-scales. A comparison of results of the macro and micro scales showed that differences in macroscale behavior are reflected in the microstructural behavior of the material.

## References

[1] S Dhara, S Basak, SK Panda, S Hazra, B Shollock, R Dashwood, Formability analysis of pre-strained AA5754-O sheet metal using Yld96 plasticity theory: Role of amount and direction of uni-axial pre-strain, J Manuf Process 24 (2016) 270-82.
https://doi.org/10.1016/j.jmapro.2016.09.014
[2] DM Collins, T Erinosho, FPE Dunne, RI Todd, T Connolley, M Mostafavi, A synchrotron X-ray diffraction study of non-proportional strain-path effects, Acta Mater 124 (2017) 290-304. https://doi.org/https://doi.org/10.1016/j.actamat.2016.11.011

[3] X Song, L Leotoing, D Guines, E Ragneau, Effect of continuous strain path changes on forming limit strains of DP600, Strain 55 (2019), e12329. https://doi.org/https://doi.org/10.1111/str.12329

[4] L Leotoing, D Guines, Investigations of the effect of strain path changes on forming limit curves using an in-plane biaxial tensile test, Int J Mech Sci 99 (2015), 21-8. https://doi.org/https://doi.org/10.1016/j.ijmecsci.2015.05.007

[5] K Yoshida, T Kuwabara, M Kuroda, Path-dependence of the forming limit stresses in a sheet metal, Int J Plast 23 (2007) 361-84. https://doi.org/https://doi.org/10.1016/j.ijplas.2006.05.005

[6] K Yoshida, T Kuwabara, Effect of strain hardening behavior on forming limit stresses of steel tube subjected to nonproportional loading paths, Int J Plast 23 (2007), 1260-84. https://doi.org/https://doi.org/10.1016/j.ijplas.2006.11.008

[7] TO Erinosho, ACF Cocks, FPE Dunne, Coupled effects of texture, hardening and non-proportionality of strain on ductility in ferritic steel, Comput Mater Sci 80 (2013) 113-22. https://doi.org/https://doi.org/10.1016/j.commatsci.2013.03.002.

[8] TO Erinosho, ACF Cocks, FPE Dunne, Texture, hardening and non-proportionality of strain in BCC polycrystal deformation, Int J Plast 50 (2013) 170-92. https://doi.org/https://doi.org/10.1016/j.ijplas.2013.04.008

[9] H Ghadbeigi, C Pinna, S Celotto, JR Yates, Local plastic strain evolution in a high strength dual-phase steel, Mater Sci Eng A 527 (2010) 5026-32. https://doi.org/https://doi.org/10.1016/j.msea.2010.04.052

[10] S Celotto, H Ghadbeigi, C Pinna, BA Shollock, P Efthymiadis, Deformation-Induced Microstructural Banding in TRIP Steels, Metall Mater Trans A 49 (2018) 2893-906. https://doi.org/10.1007/s11661-018-4650-z.

[11] C Caër, R Pesci, Local behavior of an AISI 304 stainless steel submitted to in situ biaxial loading in SEM, Mater Sci Eng A 690 (2017) 44-51. https://doi.org/https://doi.org/10.1016/j.msea.2017.02.087

[12] GOM, GOM ARAMIS. 2009. https://www.gom.com/metrology-systems/aramis (Accessed 27-03-2020)

[13] NS Small, SK Hazra, DK Williams, R Roy, Measuring and representing the formability of sheet materials in the presence of heterogeneous plastic deformation, Int. J of Adv Mat Tech 109 (2020) 397-410. https://doi.org/10.1007/s00170-020-05372-0

[14] LaVision, DaVis Strain Master Software, DaVis 10. 2017. https://www.lavision.de/en/products/davis-software/ (Accessed 26-09-2021)
[2] DM Collins, T Erinosho, FPE Dunne, RI Todd, T Connolley, M Mostafavi, A synchrotron X-ray diffraction study of non-proportional strain-path effects, Acta Mater 124 (2017) 290-304. https://doi.org/https://doi.org/10.1016/j.actamat.2016.11.011

[3] X Song, L Leotoing, D Guines, E Ragneau, Effect of continuous strain path changes on forming limit strains of DP600, Strain 55 (2019), e12329. https://doi.org/https://doi.org/10.1111/str.12329

[4] L Leotoing, D Guines, Investigations of the effect of strain path changes on forming limit curves using an in-plane biaxial tensile test, Int J Mech Sci 99 (2015), 21-8. https://doi.org/https://doi.org/10.1016/j.ijmecsci.2015.05.007

[5] K Yoshida, T Kuwabara, M Kuroda, Path-dependence of the forming limit stresses in a sheet metal, Int J Plast 23 (2007) 361-84. https://doi.org/https://doi.org/10.1016/j.ijplas.2006.05.005

[6] K Yoshida, T Kuwabara, Effect of strain hardening behavior on forming limit stresses of steel tube subjected to nonproportional loading paths, Int J Plast 23 (2007), 1260-84. https://doi.org/https://doi.org/10.1016/j.ijplas.2006.11.008

[7] TO Erinosho, ACF Cocks, FPE Dunne, Coupled effects of texture, hardening and non-proportionality of strain on ductility in ferritic steel, Comput Mater Sci 80 (2013) 113-22. https://doi.org/https://doi.org/10.1016/j.commatsci.2013.03.002.

[8] TO Erinosho, ACF Cocks, FPE Dunne, Texture, hardening and non-proportionality of strain in BCC polycrystal deformation, Int J Plast 50 (2013) 170-92. https://doi.org/https://doi.org/10.1016/j.ijplas.2013.04.008

[9] H Ghadbeigi, C Pinna, S Celotto, JR Yates, Local plastic strain evolution in a high strength dual-phase steel, Mater Sci Eng A 527 (2010) 5026-32. https://doi.org/https://doi.org/10.1016/j.msea.2010.04.052

[10] S Celotto, H Ghadbeigi, C Pinna, BA Shollock, P Efthymiadis, Deformation-Induced Microstructural Banding in TRIP Steels, Metall Mater Trans A 49 (2018) 2893-906. https://doi.org/10.1007/s11661-018-4650-z.

[11] C Caër, R Pesci, Local behavior of an AISI 304 stainless steel submitted to in situ biaxial loading in SEM, Mater Sci Eng A 690 (2017) 44-51. https://doi.org/https://doi.org/10.1016/j.msea.2017.02.087

[12] GOM, GOM ARAMIS. 2009. https://www.gom.com/metrology-systems/aramis (Accessed 27-03-2020)

[13] NS Small, SK Hazra, DK Williams, R Roy, Measuring and representing the formability of sheet materials in the presence of heterogeneous plastic deformation, Int. J of Adv Mat Tech 109 (2020) 397-410. https://doi.org/10.1007/s00170-020-05372-0

[14] LaVision, DaVis Strain Master Software, DaVis 10. 2017. https://www.lavision.de/en/products/davis-software/ (Accessed 26-09-2021)
# Molecular dynamics simulation for determining dislocation strengthening coefficient in BCC iron

MIYAZAWA Naoki¹, a* and HAMA Takayuki¹, b

¹ Graduate School of Energy Science, Kyoto University, Kyoto 606-8501, Japan

a miyazawa.naoki.3n@kyoto-u.ac.jp, b hama@energy.kyoto-u.ac.jp

**Keywords:** Molecular Dynamics Simulation, Crystal-Plasticity Analysis, Dislocation Strengthening

**Abstract.** Crystal plasticity models are promising numerical analysis methods for predicting and evaluating material forming processes. In crystal plasticity analyses, the dislocation density model, often based on the Bailey-Hirsch equation, is employed to represent the work hardening behavior of metallic materials. The dislocation strengthening coefficient is a proportional factor between the square root of dislocation density with the slip resistance. Therefore, the appropriate determination of the dislocation strengthening coefficient is crucial to perform reliable material forming analyses using crystal plasticity models. Previously, dislocation strengthening coefficients have been determined using dislocation dynamics simulations. However, dislocation dynamics simulations cannot accurately account for elastic anisotropy due to its computational cost. To address this limitation, we conducted molecular dynamics simulations to determine dislocation strengthening coefficient in bcc iron. Molecular dynamics simulations of dislocation-dislocation interaction analysis can be expected to determine the dislocation strengthening coefficient accurately, including the effects of elastic anisotropy of metallic materials.

**Introduction**

Crystal plasticity finite-element methods (CPFEM) are promising numerical tools to predict metallic material forming processes accurately. To accurately predict deformation processes through CPFEM, the selection of a work hardening model and the determination of relevant parameters are crucial. The dislocation density model is a work hardening model based on the Bailey-Hirsch equation, where slip resistance is assumed to be proportional to the square root of dislocation density. When explicitly considering interactions between different slip systems of α and β, the dislocation density model can be written, for example, as follows:

$$ \tau_{C}^{(\alpha)} = \tau_{Y}^{(\alpha)} + \mu|\mathbf{b}|\sqrt{\sum_{\beta} d_{\alpha\beta}\rho_{\beta}}, \quad (1) $$

where $τ_C$, $τ_Y$, $μ$, $b$, $ρ$ and $d_{αβ}$ are slip resistance, reference shear stress, burgers vector, shear modulus, dislocation density and dislocation strengthening coefficient matrix, respectively.

Since experimental determination of $d_{αβ}$ is difficult, dislocation dynamics (DD) simulations have been previously used to determine dislocation strengthening coefficients [1]. DD simulation is a numerical simulation method to model the dynamics of dislocation motion based on the continuum elastic theory. By precisely modeling dislocation network, DD simulation can simulate the work-hardening behavior of a metallic material in a depiction close to real materials. However, accuracy of DD simulation may be questionable because elastic anisotropy is not adequately considered due to its computational cost. For example, when the angles between slip planes are the same, the dislocation strengthening coefficients determined by DD simulation absolutely show the same value, even in the case of different combinations of slip systems. Therefore, simply introducing the dislocation strengthening coefficients obtained from DD into CPFEM does not

<figure><img src="image_929.png" /></figure>
# Molecular dynamics simulation for determining dislocation strengthening coefficient in BCC iron

MIYAZAWA Naoki¹, a* and HAMA Takayuki¹, b

¹ Graduate School of Energy Science, Kyoto University, Kyoto 606-8501, Japan

a miyazawa.naoki.3n@kyoto-u.ac.jp, b hama@energy.kyoto-u.ac.jp

**Keywords:** Molecular Dynamics Simulation, Crystal-Plasticity Analysis, Dislocation Strengthening

**Abstract.** Crystal plasticity models are promising numerical analysis methods for predicting and evaluating material forming processes. In crystal plasticity analyses, the dislocation density model, often based on the Bailey-Hirsch equation, is employed to represent the work hardening behavior of metallic materials. The dislocation strengthening coefficient is a proportional factor between the square root of dislocation density with the slip resistance. Therefore, the appropriate determination of the dislocation strengthening coefficient is crucial to perform reliable material forming analyses using crystal plasticity models. Previously, dislocation strengthening coefficients have been determined using dislocation dynamics simulations. However, dislocation dynamics simulations cannot accurately account for elastic anisotropy due to its computational cost. To address this limitation, we conducted molecular dynamics simulations to determine dislocation strengthening coefficient in bcc iron. Molecular dynamics simulations of dislocation-dislocation interaction analysis can be expected to determine the dislocation strengthening coefficient accurately, including the effects of elastic anisotropy of metallic materials.

**Introduction**

Crystal plasticity finite-element methods (CPFEM) are promising numerical tools to predict metallic material forming processes accurately. To accurately predict deformation processes through CPFEM, the selection of a work hardening model and the determination of relevant parameters are crucial. The dislocation density model is a work hardening model based on the Bailey-Hirsch equation, where slip resistance is assumed to be proportional to the square root of dislocation density. When explicitly considering interactions between different slip systems of α and β, the dislocation density model can be written, for example, as follows:

$$ \tau_C^{(\alpha)} = \tau_Y^{(\alpha)} + \mu|\mathbf{b}|\sqrt{\sum_{\beta} d_{\alpha\beta}\rho_{\beta}}, \quad (1) $$

where $τ_C$, $τ_Y$, $μ$, $b$, $ρ$ and $d_{αβ}$ are slip resistance, reference shear stress, burgers vector, shear modulus, dislocation density and dislocation strengthening coefficient matrix, respectively.

Since experimental determination of $d_{αβ}$ is difficult, dislocation dynamics (DD) simulations have been previously used to determine dislocation strengthening coefficients [1]. DD simulation is a numerical simulation method to model the dynamics of dislocation motion based on the continuum elastic theory. By precisely modeling dislocation network, DD simulation can simulate the work-hardening behavior of a metallic material in a depiction close to real materials. However, accuracy of DD simulation may be questionable because elastic anisotropy is not adequately considered due to its computational cost. For example, when the angles between slip planes are the same, the dislocation strengthening coefficients determined by DD simulation absolutely show the same value, even in the case of different combinations of slip systems. Therefore, simply introducing the dislocation strengthening coefficients obtained from DD into CPFEM does not

<figure><img src="image_936.png" /></figure>
ensure the accurate reproduction of stress-strain curve of real materials, even though they are presumed to have some validity.

Molecular dynamics (MD) simulations numerically solve the equations of motion for individual atoms constituting a material. It can accurately predict the plastic deformation behavior of metallic materials, considering the elastic anisotropy of the material within the limitation of the accuracy of interatomic potential function. Previous MD studies have investigated dislocation-dislocation interactions on fcc [2] and bcc [3] metals. While these studies extensively examined the dislocation-dislocation interactions within a specific combination of slip systems, limited MD researches have been conducted on the differences of slip resistance between various combinations of slip systems. As a result, the dislocation strengthening coefficients determined by MD simulations are largely unknown.

In the present study, MD simulations were conducted to determine the dislocation strengthening coefficients in bcc Fe, including the effect of elastic anisotropy. Dislocation-dislocation interactions were investigated for different combinations of slip systems, including {110}/{110}, {112}/{112} and {112}/{110} as active/forest dislocation interactions. The resulting dislocation reactions due to dislocation interactions were discussed in detail. The dislocation strengthening coefficients were calculated based on the critical stress ratio of dislocation reactions. Results of MD simulations were compared with previous DD simulations to investigate the changes in dislocation strengthening coefficients resulting from the influence of elastic anisotropy.

## Computational Methods

All MD simulations were performed using classical MD code LAMMPS [4] developed in Sandia National Laboratories. A computational cell of a single-crystalline bcc iron is shown in Fig. 1. After considering the limitations of MD simulations regarding the number of atoms they can handle, the simulation cell was set up with a single screw dislocation as the active slip and a single edge dislocation as the forest dislocation. The number of atoms included in the cell was about 6 million. The initial atomic coordinates around dislocations were determined

based on the elastic solution for the displacement field of the dislocation within anisotropic elastic media [5]. The employed interatomic potential was the Embedded-Atom-Method (EAM) potential of Chamati et al. [6], where the potential can accurately predict the Peierls stress in bcc Fe. The MD simulations were performed under constant number of particles, volume and temperature (NVT) ensemble. The timestep of time integration was 1 fs and the temperature was fixed at 5 K throughout simulations. The fixed boundary condition was applied in the z-direction and free boundary condition was applied in the x- and y-directions. Note that the periodic boundary conditions cannot be applied to all directions because of the discontinuity in displacement fields of dislocations within periodic boundary.

Energetically stable atomistic configuration was achieved through energy minimization with a conjugate gradient method and 100 ps-relaxation simulations. To activate the glide of active screw dislocation, shear strain at a constant strain rate (=10⁷ s⁻¹) was applied to the cell by introducing a constant velocity displacement in the x-direction of the upper outermost z-layer. According to the previous MD study [3], below 10⁷ s⁻¹ strain rate condition, the dislocation motion velocity has been reported not to exceed the speed of sound. The dislocation reactions as a result of interactions

<figure><img src="image_930.png" /><figcaption>Fig. 1 Simulation cell used in the present study.</figcaption></figure>
ensure the accurate reproduction of stress-strain curve of real materials, even though they are presumed to have some validity.

Molecular dynamics (MD) simulations numerically solve the equations of motion for individual atoms constituting a material. It can accurately predict the plastic deformation behavior of metallic materials, considering the elastic anisotropy of the material within the limitation of the accuracy of interatomic potential function. Previous MD studies have investigated dislocation-dislocation interactions on fcc [2] and bcc [3] metals. While these studies extensively examined the dislocation-dislocation interactions within a specific combination of slip systems, limited MD researches have been conducted on the differences of slip resistance between various combinations of slip systems. As a result, the dislocation strengthening coefficients determined by MD simulations are largely unknown.

In the present study, MD simulations were conducted to determine the dislocation strengthening coefficients in bcc Fe, including the effect of elastic anisotropy. Dislocation-dislocation interactions were investigated for different combinations of slip systems, including {110}/{110}, {112}/{112} and {112}/{110} as active/forest dislocation interactions. The resulting dislocation reactions due to dislocation interactions were discussed in detail. The dislocation strengthening coefficients were calculated based on the critical stress ratio of dislocation reactions. Results of MD simulations were compared with previous DD simulations to investigate the changes in dislocation strengthening coefficients resulting from the influence of elastic anisotropy.

## Computational Methods

All MD simulations were performed using classical MD code LAMMPS [4] developed in Sandia National Laboratories. A computational cell of a single-crystalline bcc iron is shown in Fig. 1. After considering the limitations of MD simulations regarding the number of atoms they can handle, the simulation cell was set up with a single screw dislocation as the active slip and a single edge dislocation as the forest dislocation. The number of atoms included in the cell was about 6 million. The initial atomic coordinates around dislocations were determined

based on the elastic solution for the displacement field of the dislocation within anisotropic elastic media [5]. The employed interatomic potential was the Embedded-Atom-Method (EAM) potential of Chamati et al. [6], where the potential can accurately predict the Peierls stress in bcc Fe. The MD simulations were performed under constant number of particles, volume and temperature (NVT) ensemble. The timestep of time integration was 1 fs and the temperature was fixed at 5 K throughout simulations. The fixed boundary condition was applied in the z-direction and free boundary condition was applied in the x- and y-directions. Note that the periodic boundary conditions cannot be applied to all directions because of the discontinuity in displacement fields of dislocations within periodic boundary.

Energetically stable atomistic configuration was achieved through energy minimization with a conjugate gradient method and 100 ps-relaxation simulations. To activate the glide of active screw dislocation, shear strain at a constant strain rate (=10⁷ s⁻¹) was applied to the cell by introducing a constant velocity displacement in the x-direction of the upper outermost z-layer. According to the previous MD study [3], below 10⁷ s⁻¹ strain rate condition, the dislocation motion velocity has been reported not to exceed the speed of sound. The dislocation reactions as a result of interactions

<figure><img src="image_937.png" /><figcaption>Fig. 1 Simulation cell used in the present study.</figcaption></figure>
between active screw and forest edge dislocations were investigated, where the combination of active/forest dislocations were {110}/{110}, {112}/{112} and {112}/{110} slip systems, respectively (Table 1). In dislocation analysis, there are two processes: immobile dislocation formation and dislocation annihilation. Among them, this study specifically investigated immobile dislocation reactions. The dislocation strengthening coefficients were calculated as the critical stress ratio of dislocation reactions for different dislocation reactions.

<table><caption>Table 1 The combinations of slip systems considered in the present study.</caption><thead><tr><th></th><th>Active/forest dislocation slip system</th></tr></thead><tbody><tr><td>{110} screw/{110} edge</td><td>[111](0)/[1](110), [111](01)/[1](101),<br>[111](01)/[1](01), [111](01)/[1](110)</td></tr><tr><td>{112} screw/{112} edge</td><td>[111](11)/[1](211), [111](11)/[1](112),<br>[111](11)/[1](12)</td></tr><tr><td>{112} screw/{110} edge</td><td>[111](12)/[111](01), [1](121)/[111](01),<br>[111](1)/[111](01), [1](2)/[111](01),<br>[1](211)/[111](01)</td></tr></tbody></table>

## Results

<figure><img src="image_931.png" /><figcaption>Fig. 2 Stress-strain curves for (a) {110}/{110}, (b) {112}/{112} and (c) {112}/{110} interactions, respectively.</figcaption></figure>

The stress-strain curves for all considered combinations of slip systems in this study are shown in Fig. 2. Unlike the stress-strain curves of real bulk pure-Fe, the curves in the present study did not show a flow curve, but exhibited linear increase followed by reaching peak and subsequent decrease. In addition, the strain until reaching the peak is very small. This is attributed to the limited number of dislocations within the computational cell, and therefore, dislocation interaction frequency was limited in the present study. Variations of critical stresses of dislocation-dislocation interactions were observed depending on different combinations of slip systems. The flow stresses of MD shear simulations were much higher than those of experimental values of pure bcc Fe, because the dislocation density of the present study was high value of about $10^{15}$ m$^{-3}$. The
between active screw and forest edge dislocations were investigated, where the combination of active/forest dislocations were {110}/{110}, {112}/{112} and {112}/{110} slip systems, respectively (Table 1). In dislocation analysis, there are two processes: immobile dislocation formation and dislocation annihilation. Among them, this study specifically investigated immobile dislocation reactions. The dislocation strengthening coefficients were calculated as the critical stress ratio of dislocation reactions for different dislocation reactions.

<table><caption>Table 1 The combinations of slip systems considered in the present study.</caption><thead><tr><th></th><th>Active/forest dislocation slip system</th></tr></thead><tbody><tr><td>{110} screw/{110} edge</td><td>[111](0)/[1](110), [111](01)/[1](101),<br>[111](01)/[1](01), [111](01)/[1](110)</td></tr><tr><td>{112} screw/{112} edge</td><td>[111](11)/[1](211), [111](11)/[1](112),<br>[111](11)/[1](12)</td></tr><tr><td>{112} screw/{110} edge</td><td>[111](12)/[111](01), [1](121)/[111](01),<br>[111](1)/[111](01), [1](2)/[111](01),<br>[1](211)/[111](01)</td></tr></tbody></table>

## Results

<figure><img src="image_938.png" /><figcaption>Fig. 2 Stress-strain curves for (a) {110}/{110}, (b) {112}/{112} and (c) {112}/{110} interactions, respectively.</figcaption></figure>

The stress-strain curves for all considered combinations of slip systems in this study are shown in Fig. 2. Unlike the stress-strain curves of real bulk pure-Fe, the curves in the present study did not show a flow curve, but exhibited linear increase followed by reaching peak and subsequent decrease. In addition, the strain until reaching the peak is very small. This is attributed to the limited number of dislocations within the computational cell, and therefore, dislocation interaction frequency was limited in the present study. Variations of critical stresses of dislocation-dislocation interactions were observed depending on different combinations of slip systems. The flow stresses of MD shear simulations were much higher than those of experimental values of pure bcc Fe, because the dislocation density of the present study was high value of about $10^{15}$ m$^{-3}$. The
{110}/{110} interactions (Fig.2a) tend to exhibit higher critical stresses compared to {112}/{112} (Fig.2b) and {112}/{110} interactions (Fig.2c).

Fig.3 shows the changes in dislocation configuration during a dislocation reaction. When shear strain was applied to the computational cell, the active screw dislocation approached to the forest edge dislocation (Fig.3a) and they underwent crossover event (Fig.3b). The active screw dislocation cross-slipped and a[001] immobile dislocation was formed (Fig.3c). The corresponding dislocation reaction can be written as follows:

<figure><img src="image_932.png" /><figcaption>Fig. 3 Dislocation configuration change during dislocation reaction when applied shear strain $γ_{zy}$ is (a) 0.0, (b) $5.0×10^{-3}$, (c) $8.0×10^{-3}$ and (d) $9.0×10^{-3}$, respectively. (e) Comparison of dislocation configuration with stress-strain curve.</figcaption></figure>

$$a[111] + a[1] \rightarrow a[001]. \qquad (2)$$

The immobile dislocation a[001] acts as a strong obstacle to active slip, resulting in work hardening from a macroscopic perspective. By continuous cross-slip of active screw dislocations, the positions of formed immobile dislocations shifted within the cell as a result of the repeated process of localized detachment from and formation of immobile dislocations. As the shear strain increased, the active screw dislocation completely escaped from the immobile dislocation and the forest edge dislocation was left within the simulation box (Fig.3d).

Fig.3e shows the comparison of the changes in dislocation configurations and corresponding stress-strain curve. As seen from the figure, the critical stress of the dislocation reaction was achieved when just before the active screw dislocation escaped from the immobile dislocation. Therefore, in this study, the critical stress can be mainly governed by the formation and detachment stress of immobile dislocations. In all combinations of slip systems, the dislocation reactions themselves were the same as Eq.2, but the length and the line directions of the formed immobile dislocations were different depending on combinations of slip systems.
{110}/{110} interactions (Fig.2a) tend to exhibit higher critical stresses compared to {112}/{112} (Fig.2b) and {112}/{110} interactions (Fig.2c).

Fig.3 shows the changes in dislocation configuration during a dislocation reaction. When shear strain was applied to the computational cell, the active screw dislocation approached to the forest edge dislocation (Fig.3a) and they underwent crossover event (Fig.3b). The active screw dislocation cross-slipped and a[001] immobile dislocation was formed (Fig.3c). The corresponding dislocation reaction can be written as follows:

<figure><img src="image_939.png" /><figcaption>Fig. 3 Dislocation configuration change during dislocation reaction when applied shear strain $\gamma_{zy}$ is (a) 0.0, (b) $5.0 \times 10^{-3}$, (c) $8.0 \times 10^{-3}$ and (d) $9.0 \times 10^{-3}$, respectively. (e) Comparison of dislocation configuration with stress-strain curve.</figcaption></figure>

$$a[111] + a[1] \rightarrow a[001]. \qquad (2)$$

The immobile dislocation a[001] acts as a strong obstacle to active slip, resulting in work hardening from a macroscopic perspective. By continuous cross-slip of active screw dislocations, the positions of formed immobile dislocations shifted within the cell as a result of the repeated process of localized detachment from and formation of immobile dislocations. As the shear strain increased, the active screw dislocation completely escaped from the immobile dislocation and the forest edge dislocation was left within the simulation box (Fig.3d).

Fig.3e shows the comparison of the changes in dislocation configurations and corresponding stress-strain curve. As seen from the figure, the critical stress of the dislocation reaction was achieved when just before the active screw dislocation escaped from the immobile dislocation. Therefore, in this study, the critical stress can be mainly governed by the formation and detachment stress of immobile dislocations. In all combinations of slip systems, the dislocation reactions themselves were the same as Eq.2, but the length and the line directions of the formed immobile dislocations were different depending on combinations of slip systems.
Discussion

<figure><img src="image_933.png" /><figcaption>Fig. 4 Dislocation strengthening coefficients calculated by MD and their comparison with previous DD simulations for (a) {110}/{110}, (b) {112}/{112} and (c) {112}/{110} interactions, respectively. The values are compared with [111](01)/[11](110) interaction.</figcaption></figure>

Fig.4 shows dislocation strengthening coefficients obtained from MD (present study) and previous DD simulations. For {110}/{110} interactions (Fig. 4a), the coefficients obtained from MD simulations were slightly smaller than DD results, but they almost showed the same values. This suggests the potential reliability of MD simulations in predicting dislocation strengthening coefficients. As discussed in results section, the critical stresses in MD simulations were mainly determined by formation and detachment stress of immobile dislocations. Thus, the dislocation strengthening coefficient can be predominantly determined by the strength of an immobile dislocation as an obstacle. The slight differences observed between the MD and DD results are attributable to the utilization of large dislocation density (=about 10¹⁵m⁻³) and high strain rate (=10⁷s⁻¹) in the current MD simulations. However, these effects were expected to be nearly canceled since dislocation strengthening coefficients are determined based on the ratio of critical stresses. For {112}/{112} interactions (Fig. 4b), MD and DD results also showed similar values, but the magnitudes of the differences were larger compared to the {110}/{110} interactions. In contrast, for {112}/{110} interactions (Fig. 4c), there were some variations in the values of MD and DD simulations.

The reason for the discrepancies in results between the MD and DD simulations when considering {112} slip systems can be their treatment of elastic anisotropy. In the DD simulation, the dislocation motion is numerically analyzed under the assumption of elastic isotropy. Therefore, shear on {110} and {112} shows the same tendency. In the MD simulation, however, elastic response on {110} and {112} shear is different, where shear modulus μ of {112} system is about 13% smaller than {110} system in the present MD simulation. In classical elastic continuum theory, the orientation-dependent dislocation line energy per unit length E can be written as follows [7]:

$$E = \frac{\mu b^2 (1 - \nu \cos^2 \beta_{dis})}{4\pi(1-\nu)} \ln(\frac{R}{b}), \quad (3)$$

where the angle βdis defines the geometrical character of the dislocation segment. In this equation, materials properties are represented by shear modulus μ and Poisson's ratio ν, indicating that the difference of μ is crucial to determine critical stress of dislocation interactions.

Note that the present study just focused on calculating the dislocation strengthening coefficients and comparing the MD results to DD results. The MD results obtained in the present study does
Discussion

<figure><img src="image_940.png" /><figcaption>Fig. 4 Dislocation strengthening coefficients calculated by MD and their comparison with previous DD simulations for (a) {110}/{110}, (b) {112}/{112} and (c) {112}/{110} interactions, respectively. The values are compared with [111](01)/[11](110) interaction.</figcaption></figure>

Fig.4 shows dislocation strengthening coefficients obtained from MD (present study) and previous DD simulations. For {110}/{110} interactions (Fig. 4a), the coefficients obtained from MD simulations were slightly smaller than DD results, but they almost showed the same values. This suggests the potential reliability of MD simulations in predicting dislocation strengthening coefficients. As discussed in results section, the critical stresses in MD simulations were mainly determined by formation and detachment stress of immobile dislocations. Thus, the dislocation strengthening coefficient can be predominantly determined by the strength of an immobile dislocation as an obstacle. The slight differences observed between the MD and DD results are attributable to the utilization of large dislocation density (=about 10¹⁵m⁻³) and high strain rate (=10⁷s⁻¹) in the current MD simulations. However, these effects were expected to be nearly canceled since dislocation strengthening coefficients are determined based on the ratio of critical stresses. For {112}/{112} interactions (Fig. 4b), MD and DD results also showed similar values, but the magnitudes of the differences were larger compared to the {110}/{110} interactions. In contrast, for {112}/{110} interactions (Fig. 4c), there were some variations in the values of MD and DD simulations.

The reason for the discrepancies in results between the MD and DD simulations when considering {112} slip systems can be their treatment of elastic anisotropy. In the DD simulation, the dislocation motion is numerically analyzed under the assumption of elastic isotropy. Therefore, shear on {110} and {112} shows the same tendency. In the MD simulation, however, elastic response on {110} and {112} shear is different, where shear modulus μ of {112} system is about 13% smaller than {110} system in the present MD simulation. In classical elastic continuum theory, the orientation-dependent dislocation line energy per unit length E can be written as follows [7]:

$$E = \frac{\mu b^2 (1 - \nu \cos^2 \beta_{dis})}{4\pi(1-\nu)} \ln\left(\frac{R}{b}\right), \quad (3)$$

where the angle βdis defines the geometrical character of the dislocation segment. In this equation, materials properties are represented by shear modulus μ and Poisson's ratio ν, indicating that the difference of μ is crucial to determine critical stress of dislocation interactions.

Note that the present study just focused on calculating the dislocation strengthening coefficients and comparing the MD results to DD results. The MD results obtained in the present study does
not immediately enhance the accuracy of stress-strain curve prediction in real materials. However, our finding, significant differences between MD and DD in the coefficients of {112}/{110} interactions due to elastic anisotropy is meaningful. To understand the effect of elastic anisotropy on work hardening of materials more deeply, macroscopic materials deformation analysis such as CPFEM is necessary. The comparison with real materials and the quantitative validation of MD results are future work.

## Summary

In summary, MD simulations were performed to determine the dislocation strengthening coefficients in bcc Fe. The interaction between active screw dislocation and forest edge dislocation was investigated using MD shear test simulations. A immobile dislocation a[001] was formed as a result of dislocation reaction with the activation of cross-slip of the screw dislocation. The peak stress in stress-strain curve was obtained just before the active dislocation escaped from the formed immobile dislocation. The critical stresses in the dislocation reactions varied depending on the combinations of slip systems between active and forest dislocations. For {110}/{110} interactions, the calculated dislocation strengthening coefficients showed values slightly smaller but closely aligned with the previous DD values. Thus, MD simulations can be reliable to determine dislocation strengthening coefficients. Similarly, MD and DD values exhibited close values, but the differences were larger in {112}/{112} than in {110}/{110} interactions. In contrast, for {112}/{110} interactions, some variations in the quantitative magnitude of dislocation strengthening coefficients was observed between MD and DD simulations. The differences in values, particularly in cases involving {112} slip systems, are attributable to the treatment of elastic anisotropy in both computational methods because the stress required for dislocation reaction is largely dependent on the shear modulus. In MD simulation, anisotropic elasticity is considered within the range of interatomic potential reliability, while isotropic elasticity is considered in DD simulation due to its computational cost. The results of the MD simulations in the present study have the potential to enhance the prediction accuracy of work hardening in CPFEM simulation through the consideration of elastic anisotropy.

## References

[1] R. Madec and L. P. Kubin, Dislocation strengthening in FCC metals and in BCC metals at high temperatures, Acta Mater. 126 (2017) 166-173. https://doi.org/10.1016/j.actamat.2016.12.040

[2] S. J. Zhou, D. L. Preston, P. S. Lomdahl and D. M. Beazley, Large-scale molecular dynamics simulations of dislocation interaction in copper, Science 279 (1998) 1525-1527. https://doi.org/10.1126/science.279.5356.1525

[3] S. M. Hafez Haghighat, R. Schaublin and D. Raabe, Atomistic simulation of the $a_0 < 100$> binary junction formation and its unzipping in body-centered cubic iron, Acta Mater. 64 (2014) 24-32. https://doi.org/10.1016/j.actamat.2013.11.037

[4] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Compt. Phys. 117 (1995) 1-19. https://doi.org/10.1006/jcph.1995.1039

[5] D. J. Bacon, D. M. Barrett and R. O. Scattergood, Anisotropic continuum theory of lattice defects, Prog. Mater. Sci. 23 (1979) 51-262. https://doi.org/10.1016/0079-6425(80)90007-9

[6] H. Chamati, N. I. Papanicolaou, Y. Mishin and D. A. Papaconstantopolous, Embedded-atom potential for Fe and its application to self-diffusion on Fe(100), Surf. Sci. 600 (2006) 1793-1803. https://doi.org/10.1016/j.susc.2006.02.010

[7] G. de Wit and J.S. Koehler, Interactions of dislocations with an applied stress in anisotropic crystals, Phys. Rev. B, 116 (1959) 1113-1120. https://doi.org/10.1103/PhysRev.116.1113
not immediately enhance the accuracy of stress-strain curve prediction in real materials. However, our finding, significant differences between MD and DD in the coefficients of {112}/{110} interactions due to elastic anisotropy is meaningful. To understand the effect of elastic anisotropy on work hardening of materials more deeply, macroscopic materials deformation analysis such as CPFEM is necessary. The comparison with real materials and the quantitative validation of MD results are future work.

## Summary

In summary, MD simulations were performed to determine the dislocation strengthening coefficients in bcc Fe. The interaction between active screw dislocation and forest edge dislocation was investigated using MD shear test simulations. A immobile dislocation a[001] was formed as a result of dislocation reaction with the activation of cross-slip of the screw dislocation. The peak stress in stress-strain curve was obtained just before the active dislocation escaped from the formed immobile dislocation. The critical stresses in the dislocation reactions varied depending on the combinations of slip systems between active and forest dislocations. For {110}/{110} interactions, the calculated dislocation strengthening coefficients showed values slightly smaller but closely aligned with the previous DD values. Thus, MD simulations can be reliable to determine dislocation strengthening coefficients. Similarly, MD and DD values exhibited close values, but the differences were larger in {112}/{112} than in {110}/{110} interactions. In contrast, for {112}/{110} interactions, some variations in the quantitative magnitude of dislocation strengthening coefficients was observed between MD and DD simulations. The differences in values, particularly in cases involving {112} slip systems, are attributable to the treatment of elastic anisotropy in both computational methods because the stress required for dislocation reaction is largely dependent on the shear modulus. In MD simulation, anisotropic elasticity is considered within the range of interatomic potential reliability, while isotropic elasticity is considered in DD simulation due to its computational cost. The results of the MD simulations in the present study have the potential to enhance the prediction accuracy of work hardening in CPFEM simulation through the consideration of elastic anisotropy.

## References

[1] R. Madec and L. P. Kubin, Dislocation strengthening in FCC metals and in BCC metals at high temperatures, Acta Mater. 126 (2017) 166-173. https://doi.org/10.1016/j.actamat.2016.12.040

[2] S. J. Zhou, D. L. Preston, P. S. Lomdahl and D. M. Beazley, Large-scale molecular dynamics simulations of dislocation interaction in copper, Science 279 (1998) 1525-1527. https://doi.org/10.1126/science.279.5356.1525

[3] S. M. Hafez Haghighat, R. Schaublin and D. Raabe, Atomistic simulation of the $a_0 < 100$> binary junction formation and its unzipping in body-centered cubic iron, Acta Mater. 64 (2014) 24-32. https://doi.org/10.1016/j.actamat.2013.11.037

[4] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Compt. Phys. 117 (1995) 1-19. https://doi.org/10.1006/jcph.1995.1039

[5] D. J. Bacon, D. M. Barrett and R. O. Scattergood, Anisotropic continuum theory of lattice defects, Prog. Mater. Sci. 23 (1979) 51-262. https://doi.org/10.1016/0079-6425(80)90007-9

[6] H. Chamati, N. I. Papanicolaou, Y. Mishin and D. A. Papaconstantopolous, Embedded-atom potential for Fe and its application to self-diffusion on Fe(100), Surf. Sci. 600 (2006) 1793-1803. https://doi.org/10.1016/j.susc.2006.02.010

[7] G. de Wit and J.S. Koehler, Interactions of dislocations with an applied stress in anisotropic crystals, Phys. Rev. B, 116 (1959) 1113-1120. https://doi.org/10.1103/PhysRev.116.1113
# Formability analyses of AA6016-T4 aluminum alloy sheets subjected to roping

TVEIT Sigbjørn¹,ª and REYES Aase¹,ᵇ*

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

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

**Keywords:** Aluminum Alloys, FEA Based FLD, Roping

**Abstract.** Due to its good formability characteristics, AA6xxx aluminum alloys have become popular as a light-weight alternative to steel for car bodywork components produced by stamping. In the design of forming operations, numerical simulations are often used, and accurate material models to describe plasticity and forming limits are important to successfully design optimized products and production processes to avoid the reliance on trial and error. For aluminum alloys used in sheet metal forming, the anisotropic plasticity characteristics caused by the directional rolling process has therefore been subject to extensive research over the last decades. A previous experimental program on AA6016-T4 sheets showed that the formability of the material is strongly affected by roping caused by rolling. In this paper, models of anisotropic plasticity, isotropic hardening, instability, and fracture, along with an approach to describe the effects of roping for proportional strain paths are used to predict forming and fracture limits in LS-DYNA.

**Introduction**

The forming limit diagram (FLD) displays which strain combinations lead to strain localizations and which do not. Similarly, it is possible to construct fracture limit curves, which shows the strain combinations at fracture. In the traditional FLD, proportional straining is assumed through the full deformation process, but the influence of non-proportional straining can be studied for two-stage strain paths where formability tests are preceded by pre-strain. Graf and Hosford [1, 2] found that a biaxial pre-strain generally lowers the forming limits, while pre-straining by uniaxial tension can increase the forming limits. However, after pre-straining, the amount of possible additional plane-strain deformation depends on the effective strain during the pre-straining regardless of strain-path. In a previous study [3, 4], formability experiments were carried out for as-received and pre-strained sheets to investigate the effects of pre-strain by rolling on the formability of AA6016 sheets in the biaxial stretching region. In these tests, a significant anisotropy in forming limit strains were uncovered, and attributed to the roping phenomenon.

Roping (also called ridging) manifests itself as a directional surface defect that can appear in many aluminum alloys, copper alloys, and stainless steels at small strains [5-10]. It can be characterized by visible lines of up to hundreds of millimeters long along the rolling direction (RD), and a few millimeters apart in the transverse direction (TD) with a height difference between ridges and valleys of about 10-30 µm when the sheet is tensioned in TD [6, 8]. It impedes the use of these alloys in visible parts of e.g. vehicles due to aesthetics, but can also affect the formability [8] (as it can serve as a stress concentration) and bending properties of the materials [5]. Early studies analyzed the development of surface roughness and found that the amount of surface roughness is proportional to the through-thickness strain, and it has a linear dependence on the grain size [5].

The origin of roping in the aluminum alloy has been believed to intimately connect to texture distribution [6], and therefore, several studies are concentrated on the effect of the microstructure on roping [8, 9]. Jin studied how to reduce roping with the break-up of texture alignment [11].

<figure><img src="image_934.png" /></figure>
# Formability analyses of AA6016-T4 aluminum alloy sheets subjected to roping

TVEIT Sigbjørn¹,ª and REYES Aase¹,ᵇ*

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

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

**Keywords:** Aluminum Alloys, FEA Based FLD, Roping

**Abstract.** Due to its good formability characteristics, AA6xxx aluminum alloys have become popular as a light-weight alternative to steel for car bodywork components produced by stamping. In the design of forming operations, numerical simulations are often used, and accurate material models to describe plasticity and forming limits are important to successfully design optimized products and production processes to avoid the reliance on trial and error. For aluminum alloys used in sheet metal forming, the anisotropic plasticity characteristics caused by the directional rolling process has therefore been subject to extensive research over the last decades. A previous experimental program on AA6016-T4 sheets showed that the formability of the material is strongly affected by roping caused by rolling. In this paper, models of anisotropic plasticity, isotropic hardening, instability, and fracture, along with an approach to describe the effects of roping for proportional strain paths are used to predict forming and fracture limits in LS-DYNA.

**Introduction**

The forming limit diagram (FLD) displays which strain combinations lead to strain localizations and which do not. Similarly, it is possible to construct fracture limit curves, which shows the strain combinations at fracture. In the traditional FLD, proportional straining is assumed through the full deformation process, but the influence of non-proportional straining can be studied for two-stage strain paths where formability tests are preceded by pre-strain. Graf and Hosford [1, 2] found that a biaxial pre-strain generally lowers the forming limits, while pre-straining by uniaxial tension can increase the forming limits. However, after pre-straining, the amount of possible additional strain deformation depends on the effective strain during the pre-straining regardless of strain-path. In a previous study [3, 4], formability experiments were carried out for as-received and pre-strained sheets to investigate the effects of pre-strain by rolling on the formability of AA6016 sheets in the biaxial stretching region. In these tests, a significant anisotropy in forming limit strains were uncovered, and attributed to the roping phenomenon.

Roping (also called ridging) manifests itself as a directional surface defect that can appear in many aluminum alloys, copper alloys, and stainless steels at small strains [5-10]. It can be characterized by visible lines of up to hundreds of millimeters long along the rolling direction (RD), and a few millimeters apart in the transverse direction (TD) with a height difference between ridges and valleys of about 10-30 µm when the sheet is tensioned in TD [6, 8]. It impedes the use of these alloys in visible parts of e.g. vehicles due to aesthetics, but can also affect the formability [8] (as it can serve as a stress concentration) and bending properties of the materials [5]. Early studies analyzed the development of surface roughness and found that the amount of surface roughness is proportional to the through-thickness strain, and it has a linear dependence on the grain size [5].

The origin of roping in the aluminum alloy has been believed to intimately connect to texture distribution [6], and therefore, several studies are concentrated on the effect of the microstructure on roping [8, 9]. Jin studied how to reduce roping with the break-up of texture alignment [11].

<figure><img src="image_941.png" /></figure>
More recently, some authors have studied the effects further. Muhammad et al. [5] studied the effects of roping (among others) on the bendability of aluminum alloys. They used crystal plasticity finite element model (CPFEM) and synthetic crystallographic textures to further investigate the relationship between through thickness clustering of similarly oriented grains, microstructure, and the observed surface topographic behavior. Hu et al. [6] introduced a roping index in the evaluation of the surface morphology and textures of four AA6XXX aluminum alloy sheets subjected to uniaxial tension. The same authors continued their investigation and applied CPFEM and developed an artificial neural network-based (ANN) model to predict thickness strain and corresponding surface roping under tensile deformation [7].

While several attempts to understand and describe the occurrence of the roping phenomenon in aluminum sheets based on crystal plasticity, less attention has been directed towards modelling the phenomenological effects of roping on forming limits and crack orientations. In this paper, based on the work of a master thesis [12], as a simple engineering approach, numerical forming and fracture limit diagrams are generated. Here, the roping is modeled by introducing perturbations to the sheet thickness based on the spatial distribution of effective plastic strains observed from experiments, and anisotropic plasticity, isotropic hardening, instability, and fracture are included in the finite element simulations in LS-DYNA.

## Material and experimental details

The material assessed in this paper is AA6016 aluminum alloy, from virgin sheets of 1.5 mm thickness. Due to its good forming properties the rolled alloy is suited for use in the bodywork of cars. The metal sheets, produced by Hydro Aluminum Rolled Products GmbH, were rolled to their nominal thickness and solution heat treated before they were stored for six months natural aging process, giving them the temper designation T4. After heat treatment and natural ageing, the pre-strain was applied by cold rolling. All information regarding the characteristics of the sheets can be found in [4]. To characterize the material's anisotropic behavior, uniaxial tensile tests were carried out in seven different directions, and disc compression and plane strain tests were also performed. In the same study [4], the formability of the material was investigated by means of Marciniak tests [13], that were performed in a BUP600 forming machine using a flat punch with a diameter of 100 mm. The width of the samples was varied between 155 and 205 mm to attain different strain paths. Because of the material's anticipated anisotropic behavior, both samples oriented parallel and normal to the rolling direction were employed. In total, seven different strain paths were tested. After the experiments, one could clearly see some sort of irregularities on the surface of the specimens. This rough surface appeared like small ridges, all oriented in the same direction (Fig. 1) and could be identified as the roping phenomenon. The forming limit strains experienced a greater anisotropy than expected, an effect attributed to the severe roping [3].

## Material models and determination of material parameters

In the present work, the numerical forming and fracture limit diagrams were generated by means of LS-DYNA [14], where *MAT_135, also known as *MAT_WTM_STM [15] was employed. This is an elasto-viscoplastic constitutive model with an anisotropic yield criterion YLD2003 [16], the associated flow rule, non-linear isotropic work hardening rule (Voce), as well as a through-thickness instability criterion (TTSIC), and the Original Cockroft Latham Fracture Criterion (OCLFC). A Non-Local Extremal Thickness Strain Deviation Criterion (NLETSDC) was established to detect the onset of local necking from the available history variables.

**Yield criterion:** The Yld2003 yield criterion was proposed by Aretz [16] and contains eight anisotropy parameters that can be fitted to experimental data. The mathematical form can be found in the original work. There, the eight parameters are found from flow-stress ratios, $r_\theta$, and transverse strain ratios, $R_\theta$, from uniaxial tensile tests with longitudinal direction inclined at $\theta =$
More recently, some authors have studied the effects further. Muhammad et al. [5] studied the effects of roping (among others) on the bendability of aluminum alloys. They used crystal plasticity finite element model (CPFEM) and synthetic crystallographic textures to further investigate the relationship between through thickness clustering of similarly oriented grains, microstructure, and the observed surface topographic behavior. Hu et al. [6] introduced a roping index in the evaluation of the surface morphology and textures of four AA6XXX aluminum alloy sheets subjected to uniaxial tension. The same authors continued their investigation and applied CPFEM and developed an artificial neural network-based (ANN) model to predict thickness strain and corresponding surface roping under tensile deformation [7].

While several attempts to understand and describe the occurrence of the roping phenomenon in aluminum sheets based on crystal plasticity, less attention has been directed towards modelling the phenomenological effects of roping on forming limits and crack orientations. In this paper, based on the work of a master thesis [12], as a simple engineering approach, numerical forming and fracture limit diagrams are generated. Here, the roping is modeled by introducing perturbations to the sheet thickness based on the spatial distribution of effective plastic strains observed from experiments, and anisotropic plasticity, isotropic hardening, instability, and fracture are included in the finite element simulations in LS-DYNA.

## Material and experimental details

The material assessed in this paper is AA6016 aluminum alloy, from virgin sheets of 1.5 mm thickness. Due to its good forming properties the rolled alloy is suited for use in the bodywork of cars. The metal sheets, produced by Hydro Aluminum Rolled Products GmbH, were rolled to their nominal thickness and solution heat treated before they were stored for six months natural aging process, giving them the temper designation T4. After heat treatment and natural ageing, the pre-strain was applied by cold rolling. All information regarding the characteristics of the sheets can be found in [4]. To characterize the material's anisotropic behavior, uniaxial tensile tests were carried out in seven different directions, and disc compression and plane strain tests were also performed. In the same study [4], the formability of the material was investigated by means of Marciniak tests [13], that were performed in a BUP600 forming machine using a flat punch with a diameter of 100 mm. The width of the samples was varied between 155 and 205 mm to attain different strain paths. Because of the material's anticipated anisotropic behavior, both samples oriented parallel and normal to the rolling direction were employed. In total, seven different strain paths were tested. After the experiments, one could clearly see some sort of irregularities on the surface of the specimens. This rough surface appeared like small ridges, all oriented in the same direction (Fig. 1) and could be identified as the roping phenomenon. The forming limit strains experienced a greater anisotropy than expected, an effect attributed to the severe roping [3].

## Material models and determination of material parameters

In the present work, the numerical forming and fracture limit diagrams were generated by means of LS-DYNA [14], where *MAT_135, also known as *MAT_WTM_STM [15] was employed. This is an elasto-viscoplastic constitutive model with an anisotropic yield criterion YLD2003 [16], the associated flow rule, non-linear isotropic work hardening rule (Voce), as well as a through-thickness instability criterion (TTSIC), and the Original Cockroft Latham Fracture Criterion (OCLFC). A Non-Local Extremal Thickness Strain Deviation Criterion (NLETSDC) was established to detect the onset of local necking from the available history variables.

**Yield criterion:** The Yld2003 yield criterion was proposed by Aretz [16] and contains eight anisotropy parameters that can be fitted to experimental data. The mathematical form can be found in the original work. There, the eight parameters are found from flow-stress ratios, $r_\theta$, and transverse strain ratios, $R_\theta$, from uniaxial tensile tests with longitudinal direction inclined at $\theta =$
0°, 45° and 90° to the sheet's rolling direction and the corresponding quantities taken from the equibiaxial flow stress and transverse strain ratios, $r_b$ and $R_b$, respectively. LS-DYNA offers a direct calibration from these experimental data points, and these were found from the tensile and compression tests in [4], except for $r_b$ which was taken as unity due to lack of experimental data. The eight data points used in the numerical analyses are given in Table 1, in addition to $m$, which is the non-quadratic yield function exponent.

**Strain hardening:** The strength of the material is defined by the flow stress, where the isotropic strain hardening is expressed by the extended Voce rule.

$$ \sigma_e = \sigma_0 + Q_{R1}[1 - \exp(-c_{R1}\varepsilon_e^p)] + Q_{R2}[1 - \exp(-c_{R2}\varepsilon_e^p)] \quad (1) $$

Here, $\varepsilon_e^p$ is the effective plastic strain, and $Q_{Ri}$ and $c_{Ri}$ are material constants. The equation was fitted to the experimental data from [4], and are given in Table 1.

**The Non-Local Extremal Thickness Strains Deviation Criterion (NLETSDC):** A simple non-local method was established to detect the onset of local necking from the available history variables in LS-DYNA. Initial geometric imperfections in terms of thickness variations are spatially distributed in the model, resulting in variable shell thicknesses among the different elements $i$. Within an area $\Omega_i$ around each element, the minimum and the maximum values of through-thickness plastic strains $\varepsilon_z^p$ were monitored. From the relationship of the two quantities, a history variable was established as

$$ \xi_i = \frac{\varepsilon_z^{p, \Omega_{min}}}{\varepsilon_z^{p, \Omega_{max}}} \quad (2) $$

The onset of local necking was assumed to occur when the history variable $\xi_i$ reaches a critical value, $\xi_{cr}$. Different critical values were tested and are shown in Table 1.

**The Through-Thickness Shear Instability Criterion (TTSIC):** This criterion was first introduced by Bressan and Williams [17]. The following equation to calculate the inclination, $\varphi$, from the critical plane based on plastic strain increments is used. Here, the established coordinate system of $x$, $y$, and $z$ is respectively aligned with the rolling, transverse, and through-thickness directions:

$$ \cos 2\varphi = - \frac{d\varepsilon_y^p}{d\varepsilon_y^p + 2d\varepsilon_x^p} \quad (3) $$

TTSIC is satisfied when the shear stress on the critical plane, $\tau_\varphi$, reaches a critical value, $\tau_{cr}$.

$$ \tau_\varphi = \frac{\sin(2\varphi)}{2} \sigma_1 \ge \tau_{cr} \quad (4) $$

The through-thickness shear instability criterion was calibrated in a one-step inverse modelling procedure in an LS-DYNA analysis of a single element with dimensions 1.2 mm by 1.2 mm, and the data from the plane strain tests (PST) from [4]. Note that it was not possible to achieve perfect
0°, 45° and 90° to the sheet's rolling direction and the corresponding quantities taken from the equibiaxial flow stress and transverse strain ratios, $r_b$ and $R_b$, respectively. LS-DYNA offers a direct calibration from these experimental data points, and these were found from the tensile and compression tests in [4], except for $r_b$ which was taken as unity due to lack of experimental data. The eight data points used in the numerical analyses are given in Table 1, in addition to $m$, which is the non-quadratic yield function exponent.

**Strain hardening:** The strength of the material is defined by the flow stress, where the isotropic strain hardening is expressed by the extended Voce rule.

$$ \sigma_e = \sigma_0 + Q_{R1}[1 - \exp(-c_{R1}\varepsilon_e^p)] + Q_{R2}[1 - \exp(-c_{R2}\varepsilon_e^p)] \quad (1) $$

Here, $\varepsilon_e^p$ is the effective plastic strain, and $Q_{Ri}$ and $c_{Ri}$ are material constants. The equation was fitted to the experimental data from [4], and are given in Table 1.

**The Non-Local Extremal Thickness Strains Deviation Criterion (NLETSDC):** A simple non-local method was established to detect the onset of local necking from the available history variables in LS-DYNA. Initial geometric imperfections in terms of thickness variations are spatially distributed in the model, resulting in variable shell thicknesses among the different elements $i$. Within an area $\Omega_i$ around each element, the minimum and the maximum values of through-thickness plastic strains $\varepsilon_z^p$ were monitored. From the relationship of the two quantities, a history variable was established as

$$ \xi_i = \frac{\varepsilon_z^{p, \Omega_{min}}}{\varepsilon_z^{p, \Omega_{max}}} \quad (2) $$

The onset of local necking was assumed to occur when the history variable $\xi_i$ reaches a critical value, $\xi_{cr}$. Different critical values were tested and are shown in Table 1.

**The Through-Thickness Shear Instability Criterion (TTSIC):** This criterion was first introduced by Bressan and Williams [17]. The following equation to calculate the inclination, $\varphi$, from the critical plane based on plastic strain increments is used. Here, the established coordinate system of $x$, $y$, and $z$ is respectively aligned with the rolling, transverse, and through-thickness directions:

$$ \cos 2\varphi = - \frac{d\varepsilon_y^p}{d\varepsilon_y^p + 2d\varepsilon_x^p} \quad (3) $$

TTSIC is satisfied when the shear stress on the critical plane, $\tau_\varphi$, reaches a critical value, $\tau_{cr}$.

$$ \tau_\varphi = \frac{\sin(2\varphi)}{2} \sigma_1 \ge \tau_{cr} \quad (4) $$

The through-thickness shear instability criterion was calibrated in a one-step inverse modelling procedure in an LS-DYNA analysis of a single element with dimensions 1.2 mm by 1.2 mm, and the data from the plane strain tests (PST) from [4]. Note that it was not possible to achieve perfect
<figure><img src="image_935.png" /><figcaption>Fig. 1 Roping and multiple local necks observed at the bottom side of a Marciniak sample. Figure rendered from [4] with permission from the author.</figcaption></figure>

<table><caption>Table 1 Input to *MAT_135, taken or identified from the experimental data in [4].</caption><tbody><tr><td rowspan="6">Strain hardening</td><td rowspan="6">Two-term Voce</td><td>σ<sub>0</sub></td><td>[MPa]</td><td>139.0</td></tr><tr><td>Q<sub>R1</sub></td><td>[MPa]</td><td>44.26</td></tr><tr><td>Q<sub>R2</sub></td><td>[MPa]</td><td>171.9</td></tr><tr><td>C<sub>R1</sub></td><td>[-]</td><td>24.20</td></tr><tr><td>C<sub>R2</sub></td><td>[-]</td><td>7.324</td></tr><tr><td>r<sub>0</sub></td><td>[-]</td><td>1.000</td></tr><tr><td rowspan="10">Yield criterion</td><td rowspan="10">Yld2003</td><td>r<sub>45</sub></td><td>[-]</td><td>0.9986</td></tr><tr><td>r<sub>90</sub></td><td>[-]</td><td>0.9910</td></tr><tr><td>r<sub>b</sub></td><td>[-]</td><td>1.000</td></tr><tr><td>R<sub>0</sub></td><td>[-]</td><td>0.6880</td></tr><tr><td>R<sub>45</sub></td><td>[-]</td><td>0.4250</td></tr><tr><td>R<sub>90</sub></td><td>[-]</td><td>1.030</td></tr><tr><td>R<sub>b</sub></td><td>[-]</td><td>0.7850</td></tr><tr><td>m</td><td>[-]</td><td>8</td></tr><tr><td rowspan="2">Instability criteria</td><td>NLETSDC</td><td>ξ<sub>cr</sub></td><td>[-]</td><td>1.5, 2.0, 3.0</td></tr><tr><td>TTSIC</td><td>τ<sub>cr</sub></td><td>[MPa]</td><td>179.8</td></tr><tr><td>Fracture criteria</td><td>OCLFC</td><td>W<sub>cr</sub></td><td>[MPa]</td><td>211.5</td></tr></tbody></table>

plane strain in the tests [4], however, the strains in the x- and y- direction were measured by Digital Image Correlation (DIC) and the data can therefore be used for calibration. As the major strain direction in the plane strain tests (PST) is in the sheet's rolling direction, the principal axis of stress coincides with the principal axis of anisotropy. Thus, the angle of the inclination of the maximum shear plane, $\varphi$, can be calculated for an isotropic material. With fracture strains for the tests at $\bar{\varepsilon}_{x,PST}^* = 0.6237$ and $\bar{\varepsilon}_{y,PST}^* = -0.1323$, the inclination of the maximum shear plane, $\varphi = 48.41^{\circ}$. A non-local proportional strain corresponding to the average strain path of the six PST tests from [4] was applied to the element, and when the FE analysis reached the average fracture strains from the experimental PST tests, the value of the major stress component $\sigma_1^* = \sigma_x^*$ was recorded. From this, the critical shear stress, $\tau_{cr}$, was found as 179.8 MPa, and is given in Table 1.

**The Original Cockcroft-Latham fracture criterion (OCLFC):** This ductile fracture criterion is assumed to be satisfied when the accumulation of principal tensile stresses over the plastic deformation reaches a critical value, and is formulated as [18]
<figure><img src="image_942.png" /><figcaption>Fig. 1 Roping and multiple local necks observed at the bottom side of a Marciniak sample. Figure rendered from [4] with permission from the author.</figcaption></figure>

<table><caption>Table 1 Input to *MAT_135, taken or identified from the experimental data in [4].</caption><tbody><tr><td rowspan="6">Strain hardening</td><td rowspan="6">Two-term Voce</td><td>σ<sub>0</sub></td><td>[MPa]</td><td>139.0</td></tr><tr><td>Q<sub>R1</sub></td><td>[MPa]</td><td>44.26</td></tr><tr><td>Q<sub>R2</sub></td><td>[MPa]</td><td>171.9</td></tr><tr><td>C<sub>R1</sub></td><td>[-]</td><td>24.20</td></tr><tr><td>C<sub>R2</sub></td><td>[-]</td><td>7.324</td></tr><tr><td>r<sub>0</sub></td><td>[-]</td><td>1.000</td></tr><tr><td rowspan="10">Yield criterion</td><td rowspan="10">Yld2003</td><td>r<sub>45</sub></td><td>[-]</td><td>0.9986</td></tr><tr><td>r<sub>90</sub></td><td>[-]</td><td>0.9910</td></tr><tr><td>r<sub>b</sub></td><td>[-]</td><td>1.000</td></tr><tr><td>R<sub>0</sub></td><td>[-]</td><td>0.6880</td></tr><tr><td>R<sub>45</sub></td><td>[-]</td><td>0.4250</td></tr><tr><td>R<sub>90</sub></td><td>[-]</td><td>1.030</td></tr><tr><td>R<sub>b</sub></td><td>[-]</td><td>0.7850</td></tr><tr><td>m</td><td>[-]</td><td>8</td></tr><tr><td rowspan="2">Instability criteria</td><td>NLETSDC</td><td>ξ<sub>cr</sub></td><td>[-]</td><td>1.5, 2.0, 3.0</td></tr><tr><td>TTSIC</td><td>τ<sub>cr</sub></td><td>[MPa]</td><td>179.8</td></tr><tr><td>Fracture criteria</td><td>OCLFC</td><td>W<sub>cr</sub></td><td>[MPa]</td><td>211.5</td></tr></tbody></table>

plane strain in the tests [4], however, the strains in the x- and y- direction were measured by Digital Image Correlation (DIC) and the data can therefore be used for calibration. As the major strain direction in the plane strain tests (PST) is in the sheet's rolling direction, the principal axis of stress coincides with the principal axis of anisotropy. Thus, the angle of the inclination of the maximum shear plane, $\phi$, can be calculated for an isotropic material. With fracture strains for the tests at $\bar{\varepsilon}_{x,PST}^* = 0.6237$ and $\bar{\varepsilon}_{y,PST}^* = -0.1323$, the inclination of the maximum shear plane, $\phi = 48.41^{\circ}$. A non-local proportional strain corresponding to the average strain path of the six PST tests from [4] was applied to the element, and when the FE analysis reached the average fracture strains from the experimental PST tests, the value of the major stress component $\sigma_1^* = \sigma_x^*$ was recorded. From this, the critical shear stress, $\tau_{cr}$, was found as 179.8 MPa, and is given in Table 1.

**The Original Cockcroft-Latham fracture criterion (OCLFC):** This ductile fracture criterion is assumed to be satisfied when the accumulation of principal tensile stresses over the plastic deformation reaches a critical value, and is formulated as [18]
$$W_1 = \int_{0}^{\overline{\varepsilon^p}} \max(\sigma_1, 0) d\overline{\varepsilon^p} \geq W_{cr} \quad (5)$$

where $\overline{\varepsilon^p}$ is the accumulated plastic strain, and $W_{cr}$ is a material constant which can be adjusted to observations from experiments where strain localization is controlled. In [4], two different plane strain tension tests specimen geometries with three parallels each were used to calculate the OCLFC. Averaging over the path, the critical plastic work was found as 211.5 MPa [4], and is also given in Table 1.

## Modeling of roping

Aiming to model the effects of roping as scaled harmonic perturbations to the sheet thickness, a sum of four sine waves can be fitted to a sheet's thickness profile obtained from measurements. As the real thickness profile was not available from [4], a fictive thickness profile based on the mean thickness and standard deviations of the sheets sampled in the experimental program was used.

In LS-DYNA, a pre-programmed module to model perturbations to the thickness of shells as one or more harmonic waves exists in the keyword *PERTURBATION. The keyword allows the user to specify the amplitude, wavelength, and periodic offset of waves though the variables AMPL, WL, and OFF, respectively. Thickness perturbations due to roping can be modelled to satisfying accuracy by adding multiple harmonic perturbations to approximate a measured thickness distribution with the formulation [15]:

$$p(x, y) = \sum_{i=1}^{n} \left( AMPL_i \left[ \sin \left( 2\pi \frac{x+XOFF_i}{XWL_i} \right) + \sin \left( 2\pi \frac{y+YOFF_i}{YWL_i} \right) \right] \right) \quad (6)$$

As roping is reported to be the manifestation of the orientation of grain's spatial distribution in the sheets' through-thickness direction [19], the observed effects in terms of thickness perturbations are accompanied with variation in material characteristics. Thus, naive of information about this variation, the modelled thickness perturbation could be scaled to obtain good correlation between the anisotropy predicted by the FE-models and the experimental results. In the MK model [13], the fundamental assumption is that a band of the sheet is somehow weaker than the rest. This weakness could be material variations or thickness variations. Aretz [20] compared the use of thickness and material imperfections to trigger localized necking in the MK model, and concluded that the two approaches are equivalent for proportional strain paths. Due to simplicity, imperfections are often thought of and modelled as a reduction in the thickness of the material. The magnitude of the imperfection is typically calibrated to match experimental observations. In the same way, a larger thickness variation than what is physically observed can be justified to account for reduced stiffness from spatial variations in material characteristics for proportional strain paths.

Limited information was available from the experimental tests about the nature of the roping phenomenon of the aluminum sheets. Thus, the distance between valleys formed by roping was approximated by visual inspections on the effective strain map from DIC for the last image prior to fracture [4]. Using the sample which most clearly showed the localizations of strains, namely the sample labelled as MK160-90-2, (see Fig. 2 (a)), the approximated distance between roping valleys was roughly measured to be 12.5 mm. With the reported non-local logarithmic fracture strain at 0.2870 for the sample, the initial distance was calculated to be approximately 8.9 mm.
$$W_1 = \int_{0}^{\overline{\varepsilon^p}} \max(\sigma_1, 0) d\overline{\varepsilon^p} \geq W_{cr} \quad (5)$$

where $\overline{\varepsilon^p}$ is the accumulated plastic strain, and $W_{cr}$ is a material constant which can be adjusted to observations from experiments where strain localization is controlled. In [4], two different plane strain tension tests specimen geometries with three parallels each were used to calculate the OCLFC. Averaging over the path, the critical plastic work was found as 211.5 MPa [4], and is also given in Table 1.

## Modeling of roping

Aiming to model the effects of roping as scaled harmonic perturbations to the sheet thickness, a sum of four sine waves can be fitted to a sheet's thickness profile obtained from measurements. As the real thickness profile was not available from [4], a fictive thickness profile based on the mean thickness and standard deviations of the sheets sampled in the experimental program was used.

In LS-DYNA, a pre-programmed module to model perturbations to the thickness of shells as one or more harmonic waves exists in the keyword *PERTURBATION. The keyword allows the user to specify the amplitude, wavelength, and periodic offset of waves though the variables AMPL, WL, and OFF, respectively. Thickness perturbations due to roping can be modelled to satisfying accuracy by adding multiple harmonic perturbations to approximate a measured thickness distribution with the formulation [15]:

$$p(x, y) = \sum_{i=1}^{n} \left( AMPL_i \left[ \sin \left( 2\pi \frac{x+XOFF_i}{XWL_i} \right) + \sin \left( 2\pi \frac{y+YOFF_i}{YWL_i} \right) \right] \right) \quad (6)$$

As roping is reported to be the manifestation of the orientation of grain's spatial distribution in the sheets' through-thickness direction [19], the observed effects in terms of thickness perturbations are accompanied with variation in material characteristics. Thus, naive of information about this variation, the modelled thickness perturbation could be scaled to obtain good correlation between the anisotropy predicted by the FE-models and the experimental results. In the MK model [13], the fundamental assumption is that a band of the sheet is somehow weaker than the rest. This weakness could be material variations or thickness variations. Aretz [20] compared the use of thickness and material imperfections to trigger localized necking in the MK model, and concluded that the two approaches are equivalent for proportional strain paths. Due to simplicity, imperfections are often thought of and modelled as a reduction in the thickness of the material. The magnitude of the imperfection is typically calibrated to match experimental observations. In the same way, a larger thickness variation than what is physically observed can be justified to account for reduced stiffness from spatial variations in material characteristics for proportional strain paths.

Limited information was available from the experimental tests about the nature of the roping phenomenon of the aluminum sheets. Thus, the distance between valleys formed by roping was approximated by visual inspections on the effective strain map from DIC for the last image prior to fracture [4]. Using the sample which most clearly showed the localizations of strains, namely the sample labelled as MK160-90-2, (see Fig. 2 (a)), the approximated distance between roping valleys was roughly measured to be 12.5 mm. With the reported non-local logarithmic fracture strain at 0.2870 for the sample, the initial distance was calculated to be approximately 8.9 mm.
<figure><img src="image_936.png" /><figcaption>Fig. 2 (a) Effective strain map at the last frame prior to fracture of the sample labelled as MK160-90-2 of virgin AA6016-T4 (ID=6), rendered from [4] with permission from the author. (b) Thickness perturbation along the length in the transverse direction.</figcaption></figure>

<table><caption>Table 2 Fitted parameters of the four-term sine wave roping perturbation.</caption><thead><tr><th></th><th>AMPL<sub>i</sub>[mm]</th><th>YWL<sub>i</sub>[mm]</th><th>YOFF<sub>i</sub>[mm]</th></tr></thead><tbody><tr><td>i = 1</td><td>0.00168</td><td>4.462</td><td>3.382</td></tr><tr><td>i = 2</td><td>0.00130</td><td>2.976</td><td>-0.692</td></tr><tr><td>i = 3</td><td>0.00436</td><td>8.907</td><td>-2.267</td></tr><tr><td>i = 4</td><td>-5.2E-6</td><td>8.825</td><td>-5.549</td></tr></tbody></table>

From this estimated distance between grooves, along with the mean measured thickness of 1.496 mm and standard deviation of 0.004 mm from thickness measurements performed in the experimental study, a fictive thickness profile was established- The parameters of a four-term sine wave roping perturbation model was fitted to it, using the Sum of Sines Models-tool in MATLAB R2020a [20]. Fig. 2 (b) shows the fictive perturbation measurements along with the fitted four-term sine wave model, while Table 2 displays the amplitude, *AMPL*<sub>*i*</sub>, wavelength, *YWL*<sub>*i*</sub>, and offset, *YOFF*<sub>*i*</sub>, for each of the four harmonic functions, *i*.

With roping modelled as a four-term harmonic function in the transverse direction, no thickness variations are introduced in the sheet's rolling direction. In a perturbation plot generated from white-light interferometry images by Engler et al. [8], the valleys and ridges were accompanied by unstructured thickness deviations in the rolling directions. Attempting to simulate this, a FORTRAN-program which generated random unstructured thicknesses perturbations based on a mean thickness and a standard deviation, was used. As the random perturbation is added to the existing four-term sine wave perturbation, a rather moderate standard deviation of 0.002 mm was used along with the mean thickness of 1.496 mm. A scaling factor of six was used for the four-term sine wave perturbation. Fig. 3 (a) shows the thickness perturbation of the undeformed patch.

## Formability and Fracture Diagrams with roping model

To construct the FEA-generated forming and fracture limit diagrams in LS-DYNA, a 60 mm by 60 mm squared patch of 50 by 50 Belytschko-Tsay shell elements was subjected to 15 different proportional biaxial deformations. The four individual edges were continuously constrained according to the illustration in Fig. 3 (b). A controlled displacement was then assigned to the right and the top edge in their normal directions to create the non-local, proportional, biaxial strain paths.
<figure><img src="image_943.png" /><figcaption>Fig. 2 (a) Effective strain map at the last frame prior to fracture of the sample labelled as MK160-90-2 of virgin AA6016-T4 (ID=6), rendered from [4] with permission from the author. (b) Thickness perturbation along the length in the transverse direction.</figcaption></figure>

<table><caption>Table 2 Fitted parameters of the four-term sine wave roping perturbation.</caption><thead><tr><th></th><th>AMPL<sub>i</sub> [mm]</th><th>YWL<sub>i</sub> [mm]</th><th>YOFF<sub>i</sub> [mm]</th></tr></thead><tbody><tr><td>i = 1</td><td>0.00168</td><td>4.462</td><td>3.382</td></tr><tr><td>i = 2</td><td>0.00130</td><td>2.976</td><td>-0.692</td></tr><tr><td>i = 3</td><td>0.00436</td><td>8.907</td><td>-2.267</td></tr><tr><td>i = 4</td><td>-5.2E-6</td><td>8.825</td><td>-5.549</td></tr></tbody></table>

From this estimated distance between grooves, along with the mean measured thickness of 1.496 mm and standard deviation of 0.004 mm from thickness measurements performed in the experimental study, a fictive thickness profile was established- The parameters of a four-term sine wave roping perturbation model was fitted to it, using the Sum of Sines Models-tool in MATLAB R2020a [20]. Fig. 2 (b) shows the fictive perturbation measurements along with the fitted four-term sine wave model, while Table 2 displays the amplitude, *AMPL*<sub>*i*</sub>, wavelength, *YWL*<sub>*i*</sub>, and offset, *YOFF*<sub>*i*</sub>, for each of the four harmonic functions, *i*.

With roping modelled as a four-term harmonic function in the transverse direction, no thickness variations are introduced in the sheet's rolling direction. In a perturbation plot generated from white-light interferometry images by Engler et al. [8], the valleys and ridges were accompanied by unstructured thickness deviations in the rolling directions. Attempting to simulate this, a FORTRAN-program which generated random unstructured thicknesses perturbations based on a mean thickness and a standard deviation, was used. As the random perturbation is added to the existing four-term sine wave perturbation, a rather moderate standard deviation of 0.002 mm was used along with the mean thickness of 1.496 mm. A scaling factor of six was used for the four-term sine wave perturbation. Fig. 3 (a) shows the thickness perturbation of the undeformed patch.

## Formability and Fracture Diagrams with roping model

To construct the FEA-generated forming and fracture limit diagrams in LS-DYNA, a 60 mm by 60 mm squared patch of 50 by 50 Belytschko-Tsay shell elements was subjected to 15 different proportional biaxial deformations. The four individual edges were continuously constrained according to the illustration in Fig. 3 (b). A controlled displacement was then assigned to the right and the top edge in their normal directions to create the non-local, proportional, biaxial strain paths.
Along with the Yld2003 yield function, the NLETSDC, TTSIC, and OCLFC were used to predict local necking and/or fracture. Through the material keyword *MAT_135, TTSIC and OCLFC was employed as uncoupled criteria where satisfaction of either criterion in three out of the five through-thickness integration points in an element caused element erosion. As the

<figure><img src="image_937.png" /><figcaption>Fig. 3 (a) Thickness variation of the FEA-model with four-term sine wave perturbation (Sc. = 6) and random perturbation with SD = 0.002 mm. (b) Boundary conditions of patch model subjected to biaxial strains paths for the construction of FEA-generated forming and fracture limit diagrams. The rolling supports should be considered continuous along the edges.</figcaption></figure>

<figure><img src="image_938.png" /><figcaption>Fig. 4 FLDs from NLETSDC for (a) different critical extremal thickness strain ratio and (b) with roping model scaled by a factor of six, with and without additional random thickness perturbation. The points are the experimental necking (crosses) and fracture (stars) strains from [4].</figcaption></figure>

NLETSDC was not implemented in the material model, fulfillment of the criterion was manually monitored. When either criterion was satisfied, the displacements in the two directions were recorded, and the non-local strain values were calculated as the necking or fracture strains.
Along with the Yld2003 yield function, the NLETSDC, TTSIC, and OCLFC were used to predict local necking and/or fracture. Through the material keyword *MAT_135, TTSIC and OCLFC was employed as uncoupled criteria where satisfaction of either criterion in three out of the five through-thickness integration points in an element caused element erosion. As the

<figure><img src="image_944.png" /><figcaption>Fig. 3 (a) Thickness variation of the FEA-model with four-term sine wave perturbation (Sc. = 6) and random perturbation with SD = 0.002 mm. (b) Boundary conditions of patch model subjected to biaxial strains paths for the construction of FEA-generated forming and fracture limit diagrams. The rolling supports should be considered continuous along the edges.</figcaption></figure>

<figure><img src="image_945.png" /><figcaption>Fig. 4 FLDs from NLETSDC for (a) different critical extremal thickness strain ratio and (b) with roping model scaled by a factor of six, with and without additional random thickness perturbation. The points are the experimental necking (crosses) and fracture (stars) strains from [4].</figcaption></figure>

NLETSDC was not implemented in the material model, fulfillment of the criterion was manually monitored. When either criterion was satisfied, the displacements in the two directions were recorded, and the non-local strain values were calculated as the necking or fracture strains.
To test NLETSDC, a sensitivity analysis of the critical extremal thickness strain ratio, $\xi_{cr}$ was carried out. On the model with unscaled roping perturbations and no random perturbation in the rolling direction, forming limit curves were plotted with the critical values set to 1.5, 2.0 and 3.0. The different curves, displayed in Fig. 4 (a), shows that variations in the critical value yields small differences in forming limits. For the further analyses, $\xi_{cr} = 2.0$ was used as the critical value. With the four-term sine wave perturbation model unscaled (Sc. = 1.0), the forming limit diagram exhibited moderate anisotropy. The two plane strain configurations exhibited forming limits close to the measured values but were clearly overestimated as the strains approached biaxial symmetry.

Because roping is the manifestation of band clusters of grains with the same orientation, the material itself could be assumed to have varying material characteristics that are spatially distributed according to the sheet thickness. Thus, lacking information about this distribution of plasticity characteristics, a scaling of the four-wave sine perturbation was performed to simulate further reduced stiffness, to meet the strong anisotropy of forming limits observed in experiments for portional strain paths. The scaling was performed by multiplying the amplitude of each curve in the four-term sine wave model with a scalar (Sc. = 2.0 and 6.0) to amplify the effect of roping. From Fig. 5 (a), which displays the FLDs predicted by NLETSDC, it is evident that the effect of amplifying the roping perturbation model yielded significant reduction in forming limit strains in the sheet's transverse direction. With a scaling factor Sc. = 6, the predicted and measured results coincided very well for configurations where $\epsilon_{TD} > \epsilon_{RD}$. Nevertheless, the forming limit strains were still exaggerated by the FEA-model for its symmetric counterpart.

The FLDs with the roping model, with and without the random perturbation with standard deviation of 0.002 mm, are compared in Fig. 4 (b). When the harmonic perturbation was accompanied with the random perturbation, the forming limits for strain paths when $\epsilon_{TD} > \epsilon_{RD}$

<figure><img src="image_939.png" /><figcaption>Fig. 5 Forming and fracture limit curves with and without the roping model, for different scales of the four-term sine wave perturbation for the (a) NLETSDC, (b) TTSIC, and (c) OCLFC. The points represent the experimental necking (crosses) and fracture (stars) strains from [4].</figcaption></figure>

remained nearly unchanged, while its symmetric counterpart predicted drastically reduced necking strains, just on the conservative side of the experimentally detected forming limit strains.

Fig. 5 (b) and (c) displays the forming and fracture limit diagram generated for TTSIC and OCLFC, respectively, with different scaling factors to the harmonic perturbations. Evidently, the model with perfectly uniform thickness was not able to produce the shape nor the magnitude of
To test NLETSDC, a sensitivity analysis of the critical extremal thickness strain ratio, $\xi_{cr}$ was carried out. On the model with unscaled roping perturbations and no random perturbation in the rolling direction, forming limit curves were plotted with the critical values set to 1.5, 2.0 and 3.0. The different curves, displayed in Fig. 4 (a), shows that variations in the critical value yields small differences in forming limits. For the further analyses, $\xi_{cr} = 2.0$ was used as the critical value. With the four-term sine wave perturbation model unscaled (Sc. = 1.0), the forming limit diagram exhibited moderate anisotropy. The two plane strain configurations exhibited forming limits close to the measured values but were clearly overestimated as the strains approached biaxial symmetry.

Because roping is the manifestation of band clusters of grains with the same orientation, the material itself could be assumed to have varying material characteristics that are spatially distributed according to the sheet thickness. Thus, lacking information about this distribution of plasticity characteristics, a scaling of the four-wave sine perturbation was performed to simulate further reduced stiffness, to meet the strong anisotropy of forming limits observed in experiments for portional strain paths. The scaling was performed by multiplying the amplitude of each curve in the four-term sine wave model with a scalar (Sc. = 2.0 and 6.0) to amplify the effect of roping. From Fig. 5 (a), which displays the FLDs predicted by NLETSDC, it is evident that the effect of amplifying the roping perturbation model yielded significant reduction in forming limit strains in the sheet's transverse direction. With a scaling factor Sc. = 6, the predicted and measured results coincided very well for configurations where $\epsilon_{TD} > \epsilon_{RD}$. Nevertheless, the forming limit strains were still exaggerated by the FEA-model for its symmetric counterpart.

The FLDs with the roping model, with and without the random perturbation with standard deviation of 0.002 mm, are compared in Fig. 4 (b). When the harmonic perturbation was accompanied with the random perturbation, the forming limits for strain paths when $\epsilon_{TD} > \epsilon_{RD}$

<figure><img src="image_946.png" /><figcaption>Fig. 5 Forming and fracture limit curves with and without the roping model, for different scales of the four-term sine wave perturbation for the (a) NLETSDC, (b) TTSIC, and (c) OCLFC. The points represent the experimental necking (crosses) and fracture (stars) strains from [4].</figcaption></figure>

remained nearly unchanged, while its symmetric counterpart predicted drastically reduced necking strains, just on the conservative side of the experimentally detected forming limit strains.

Fig. 5 (b) and (c) displays the forming and fracture limit diagram generated for TTSIC and OCLFC, respectively, with different scaling factors to the harmonic perturbations. Evidently, the model with perfectly uniform thickness was not able to produce the shape nor the magnitude of
forming and fracture limits from experiments for neither criteria. However, with *Sc.* = 6, the TTSIC criterion can be said to predict conservative results compared to the experimental fracture strains, and the shape of the predicted fracture curves seems reasonable. For the OCLFC, *Sc.* = 2 was sufficient to obtain a reasonable shape for the fracture limit diagram and predict conservative results. Larger scale factors were therefore not tested for OCLFC. The magnitudes of the predicted fracture strains were clearly underestimated for strain paths in vicinity to the two plane-strain axes, while strain paths near the equibiaxial axis displayed better agreement with the experimental data. Although the introduction of thickness perturbations helped, the OCLFC was not able to recreate the shape of the fracture strain curves in a satisfying way.

## Conclusions

With the introduction of the harmonic thickness perturbations, together with the anisotropic yield criterion Yld2003, and the isotropic two-term Voce hardening law, experimental forming limits were recreated with good accuracy for proportional strain paths using one local necking instability criterion and two different fracture criteria. Without the introduction of the harmonic thickness perturbations, all three criteria produced forming and fracture limits very different to what was observed in experiments. The best fit to experiments was achieved when the four-term sine wave perturbation, which was initially fitted to the amplitudes of measured thickness deviations, was scaled with a factor of six.

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