was noted at large rotation angles, a phenomenon that warrants further investigation in future studies.

**Acknowledgement**

The authors gratefully acknowledge the support from the Research Fund for Coal and Steel under grant agreement No 888153 (EU-RFCS 2019 project No. 888153 | vForm-xSteels).

**Disclaimer**

The results reflect only the authors' view, and the European Commission is not responsible for any use that may be made of the information it contains.

**References**

[1] Raabe D, Textures of strip cast and hot rolled ferritic and austenitic stainless steel. Mater. Sci. Tech. Ser. 11 (1995) 461–468. https://doi.org/10.1179/026708395790165192

[2] Barsom J, Korvink S, Through-thickness properties of structural steels, J. Struct. Eng. (1998) 727–735. https://doi.org/10.1061/(ASCE)0733-9445(1998)124:7(727)

[3] Sohn SS, Han SY, Bae J-H, Kim HS, Lee S, Effects of microstructure and pipe forming strain on yield strength before and after spiral pipe forming of api x70 and x80 linepipe steel sheets. Mat. Sci. Eng. A 573 (2013) 18–26. https://doi.org/10.1016/j.msea.2013.02.050

[4] Zhang H, Coppieters S, Jiménez-Peña C, Debruyne D, Inverse identification of the post-necking work hardening behaviour of thick HSS through full-field strain measurements during diffuse necking. Mechanics of Materials, 2019, 129 Pages 361-374. https://doi.org/10.1016/j.mechmat.2018.12.014

[5] Coppieters S, Zhang Y, Vancraeynest N, Lambrughi A, Cooreman S, Inverse Identification of a 3D Anisotropic Yield Function Through an Information-Rich Tensile Test and Multi-sDIC. Proceedings of the 14th International Conference on the Technology of Plasticity - Current Trends in the Technology of Plasticity. ICTP 2023. https://doi.org/10.1007/978-3-031-42093-1_66

[6] Cholewa N, Summers PT, Feih S, Mouritz AP, Lattimer BY, Case SW, A Technique for Coupled Thermomechanical Response Measurement Using Infrared Thermography and Digital Image Correlation (TDIC), Experimental Mechanics, Volume 56, 2016, Pages 145-164. https://doi.org/10.1007/s11340-015-0086-1

[7] Louche H, Analyse par thermographie infrarouge des effets dissipatifs de la localisation dans des aciers, thèse de doctorat, Université Montpellier II - Sciences et Techniques du Languedoc, 1999.

[8] Vancraeynest N, Cooreman S, Coppieters S, Identification of the large strain flow curve of high strength steel via the torsion test and FEMU. In: Material Forming – ESAFORM 2023: vol. 28, 1167-1174. https://doi.org/10.21741/9781644902479-127
# Measurement of shear deformation behavior in thickness direction for a mild steel sheet

HAKOYAMA Tomoyuki¹,ᵃ*, HAKOYAMA Chiharu²,ᵇ and FURUSATO Daichi³,ᶜ

¹Tokai National Higher Education and Research System, Gifu University, 1-1 Yanagido, Gifu, 501-1193, Japan

²Chubu University, 1200 Matsumoto-cho, Kasugai-shi, Aichi, 487-8501, Japan

³Graduate student, Tokai National Higher Education and Research System, Gifu University, 1-1 Yanagido, Gifu, 501-1193, Japan

ᵃhakoyama.tomoyuki.s7@f.gifu-u.ac.jp, ᵇc_hakoyama@isc.chubu.ac.jp,

ᶜfurusato.daichi.u8@s.gifu-u.ac.jp

**Keywords:** Material Modeling, Plate Forging, Shear Deformation

**Abstract.** The anisotropy parameter of the through-thickness shear for a hot-rolled mild steel sheet was determined by comparing the measured and simulated deformation behaviors. A tensile test using a strip specimen with a stepped shape in the thickness direction to apply the through-thickness shear deformation at the center of the specimen was conducted. Finite element analyses were performed by changing the anisotropy parameter *M* of the Hill '48 yield function. The measured through-thickness shear strain γₓₓ was compared with the simulated one to identify the parameter *M*. In addition, plate compression simulations were performed to evaluate the effect of the through-thickness shear anisotropy on the deformation behavior during plate compression.

**Introduction**

Plate forging (sheet-bulk metal forming), which is a hybrid process of sheet metal forming and forging, is used to manufacture parts with complex shapes. In plate forging, the same accuracy as that of machining is required [1]. The use of finite element analysis is crucial for realizing trial-and-error-less manufacturing in plate forging.

One of the authors [2] investigated the deformation behavior during a plate compression test with a large-diameter circular plate relative to its thickness. The measured thickness distribution and change in diameter during plate compression are in qualitative agreement with the simulated values. On the other hand, there are some quantitative discrepancies between the experiment and simulation results. Improving the representation accuracy of material models is vital for enhancing the precision of plasticity simulations [3].

A crucial deformation mode in the forging process is shear deformation. Since the surface of the workpiece is constrained by friction between the workpiece and the tool, large through-thickness shear deformation occurs during plate forging. Fig. 1 shows the simulated through-thickness shear strain distribution during plate compression. The simulation conditions for Fig. 1 are the same as those in the reference [2]. The constraint by friction results in a maximum shear strain of approximately 0.15 in the through-thickness direction on the material surface near the plate edge when the plate is compressed by approximately 10%.

Lattanzi et al. [4] measured the strain distribution during the shearing process using digital image correlation (DIC) and identified a through-thickness shear anisotropy parameter of the Hill '48 yield function [5] based on finite element model updating (FEMU). In the identification using shearing processing, since the effect of the constraint at the material edge on the strain distribution is significant, a long test specimen is required. Then the shear strain attained is relatively small.

<figure><img src="image_1095.png" /></figure>
# Measurement of shear deformation behavior in thickness direction for a mild steel sheet

HAKOYAMA Tomoyuki¹,ᵃ*, HAKOYAMA Chiharu²,ᵇ and FURUSATO Daichi³,ᶜ

¹Tokai National Higher Education and Research System, Gifu University, 1-1 Yanagido, Gifu, 501-1193, Japan

²Chubu University, 1200 Matsumoto-cho, Kasugai-shi, Aichi, 487-8501, Japan

³Graduate student, Tokai National Higher Education and Research System, Gifu University, 1-1 Yanagido, Gifu, 501-1193, Japan

ᵃhakoyama.tomoyuki.s7@f.gifu-u.ac.jp, ᵇc_hakoyama@isc.chubu.ac.jp,

ᶜfurusato.daichi.u8@s.gifu-u.ac.jp

**Keywords:** Material Modeling, Plate Forging, Shear Deformation

**Abstract.** The anisotropy parameter of the through-thickness shear for a hot-rolled mild steel sheet was determined by comparing the measured and simulated deformation behaviors. A tensile test using a strip specimen with a stepped shape in the thickness direction to apply the through-thickness shear deformation at the center of the specimen was conducted. Finite element analyses were performed by changing the anisotropy parameter *M* of the Hill '48 yield function. The measured through-thickness shear strain γzx was compared with the simulated one to identify the parameter *M*. In addition, plate compression simulations were performed to evaluate the effect of the through-thickness shear anisotropy on the deformation behavior during plate compression.

**Introduction**

Plate forging (sheet-bulk metal forming), which is a hybrid process of sheet metal forming and forging, is used to manufacture parts with complex shapes. In plate forging, the same accuracy as that of machining is required [1]. The use of finite element analysis is crucial for realizing trial-and-error-less manufacturing in plate forging.

One of the authors [2] investigated the deformation behavior during a plate compression test with a large-diameter circular plate relative to its thickness. The measured thickness distribution and change in diameter during plate compression are in qualitative agreement with the simulated values. On the other hand, there are some quantitative discrepancies between the experiment and simulation results. Improving the representation accuracy of material models is vital for enhancing the precision of plasticity simulations [3].

A crucial deformation mode in the forging process is shear deformation. Since the surface of the workpiece is constrained by friction between the workpiece and the tool, large through-thickness shear deformation occurs during plate forging. Fig. 1 shows the simulated through-thickness shear strain distribution during plate compression. The simulation conditions for Fig. 1 are the same as those in the reference [2]. The constraint by friction results in a maximum shear strain of approximately 0.15 in the through-thickness direction on the material surface near the plate edge when the plate is compressed by approximately 10%.

Lattanzi et al. [4] measured the strain distribution during the shearing process using digital image correlation (DIC) and identified a through-thickness shear anisotropy parameter of the Hill '48 yield function [5] based on finite element model updating (FEMU). In the identification using shearing processing, since the effect of the constraint at the material edge on the strain distribution is significant, a long test specimen is required. Then the shear strain attained is relatively small.

<figure><img src="image_1097.png" /></figure>
In this study a through-thickness shear anisotropy parameter of the Hill '48 yield function was identified using a testing method with the tensile test type specimen. A tensile test using a strip specimen with a stepped shape in the thickness direction to apply the through-thickness shear deformation at the center of the specimen was conducted. Finite element analyses (FEAs) of the tensile test were performed by changing the anisotropy parameter *M* of the Hill '48 yield function. The variation of the measured through-thickness shear strain γ<sub>zx</sub> during tensile test was compared with the simulated one to identify the parameter *M*. In addition, plate compression simulations using the best parameter of *M* were performed to evaluate the influence of the through-thickness shear anisotropy on the deformation behavior of plate compression.

<figure><img src="image_1096.png" /><figcaption>Figure 1 Through-thickness shear strain distribution during the plate compression test.</figcaption></figure>

Experimental conditions

Test material. The material used in this study was a commercial hot-rolled mild steel sheet SPHC (nominal thickness *t*₀ = 2.3 mm). In order to investigate the work hardening behavior of this material, a uniaxial tensile test was conducted. The testing apparatus used was the Autograph AG-50kN X plus (Shimadzu corporation). A strain gage was attached to the center of the test specimen to measure the longitudinal strain. The stress–strain curve measured by the tensile test is shown in Fig. 2. Swift’s work hardening law was approximated from the testing result (eq. (1)).

Hereafter, the rolling direction is referred to as *x*, the transverse direction as *y*, and the thickness direction as *z*.

$$
\sigma = \begin{cases} 383 \text{ MPa} & \text{for } \varepsilon_0^p < 0.025 \\ 721(\varepsilon_0^p + 0.0007)^{0.17} \text{ MPa} & \text{for } \varepsilon_0^p \ge 0.025 \end{cases} \quad (1)
$$

where ε₀p is the reference plastic strain.
In this study a through-thickness shear anisotropy parameter of the Hill '48 yield function was identified using a testing method with the tensile test type specimen. A tensile test using a strip specimen with a stepped shape in the thickness direction to apply the through-thickness shear deformation at the center of the specimen was conducted. Finite element analyses (FEAs) of the tensile test were performed by changing the anisotropy parameter *M* of the Hill '48 yield function. The variation of the measured through-thickness shear strain γ<sub>zx</sub> during tensile test was compared with the simulated one to identify the parameter *M*. In addition, plate compression simulations using the best parameter of *M* were performed to evaluate the influence of the through-thickness shear anisotropy on the deformation behavior of plate compression.

<figure><img src="image_1098.png" /><figcaption>Figure 1 Through-thickness shear strain distribution during the plate compression test.</figcaption></figure>

Experimental conditions

Test material. The material used in this study was a commercial hot-rolled mild steel sheet SPHC (nominal thickness *t*₀ = 2.3 mm). In order to investigate the work hardening behavior of this material, a uniaxial tensile test was conducted. The testing apparatus used was the Autograph AG-50kN X plus (Shimadzu corporation). A strain gage was attached to the center of the test specimen to measure the longitudinal strain. The stress–strain curve measured by the tensile test is shown in Fig. 2. Swift’s work hardening law was approximated from the testing result (eq. (1)).

Hereafter, the rolling direction is referred to as *x*, the transverse direction as *y*, and the thickness direction as *z*.

$$
\sigma = \begin{cases} 383 \text{ MPa} & \text{for } \varepsilon_0^p < 0.025 \\ 721(\varepsilon_0^p + 0.0007)^{0.17} \text{ MPa} & \text{for } \varepsilon_0^p \ge 0.025 \end{cases} \quad (1)
$$

where ε₀p is the reference plastic strain.
<figure><img src="image_1097.png" /><figcaption>Figure 2 True stress-logarithmic plastic strain curves measured and approximated.</figcaption></figure>

Specimen for the measurement of shear deformation behavior. The shape of the specimen used in this study is shown in Fig. 3. By applying tensile force to both ends of the specimen, shear deformation can be probed in the centoral zone of the specimen. Based on the FEA results, the dimensions of the specimen were optimized to achieve the largest possible shear strain within the manufacturing constraints (L = 50 mm, l = 1.0 mm, t = t₀ = 2.22 mm, and R = t' = 0.75 mm; width in y-direction is 10 mm). The specimen was cut out using an electrical discharge machine.

<figure><img src="image_1098.png" /><figcaption>Figure 3 Geometry of the specimen.</figcaption></figure>

Experimental condition. The testing apparatus used was the Autograph AG-50kN X plus (Shimadzu corporation). Fig. 3 shows the schematic of the experiment setup. A DIC system was used to measure the strain on the x-z plane of the specimen. Two cameras were used to capture the images. MultiDIC software [6] was used to calculate the strain distributions. The displacement *d* at ±10 mm from the center of the specimen in the longitudinal (*x*) direction was also calculated.
<figure><img src="image_1099.png" /><figcaption>Figure 2 True stress-logarithmic plastic strain curves measured and approximated.</figcaption></figure>

Specimen for the measurement of shear deformation behavior. The shape of the specimen used in this study is shown in Fig. 3. By applying tensile force to both ends of the specimen, shear deformation can be probed in the centoral zone of the specimen. Based on the FEA results, the dimensions of the specimen were optimized to achieve the largest possible shear strain within the manufacturing constraints (L = 50 mm, l = 1.0 mm, t = t₀ = 2.22 mm, and R = t' = 0.75 mm; width in y-direction is 10 mm). The specimen was cut out using an electrical discharge machine.

<figure><img src="image_1100.png" /><figcaption>Figure 3 Geometry of the specimen.</figcaption></figure>

Experimental condition. The testing apparatus used was the Autograph AG-50kN X plus (Shimadzu corporation). Fig. 3 shows the schematic of the experiment setup. A DIC system was used to measure the strain on the x-z plane of the specimen. Two cameras were used to capture the images. MultiDIC software [6] was used to calculate the strain distributions. The displacement *d* at ±10 mm from the center of the specimen in the longitudinal (*x*) direction was also calculated.
<figure><img src="image_1099.png" /><figcaption>Figure 4 Schematic of the experimental setup.</figcaption></figure>

## Simulation conditions

3D-FEAs were performed to compare the deformation behavior with the experimental one. The shape of the specimen is affected by the precision of manufacturing process by electrical discharge machining. The contour shape of the specimen was measured using a microscope and reproduced by CAD. The software used in FEAs was Simufact Forming 2023 (Hexagon). A mesh window was set at the vicinity of the center of the specimen.

The simulation model is shown in Fig. 5. The material was assumed to be elastoplastic. The elastic properties were set as Young's modulus of 200 GPa and Poisson's ratio of 0.3. For the plastic properties, the work hardening law was eq. (1) and the Hill '48 yield function was used.

Only the parameter *M* of the Hill '48 yield function (Eq. (2)) was changed from 2.0 to 5.0 at the interval of 0.1 to find the the through-thickness shear anisotropy parameter to minimize the error between experiment and simulation. All other anisotropy coefficients were assumed to be isotropic ($F=G=H=1$ and $L=N=3$).

$$f = \frac{1}{2} \{ F(\sigma_y - \sigma_z)^2 + G(\sigma_z - \sigma_x)^2 + H(\sigma_x - \sigma_y)^2 + 2L\sigma_{yz}^2 + 2M\sigma_{zx}^2 + 2N\sigma_{xy}^2 \} \quad (2)$$

<figure><img src="image_1100.png" /><figcaption>Figure 5 Simulation model.</figcaption></figure>
<figure><img src="image_1101.png" /><figcaption>Figure 4 Schematic of the experimental setup.</figcaption></figure>

## Simulation conditions

3D-FEAs were performed to compare the deformation behavior with the experimental one. The shape of the specimen is affected by the precision of manufacturing process by electrical discharge machining. The contour shape of the specimen was measured using a microscope and reproduced by CAD. The software used in FEAs was Simufact Forming 2023 (Hexagon). A mesh window was set at the vicinity of the center of the specimen.

The simulation model is shown in Fig. 5. The material was assumed to be elastoplastic. The elastic properties were set as Young's modulus of 200 GPa and Poisson's ratio of 0.3. For the plastic properties, the work hardening law was eq. (1) and the Hill '48 yield function was used.

Only the parameter *M* of the Hill '48 yield function (Eq. (2)) was changed from 2.0 to 5.0 at the interval of 0.1 to find the the through-thickness shear anisotropy parameter to minimize the error between experiment and simulation. All other anisotropy coefficients were assumed to be isotropic ($F=G=H=1$ and $L=N=3$).

$$f = \frac{1}{2} \{ F(\sigma_y - \sigma_z)^2 + G(\sigma_z - \sigma_x)^2 + H(\sigma_x - \sigma_y)^2 + 2L\sigma_{yz}^2 + 2M\sigma_{zx}^2 + 2N\sigma_{xy}^2 \} \quad (2)$$

<figure><img src="image_1102.png" /><figcaption>Figure 5 Simulation model.</figcaption></figure>
## Results and discussion

Shear strain-displacement curves. The variations of the through-thickness engineering shear strain $\gamma_{xz}$ at the center of the specimen with increasing displacement $d$ up to 0.5 mm are shown in Fig. 6. A shear strain of 0.02 can be attained in the experiment.

While the shear strain-displacement curve calculated using the parameter $M = 3.5$ agrees well with the one measured up to a displacement of 0.3, the calculated curve using $M = 3.9$ agrees well with the measured one over a displacement of 0.3. This behavior might be caused by the differential hardening [7].

<figure><img src="image_1101.png" /><figcaption>Figure 6 Measured and simulated shear strain-displacement curves.</figcaption></figure>

Influence of shear anisotropy on the deformation behavior of plate forging. Plate compression analyses were performed using the simulation model shown in Fig. 1 and the conditions in the reference [2]. The yield function used is Hill ‘48 yield function. Two simulation was performed to evaluate the influence of shear anisotropy on the deformation behavior of plate compression. One is the through-thickness shear parameter $M$ is set as the isotropic condition ($M = 3.0$). Another is $M$ is set as the best parameter identified in the previous section ($M = 3.9$). The anisotropy parameters $F, G, H, L$, and $N$ are set as the isotropic condition ($F = G = H = 1$ and $L = N = 3$).

Fig. 7 shows the distribution of the shear strain $\gamma_{zx}$ at the compression ratio at the edge of 10%. The maximum through-thickness shear strain for $M = 3.9$ is twice as large as that for $M = 3.0$. The anisotropy of through-thickness shear affects the deformation behavior of plate compression.

<figure><img src="image_1102.png" /><figcaption>Figure 7 Distribution of engineering shear strain $\gamma_{zx}$ at the compression ratio at the edge of 10%.</figcaption></figure>
## Results and discussion

Shear strain-displacement curves. The variations of the through-thickness engineering shear strain $\gamma_{xz}$ at the center of the specimen with increasing displacement $d$ up to 0.5 mm are shown in Fig. 6. A shear strain of 0.02 can be attained in the experiment.

While the shear strain-displacement curve calculated using the parameter $M = 3.5$ agrees well with the one measured up to a displacement of 0.3, the calculated curve using $M = 3.9$ agrees well with the measured one over a displacement of 0.3. This behavior might be caused by the differential hardening [7].

<figure><img src="image_1103.png" /><figcaption>Figure 6 Measured and simulated shear strain-displacement curves.</figcaption></figure>

Influence of shear anisotropy on the deformation behavior of plate forging. Plate compression analyses were performed using the simulation model shown in Fig. 1 and the conditions in the reference [2]. The yield function used is Hill ‘48 yield function. Two simulation was performed to evaluate the influence of shear anisotropy on the deformation behavior of plate compression. One is the through-thickness shear parameter $M$ is set as the isotropic condition ($M = 3.0$). Another is $M$ is set as the best parameter identified in the previous section ($M = 3.9$). The anisotropy parameters $F, G, H, L$, and $N$ are set as the isotropic condition ($F = G = H = 1$ and $L = N = 3$).

Fig. 7 shows the distribution of the shear strain $\gamma_{zx}$ at the compression ratio at the edge of 10%. The maximum through-thickness shear strain for $M = 3.9$ is twice as large as that for $M = 3.0$. The anisotropy of through-thickness shear affects the deformation behavior of plate compression.

<figure><img src="image_1104.png" /><figcaption>Figure 7 Distribution of engineering shear strain $\gamma_{zx}$ at the compression ratio at the edge of 10%.</figcaption></figure>
## Conclusions

(1) A through-thickness shear strain up to 0.02 can be attained by a tensile testing type material testing method.

(2) The parameters of the Hill '48 yield function were identified using the shear strain-displacement curves. The shear strain-displacement curve calculated using the parameter $M = 3.5$ agrees well with the one measured up to a displacement of 0.3. The calculated curve using $M = 3.9$ agrees well with the measured one over a displacement of 0.3.

(3) The anisotropy in the thickness direction affects the deformation behavior of plate compression.

Some material requires a high order yield function such as the Yld2004 yield function [8]. However since the number of the parameters increases to 2 for the Yld2004, it might be better to additionally evaluate the strain distribution in the specimen to identify the anisotropy parameters. The application of other materials is the future work.

## Acknowledgment

This work was supported by the grants of THE AMADA FOUNDATION and the Ihara Science Nakano Memorial Foundation.

## References

[1] M. Merklein, J.M. Allwood, B-A. Behrens, A. Brosius, H. Hagenah, K. Kuzman, K. Mori, A.E. Tekkaya, A. Weckenmann, Bulk forming of sheet metal. CIRP Ann. 61-2 (2012) 725-745. https://doi.org/10.1016/j.cirp.2012.05.007

[2] Z.G. Wang, T. Hakoyama, Y. Yoshikawa, Plastic deformation of workpiece during unloading in plate compression, CIRP Ann.-Manuf. Tech. 70 (2021) 223-226. https://doi.org/10.1016/j.cirp.2021.04.005

[3] T. Kuwabara, Advances in experiments on metal sheets and tubes in support of constitutive modeling and forming simulations, Int. J. Plast. 23-3 (2007) 385-419. https://doi.org/10.1016/j.ijplas.2006.06.003

[4] A. Lattanzi, M. Rossi, A. Baldi, D. Amodio, Development of new experiment test for sheet metals through-thickness behaviour characterization, Journal of Physics: Conf. Series 1063 (2018) 012040. https://doi.org/10.1088/1742-6596/1063/1/012040

[5] R. Hill, A theory of the yielding and plastic flow of anisotropic materials, Proceedings of the Royal Society A 193-1033 (1948) 281-297. https://doi.org/10.1098/rspa.1948.0045

[6] D. Solav, K. M. Moerman, A. M. Jaeger, K. Genovese and H. M. Herr, MultiDIC: An Open-Source Toolbox for Multi-View 3D Digital Image Correlation, IEEE Access 6 (2018) 30520-30535. https://doi.org/10.1109/ACCESS.2018.2843725

[7] R. Hill, J.W. Hutchinson, Differential Hardening in Sheet Metal Under Biaxial Loading: A Theoretical Framework, ASME. J. Appl. Mech 59-2S (1992) S1-S9. https://doi.org/10.1115/1.2899489

[8] F. Barlat, J.C. Bream, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.-H. Choi and E. Chu, Plane stress yield function for aluminum alloy sheets—part 1: theory, Int. J. Plasticity 19-9 (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0
## Conclusions

(1) A through-thickness shear strain up to 0.02 can be attained by a tensile testing type material testing method.

(2) The parameters of the Hill '48 yield function were identified using the shear strain-displacement curves. The shear strain-displacement curve calculated using the parameter $M = 3.5$ agrees well with the one measured up to a displacement of 0.3. The calculated curve using $M = 3.9$ agrees well with the measured one over a displacement of 0.3.

(3) The anisotropy in the thickness direction affects the deformation behavior of plate compression.

Some material requires a high order yield function such as the Yld2004 yield function [8]. However since the number of the parameters increases to 2 for the Yld2004, it might be better to additionally evaluate the strain distribution in the specimen to identify the anisotropy parameters. The application of other materials is the future work.

## Acknowledgment

This work was supported by the grants of THE AMADA FOUNDATION and the Ihara Science Nakano Memorial Foundation.

## References

[1] M. Merklein, J.M. Allwood, B-A. Behrens, A. Brosius, H. Hagenah, K. Kuzman, K. Mori, A.E. Tekkaya, A. Weckenmann, Bulk forming of sheet metal. CIRP Ann. 61-2 (2012) 725-745. https://doi.org/10.1016/j.cirp.2012.05.007

[2] Z.G. Wang, T. Hakoyama, Y. Yoshikawa, Plastic deformation of workpiece during unloading in plate compression, CIRP Ann.-Manuf. Tech. 70 (2021) 223-226. https://doi.org/10.1016/j.cirp.2021.04.005

[3] T. Kuwabara, Advances in experiments on metal sheets and tubes in support of constitutive modeling and forming simulations, Int. J. Plast. 23-3 (2007) 385-419. https://doi.org/10.1016/j.ijplas.2006.06.003

[4] A. Lattanzi, M. Rossi, A. Baldi, D. Amodio, Development of new experiment test for sheet metals through-thickness behaviour characterization, Journal of Physics: Conf. Series 1063 (2018) 012040. https://doi.org/10.1088/1742-6596/1063/1/012040

[5] R. Hill, A theory of the yielding and plastic flow of anisotropic materials, Proceedings of the Royal Society A 193-1033 (1948) 281-297. https://doi.org/10.1098/rspa.1948.0045

[6] D. Solav, K. M. Moerman, A. M. Jaeger, K. Genovese and H. M. Herr, MultiDIC: An Open-Source Toolbox for Multi-View 3D Digital Image Correlation, IEEE Access 6 (2018) 30520-30535. https://doi.org/10.1109/ACCESS.2018.2843725

[7] R. Hill, J.W. Hutchinson, Differential Hardening in Sheet Metal Under Biaxial Loading: A Theoretical Framework, ASME. J. Appl. Mech 59-2S (1992) S1-S9. https://doi.org/10.1115/1.2899489

[8] F. Barlat, J.C. Bream, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.-H. Choi and E. Chu, Plane stress yield function for aluminum alloy sheets—part 1: theory, Int. J. Plasticity 19-9 (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0
# The experimental full-field method (EFFM) for parameter calibration applied on an anisotropic constitutive model

ILG Christian¹,ᵃ*, SHETTY Mayank¹,ᵇ, KARADOGAN Celalettin¹,ᶜ, HAUFE André¹,ᵈ and LIEWALD Mathias²,ᵉ

¹DYNAmore an ANSYS Company, Industriestraße 2, 70565 Stuttgart, Germany

²Institute for Metal Forming Technology (IFU), Holzgartenstraße 17, 70174 Stuttgart, Germany

ᵃchristian.ILG@ansys.com, ᵇmayank.shetty@ansys.com, ᶜcelal.karadogan@ansys.com,

ᵈandre.haufe@ansys.com, ᶜmathias.liewald@ifu.uni-stuttgart.de

**Keywords:** FEA, Parameter Identification, DIC, Optimization

**Abstract.** Accurate characterization of material models is essential to ensure a higher prediction quality in Finite Element Analysis (FEA) under general loading conditions in sheet metal forming. Achieving accurate material model data frequently involves intricate inverse analysis techniques and numerous experimental tests. To overcome the complexities associated with calibration processes, the adoption of optical measuring systems like Digital Image Correlation (DIC) is widespread in material model calibration. The rich information obtained from DIC measurements is often used by material model calibration strategies to calibrate the values of material or model parameters, such as extrapolation of stress-strain curves or Lankford parameters that are not necessarily constant over the entire range of plastic deformation. This study presents the Experimental Full-Field Method (EFFM) as an innovative iterative approach for the calibration of material properties. As a special implementation of Finite Element Model Update (FEMU) [1], the EFFM uses the whole deformation field gained from DIC as boundary conditions [2]. This is then used in an inverse optimization procedure to determine parameters of complex material models. In this research, the EFFM is applied to an anisotropic constitutive model [3] to optimize three flow curves in 0°, 45° and 90° directions w.r.t. the rolling direction and the yield surface exponent, which reflects the polycrystal structure of the sheet material, to define the shape of the evolving yield locus in stress space. This is achieved by a modified tensile test specimen with L-shaped cut-outs which allows a distribution of higher strain values over a wider range of triaxiality values. With the direct use of the experimental deformation field in the FE simulation, displacements and strains are not any more objects of the optimization but only stresses. This also eliminates the step of mapping between experiments and simulations. Moreover, by using implicit time integration the inversion of the stiffness matrix becomes redundant, as positions of all nodes are already predetermined at each time step. These aspects make EFFM faster and more accurate than conventional FEMU.

**Background**

Accurate representation of material behavior within FEA is a cornerstone for predicting the structural response of components and systems under various loading conditions. The fidelity of FEA predictions directly hinges on the precision of the material models employed. These models encapsulate the intricate relationships between applied forces, deformation, and the resulting stress and strain distributions within a material.

However, characterizing material behavior for FEA purposes is a complex task laden with challenges. Materials often exhibit nonlinear, anisotropic, and time-dependent behaviors under various loading conditions. Traditional approaches to material characterization often rely on simplistic models or standardized test data, which may inadequately capture the nuanced behavior

<figure><img src="image_1103.png" /></figure>
# The experimental full-field method (EFFM) for parameter calibration applied on an anisotropic constitutive model

ILG Christian¹,ᵃ*, SHETTY Mayank¹,ᵇ, KARADOGAN Celalettin¹,ᶜ, HAUFE André¹,ᵈ and LIEWALD Mathias²,ᵉ

¹DYNAmore an ANSYS Company, Industriestraße 2, 70565 Stuttgart, Germany

²Institute for Metal Forming Technology (IFU), Holzgartenstraße 17, 70174 Stuttgart, Germany

ᵃchristian.ilg@ansys.com, ᵇmayank.shetty@ansys.com, ᶜcelal.karadogan@ansys.com,

ᵈandre.haufe@ansys.com, ᶜmathias.liewald@ifu.uni-stuttgart.de

**Keywords:** FEA, Parameter Identification, DIC, Optimization

**Abstract.** Accurate characterization of material models is essential to ensure a higher prediction quality in Finite Element Analysis (FEA) under general loading conditions in sheet metal forming. Achieving accurate material model data frequently involves intricate inverse analysis techniques and numerous experimental tests. To overcome the complexities associated with calibration processes, the adoption of optical measuring systems like Digital Image Correlation (DIC) is widespread in material model calibration. The rich information obtained from DIC measurements is often used by material model calibration strategies to calibrate the values of material or model parameters, such as extrapolation of stress-strain curves or Lankford parameters that are not necessarily constant over the entire range of plastic deformation. This study presents the Experimental Full-Field Method (EFFM) as an innovative iterative approach for the calibration of material properties. As a special implementation of Finite Element Model Update (FEMU) [1], the EFFM uses the whole deformation field gained from DIC as boundary conditions [2]. This is then used in an inverse optimization procedure to determine parameters of complex material models. In this research, the EFFM is applied to an anisotropic constitutive model [3] to optimize three flow curves in 0°, 45° and 90° directions w.r.t. the rolling direction and the yield surface exponent, which reflects the polycrystal structure of the sheet material, to define the shape of the evolving yield locus in stress space. This is achieved by a modified tensile test specimen with L-shaped cut-outs which allows a distribution of higher strain values over a wider range of triaxiality values. With the direct use of the experimental deformation field in the FE simulation, displacements and strains are not any more objects of the optimization but only stresses. This also eliminates the step of mapping between experiments and simulations. Moreover, by using implicit time integration the inversion of the stiffness matrix becomes redundant, as positions of all nodes are already predetermined at each time step. These aspects make EFFM faster and more accurate than conventional FEMU.

**Background**

Accurate representation of material behavior within FEA is a cornerstone for predicting the structural response of components and systems under various loading conditions. The fidelity of FEA predictions directly hinges on the precision of the material models employed. These models encapsulate the intricate relationships between applied forces, deformation, and the resulting stress and strain distributions within a material.

However, characterizing material behavior for FEA purposes is a complex task laden with challenges. Materials often exhibit nonlinear, anisotropic, and time-dependent behaviors under various loading conditions. Traditional approaches to material characterization often rely on simplistic models or standardized test data, which may inadequately capture the nuanced behavior

<figure><img src="image_1105.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
of materials in real-world applications. This discrepancy between actual material behavior and model representation poses a significant challenge in achieving accurate FEA predictions.

This is where Optical Measuring Systems, particularly DIC, have emerged as invaluable tools in material model calibration. DIC systems enable high-resolution, full-field measurements of surface deformation on materials subjected to loading. They provide detailed information about the displacement and strain fields across the material's surface, offering a wealth of data for characterizing material behavior. This detailed deformation information, when integrated within FEA, allows for a more accurate calibration of material models. Avril et al. [4] wrote a valuable review on parameter identification methods.

## Introduction of the EFFM

Utilizing a DIC system (12 mega pixels GOM ARAMIS system) the complete displacement field of a specimen can be acquired, exported as XML-file, and transformed into boundary conditions using the Envyo® mapping software. These imposed boundary conditions are then applied to all nodes within the corresponding LS-DYNA® simulation model, resulting in the creation of a digital replica (digital twin) of the specimen's geometry and experimental kinematics.

Using the digital twin as a basis, an iterative optimization process in LS-OPT® can be initiated to calibrate miscellaneous parameters of all material models available in LS-DYNA across various materials. Global force-time curves integrated over many hypothetical cross sections over the specimen or local force equilibriums at different nodes can be employed as objective functions for the optimization process. Prior to this, one must be aware of the experimental data or digital twin that should be used for the calibration of a specific material or parameters in a material model.

Utilizing experimental displacements directly obviates the necessity of employing displacements or strains as the optimization target. The optimization is based on stresses that are calculated from inherited strains. A significant advantage of directly feeding the measured displacements as boundary conditions in simulations, as opposed to classical FEMU, is the enhanced speed of conducting simulations. When calculating a digital twin with implicit time integration, the inversion of the stiffness matrix becomes unnecessary, as node positions are already predetermined at each time step.

## Application of the EFFM

The following chapter explores the application of the EFFM using the sheet metal DP800 and the calibration of its material parameters as an example and its calibration for an anisotropic material model as an example. The application of the EFFM is partly derived from prior works that have influenced aspects such as data filtering, the selection of the material model, and the shape of the specimen geometry [2]. The thickness of the DP800 sheet is 1.5 mm, and the specimens used were obtained through waterjet cutting at 0°, 45°, and 90° angles relative to the rolling direction of the steel sheet. The specimens were subsequently printed using a UV LED printer to ensure a high-quality speckle pattern across the entire specimen. This helps to avoid reflections and DIC errors caused by unsuitable patterns. Finally, the specimens are tested using a universal tensile testing machine.

*Material model.* Prior experiments employing the specified methodology based on the experimental deformation field have demonstrated that optimizing basic material models, such as the von Mises yield criterion (*MAT_024 in LS-DYNA) [5], yields unsatisfactory outcomes. Even standard tensile specimen geometries with a rectangular gauge shape manifest intricate material behaviors, for example in the localization zone, which cannot be realistically replicated using basic material models.

Consequently, an orthotropic plasticity model, specifically the material model denoted as *MAT_036E in LS-DYNA [5], was chosen for this study. This model is created based on the
of materials in real-world applications. This discrepancy between actual material behavior and model representation poses a significant challenge in achieving accurate FEA predictions.

This is where Optical Measuring Systems, particularly DIC, have emerged as invaluable tools in material model calibration. DIC systems enable high-resolution, full-field measurements of surface deformation on materials subjected to loading. They provide detailed information about the displacement and strain fields across the material's surface, offering a wealth of data for characterizing material behavior. This detailed deformation information, when integrated within FEA, allows for a more accurate calibration of material models. Avril et al. [4] wrote a valuable review on parameter identification methods.

## Introduction of the EFFM

Utilizing a DIC system (12 mega pixels GOM ARAMIS system) the complete displacement field of a specimen can be acquired, exported as XML-file, and transformed into boundary conditions using the Envyo® mapping software. These imposed boundary conditions are then applied to all nodes within the corresponding LS-DYNA® simulation model, resulting in the creation of a digital replica (digital twin) of the specimen's geometry and experimental kinematics.

Using the digital twin as a basis, an iterative optimization process in LS-OPT® can be initiated to calibrate miscellaneous parameters of all material models available in LS-DYNA across various materials. Global force-time curves integrated over many hypothetical cross sections over the specimen or local force equilibriums at different nodes can be employed as objective functions for the optimization process. Prior to this, one must be aware of the experimental data or digital twin that should be used for the calibration of a specific material or parameters in a material model.

Utilizing experimental displacements directly obviates the necessity of employing displacements or strains as the optimization target. The optimization is based on stresses that are calculated from inherited strains. A significant advantage of directly feeding the measured displacements as boundary conditions in simulations, as opposed to classical FEMU, is the enhanced speed of conducting simulations. When calculating a digital twin with implicit time integration, the inversion of the stiffness matrix becomes unnecessary, as node positions are already predetermined at each time step.

## Application of the EFFM

The following chapter explores the application of the EFFM using the sheet metal DP800 and the calibration of its material parameters as an example and its calibration for an anisotropic material model as an example. The application of the EFFM is partly derived from prior works that have influenced aspects such as data filtering, the selection of the material model, and the shape of the specimen geometry [2]. The thickness of the DP800 sheet is 1.5 mm, and the specimens used were obtained through waterjet cutting at 0°, 45°, and 90° angles relative to the rolling direction of the steel sheet. The specimens were subsequently printed using a UV LED printer to ensure a high-quality speckle pattern across the entire specimen. This helps to avoid reflections and DIC errors caused by unsuitable patterns. Finally, the specimens are tested using a universal tensile testing machine.

*Material model.* Prior experiments employing the specified methodology based on the experimental deformation field have demonstrated that optimizing basic material models, such as the von Mises yield criterion (*MAT_024 in LS-DYNA) [5], yields unsatisfactory outcomes. Even standard tensile specimen geometries with a rectangular gauge shape manifest intricate material behaviors, for example in the localization zone, which cannot be realistically replicated using basic material models.

Consequently, an orthotropic plasticity model, specifically the material model denoted as *MAT_036E in LS-DYNA [5], was chosen for this study. This model is created based on the
framework proposed by Barlat and Lian in 1989 [6]. To enhance its versatility, the model was extended to accommodate distinct strain hardening curves applicable to the three in-plane rolling directions (0°, 45°, and 90°), as well as for shear and equibiaxial tension. Whereby the flow curves in shear and equibiaxial tension are not obligatory. Additionally, the HOSF=1 option was employed, mitigating the presence of a concave yield locus. This is achieved by utilizing solely the information pertaining to anisotropy in the flow curves and incorporating Lankford parameters exclusively for the plastic flow rule, as detailed in the paper by Andrade et al. [3]. Therefore, if the principal and material axes are coincident, the yield locus can be described as

$$ \Phi(\boldsymbol{\sigma}) = \frac{1}{2}(|\sigma_1|^m + |\sigma_2|^m + |\sigma_1 - \sigma_2|^m) - \sigma_y^m(\sigma_y^0, \sigma_y^{45}, \sigma_y^{90}, \sigma_y^s, \sigma_y^b) = 0. \quad (1) $$

The exponent m in Eq. 1 determines the curvature near points of deviatoric uniaxial tensile and compressive stress and represents the polycrystal structure of the sheet material.

In the material card calibration carried out in this work, the exponent of the yield surface and three parameterized flow curves are optimized for the different extraction angles relative to the rolling direction. The initial estimation of the flow curve is based on experimental tensile tests and the conversion of the extracted engineering stress strain curves [7]. Retaining the experimental yield stresses up to the point of diffuse necking and ensuring C1-continuity thereafter, the Hockett-Sherby hardening law [8] is employed to calculate the subsequent yield stress values:

$$ \sigma_y(\epsilon_{pl}) = A - B e^{(-c\epsilon_{pl}^n)}. \quad (2) $$

In the Hockett-Sherby equation, due to the C1-continuity condition, the parameters A and B are excluded from the variable parameter list of the optimization and only the c and n are used as variables to be optimized.

*Specimen geometry.* In order to obtain as much data as possible for the calibration of an anisotropic material model while minimizing the number of experiments, novel specimen geometries can be employed [9,10]. Ilg et al. [11] introduced an innovative method for optimizing tensile specimen geometries, based on the distribution of plastic strains over stress triaxiality. This approach ensures coverage of all possible loading scenarios by the specimen geometry, with the ability to selectively emphasize or neglect specific regions in the diagram.

During the computational optimization of such a specimen geometry using the aforementioned methodology, several considerations must be taken into account: Firstly, the simulation must already possess high predictive accuracy. This implies that the material model used for specimen geometry optimization should already be characterized to a certain accuracy based on the experimental data. Additionally, when optimizing with shell elements, it is crucial to use at least three integration points across the shell thickness to accurately capture twisting/warping of the tensile specimen during virtual load application. Furthermore, attention should be given to the manufacturability of the specimen, avoiding small cutouts, angles, or radii. Such extreme shapes can distort the optimization results, as very short sides may cause irregular elements with poor aspect ratio may need to be fitted in these areas, leading to unphysical stress states or extremely high strains.

Another important factor for the optimization is the vertical symmetry of the tensile specimen. Testing vertically asymmetric tensile specimens results in transversal displacements at clamping jaws during testing, leading also to transversal forces that most load cells cannot measure. If the optimization target of the material calibration involves the global force, only vertically symmetric tensile specimens should be used.
framework proposed by Barlat and Lian in 1989 [6]. To enhance its versatility, the model was extended to accommodate distinct strain hardening curves applicable to the three in-plane rolling directions (0°, 45°, and 90°), as well as for shear and equibiaxial tension. Whereby the flow curves in shear and equibiaxial tension are not obligatory. Additionally, the HOSF=1 option was employed, mitigating the presence of a concave yield locus. This is achieved by utilizing solely the information pertaining to anisotropy in the flow curves and incorporating Lankford parameters exclusively for the plastic flow rule, as detailed in the paper by Andrade et al. [3]. Therefore, if the principal and material axes are coincident, the yield locus can be described as

$$ \Phi(\sigma) = \frac{1}{2}(|\sigma_1|^m + |\sigma_2|^m + |\sigma_1 - \sigma_2|^m) - \sigma_y^m(\sigma_y^0, \sigma_y^{45}, \sigma_y^{90}, \sigma_y^s, \sigma_y^b) = 0. \quad (1) $$

The exponent m in Eq. 1 determines the curvature near points of deviatoric uniaxial tensile and compressive stress and represents the polycrystal structure of the sheet material.

In the material card calibration carried out in this work, the exponent of the yield surface and three parameterized flow curves are optimized for the different extraction angles relative to the rolling direction. The initial estimation of the flow curve is based on experimental tensile tests and the conversion of the extracted engineering stress strain curves [7]. Retaining the experimental yield stresses up to the point of diffuse necking and ensuring C1-continuity thereafter, the Hockett-Sherby hardening law [8] is employed to calculate the subsequent yield stress values:

$$ \sigma_y(\epsilon_{pl}) = A - B e^{(-c\epsilon_{pl}^n)}. \quad (2) $$

In the Hockett-Sherby equation, due to the C1-continuity condition, the parameters A and B are excluded from the variable parameter list of the optimization and only the c and n are used as variables to be optimized.

*Specimen geometry.* In order to obtain as much data as possible for the calibration of an anisotropic material model while minimizing the number of experiments, novel specimen geometries can be employed [9,10]. Ilg et al. [11] introduced an innovative method for optimizing tensile specimen geometries, based on the distribution of plastic strains over stress triaxiality. This approach ensures coverage of all possible loading scenarios by the specimen geometry, with the ability to selectively emphasize or neglect specific regions in the diagram.

During the computational optimization of such a specimen geometry using the aforementioned methodology, several considerations must be taken into account: Firstly, the simulation must already possess high predictive accuracy. This implies that the material model used for specimen geometry optimization should already be characterized to a certain accuracy based on the experimental data. Additionally, when optimizing with shell elements, it is crucial to use at least three integration points across the shell thickness to accurately capture twisting/warping of the tensile specimen during virtual load application. Furthermore, attention should be given to the manufacturability of the specimen, avoiding small cutouts, angles, or radii. Such extreme shapes can distort the optimization results, as very short sides may cause irregular elements with poor aspect ratio may need to be fitted in these areas, leading to unphysical stress states or extremely high strains.

Another important factor for the optimization is the vertical symmetry of the tensile specimen. Testing vertically asymmetric tensile specimens results in transversal displacements at clamping jaws during testing, leading also to transversal forces that most load cells cannot measure. If the optimization target of the material calibration involves the global force, only vertically symmetric tensile specimens should be used.
For this reason, the specimen geometry with L-shaped holes used in a previous work [2] was modified to achieve vertical symmetry. Fig. 1 illustrates the technical drawing of the specimen geometry used in this study.

<figure><img src="image_1104.png" /><figcaption>Fig. 1. Technical drawing of the specimen geometry with 8-L-shaped cut-outs and vertical symmetry.</figcaption></figure>

The tensile specimen with vertical symmetry does not achieve the wide-ranging distribution of stress triaxiality exhibited by the asymmetric specimen; however, it still demonstrates a superior distribution compared to conventional tensile specimen geometries. Fig. 2 displays the traces of load paths of the elements in stress space (left) and the plastic strain over stress triaxiality (right) just before the deletion of the first shell element. These diagrams not only capture a snapshot just prior the simulations terminate but also include the history of load paths.

<figure><img src="image_1105.png" /><figcaption>Fig. 2. Distribution of the equivalent plastic strain in the principal stress space (left) and via stress triaxiality (right).</figcaption></figure>

Fig. 3 exemplifies displacement-time curves for various facet mid-points respectively nodal points within different loading zones. The digital twin, presented in the upper left corner of the diagram, illustrates the regions from which the node information for the curves was extracted. The green region lies directly at the upper clamping device within the force initiation zone, the orange
For this reason, the specimen geometry with L-shaped holes used in a previous work [2] was modified to achieve vertical symmetry. Fig. 1 illustrates the technical drawing of the specimen geometry used in this study.

<figure><img src="image_1106.png" /><figcaption>Fig. 1. Technical drawing of the specimen geometry with 8-L-shaped cut-outs and vertical symmetry.</figcaption></figure>

The tensile specimen with vertical symmetry does not achieve the wide-ranging distribution of stress triaxiality exhibited by the asymmetric specimen; however, it still demonstrates a superior distribution compared to conventional tensile specimen geometries. Fig. 2 displays the traces of load paths of the elements in stress space (left) and the plastic strain over stress triaxiality (right) just before the deletion of the first shell element. These diagrams not only capture a snapshot just prior the simulations terminate but also include the history of load paths.

<figure><img src="image_1107.png" /><figcaption>Fig. 2. Distribution of the equivalent plastic strain in the principal stress space (left) and via stress triaxiality (right).</figcaption></figure>

Fig. 3 exemplifies displacement-time curves for various facet mid-points respectively nodal points within different loading zones. The digital twin, presented in the upper left corner of the diagram, illustrates the regions from which the node information for the curves was extracted. The green region lies directly at the upper clamping device within the force initiation zone, the orange
region is situated in the middle of the tensile specimen, and the blue region is near the lower clamping device.

The displacements of all facet mid-points only in the vertical and horizontal directions were converted into boundary conditions for the simulation, with no consideration given to displacements in the thickness direction.

<figure><img src="image_1106.png" /><figcaption>Fig. 3. Displacement-time curves of various nodal points in different load areas and the digital twin, undeformed and deformed, in the top left corner.</figcaption></figure>

*Definition of optimization targets.* Based on the digital twin, an iterative optimization process can now be initiated. In this study, two optimization criteria were applied to assess their utility in this methodology. On the one hand, a force-time criterion, further termed “Global Force”, was employed, wherein the global force is integrated using three different cross sections through the specimen and calibrated to match the experimentally measured global force. A drawback of this criterion is the loss of information at the edge of the specimen by the optical measurement system, necessitating compensation through scaling of the simulated force-time curves. The locations of the hypothetical sections in the digital twin are depicted in Fig. 4, positioned in the lower, middle, and upper regions of the specimen.

<figure><img src="image_1107.png" /><figcaption>Fig. 4. Visualization of the three cross sections (black circles) and the nodes highlighted with orange circles which are used for the different methodologies.</figcaption></figure>
region is situated in the middle of the tensile specimen, and the blue region is near the lower clamping device.

The displacements of all facet mid-points only in the vertical and horizontal directions were converted into boundary conditions for the simulation, with no consideration given to displacements in the thickness direction.

<figure><img src="image_1108.png" /><figcaption>Fig. 3. Displacement-time curves of various nodal points in different load areas and the digital twin, undeformed and deformed, in the top left corner.</figcaption></figure>

*Definition of optimization targets.* Based on the digital twin, an iterative optimization process can now be initiated. In this study, two optimization criteria were applied to assess their utility in this methodology. On the one hand, a force-time criterion, further termed “Global Force”, was employed, wherein the global force is integrated using three different cross sections through the specimen and calibrated to match the experimentally measured global force. A drawback of this criterion is the loss of information at the edge of the specimen by the optical measurement system, necessitating compensation through scaling of the simulated force-time curves. The locations of the hypothetical sections in the digital twin are depicted in Fig. 4, positioned in the lower, middle, and upper regions of the specimen.

<figure><img src="image_1109.png" /><figcaption>Fig. 4. Visualization of the three cross sections (black circles) and the nodes highlighted with orange circles which are used for the different methodologies.</figcaption></figure>
On the other hand, a criterion based on the force equilibrium at nodes, further termed “Split Nodes”, is applied. Individual nodes are split, and their direction-dependent forces are summed. The resulting force of these nodes must be equal to zero. In Fig. 4, three positions where this criterion was applied are marked with orange circles. When selecting nodes, it is crucial to ensure they are in regions of high plastic strains, at least above the uniform strain, as the flow curve is partially predefined and only extrapolated part is fitted, as outlined in Eq. 2. A schematic drawing illustrating the node splitting process for a single nodal point is presented in Fig. 5. Activating the option to split nodes of the digital twin can be activated in the Envyo mapping software.

<figure><img src="image_1108.png" /><figcaption>Fig. 5. Schematic illustration of the splitting nodes option for a single nodal point.</figcaption></figure>

## Results of the EFFM

In this section, the results of the optimization using the chosen strategies, "Global Force" and "Split Nodes," are revealed. The optimization results using the "Global Force" method are depicted in Fig. 6. The lines marked with crosses represent the global force-time curves from the experimental tests at 0°, 45°, and 90° to the rolling direction. The solid curves illustrate the integrated force from the middle cross-section of each simulation. The blue curves represent the force-time curves of the baseline run, which does not involve parameter optimization, and the red curves represent the force-time curves after 15 iterations.

The results indicate no significant improvement after the parameter optimization. The deviations from the experimental force-time curves only marginally decrease compared to the first iteration. The reason for this lies in the selection of the locations of the cross sections in the digital twin. The regions traversed by these cuts exhibit no plastic strains exceeding the uniform elongation strain. This implies that there is no information in the cuts necessary for flow curve extrapolation.

Furthermore, it is noteworthy that the force curves are highly noisy. Despite filtering the experimental data and the seemingly smooth displacement-time curves of individual facet mid-points, noise-free curves cannot be obtained in the simulation.

<figure><img src="image_1109.png" /><figcaption>Fig. 6. Optimization results showing the global force-time curves from the experimental tests at 0°, 45°, and 90° to the rolling direction.</figcaption></figure>
On the other hand, a criterion based on the force equilibrium at nodes, further termed “Split Nodes”, is applied. Individual nodes are split, and their direction-dependent forces are summed. The resulting force of these nodes must be equal to zero. In Fig. 4, three positions where this criterion was applied are marked with orange circles. When selecting nodes, it is crucial to ensure they are in regions of high plastic strains, at least above the uniform strain, as the flow curve is partially predefined and only extrapolated part is fitted, as outlined in Eq. 2. A schematic drawing illustrating the node splitting process for a single nodal point is presented in Fig. 5. Activating the option to split nodes of the digital twin can be activated in the Envyo mapping software.

<figure><img src="image_1110.png" /><figcaption>Fig. 5. Schematic illustration of the splitting nodes option for a single nodal point.</figcaption></figure>

## Results of the EFFM

In this section, the results of the optimization using the chosen strategies, "Global Force" and "Split Nodes," are revealed. The optimization results using the "Global Force" method are depicted in Fig. 6. The lines marked with crosses represent the global force-time curves from the experimental tests at 0°, 45°, and 90° to the rolling direction. The solid curves illustrate the integrated force from the middle cross-section of each simulation. The blue curves represent the force-time curves of the baseline run, which does not involve parameter optimization, and the red curves represent the force-time curves after 15 iterations.

The results indicate no significant improvement after the parameter optimization. The deviations from the experimental force-time curves only marginally decrease compared to the first iteration. The reason for this lies in the selection of the locations of the cross sections in the digital twin. The regions traversed by these cuts exhibit no plastic strains exceeding the uniform elongation strain. This implies that there is no information in the cuts necessary for flow curve extrapolation.

Furthermore, it is noteworthy that the force curves are highly noisy. Despite filtering the experimental data and the seemingly smooth displacement-time curves of individual facet mid-points, noise-free curves cannot be obtained in the simulation.

<figure><img src="image_1111.png" /><figcaption>Fig. 6. Optimization results showing the global force-time curves from the experimental tests at 0°, 45°, and 90° to the rolling direction.</figcaption></figure>
The results of the optimization using the "Split Nodes" method are presented in Fig 7. The diagram illustrates the resulting force-time curves in loading and transversal direction, exemplified at the first node (Fig. 4), of the digital twin extracted in the rolling direction. The blue curves represent the resulting force-time curves of the baseline run, which does not involve parameter optimization, and the red curves represent the resulting force-time curves after 15 iterations. The results show a substantial improvement in the force equilibrium after the parameter optimization. However, a complete force equilibrium could not be achieved.

Upon examining the curve plots, it is noticeable that the noisy measurement data is also reflected in noisy simulation data.

<figure><img src="image_1110.png" /><figcaption>Fig. 7. Optimization results showing the force equilibrium over time in loading direction (left) and in transversal direction (right).</figcaption></figure>

## Summary

In the present study, a parameter-identification procedure called EFFM was carried out for the calibration of an anisotropic material model. Containing the calibration of three flow curves extrapolated according to a modified Hockett-Sherby approach and the exponent of the yield surface based on the optical full-field measurement. Within the EFFM one is able to build a digital twin by using local displacement measurements during the test and using this information as boundary conditions for a corresponding FE mesh.

For the method a modified L-shaped specimen geometry was used to enable more accurate and representative measurements of the strain field under complex loading conditions, including uniaxial tension, plane strain, and equibiaxial tension.

The optimization of the variables used in the modified Hockett-Sherby extrapolation approach and the exponent of the yield surface based on the new tensile test specimen and with the global force as an optimization target shows a poor agreement regarding the experiment. The results obtained can be attributed to multiple factors: Noise in the experimental measurement data, the conversion of the experimental data and inadequate spatial discretization. Noise in experimental measurement data, if not correctly accounted for, may also lead to inaccurate simulation results. Furthermore, a deviation already occurs when converting the optical measurement data into a simulation model, as the optical measurement is performed on the sample surface and projected into the shell mid-plane. Additionally, the chosen shell formulation for spatial discretization was not capable of capturing stress in the thickness direction, which may result as well in inaccurate simulation results w.r.t. three-dimensional stress states developing in necking and localization regions.
The results of the optimization using the "Split Nodes" method are presented in Fig 7. The diagram illustrates the resulting force-time curves in loading and transversal direction, exemplified at the first node (Fig. 4), of the digital twin extracted in the rolling direction. The blue curves represent the resulting force-time curves of the baseline run, which does not involve parameter optimization, and the red curves represent the resulting force-time curves after 15 iterations. The results show a substantial improvement in the force equilibrium after the parameter optimization. However, a complete force equilibrium could not be achieved.

Upon examining the curve plots, it is noticeable that the noisy measurement data is also reflected in noisy simulation data.

<figure><img src="image_1112.png" /><figcaption>Fig. 7. Optimization results showing the force equilibrium over time in loading direction (left) and in transversal direction (right).</figcaption></figure>

## Summary

In the present study, a parameter-identification procedure called EFFM was carried out for the calibration of an anisotropic material model. Containing the calibration of three flow curves extrapolated according to a modified Hockett-Sherby approach and the exponent of the yield surface based on the optical full-field measurement. Within the EFFM one is able to build a digital twin by using local displacement measurements during the test and using this information as boundary conditions for a corresponding FE mesh.

For the method a modified L-shaped specimen geometry was used to enable more accurate and representative measurements of the strain field under complex loading conditions, including uniaxial tension, plane strain, and equibiaxial tension.

The optimization of the variables used in the modified Hockett-Sherby extrapolation approach and the exponent of the yield surface based on the new tensile test specimen and with the global force as an optimization target shows a poor agreement regarding the experiment. The results obtained can be attributed to multiple factors: Noise in the experimental measurement data, the conversion of the experimental data and inadequate spatial discretization. Noise in experimental measurement data, if not correctly accounted for, may also lead to inaccurate simulation results. Furthermore, a deviation already occurs when converting the optical measurement data into a simulation model, as the optical measurement is performed on the sample surface and projected into the shell mid-plane. Additionally, the chosen shell formulation for spatial discretization was not capable of capturing stress in the thickness direction, which may result as well in inaccurate simulation results w.r.t. three-dimensional stress states developing in necking and localization regions.
The optimization based on a criterion with split nodes shows a more satisfactory result. But the noise in the measurement data is as well visible in the resulting simulative force-time curves. This underscores the importance of data quality and preprocessing of the optical measurement data.

There are further factors that influence the results of both methods: the exclusion of displacements in the thickness direction and the absence of strain rate effects in the material model that was used. To improve the accuracy of the method, a conversion which results in boundary conditions on the shell surface, with displacements in thickness direction should be implemented, DIC data smoothing and filtering techniques in space and time to fight the noise in the measurement data, and alternative shell formulations or the use of multiple solid elements across the sheet metal thickness should be considered. To investigate the effect of noise, still images can be used to understand the ground noise and eliminate it before using the displacement field with the Envyo mapping software. Clearly, both of which require further assumptions about the section deformation through the thickness direction. Hence, an improved approach, with careful enhancements of the experimental data in lateral direction, could be to map the deformation field onto a regular, very fine 3D mesh as already proposed by Liebold et al. [12]. Furthermore, optimization criteria that include the vanishing of stress divergence at multiple or even all nodes can be tested [13].

The method is also promising when it is coupled with Digital Volume Correlation (DVC) for materials which allow full 3D computerized tomography (CT) to capture internal displacement fields.

## Acknowledgments

The authors would like to thank Stefan Wacker, Christian Liebold and Dr. Nils Karajan for performing the experiments at the laboratory of the Material Competence Center in Stuttgart, Germany, for code development and for their advice. Their help and support are greatly acknowledged.

This work has been carried out within the research project AIMM (Artificial Intelligence for Material Models) funded by the Federal Ministry for Economic Affairs and Climate Action (BMWK) under the grant number 19 | 20024A.

## References

[1] R. Mahnken, E. Stein, A unified approach for parameter identification of inelastic material models in the frame of the finite element method, J. Computer Methods in Applied Mechanics and Engineering 136, 1996, pp. 225-258. https://doi.org/10.1016/0045-7825(96)00991-7

[2] C. Ilg, C. Liebold, A. Haufe, V. Sreenivasa, C. Karadogan, M. Liewald, Displacement driven simulation and material calibration based on digital image correlation Part II – Application, IOP Conf. Ser.: Mater. Sci. Eng. 1284, 2023. https://doi.org/10.1088/1757-899X/1284/1/012056

[3] F. Andrade, T. Borrvall, P. DuBois, M. Feucht, A Hosford-based orthotropic plasticity model in LS-DYNA, 12th European LS-DYNA Users' Conference, 2019

[4] S. Avril, M. Bonnet, A.-S. Bretelle, M. Grèdiac, F. Hild, P. Ienny, F. Latourte, D. Lemosse, S. Pagano, E. Pagnacco, F. Pierron, Overview of identification methods of mechanical parameters based on full-field measurements, Experimental Mechanics 48, 2008, pp. 381-402. https://doi.org/10.1007/s11340-008-9148-y

[5] Livermore Software Technology 2020 LS-DYNA Keyword User's Manual vol 2 R12

[6] F. Barlat, K. Lian, Plastic behavior and stretchability of sheet metals. Part I: A yield function for orthotropic sheets under plane stress conditions, Int. J. Plasticity 5, 1989, pp. 51-56. https://doi.org/10.1016/0749-6419(89)90019-3
The optimization based on a criterion with split nodes shows a more satisfactory result. But the noise in the measurement data is as well visible in the resulting simulative force-time curves. This underscores the importance of data quality and preprocessing of the optical measurement data.

There are further factors that influence the results of both methods: the exclusion of displacements in the thickness direction and the absence of strain rate effects in the material model that was used. To improve the accuracy of the method, a conversion which results in boundary conditions on the shell surface, with displacements in thickness direction should be implemented, DIC data smoothing and filtering techniques in space and time to fight the noise in the measurement data, and alternative shell formulations or the use of multiple solid elements across the sheet metal thickness should be considered. To investigate the effect of noise, still images can be used to understand the ground noise and eliminate it before using the displacement field with the Envyo mapping software. Clearly, both of which require further assumptions about the section deformation through the thickness direction. Hence, an improved approach, with careful enhancements of the experimental data in lateral direction, could be to map the deformation field onto a regular, very fine 3D mesh as already proposed by Liebold et al. [12]. Furthermore, optimization criteria that include the vanishing of stress divergence at multiple or even all nodes can be tested [13].

The method is also promising when it is coupled with Digital Volume Correlation (DVC) for materials which allow full 3D computerized tomography (CT) to capture internal displacement fields.

## Acknowledgments

The authors would like to thank Stefan Wacker, Christian Liebold and Dr. Nils Karajan for performing the experiments at the laboratory of the Material Competence Center in Stuttgart, Germany, for code development and for their advice. Their help and support are greatly acknowledged.

This work has been carried out within the research project AIMM (Artifical Intelligence for Material Models) funded by the Federal Ministry for Economic Affairs and Climate Action (BMWK) under the grant number 19 | 20024A.

## References

[1] R. Mahnken, E. Stein, A unified approach for parameter identification of inelastic material models in the frame of the finite element method, J. Computer Methods in Applied Mechanics and Engineering 136, 1996, pp. 225-258. https://doi.org/10.1016/0045-7825(96)00991-7

[2] C. Ilg, C. Liebold, A. Haufe, V. Sreenivasa, C. Karadogan, M. Liewald, Displacement driven simulation and material calibration based on digital image correlation Part II – Application, IOP Conf. Ser.: Mater. Sci. Eng. 1284, 2023. https://doi.org/10.1088/1757-899X/1284/1/012056

[3] F. Andrade, T. Borrvall, P. DuBois, M. Feucht, A Hosford-based orthotropic plasticity model in LS-DYNA, 12th European LS-DYNA Users' Conference, 2019

[4] S. Avril, M. Bonnet, A.-S. Bretelle, M. Grèdiac, F. Hild, P. Ienny, F. Latourte, D. Lemosse, S. Pagano, E. Pagnacco, F. Pierron, Overview of identification methods of mechanical parameters based on full-field measurements, Experimental Mechanics 48, 2008, pp. 381-402. https://doi.org/10.1007/s11340-008-9148-y

[5] Livermore Software Technology 2020 LS-DYNA Keyword User's Manual vol 2 R12

[6] F. Barlat, K. Lian, Plastic behavior and stretchability of sheet metals. Part I: A yield function for orthotropic sheets under plane stress conditions, Int. J. Plasticity 5, 1989, pp. 51-56. https://doi.org/10.1016/0749-6419(89)90019-3
[7] C. Ilg, A. Haufe, K. Witowski, D. Koch, M. Liewald, Parameter identification using full-field calibration (FFC), IOP Conf. Ser.: Mater. Sci. Eng. 651, 2019, doi:10.1088/1757-899X/651/1/012070

[8] J.E. Hockett, O.D. Sherby, Large strain deformation of polycrystalline metals at low homologous temperatures, J. of the Mech. and Phys. of Solids 23, 1975, pp. 87-98.
https://doi.org/10.1016/0022-5096(75)90018-6

[9] M. Gonçalves, M. G. Oliveira, S. Thuillier, A. Andrade-Campos, On the comparison of heterogeneous mechanical tests for sheet metal characterization, Materials Research Proc. 28, 2023, pp. 1121-1130. https://doi.org/10.21741/9781644902479-123

[10] B. Williams, C. Simha, On the design of novel multi-failure specimens for ductile failure testing, IOP Conf. Ser.: Mater. Sci. Eng. 651, 2019, https://doi.org/10.1088/1757-899X/651/1/012091

[11] C. Ilg, A. Haufe, V. Sreenivasa, C. Karadogan, M. Liewald, Parameter identification applying full-field calibration (FFC) techniques, Proc. of the 14th Int. Conf. on the Techn. of Plast., 2023, pp. 578-585. https://doi.org/10.1007/978-3-031-42093-1_55

[12] C. Liebold, C. Ilg, A. Haufe, T. Usta, V. Sreenivasa, Displacement based simulation and material calibration based on digital image correlation Part I – Theory, IOP Conf. Ser.: Mater. Sci. Eng. 1284, 2023, https://doi.org/10.1088/1757-899X/1284/1/012053

[13] P. Böhringer, D. Sommer, T. Haase, M. Barteczko, J. Sprave, M. Stoll, C. Karadogan, D. Koch, P. Middendorf, M. Liewald, A strategy to train machine learning material models for finite element simulations on data acquirable from physical experiments, Comp. Meth. in Applied Mech. and Eng. 406, 2023, https://doi.org/10.1016/j.cma.2023.115894
[7] C. Ilg, A. Haufe, K. Witowski, D. Koch, M. Liewald, Parameter identification using full-field calibration (FFC), IOP Conf. Ser.: Mater. Sci. Eng. 651, 2019, doi:10.1088/1757-899X/651/1/012070

[8] J.E. Hockett, O.D. Sherby, Large strain deformation of polycrystalline metals at low homologous temperatures, J. of the Mech. and Phys. of Solids 23, 1975, pp. 87-98.
https://doi.org/10.1016/0022-5096(75)90018-6

[9] M. Gonçalves, M. G. Oliveira, S. Thuillier, A. Andrade-Campos, On the comparison of heterogeneous mechanical tests for sheet metal characterization, Materials Research Proc. 28, 2023, pp. 1121-1130. https://doi.org/10.21741/9781644902479-123

[10] B. Williams, C. Simha, On the design of novel multi-failure specimens for ductile failure testing, IOP Conf. Ser.: Mater. Sci. Eng. 651, 2019, https://doi.org/10.1088/1757-899X/651/1/012091

[11] C. Ilg, A. Haufe, V. Sreenivasa, C. Karadogan, M. Liewald, Parameter identification applying full-field calibration (FFC) techniques, Proc. of the 14th Int. Conf. on the Techn. of Plast., 2023, pp. 578-585. https://doi.org/10.1007/978-3-031-42093-1_55

[12] C. Liebold, C. Ilg, A. Haufe, T. Usta, V. Sreenivasa, Displacement based simulation and material calibration based on digital image correlation Part I – Theory, IOP Conf. Ser.: Mater. Sci. Eng. 1284, 2023, https://doi.org/10.1088/1757-899X/1284/1/012053

[13] P. Böhringer, D. Sommer, T. Haase, M. Barteczko, J. Sprave, M. Stoll, C. Karadogan, D. Koch, P. Middendorf, M. Liewald, A strategy to train machine learning material models for finite element simulations on data acquirable from physical experiments, Comp. Meth. in Applied Mech. and Eng. 406, 2023, https://doi.org/10.1016/j.cma.2023.115894
# Warm forming of AA7075-T6: optimizing the heating time to maintain T6 condition

LAURENT Hervé¹,ᵃ*, ROYNE Sylvain¹,ᵇ and MAILLARD André²,ᶜ

¹Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France

²CETIM, F-60300 Senlis, France

ᵃherve.laurent@univ-ubs.fr, ᵇsylvain.royne@univ-ubs.fr, ᶜAndre.Maillard@cetim.fr

**Keywords:** Warm Forming, AA7075-T6, Heating Time, Aging, Precipitation

**Abstract.** Warm forming at 200°C is an interesting way of improving the formability of AA7075-T6. The T6 condition offers the highest ultimate and yield strengths of this aluminium alloy. It is therefore important to maintain these excellent mechanical properties at the end of the forming process. With the hot forming process at temperature above 450°C, it is necessary to add a heat recovery treatment obtained during paint baking to keep this T6 state, which is costly in terms of time and energy. In warm forming process, it is possible to maintain the T6 condition by controlling the heating time to avoid precipitation changes. The objective of this study is to find these optimal heating time conditions to maintain the T6 state during warm forming multi-step process. Different heating times were reproduced using a Gleeble 3500 machine. Electrical conductivity and hardness were measured after these different conditions to make the link with the evolutions of precipitates of AA7075-T6. Tensile tests were also performed to characterize the mechanical behavior at the end of these heating cycles. A holding time of less than 10 seconds is determined to maintain the T6 state at 200°C. Two multi-step warm forming devices (a cylindrical cup in two steps and a U-channel part) were finally tested to validate these optimal time forming conditions.

**Introduction**

7xxx series aluminium alloys (Al-Zn-Mg-Cu) have very good mechanical characteristics and are therefore very interesting candidates for replacing steels and thereby reducing the weight of automobile parts and therefore CO2 emissions [1]. Unfortunately, these alloys are difficult to form at room temperature and have a high springback. To overcome this difficulty, it is increasingly being considered to form them at high temperature using a multi-stage forming process to increase the depth of draw.

Regarding the possible forming processes in temperature, AA7075 sheet can be stamped either by hot forming or by warm forming [2]. After hot forming at 480°C (immediately above its solvus temperature) which needs simultaneous forming and quenching, the material requires an artificial aging cycle to reach the T6 state which gives the highest mechanical strength [3]. Because of all these steps, this makes this process costly in time and energy. To avoid this drawback, warm forming technology is a promising approach to improve the formability of peak-aged 7075 aluminium alloys (AA7075-T6) [4]. However, to avoid any paint baking operation and changes in precipitation state due to the temperature [5], it is important to control the heating time. For example, the precipitation of the η phase in η' appears at a temperature of 200 °C and leads to a reduction in the mechanical properties if the temperature holding time is too long [6]. In the literature, several authors have analysed the deterioration of mechanical properties and the microstructural evolution of 7xxx alloys during long-term exposure to high temperature [7,8]. However, these studies focus mainly on the influence of temperature on extended holding times of more than a minute or even an hour. But with the development of warm multi-stage forming

<figure><img src="image_1111.png" /></figure>
# Warm forming of AA7075-T6: optimizing the heating time to maintain T6 condition

LAURENT Hervé¹,ᵃ*, ROYNE Sylvain¹,ᵇ and MAILLARD André²,ᶜ

¹Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France

²CETIM, F-60300 Senlis, France

ᵃherve.laurent@univ-ubs.fr, ᵇsylvain.royne@univ-ubs.fr, ᶜAndre.Maillard@cetim.fr

**Keywords:** Warm Forming, AA7075-T6, Heating Time, Aging, Precipitation

**Abstract.** Warm forming at 200°C is an interesting way of improving the formability of AA7075-T6. The T6 condition offers the highest ultimate and yield strengths of this aluminium alloy. It is therefore important to maintain these excellent mechanical properties at the end of the forming process. With the hot forming process at temperature above 450°C, it is necessary to add a heat recovery treatment obtained during paint baking to keep this T6 state, which is costly in terms of time and energy. In warm forming process, it is possible to maintain the T6 condition by controlling the heating time to avoid precipitation changes. The objective of this study is to find these optimal heating time conditions to maintain the T6 state during warm forming multi-step process. Different heating times were reproduced using a Gleeble 3500 machine. Electrical conductivity and hardness were measured after these different conditions to make the link with the evolutions of precipitates of AA7075-T6. Tensile tests were also performed to characterize the mechanical behavior at the end of these heating cycles. A holding time of less than 10 seconds is determined to maintain the T6 state at 200°C. Two multi-step warm forming devices (a cylindrical cup in two steps and a U-channel part) were finally tested to validate these optimal time forming conditions.

**Introduction**

7xxx series aluminium alloys (Al-Zn-Mg-Cu) have very good mechanical characteristics and are therefore very interesting candidates for replacing steels and thereby reducing the weight of automobile parts and therefore CO2 emissions [1]. Unfortunately, these alloys are difficult to form at room temperature and have a high springback. To overcome this difficulty, it is increasingly being considered to form them at high temperature using a multi-stage forming process to increase the depth of draw.

Regarding the possible forming processes in temperature, AA7075 sheet can be stamped either by hot forming or by warm forming [2]. After hot forming at 480°C (immediately above its solvus temperature) which needs simultaneous forming and quenching, the material requires an artificial aging cycle to reach the T6 state which gives the highest mechanical strength [3]. Because of all these steps, this makes this process costly in time and energy. To avoid this drawback, warm forming technology is a promising approach to improve the formability of peak-aged 7075 aluminium alloys (AA7075-T6) [4]. However, to avoid any paint baking operation and changes in precipitation state due to the temperature [5], it is important to control the heating time. For example, the precipitation of the η phase in η' appears at a temperature of 200 °C and leads to a reduction in the mechanical properties if the temperature holding time is too long [6]. In the literature, several authors have analysed the deterioration of mechanical properties and the microstructural evolution of 7xxx alloys during long-term exposure to high temperature [7,8]. However, these studies focus mainly on the influence of temperature on extended holding times of more than a minute or even an hour. But with the development of warm multi-stage forming

<figure><img src="image_1113.png" /></figure>
operations compatible with industrial rates, we need to consider times of the order of 2 to 20 seconds for a complete heating and drawing cycle.

To know the influence of the heating cycle time, the objective of this study is to determine the evolution of the mechanical properties for very short heating cycle times to maintain the T6 state of 7075 alloy. For this purpose, different heating time were reproduced with a Gleeble 3500 machine on samples of AA7075-T6. The electrical conductivity and hardness were then measured and compared to the standard values to respect the T6 state. The optimum time values obtained were then validated by analysing the hardness and conductivity values on the final stamped parts of a warm-formed cylindrical cup [9,10] and a U-shaped channel [11,12].

## Material

Aluminium alloy sheet AA7075 with a thickness of 0.8mm is considered in this work. The commercially purchased aluminium is directly received in the T6 state. The chemical composition of this alloy is presented in Table 1. The mechanical properties (Yield Strength YS, Ultimate tensile strength UTS and elongation at break E%) and the conductivity and hardness properties measured under standard conditions to meet the T6 condition and the reception conditions for the alloy are given in Table 2.

<table><caption>Table 1 Chemical composition of AA7075-T6 sheet in %wt.</caption><thead><tr><th>Zn</th><th>Mg</th><th>Cu</th><th>Si</th><th>Cr</th><th>Fe</th><th>Mn</th><th>Ti</th><th>Al</th></tr></thead><tbody><tr><td>5.1-6.1</td><td>2.1-2.9</td><td>1.2-2.0</td><td>0.4</td><td>0.18-0.28</td><td>0.5</td><td>0.3</td><td>0.2</td><td>Balance</td></tr></tbody></table>

<table><caption>Table 2 Mechanical and electrical properties of AA7075-T6 to comply with the standard and those obtained at reception.</caption><thead><tr><th>Condition</th><th>YS [MPa]</th><th>UTS [MPa]</th><th>E [%]</th><th>Conductivity [MS/m]</th><th>Hardness [HV10]</th></tr></thead><tbody><tr><td>Standard</td><td>≥ 460</td><td>≥ 525</td><td>≥ 6</td><td>17-21</td><td>-</td></tr><tr><td>Reception</td><td>500</td><td>557</td><td>14.6</td><td>18.5</td><td>188</td></tr></tbody></table>

## Experimental conditions to control the T6-state

Different conditions of holding time at 200°C were reproduced using a Gleeble 3500 machine in order to reproduce the conventional conditions used in the multi-step warm forming process. With the Gleeble 3500 direct resistance heating system, a dog-bone specimen was heated by temperature with a thermocouple welded to the center of the specimen. The thermal cycles, presented in Table 3, consist of initial heating at 250 °C/s (200°C in 0.8 s), which corresponds to heating obtained by direct contact with heating tools. Four different holding times of 2, 10, 20 and 30 s were then applied, followed by cooling at 2°C/s, identical to cooling in free air.

<table><caption>Table 3 Conditions of heating rate, holding times and cooling rate.</caption><thead><tr><th>Heating rate [°C/s]</th><th>Holding time[s]</th><th>Cooling rate [°C/s]</th></tr></thead><tbody><tr><td>250.</td><td>2., 10., 20., 30.</td><td>2.</td></tr></tbody></table>

After these thermal cycles, tensile tests at room temperature (RT) were performed on an Instron 5969 tensile machine with a strain rate of 2.10⁻⁴s⁻¹. Strain measurement was obtained by using the GOM Aramis DIC system in the central zone of the specimen (6mmx10mm rectangle), which corresponds to the homogenous temperature of the sample. The tests were repeated 3 times for each condition. The YS, UTS and E values for all these tensile tests were then determined from the nominal stress-strain curves. Finally, at this stage, conductivity and hardness tests were carried out in this temperature-homogeneous zone of specimens.
operations compatible with industrial rates, we need to consider times of the order of 2 to 20 seconds for a complete heating and drawing cycle.

To know the influence of the heating cycle time, the objective of this study is to determine the evolution of the mechanical properties for very short heating cycle times to maintain the T6 state of 7075 alloy. For this purpose, different heating time were reproduced with a Gleeble 3500 machine on samples of AA7075-T6. The electrical conductivity and hardness were then measured and compared to the standard values to respect the T6 state. The optimum time values obtained were then validated by analysing the hardness and conductivity values on the final stamped parts of a warm-formed cylindrical cup [9,10] and a U-shaped channel [11,12].

## Material

Aluminium alloy sheet AA7075 with a thickness of 0.8mm is considered in this work. The commercially purchased aluminium is directly received in the T6 state. The chemical composition of this alloy is presented in Table 1. The mechanical properties (Yield Strength YS, Ultimate tensile strength UTS and elongation at break E%) and the conductivity and hardness properties measured under standard conditions to meet the T6 condition and the reception conditions for the alloy are given in Table 2.

<table><caption>Table 1 Chemical composition of AA7075-T6 sheet in %wt.</caption><thead><tr><th>Zn</th><th>Mg</th><th>Cu</th><th>Si</th><th>Cr</th><th>Fe</th><th>Mn</th><th>Ti</th><th>Al</th></tr></thead><tbody><tr><td>5.1-6.1</td><td>2.1-2.9</td><td>1.2-2.0</td><td>0.4</td><td>0.18-0.28</td><td>0.5</td><td>0.3</td><td>0.2</td><td>Balance</td></tr></tbody></table>

<table><caption>Table 2 Mechanical and electrical properties of AA7075-T6 to comply with the standard and those obtained at reception.</caption><thead><tr><th>Condition</th><th>YS [MPa]</th><th>UTS [MPa]</th><th>E [%]</th><th>Conductivity [MS/m]</th><th>Hardness [HV10]</th></tr></thead><tbody><tr><td>Standard</td><td>≥ 460</td><td>≥ 525</td><td>≥ 6</td><td>17-21</td><td>-</td></tr><tr><td>Reception</td><td>500</td><td>557</td><td>14.6</td><td>18.5</td><td>188</td></tr></tbody></table>

## Experimental conditions to control the T6-state

Different conditions of holding time at 200°C were reproduced using a Gleeble 3500 machine in order to reproduce the conventional conditions used in the multi-step warm forming process. With the Gleeble 3500 direct resistance heating system, a dog-bone specimen was heated by temperature with a thermocouple welded to the center of the specimen. The thermal cycles, presented in Table 3, consist of initial heating at 250 °C/s (200°C in 0.8 s), which corresponds to heating obtained by direct contact with heating tools. Four different holding times of 2, 10, 20 and 30 s were then applied, followed by cooling at 2°C/s, identical to cooling in free air.

<table><caption>Table 3 Conditions of heating rate, holding times and cooling rate.</caption><thead><tr><th>Heating rate [°C/s]</th><th>Holding time[s]</th><th>Cooling rate [°C/s]</th></tr></thead><tbody><tr><td>250.</td><td>2., 10., 20., 30.</td><td>2.</td></tr></tbody></table>

After these thermal cycles, tensile tests at room temperature (RT) were performed on an Instron 5969 tensile machine with a strain rate of 2.10⁻⁴s⁻¹. Strain measurement was obtained by using the GOM Aramis DIC system in the central zone of the specimen (6mmx10mm rectangle), which corresponds to the homogenous temperature of the sample. The tests were repeated 3 times for each condition. The YS, UTS and E values for all these tensile tests were then determined from the nominal stress-strain curves. Finally, at this stage, conductivity and hardness tests were carried out in this temperature-homogeneous zone of specimens.
## Results after tensile tests

Table 4 shows the results for electrical conductivity, hardness and mechanical properties obtained after various holding times at 200°C. These results must be compared with the values obtained at reception and with the standard values for maintaining the T6 state presented in Table 2.

<table><caption>Table 4 Mechanical and electrical properties of the AA7075-T6 alloy after various holding times.</caption><thead><tr><th>Holding time [s]</th><th>YS [MPa]</th><th>UTS [MPa]</th><th>E [%]</th><th>Conductivity [MS/m]</th><th>Hardness [HV10]</th></tr></thead><tbody><tr><td>2.</td><td>488±7.78</td><td>551±0.71</td><td>12.22±0.31</td><td>18.48±0.094</td><td>180±1.56</td></tr><tr><td>10.</td><td>459±8.49</td><td>538±2.69</td><td>11.58±0.08</td><td>18.55±0.090</td><td>175±1.42</td></tr><tr><td>20.</td><td>448±8.08</td><td>533±3.21</td><td>11.27±0.08</td><td>18.66±0.067</td><td>174±1.68</td></tr><tr><td>30.</td><td>442±1.41</td><td>526±0.67</td><td>12.04±1.77</td><td>18.71±0.079</td><td>172±1.07</td></tr></tbody></table>

The overall trend in conductivity and hardness presented in Table 4 shows an increase in conductivity and a decrease in hardness. However, the conductivity did not change between 2 and 10s compared with the values measured in reception condition. For longer holding times, an increase in conductivity of 18.66 and 18.71 MS/m was observed at 20 and 30s respectively, indicating an evolution of the metallurgical after a holding time of more than 10s. In the case of hardness, a continuous decrease was noted from 188 HV10 in reception condition to 172 HV10 after 30s of holding time. This decrease also indicates metallurgical changes which is in contradiction with the results of conductivity between 2 and 10s, i.e. decrease in hardness while conductivity does not change. This decrease in hardness can be associated with the phenomenon of coalescence of precipitates and can therefore imply a reduction in the mechanical characteristics.

Concerning these mechanical properties, there was no evolution from the reception condition after 2s of holding time. After 10s, YS and UTS decrease respectively of 29MPa and 13MPa and of 46MPa and 25MPa, respectively between 2 and 30s.

In comparison with the values required to comply the condition T6 (Table 2), for the YS, hardness and conductivity values, 10s appears to be the limit value for the holding time. Concerning the UTS, after 30s, its value is close to the standard value of 525MPa. It is also important to note that all the values of the total elongation are higher than the limit given by the standard (≥ 6%). However, this elongation condition is very difficult to control due to the thermocouple welded to the specimen to control temperature, which can lead to a premature rupture, and it is therefore not relevant to analyse this parameter. Finally, it appears that 10s seems an optimum time to keep the material in a T6 state.

To verify this optimum time, two warm forming process are tested in the following with different heating and forming times. Conductivity and hardness measurements are compared to control the validity of this time.

## Warm forming of a cylindrical cup in two steps

The first process involves warm forming of a cylindrical cup in two successively drawing steps on a 200 KN press. Fig. 1 presents the device developed by the CETIM including a heating step and two forming steps [9,10]. Fig. 2 shows the theoretical dimensions and the actual cup after each forming step, obtained from an initial diameter of the blank of 157 mm.
## Results after tensile tests

Table 4 shows the results for electrical conductivity, hardness and mechanical properties obtained after various holding times at 200°C. These results must be compared with the values obtained at reception and with the standard values for maintaining the T6 state presented in Table 2.

<table><caption>Table 4 Mechanical and electrical properties of the AA7075-T6 alloy after various holding times.</caption><thead><tr><th>Holding time [s]</th><th>YS [MPa]</th><th>UTS [MPa]</th><th>E [%]</th><th>Conductivity [MS/m]</th><th>Hardness [HV10]</th></tr></thead><tbody><tr><td>2.</td><td>488±7.78</td><td>551±0.71</td><td>12.22±0.31</td><td>18.48±0.094</td><td>180±1.56</td></tr><tr><td>10.</td><td>459±8.49</td><td>538±2.69</td><td>11.58±0.08</td><td>18.55±0.090</td><td>175±1.42</td></tr><tr><td>20.</td><td>448±8.08</td><td>533±3.21</td><td>11.27±0.08</td><td>18.66±0.067</td><td>174±1.68</td></tr><tr><td>30.</td><td>442±1.41</td><td>526±0.67</td><td>12.04±1.77</td><td>18.71±0.079</td><td>172±1.07</td></tr></tbody></table>

The overall trend in conductivity and hardness presented in Table 4 shows an increase in conductivity and a decrease in hardness. However, the conductivity did not change between 2 and 10s compared with the values measured in reception condition. For longer holding times, an increase in conductivity of 18.66 and 18.71 MS/m was observed at 20 and 30s respectively, indicating an evolution of the metallurgical after a holding time of more than 10s. In the case of hardness, a continuous decrease was noted from 188 HV10 in reception condition to 172 HV10 after 30s of holding time. This decrease also indicates metallurgical changes which is in contradiction with the results of conductivity between 2 and 10s, i.e. decrease in hardness while conductivity does not change. This decrease in hardness can be associated with the phenomenon of coalescence of precipitates and can therefore imply a reduction in the mechanical characteristics.

Concerning these mechanical properties, there was no evolution from the reception condition after 2s of holding time. After 10s, YS and UTS decrease respectively of 29MPa and 13MPa and of 46MPa and 25MPa, respectively between 2 and 30s.

In comparison with the values required to comply the condition T6 (Table 2), for the YS, hardness and conductivity values, 10s appears to be the limit value for the holding time. Concerning the UTS, after 30s, its value is close to the standard value of 525MPa. It is also important to note that all the values of the total elongation are higher than the limit given by the standard (≥ 6%). However, this elongation condition is very difficult to control due to the thermocouple welded to the specimen to control temperature, which can lead to a premature rupture, and it is therefore not relevant to analyse this parameter. Finally, it appears that 10s seems an optimum time to keep the material in a T6 state.

To verify this optimum time, two warm forming process are tested in the following with different heating and forming times. Conductivity and hardness measurements are compared to control the validity of this time.

## Warm forming of a cylindrical cup in two steps

The first process involves warm forming of a cylindrical cup in two successively drawing steps on a 200 KN press. Fig. 1 presents the device developed by the CETIM including a heating step and two forming steps [9,10]. Fig. 2 shows the theoretical dimensions and the actual cup after each forming step, obtained from an initial diameter of the blank of 157 mm.
<figure><img src="image_1112.png" /><figcaption>Fig. 1 Principle of press configuration with the heating step and the two forming cylindrical cup stages steps.</figcaption></figure>

<figure><img src="image_1113.png" /><figcaption>Fig. 2 a) Dimensions (in mm) of the part after step 1 and b) after step 2 [9].</figcaption></figure>

The heating stage is obtained by contact clamping of the sheet to achieve a uniform temperature of 200°C in 0.45s. The blank is then manually transferred to the first and second forming stations. These two forming stations are placed next to each other to limit the transfer time, but the total time to perform the total cycle time (heating and the two forming stages) is around 20s. Heating cartridges are used to heat all the tools and K-type thermocouples control their temperature. During the first forming, a clamping force of 66kN is applied while no force is used in the second stage. A force sensor measures the reaction force under the punch during both stamping processes. A pyrometer, placed 20 mm from the blank, measures the temperature of the blank during these 2 stages. A JELT 5411 lubricant is applied to the sheet before the tests.

The tests are carried out at RT and at 200°C. At RT, the sheet metal breaks during the first stage. This is not the case at 200°C where the drawing part does not show any breakage. Four tests are performed at 200°C to check the reproducibility. At the end of the test, conductivity and hardness are measured at the bottom of the cup.

The data obtained are presented in Table 5. It is difficult to assess the T6 condition uniquely with the conductivity and the hardness measurements alone. However, by comparing hardness results with the values of UTS and YS presented in Table 4, it is possible to make the link with the conservation of the T6 condition. Here, the conductivity of 18.42MS/m remains of the same order of magnitude compared to the reception state (Table 2). The hardness of 177HV10 is reduced
<figure><img src="image_1114.png" /><figcaption>Fig. 1 Principle of press configuration with the heating step and the two forming cylindrical cup stages steps.</figcaption></figure>

<figure><img src="image_1115.png" /><figcaption>Fig. 2 a) Dimensions (in mm) of the part after step 1 and b) after step 2 [9].</figcaption></figure>

The heating stage is obtained by contact clamping of the sheet to achieve a uniform temperature of 200°C in 0.45s. The blank is then manually transferred to the first and second forming stations. These two forming stations are placed next to each other to limit the transfer time, but the total time to perform the total cycle time (heating and the two forming stages) is around 20s. Heating cartridges are used to heat all the tools and K-type thermocouples control their temperature. During the first forming, a clamping force of 66kN is applied while no force is used in the second stage. A force sensor measures the reaction force under the punch during both stamping processes. A pyrometer, placed 20 mm from the blank, measures the temperature of the blank during these 2 stages. A JELT 5411 lubricant is applied to the sheet before the tests.

The tests are carried out at RT and at 200°C. At RT, the sheet metal breaks during the first stage. This is not the case at 200°C where the drawing part does not show any breakage. Four tests are performed at 200°C to check the reproducibility. At the end of the test, conductivity and hardness are measured at the bottom of the cup.

The data obtained are presented in Table 5. It is difficult to assess the T6 condition uniquely with the conductivity and the hardness measurements alone. However, by comparing hardness results with the values of UTS and YS presented in Table 4, it is possible to make the link with the conservation of the T6 condition. Here, the conductivity of 18.42MS/m remains of the same order of magnitude compared to the reception state (Table 2). The hardness of 177HV10 is reduced
compared to the reception state but it is still close to that of the standard values of YS and UTS to respect the T6 material (Table 4). The total cycle time of warm forming of 20s is therefore an ultimate time to maintain the T6-state in this case of warm forming.

<table><caption>Table 5 Results of hardness and conductivity obtained at the end of cylindrical cup forming in two steps</caption><thead><tr><td>Total cycle time [s]</td><td>Conductivity [MS/m]</td><td>Hardness [HV10]</td></tr></thead><tbody><tr><td>20</td><td>18.42±0.075</td><td>177±2.50</td></tr></tbody></table>

## Warm forming of a U-channel part

The second test corresponds to the multi-step warm forming of a U-Channel part whose dimensions of the final part are presented in Fig. 2.a-b [11,12]. After two blanking steps (depicted in blue), the blank is heated by clamping contact (in red) to obtain a uniform temperature of 200°C in 0.8s. The U-channel is then formed with tools heated by heating cartridges with an industrial rate of around 30 strokes/min. Based on the technique of the progressive die, between each of these steps, the part is transferred automatically without any manual handling. The part remains at the target temperature for approximately 2.6s which includes the part transfer and the forming operation. During the transfer from the heating step to the forming step, the heat loss is around 5°C and is therefore considered negligible.

The forming process is performed three times at RT and at 200°C to check the reproducibility. Springback is greatly reduced at 200°C compared to the part obtained at RT but it is still present (see Fig. 3-c).

<figure><img src="image_1114.png" /><figcaption>Fig.</figcaption></figure>

3 a) Three stages of the multi-step device in the industrial press with blank before and after cutting stages; b) Theoretical dimensions of the U-channel part in mm (width=60 mm); c) Final U-channel part at RT and at 200°C after springback.

To control the metallurgical condition after the two steps of forming, hardness and conductivity are measured three times in the undeformed areas indicated in Fig. 3.c. Table 6 present the hardness and conductivity results obtained at RT and 200°C.

<table><caption>Table 6 Hardness and conductivity results obtained at RT and 200°C on the U-channel part</caption><thead><tr><td>Temperature [°C]</td><td>Conductivity [MS/m]</td><td>Hardness [HV10]</td></tr></thead><tbody><tr><td>20</td><td>18.99±0.14</td><td>188±2.22</td></tr><tr><td>200</td><td>18.94±0.13</td><td>182±2.55</td></tr></tbody></table>
compared to the reception state but it is still close to that of the standard values of YS and UTS to respect the T6 material (Table 4). The total cycle time of warm forming of 20s is therefore an ultimate time to maintain the T6-state in this case of warm forming.

<table><caption>Table 5 Results of hardness and conductivity obtained at the end of cylindrical cup forming in two steps</caption><thead><tr><td>Total cycle time [s]</td><td>Conductivity [MS/m]</td><td>Hardness [HV10]</td></tr></thead><tbody><tr><td>20</td><td>18.42±0.075</td><td>177±2.50</td></tr></tbody></table>

## Warm forming of a U-channel part

The second test corresponds to the multi-step warm forming of a U-Channel part whose dimensions of the final part are presented in Fig. 2.a-b [11,12]. After two blanking steps (depicted in blue), the blank is heated by clamping contact (in red) to obtain a uniform temperature of 200°C in 0.8s. The U-channel is then formed with tools heated by heating cartridges with an industrial rate of around 30 strokes/min. Based on the technique of the progressive die, between each of these steps, the part is transferred automatically without any manual handling. The part remains at the target temperature for approximately 2.6s which includes the part transfer and the forming operation. During the transfer from the heating step to the forming step, the heat loss is around 5°C and is therefore considered negligible.

The forming process is performed three times at RT and at 200°C to check the reproducibility. Springback is greatly reduced at 200°C compared to the part obtained at RT but it is still present (see Fig. 3-c).

<figure><img src="image_1116.png" /><figcaption>Fig.</figcaption></figure>

3 a) Three stages of the multi-step device in the industrial press with blank before and after cutting stages; b) Theoretical dimensions of the U-channel part in mm (width=60 mm); c) Final U-channel part at RT and at 200°C after springback.

To control the metallurgical condition after the two steps of forming, hardness and conductivity are measured three times in the undeformed areas indicated in Fig. 3.c. Table 6 present the hardness and conductivity results obtained at RT and 200°C.

<table><caption>Table 6 Hardness and conductivity results obtained at RT and 200°C on the U-channel part</caption><thead><tr><td>Temperature [°C]</td><td>Conductivity [MS/m]</td><td>Hardness [HV10]</td></tr></thead><tbody><tr><td>20</td><td>18.99±0.14</td><td>188±2.22</td></tr><tr><td>200</td><td>18.94±0.13</td><td>182±2.55</td></tr></tbody></table>
Hardness is similar to the value obtained for the RT (Table 2) and with value measured after the Gleeble test with a holding time of 2s (Table 4) which is consistent with the total forming process time of 2.6s of this test. This is not the case for the conductivity, but this conductivity doesn't really evolve at 200°C in comparison with the values measured at RT, in the same condition after forming. Globally, there is no evolution of hardness and at this value of 182HV10, the YS and UTS are in the T6 state (see Table 2). This seems to confirm that the T6 state is preserved with this warm forming time in this case. To avoid any risk, all of these data indicate that the maximum temperature holding time at 200°C can be estimated at less 10s in order to preserve the mechanical characteristics of the T6 temper.

## Conclusion

After reproducing different heating cycles using a Gleeble machine, electrical conductivity, hardness and tensile tests made it possible to determine the temperature holding time at 200°C to maintain the T6 state during warm forming. A total time of less 10s must be used to be sure to respect the T6 state. Two warm forming tests have confirmed this cycle time to respect. However, this value constitutes a safety limit to avoid any evolution due for example to the composition of the alloy considered and in the case of any plastic deformation are applied on the part.

## Acknowledgements

The authors gratefully acknowledge the financial support of the Brittany Region (France) via the program RB EMBHOTAL.

## References

[1] T.Rahmaan, C.Butcher, K.J. Daun, J. Imbert, M.J. Worswick, High strain rate constitutive and fracture characterization of AA7075-T6 sheet under various stress states. International Journal of Impact Engineering, (2023).104812 https://doi.org/10.1016/j.ijimpeng.2023.104812

[2] Zheng K, Politis DJ, Wang L, Lin J. A review on forming techniques for manufacturing lightweight complex-shaped aluminium panel components. International Journal of Lightweight Materials and Manufacture (2018) 1(2):55-80. https://doi.org/10.1016/j.ijlmm.2018.03.006

[3] L. Chen, S. Fang, K. Zhao, R. Xiao, H. Zhai, Forming limit prediction using an integrated model for 7075 aluminum alloy sheets at an elevated temperature, International Journal of Solids and Structures. (2020) 202:475-485. https://doi.org/10.1016/j.ijsolstr.2020.06.030

[4] J. Lin, X. Bao, Y. Hou, J. Min, X. Qu, X. Tao, J. Chen, Investigation on Yield Behavior of 7075-T6 Aluminum Alloy at Elevated Temperatures. Chinese Journal of Mechanical Engineering. (2020) 33(1):76. https://doi.org/10.1186/s10033-020-00494-8

[5] J.A. Österreicher, M.A. Tunes, F. Grabner, A. Arnoldt, T. Kremmer, S. Pogatscher, C.M. Schlögl, Warm-forming of pre-aged Al-Zn-Mg-Cu alloy sheet. Materials & Design (2020) 193:108837. https://doi.org/10.1016/j.matdes.2020.108837

[6] A. Deschamps, G. Texier, S. Ringeval, L.Delfaut-Durut, Influence of cooling rate on the precipitation microstructure in a medium strength Al-Zn-Mg alloy. Materials Science and Engineering: A. (2009) 501(1):133-139. https://doi.org/10.1016/j.msea.2008.09.067.

[7] J. Zang, P. Dai, Y. Yang, S. Liu, B. Huang, J. Ru, X. Luo, Study on the Relationship between High Temperature Mechanical Properties and Precipitates Evolution of 7085 Al Alloy after Long Time Thermal Exposures. Metals (2021) 11(9), 1483. https://doi.org/10.3390/met11091483

[8] J.K. Zheng, X. Xu, R. Luo, X. Zeng, B. Chen, Degradation of precipitation hardening in 7075 alloy subject to thermal exposure: A Cs-corrected STEM study. Journal of Alloys and Compounds. (2018) 741:656-660. https://doi.org/10.1016/j.jallcom.2018.01.172
Hardness is similar to the value obtained for the RT (Table 2) and with value measured after the Gleeble test with a holding time of 2s (Table 4) which is consistent with the total forming process time of 2.6s of this test. This is not the case for the conductivity, but this conductivity doesn't really evolve at 200°C in comparison with the values measured at RT, in the same condition after forming. Globally, there is no evolution of hardness and at this value of 182HV10, the YS and UTS are in the T6 state (see Table 2). This seems to confirm that the T6 state is preserved with this warm forming time in this case. To avoid any risk, all of these data indicate that the maximum temperature holding time at 200°C can be estimated at less 10s in order to preserve the mechanical characteristics of the T6 temper.

## Conclusion

After reproducing different heating cycles using a Gleeble machine, electrical conductivity, hardness and tensile tests made it possible to determine the temperature holding time at 200°C to maintain the T6 state during warm forming. A total time of less 10s must be used to be sure to respect the T6 state. Two warm forming tests have confirmed this cycle time to respect. However, this value constitutes a safety limit to avoid any evolution due for example to the composition of the alloy considered and in the case of any plastic deformation are applied on the part.

## Acknowledgements

The authors gratefully acknowledge the financial support of the Brittany Region (France) via the program RB EMBHOTAL.

## References

[1] T.Rahmaan, C.Butcher, K.J. Daun, J. Imbert, M.J. Worswick, High strain rate constitutive and fracture characterization of AA7075-T6 sheet under various stress states. International Journal of Impact Engineering, (2023).104812 https://doi.org/10.1016/j.ijimpeng.2023.104812

[2] Zheng K, Politis DJ, Wang L, Lin J. A review on forming techniques for manufacturing lightweight complex-shaped aluminium panel components. International Journal of Lightweight Materials and Manufacture (2018) 1(2):55-80. https://doi.org/10.1016/j.ijlmm.2018.03.006

[3] L. Chen, S. Fang, K. Zhao, R. Xiao, H. Zhai, Forming limit prediction using an integrated model for 7075 aluminum alloy sheets at an elevated temperature, International Journal of Solids and Structures. (2020) 202:475-485. https://doi.org/10.1016/j.ijsolstr.2020.06.030

[4] J. Lin, X. Bao, Y. Hou, J. Min, X. Qu, X. Tao, J. Chen, Investigation on Yield Behavior of 7075-T6 Aluminum Alloy at Elevated Temperatures. Chinese Journal of Mechanical Engineering. (2020) 33(1):76. https://doi.org/10.1186/s10033-020-00494-8

[5] J.A. Österreicher, M.A. Tunes, F. Grabner, A. Arnoldt, T. Kremmer, S. Pogatscher, C.M. Schlögl, Warm-forming of pre-aged Al-Zn-Mg-Cu alloy sheet. Materials & Design (2020) 193:108837. https://doi.org/10.1016/j.matdes.2020.108837

[6] A. Deschamps, G. Texier, S. Ringeval, L.Delfaut-Durut, Influence of cooling rate on the precipitation microstructure in a medium strength Al-Zn-Mg alloy. Materials Science and Engineering: A. (2009) 501(1):133-139. https://doi.org/10.1016/j.msea.2008.09.067.

[7] J. Zang, P. Dai, Y. Yang, S. Liu, B. Huang, J. Ru, X. Luo, Study on the Relationship between High Temperature Mechanical Properties and Precipitates Evolution of 7085 Al Alloy after Long Time Thermal Exposures. Metals (2021) 11(9), 1483. https://doi.org/10.3390/met11091483

[8] J.K. Zheng, X. Xu, R. Luo, X. Zeng, B. Chen, Degradation of precipitation hardening in 7075 alloy subject to thermal exposure: A Cs-corrected STEM study. Journal of Alloys and Compounds. (2018) 741:656-660. https://doi.org/10.1016/j.jallcom.2018.01.172
[9] S. Royne, H. Laurent, A. Maillard, Numerical and Experimental Forming of a Cylindrical Cup in 7075-T6 Under Warm Temperature Conditions, Proceedings of the 14th International Conference on the Technology of Plasticity - Current Trends in the Technology of Plasticity, (2023) pp. 244-251. https://doi.org/10.1007/978-3-031-40920-2_26

[10] S. Royne, H. Laurent, A. Maillard, Forming of a cylindrical cup in 7075-T6 under warm temperature conditions, IOP Conference Series: Materials Science and Engineering, (2023) 1284(1), p. 012026. https://doi.org/10.1088/1757-899X/1284/1/012026.

[11] S. Royne, H. Laurent, A. Maillard, U-channel forming of an 7075-T6 in warm conditions. IOP Conference Series: Materials Science and Engineering (2022) 1238(1), 012011. https://dx.doi.org/10.1088/1757-899X/1238/1/012011.

[12] A. Maillard, C. Piat, A study of a progressive die for warm forming of aluminium. IDDRG Conference Proceedings - Innovations for the sheet metal industry, (2014) 300-306
[9] S. Royne, H. Laurent, A. Maillard, Numerical and Experimental Forming of a Cylindrical Cup in 7075-T6 Under Warm Temperature Conditions, Proceedings of the 14th International Conference on the Technology of Plasticity - Current Trends in the Technology of Plasticity, (2023) pp. 244-251. https://doi.org/10.1007/978-3-031-40920-2_26

[10] S. Royne, H. Laurent, A. Maillard, Forming of a cylindrical cup in 7075-T6 under warm temperature conditions, IOP Conference Series: Materials Science and Engineering, (2023) 1284(1), p. 012026. https://doi.org/10.1088/1757-899X/1284/1/012026.

[11] S. Royne, H. Laurent, A. Maillard, U-channel forming of an 7075-T6 in warm conditions. IOP Conference Series: Materials Science and Engineering (2022) 1238(1), 012011. https://dx.doi.org/10.1088/1757-899X/1238/1/012011.

[12] A. Maillard, C. Piat, A study of a progressive die for warm forming of aluminium. IDDRG Conference Proceedings - Innovations for the sheet metal industry, (2014) 300-306
# The plane strain compression test, an alternative to large strain hardening characterization

NEAG Adriana¹,²,a, BALAN Tudor¹,b*

¹Arts et Metiers Institute of Technology, Université de Lorraine, LCFC, F-57070 Metz, France

²Dept of Materials Science and Engineering, Technical University, Cluj-Napoca 400641, Romania

aadriana.neag@ipm.utcluj.ro, btudor.balan@ensam.eu

**Keywords:** Plane Strain Compression, Characterization, Large Strain

**Abstract.** Large strain characterization of sheet metals has become increasingly important with the generalization of advanced high strength steels, for which the tensile test provides data over a very reduced strain range. Among the numerous alternative characterization tests, the plane strain compression test (PSCT) requires a small amount of material and classical testing machine and acquisition. PSCT was mainly used for hot forming characterization, but recently it has been proved sufficiently accurate for application in cold metal forming. The aim of this work is to provide an in-depth validation of the PSCT by means of the finite element method. When converting the PSCT force-displacement curve into a stress-strain curve (flow curve), several analytical corrections are applied. The FE simulation of the test was used in order to validate these correction terms and their hypotheses. The originality of the approach is the design of a sequence of test configurations which allow for the individual validation of each and every one of the correction terms concerning the effect of: yield surface; friction; variation of sample width; variation of contact zone length; edge effects. The FE simulations showed that the analytical exploitation of the PSCT provides a very good accuracy. They help identifying the most suitable correction to consider the effects of yield surface shape, friction, edge effects. The traditional correction of the sample width as well as the more recent correction of the effective tool width were validated. The results show that the PSCT provides accurate flow curves and designate the best analytical expressions to be used.

**Introduction**

Stress-strain curves of sheet metal are crucial for realistic simulations of sheet metal forming. They are traditionally extracted from the uniaxial tensile test, which is limited to the uniform elongation. Thus, alternative methods are required to generate an accurate flow curve. Common alternatives in the literature are the bulge test, the simple shear test, the stack compression test, the in-plane torsion test, and the plane strain compression test. The plane strain compression test (PSCT) is one of the simplest, since it uses a simple test device and a universal tension-compression machine and standard force-displacement acquisition (see Figure 1). Moreover, it requires small amounts of sample material. It is thus an appealing test for industrial application. Traditionally applied for hot rolling simulations, several authors show that the test is sufficiently accurate for the flow curve determination at room temperature [1][2].

<figure><img src="image_1115.png" /></figure>
# The plane strain compression test, an alternative to large strain hardening characterization

NEAG Adriana¹,²,a, BALAN Tudor¹,b*

¹Arts et Metiers Institute of Technology, Université de Lorraine, LCFC, F-57070 Metz, France

²Dept of Materials Science and Engineering, Technical University, Cluj-Napoca 400641, Romania

aadriana.neag@ipm.utcluj.ro, btudor.balan@ensam.eu

**Keywords:** Plane Strain Compression, Characterization, Large Strain

**Abstract.** Large strain characterization of sheet metals has become increasingly important with the generalization of advanced high strength steels, for which the tensile test provides data over a very reduced strain range. Among the numerous alternative characterization tests, the plane strain compression test (PSCT) requires a small amount of material and classical testing machine and acquisition. PSCT was mainly used for hot forming characterization, but recently it has been proved sufficiently accurate for application in cold metal forming. The aim of this work is to provide an in-depth validation of the PSCT by means of the finite element method. When converting the PSCT force-displacement curve into a stress-strain curve (flow curve), several analytical corrections are applied. The FE simulation of the test was used in order to validate these correction terms and their hypotheses. The originality of the approach is the design of a sequence of test configurations which allow for the individual validation of each and every one of the correction terms concerning the effect of: yield surface; friction; variation of sample width; variation of contact zone length; edge effects. The FE simulations showed that the analytical exploitation of the PSCT provides a very good accuracy. They help identifying the most suitable correction to consider the effects of yield surface shape, friction, edge effects. The traditional correction of the sample width as well as the more recent correction of the effective tool width were validated. The results show that the PSCT provides accurate flow curves and designate the best analytical expressions to be used.

**Introduction**

Stress-strain curves of sheet metal are crucial for realistic simulations of sheet metal forming. They are traditionally extracted from the uniaxial tensile test, which is limited to the uniform elongation. Thus, alternative methods are required to generate an accurate flow curve. Common alternatives in the literature are the bulge test, the simple shear test, the stack compression test, the in-plane torsion test, and the plane strain compression test. The plane strain compression test (PSCT) is one of the simplest, since it uses a simple test device and a universal tension-compression machine and standard force-displacement acquisition (see Figure 1). Moreover, it requires small amounts of sample material. It is thus an appealing test for industrial application. Traditionally applied for hot rolling simulations, several authors show that the test is sufficiently accurate for the flow curve determination at room temperature [1][2].

<figure><img src="image_1117.png" /></figure>
<figure><img src="image_1116.png" /><figcaption>Figure 1. Schematic description of the PSCT test and corresponding notations</figcaption></figure>

The PSCT configuration is simple: a rectangular sample is compressed between two rectangular dies, wider than the part. The displacement of the upper tool is imposed and the force is measured. A stress-strain curve under the plane strain stress state is derived. The tools are lubricated based on graphite, glass, polymer (Teflon), oil... depending on the test temperature. The accuracy of the PSCT-based flow curve is affected by several factors: tool misalignment, stress concentration, heterogeneous strain distribution, friction. Becker and Pöhlandt [1] used a tool with a radius to diminish stress concentration. The effect of factors that cannot be entirely eliminated, like friction, require a correction of the flow curve calculation equation [4].

This paper investigates the various correction terms for PSCT flow curve calculation, by means of finite element simulations. A reference, known material law is selected and PSCT simulations are performed under different conditions. The upper die force is plotted as a function of displacement, and stress-strain curves are calculated by applying the PSCT analytical formulas to these FE-based force-displacement recordings. These curves can be validated with respect to the reference flow curve. Also, simulation allows to decouple the validation of each correction terms separately, which is impossible experimentally. For this, several 2D and 3D test configurations were simulated.

Finite element simulation of the PSCT

The FE simulation of the PSCT is performed using the Forge® commercial software. The selected FE code must provide a remeshing function since significant element distortion may be induced by the material flow in the area of the die corner. Two-dimensional and three-dimensional simulations allowed to validate the various hypotheses independently. Indeed, 2D simulations respect the plane strain hypothesis utilized in the analytical calculation, thus allowing for the validation of the friction and shear corrections. Then, 3D simulations allowed for the verification of the width increase correction. Figure 2a-b illustrates the increase of the sample width as the plane strain compression proceeds, in particular when large strain levels are reached.

Fine mesh density is required in order to describe the strain gradients that develop in the active area of the sample (see for example Figure 2c). Furthermore, sudden variations in the force evolution are observed when a node loses contact with the die as the lateral flow progresses. As seen in Figure 3, even when the average force evolution curve becomes independent of the mesh size, the amplitude of the force variation due to contact evolution continues du diminish as the mesh size diminishes. An average element size of 0.05 mm was selected in the active area of the sample. Three symmetry planes were used in order to reduce computing time
<figure><img src="image_1118.png" /><figcaption>Figure 1. Schematic description of the PSCT test and corresponding notations</figcaption></figure>

The PSCT configuration is simple: a rectangular sample is compressed between two rectangular dies, wider than the part. The displacement of the upper tool is imposed and the force is measured. A stress-strain curve under the plane strain stress state is derived. The tools are lubricated based on graphite, glass, polymer (Teflon), oil... depending on the test temperature. The accuracy of the PSCT-based flow curve is affected by several factors: tool misalignment, stress concentration, heterogeneous strain distribution, friction. Becker and Pöhlandt [1] used a tool with a radius to diminish stress concentration. The effect of factors that cannot be entirely eliminated, like friction, require a correction of the flow curve calculation equation [4].

This paper investigates the various correction terms for PSCT flow curve calculation, by means of finite element simulations. A reference, known material law is selected and PSCT simulations are performed under different conditions. The upper die force is plotted as a function of displacement, and stress-strain curves are calculated by applying the PSCT analytical formulas to these FE-based force-displacement recordings. These curves can be validated with respect to the reference flow curve. Also, simulation allows to decouple the validation of each correction terms separately, which is impossible experimentally. For this, several 2D and 3D test configurations were simulated.

Finite element simulation of the PSCT

The FE simulation of the PSCT is performed using the Forge® commercial software. The selected FE code must provide a remeshing function since significant element distortion may be induced by the material flow in the area of the die corner. Two-dimensional and three-dimensional simulations allowed to validate the various hypotheses independently. Indeed, 2D simulations respect the plane strain hypothesis utilized in the analytical calculation, thus allowing for the validation of the friction and shear corrections. Then, 3D simulations allowed for the verification of the width increase correction. Figure 2a-b illustrates the increase of the sample width as the plane strain compression proceeds, in particular when large strain levels are reached.

Fine mesh density is required in order to describe the strain gradients that develop in the active area of the sample (see for example Figure 2c). Furthermore, sudden variations in the force evolution are observed when a node loses contact with the die as the lateral flow progresses. As seen in Figure 3, even when the average force evolution curve becomes independent of the mesh size, the amplitude of the force variation due to contact evolution continues du diminish as the mesh size diminishes. An average element size of 0.05 mm was selected in the active area of the sample. Three symmetry planes were used in order to reduce computing time
<figure><img src="image_1117.png" /><figcaption>Figure 2. Typical results of 3D PSCT simulations: a) displacement of material points in the punch travel (z) direction; b) displacement in the y direction; c) equivalent strain distribution.</figcaption></figure>

<figure><img src="image_1118.png" /><figcaption>Figure 3. Evolution of the die force for two converged mesh sizes.</figcaption></figure>

## Results and discussion

The PSCT flow curve analytical formulas are based on the slab method analysis of the deformation zone. The boundary condition at the die exit surfaces has an influence on the results and was questioned repeatedly in the literature. Becker and Pöhland [1] introduced a so-called shear correction factor in this respect. According to Figure 4, this shear correction slightly improves the flow curve calculation, provided that the friction coefficient (and law) is exactly known, as is the
<figure><img src="image_1119.png" /><figcaption>Figure 2. Typical results of 3D PSCT simulations: a) displacement of material points in the punch travel (z) direction; b) displacement in the y direction; c) equivalent strain distribution.</figcaption></figure>

<figure><img src="image_1120.png" /><figcaption>Figure 3. Evolution of the die force for two converged mesh sizes.</figcaption></figure>

## Results and discussion

The PSCT flow curve analytical formulas are based on the slab method analysis of the deformation zone. The boundary condition at the die exit surfaces has an influence on the results and was questioned repeatedly in the literature. Becker and Pöhland [1] introduced a so-called shear correction factor in this respect. According to Figure 4, this shear correction slightly improves the flow curve calculation, provided that the friction coefficient (and law) is exactly known, as is the
case here. However, when a real experiment is analyzed, this improvement may not be systematically observed.

Another classical correction term aims to take into consideration the increase of the sample width, which induces an increase in the contact surface between dies and sample, thus affecting the calculation of the average normal stress. The simulation of the width increase required a full 3D simulation. Figure 5 shows that this correction term compensates very well the influence of the width increase on the flow curve calculation. The same conclusions were obtained with small to moderate friction conditions. Finally, three different friction correction terms were confronted to each other using both 2D and 3D simulations of the PSCT. In both cases, the friction correction is necessary and the three proposed correction terms are almost identical.

<figure><img src="image_1119.png" /><figcaption>Figure 4. Influence of the shear correction factor on the PSCT-based flow curve.</figcaption></figure>

The influence of a tool misalignment was also investigated using 2D simulations. While no correction term can compensate for an accidental (thus unknown) misalignment, the simulation results show that the error can be neglected for tool misalignments inferior to 0.2 mm. The error approaches 10% when the tool misalignment reaches 0.5 mm for the given sheet and die geometries (Figure 6).

## Conclusions

The plane strain compression test can be used as an economical method for the calculation of flow curve for sheet metals at large strains and at room temperature. The flow curve calculation includes several correction factors to compensate for edge effects, friction effects etc. The numerical simulation of the test provides a means for their independent validation, very difficult or even impossible to achieve experimentally. The investigation revealed that the so-called shear correction of the edge effects does not improve significantly the results and can be avoided. In turn, the classical compensation terms for the specimen width increase and the friction influence proved both necessary and accurate. In particular, the performed simulations could not show any difference between the correction terms using Coulomb and Tresca friction laws. In any case, these corrections are only efficient for low to moderate friction conditions, thus friction should be carefully reduced in the experiments.
case here. However, when a real experiment is analyzed, this improvement may not be systematically observed.

Another classical correction term aims to take into consideration the increase of the sample width, which induces an increase in the contact surface between dies and sample, thus affecting the calculation of the average normal stress. The simulation of the width increase required a full 3D simulation. Figure 5 shows that this correction term compensates very well the influence of the width increase on the flow curve calculation. The same conclusions were obtained with small to moderate friction conditions. Finally, three different friction correction terms were confronted to each other using both 2D and 3D simulations of the PSCT. In both cases, the friction correction is necessary and the three proposed correction terms are almost identical.

<figure><img src="image_1121.png" /><figcaption>Figure 4. Influence of the shear correction factor on the PSCT-based flow curve.</figcaption></figure>

The influence of a tool misalignment was also investigated using 2D simulations. While no correction term can compensate for an accidental (thus unknown) misalignment, the simulation results show that the error can be neglected for tool misalignments inferior to 0.2 mm. The error approaches 10% when the tool misalignment reaches 0.5 mm for the given sheet and die geometries (Figure 6).

## Conclusions

The plane strain compression test can be used as an economical method for the calculation of flow curve for sheet metals at large strains and at room temperature. The flow curve calculation includes several correction factors to compensate for edge effects, friction effects etc. The numerical simulation of the test provides a means for their independent validation, very difficult or even impossible to achieve experimentally. The investigation revealed that the so-called shear correction of the edge effects does not improve significantly the results and can be avoided. In turn, the classical compensation terms for the specimen width increase and the friction influence proved both necessary and accurate. In particular, the performed simulations could not show any difference between the correction terms using Coulomb and Tresca friction laws. In any case, these corrections are only efficient for low to moderate friction conditions, thus friction should be carefully reduced in the experiments.
<figure><img src="image_1120.png" /><figcaption>Figure 5. Influence of the width correction factor on the PSCT-based flow curve using 3D simulations of the PSCT test.</figcaption></figure>

<figure><img src="image_1121.png" /><figcaption>Figure 6. Error in the predicted flow curves for tool misalignment of 0.1 mm (blue) and 0.5 mm (orange).</figcaption></figure>

## References

[1] Becker N, Pöhlandt K, Improvement of the plane-strain compression test for determining flow curves. Ann CIRP 38 (1989) 227-230. https://doi.org/10.1016/S0007-8506(07)62691-2

[2] Chermette C, Unruh K, Peshekhodov I, Chottin J, Balan T, A new analytical method for determination of the flow curve for high-strength sheet steels using the PSCT, Int J Material Forming 13 (2020) 269-292. https://doi.org/10.1007/s12289-019-01485-4

[3] Gelin JC, Ghouati O, Shahani R (1994) Modelling the plane strain compression test to obtain constitutive equations of aluminium alloys. Int J Mech Sci 36(9):773-796.
https://doi.org/10.1016/0020-7403(94)90009-4

[4] Aksenov S, Kliber J, Puzino Y, Bober S, Processing of PSCT results for investigation of AISI-304 stainless steel constitutive behavior. J Chem Techn Metallurgy 50 (2015) 644-650
<figure><img src="image_1122.png" /><figcaption>Figure 5. Influence of the width correction factor on the PSCT-based flow curve using 3D simulations of the PSCT test.</figcaption></figure>

<figure><img src="image_1123.png" /><figcaption>Figure 6. Error in the predicted flow curves for tool misalignment of 0.1 mm (blue) and 0.5 mm (orange).</figcaption></figure>

## References

[1] Becker N, Pöhlandt K, Improvement of the plane-strain compression test for determining flow curves. Ann CIRP 38 (1989) 227-230. https://doi.org/10.1016/S0007-8506(07)62691-2

[2] Chermette C, Unruh K, Peshekhodov I, Chottin J, Balan T, A new analytical method for determination of the flow curve for high-strength sheet steels using the PSCT, Int J Material Forming 13 (2020) 269-292. https://doi.org/10.1007/s12289-019-01485-4

[3] Gelin JC, Ghouati O, Shahani R (1994) Modelling the plane strain compression test to obtain constitutive equations of aluminium alloys. Int J Mech Sci 36(9):773-796. https://doi.org/10.1016/0020-7403(94)90009-4

[4] Aksenov S, Kliber J, Puzino Y, Bober S, Processing of PSCT results for investigation of AISI-304 stainless steel constitutive behavior. J Chem Techn Metallurgy 50 (2015) 644-650
# Impact of self-heating during hot deformation in the post-dynamic recrystallization and grain growth of an Fe-based superalloy

POTENCIANO Antonio¹, a*, BERNACKI Marc¹, b, NICOLAY Alexis¹, c,
COLLIN Cyrille¹, d, DAIRON Jonathan², e, DANYLOVA Olena², f and
BOZZOLO Nathalie¹, g

¹Mines Paris, PSL University, Centre for material forming (CEMEF), UMR CNRS, 06904 Sophia Antipolis, France

²Aperam Alloys, 58160 Imphy, France

aantonio.potenciano_carpintero@minesparis.psl.eu, bmarc.bernacki@minesparis.psl.eu,

calexis.nicolay@minesparis.psl.eu, dcyrille.collin@minesparis.psl.eu,

ejonathan.dairon@aperam.com, folena.danylova@aperam.com,

gnathalie.bozzolo@minesparis.psl.eu

**Keywords:** Self-Heating, Microstructure, Recrystallization, Superalloy, Gleeble

**Abstract.** During hot working, strain rate, strain level and the temperature of deformation, and hence the Zener-Hollomon parameter, become the regulatory thermomechanical parameters of the resultant stored energy value and distribution. An accurate estimation of the aforementioned parameters is thus required to control microstructural homogeneity in the post-dynamic state, since stored energy value and distribution directly affects post-dynamic recrystallization and grain growth mechanisms. In order to do so, the self-heating phenomena that becomes significant at high strain rate hot deformation must be taken into consideration. In this work, several cylindrical samples of the A-286 alloy are submitted to hot compression tests performed in a Gleeble 3500 machine. The microstructure in the deformed-state is analyzed at the center, and the effect of the thermomechanical parameters of hot deformation on the resulting microstructure is discussed. In addition to this, the local thermomechanical testing conditions are estimated considering self-heating and local strain level and strain rate fields. Hence, the relationship between these newly estimated parameters and the microstructure can be assessed.

**Introduction**

Fe-based superalloys constitute a cost-effective alternative to Ni-based superalloys, operating in a range of less extreme temperatures. In this context, the A-286 alloy is a Fe-based superalloy employed in the manufacture of various engine and gas turbine components for a wide range of sectors, including aeronautics, automotive, hydrogen power generation, and oil & gas [1-2], for applications in harsh environments. Hence, fine and homogeneous microstructures are required in order to guarantee mechanical properties stability, optimizing the in-use performance and durability of mechanical components. Along the manufacturing route, after successive hot deformations and a subsequent quenching as part of a hot rolling process, this alloy is submitted to a solution heat treatment. This annealing provides good formability, for the sake of recrystallization for further deformation steps, or for the sake of alloying element solutioning prior to controlled precipitation hardening. Under certain specific conditions, heterogeneous grain growth can be activated during the aforementioned heat treatment.

It is known that static recrystallization can be onset in the presence of stored energy, whose value and distribution have a direct impact on the resulting grain size. Store energy is, in turn, dependent on the thermomechanical parameters of hot deformation. During high strain rate hot deformation, significant self-heating can occur. Several hot compression tests, in adiabatic

<figure><img src="image_1122.png" /></figure>
# Impact of self-heating during hot deformation in the post-dynamic recrystallization and grain growth of an Fe-based superalloy

POTENCIANO Antonio<sup>1,a*</sup>, BERNACKI Marc<sup>1,b</sup>, NICOLAY Alexis<sup>1,c</sup>, COLLIN Cyrille<sup>1,d</sup>, DAIRON Jonathan<sup>2,e</sup>, DANYLOVA Olena<sup>2,f</sup> and BOZZOLO Nathalie<sup>1,g</sup>

<sup>1</sup>Mines Paris, PSL University, Centre for material forming (CEMEF), UMR CNRS, 06904 Sophia Antipolis, France

<sup>2</sup>Aperam Alloys, 58160 Imphy, France

<sup>a</sup>antonio.potenciano_carpintero@minesparis.psl.eu, <sup>b</sup>marc.bernacki@minesparis.psl.eu,

<sup>c</sup>alexis.nicolay@minesparis.psl.eu, <sup>d</sup>cyrille.collin@minesparis.psl.eu,

<sup>e</sup>jonathan.dairon@aperam.com, <sup>f</sup>olena.danylova@aperam.com,

<sup>g</sup>nathalie.bozzolo@minesparis.psl.eu

**Keywords:** Self-Heating, Microstructure, Recrystallization, Superalloy, Gleeble

**Abstract.** During hot working, strain rate, strain level and the temperature of deformation, and hence the Zener-Hollomon parameter, become the regulatory thermomechanical parameters of the resultant stored energy value and distribution. An accurate estimation of the aforementioned parameters is thus required to control microstructural homogeneity in the post-dynamic state, since stored energy value and distribution directly affects post-dynamic recrystallization and grain growth mechanisms. In order to do so, the self-heating phenomena that becomes significant at high strain rate hot deformation must be taken into consideration. In this work, several cylindrical samples of the A-286 alloy are submitted to hot compression tests performed in a Gleeble 3500 machine. The microstructure in the deformed-state is analyzed at the center, and the effect of the thermomechanical parameters of hot deformation on the resulting microstructure is discussed. In addition to this, the local thermomechanical testing conditions are estimated considering self-heating and local strain level and strain rate fields. Hence, the relationship between these newly estimated parameters and the microstructure can be assessed.

**Introduction**

Fe-based superalloys constitute a cost-effective alternative to Ni-based superalloys, operating in a range of less extreme temperatures. In this context, the A-286 alloy is a Fe-based superalloy employed in the manufacture of various engine and gas turbine components for a wide range of sectors, including aeronautics, automotive, hydrogen power generation, and oil & gas [1-2], for applications in harsh environments. Hence, fine and homogeneous microstructures are required in order to guarantee mechanical properties stability, optimizing the in-use performance and durability of mechanical components. Along the manufacturing route, after successive hot deformations and a subsequent quenching as part of a hot rolling process, this alloy is submitted to a solution heat treatment. This annealing provides good formability, for the sake of recrystallization for further deformation steps, or for the sake of alloying element solutioning prior to controlled precipitation hardening. Under certain specific conditions, heterogeneous grain growth can be activated during the aforementioned heat treatment.

It is known that static recrystallization can be onset in the presence of stored energy, whose value and distribution have a direct impact on the resulting grain size. Store energy is, in turn, dependent on the thermomechanical parameters of hot deformation. During high strain rate hot deformation, significant self-heating can occur. Several hot compression tests, in adiabatic

<figure><img src="image_1124.png" /></figure>
conditions [3-4], are performed in a Gleeble 3500 machine, covering a wide thermomechanical window where noticeable self-heating phenomena and an acceleration of dynamic recrystallization kinetics is expected. With the aim to reproduce industrial processing conditions, Joule heating can be stopped during the compression stage, due to the compensatory effect of self-heating phenomena at the tested conditions and due to the high strain rate at which the tests are conducted which results in a fast deformation with no time for a cool down.

In order to assess the thermomechanical parameters of hot deformation that will be discussed in the present work, a single parameter is often used to represent the combined effects of strain rate (ε̇) and deformation temperature (T, in K): the Zener-Hollomon parameter (Z) [5], defined as:

$$Z = \dot{\epsilon} \cdot \exp\left(\frac{Q}{RT}\right), \quad (1)$$

where R (J·mol⁻¹·K⁻¹) is the gas constant and Q (J·mol⁻¹) stands for the activation energy for hot deformation.

Dynamic and post-dynamic recrystallized grain size is a direct consequence of plastic stored energy [6]. Recrystallized grain size is then a microstructural parameter of interest in which the effects of the thermomechanical parameters of hot deformation can be analyzed. Derby's law presents a power-law relationship between recrystallized grain size and Zener-Hollomon parameter [7]:

$$D_{RX} = aZ^b, \quad (2)$$

with a, b two constant coefficients.

Hence, this article presents an analysis of Derby's law with respect to different Zener-Hollomon parameter estimations (considering self-heating or not), by which it would be possible to discuss the impact of self-heating on the deformed-state microstructure of the A-286 alloy and the relevance of considering it when describing dynamic and post-dynamic microstructural evolutions. Therefore, this article aims to establish a link between the thermomechanical processing conditions and the metallurgical parameters of interest for heterogeneous grain growth in static conditions.

A specific nomenclature will be used to identify whether self-heating is taken into account or not in parameter value estimation:

* Z₀(ε̇n, Tn) and εn -also expressed as Z₀(ε̇nominal, Tnominal) and εnominal- refer to the nominal Zener-Hollomon parameter value and nominal strain level, respectively, at which the thermomechanical test is desired to be performed.

* Z₁(ε̇f, TSH) and εSH represent the estimated value at the end of deformation, considering self-heating, with ε̇f = εSH/tcompression. They correspond to the local values at the same central region of the sample at which microstructural observations are performed (Fig. 2).

* Z₂(ε̇n, Ti) refers to the local values at the same central region of the sample as before, but Ti is the temperature reached at the last deformation-free stage previous to the compression, without any possible self-heating effect.
conditions [3-4], are performed in a Gleeble 3500 machine, covering a wide thermomechanical window where noticeable self-heating phenomena and an acceleration of dynamic recrystallization kinetics is expected. With the aim to reproduce industrial processing conditions, Joule heating can be stopped during the compression stage, due to the compensatory effect of self-heating phenomena at the tested conditions and due to the high strain rate at which the tests are conducted which results in a fast deformation with no time for a cool down.

In order to assess the thermomechanical parameters of hot deformation that will be discussed in the present work, a single parameter is often used to represent the combined effects of strain rate (ε̇) and deformation temperature (T, in K): the Zener-Hollomon parameter (Z) [5], defined as:

$$Z = \dot{\epsilon} \cdot \exp\left(\frac{Q}{RT}\right), \quad (1)$$

where R (J·mol⁻¹·K⁻¹) is the gas constant and Q (J·mol⁻¹) stands for the activation energy for hot deformation.

Dynamic and post-dynamic recrystallized grain size is a direct consequence of plastic stored energy [6]. Recrystallized grain size is then a microstructural parameter of interest in which the effects of the thermomechanical parameters of hot deformation can be analyzed. Derby's law presents a power-law relationship between recrystallized grain size and Zener-Hollomon parameter [7]:

$$D_{RX} = aZ^b, \quad (2)$$

with a, b two constant coefficients.

Hence, this article presents an analysis of Derby's law with respect to different Zener-Hollomon parameter estimations (considering self-heating or not), by which it would be possible to discuss the impact of self-heating on the deformed-state microstructure of the A-286 alloy and the relevance of considering it when describing dynamic and post-dynamic microstructural evolutions. Therefore, this article aims to establish a link between the thermomechanical processing conditions and the metallurgical parameters of interest for heterogeneous grain growth in static conditions.

A specific nomenclature will be used to identify whether self-heating is taken into account or not in parameter value estimation:

* Z₀(ε̇n, Tn) and εn -also expressed as Z₀(ε̇nominal, Tnominal) and εnominal- refer to the nominal Zener-Hollomon parameter value and nominal strain level, respectively, at which the thermomechanical test is desired to be performed.

* Z₁(ε̇f, TSH) and εSH represent the estimated value at the end of deformation, considering self-heating, with ε̇f = εSH/tcompression. They correspond to the local values at the same central region of the sample at which microstructural observations are performed (Fig. 2).

* Z₂(ε̇n, Ti) refers to the local values at the same central region of the sample as before, but Ti is the temperature reached at the last deformation-free stage previous to the compression, without any possible self-heating effect.
* $Z_3(\dot{\epsilon}_f, T_i)$ differs from $Z_2(\dot{\epsilon}_n, T_i)$ in the strain rate value considered. In this case, local strain rate at the end of the deformation ($\dot{\epsilon}_f = \epsilon_{SH}/t_{compression}$) is used.

## Material and experimental procedure

All experiments were conducted on cylindrical specimens (ø10 mm x L12 mm) sectioned from a 10mm diameter hot rolled bar of A-286 alloy, whose chemical composition is given in Table 1.

<table><caption>Table 1. A-286 superalloy chemical composition.</caption><thead><tr><th>Elem.</th><th>Fe</th><th>Ni</th><th>Cr</th><th>Ti</th><th>Mo</th><th>V</th><th>Mn</th><th>Si</th><th>Al</th><th>C</th><th>S</th><th>B</th></tr></thead><tbody><tr><th>% Wt.</th><td>Base</td><td>27.00 – 24.00</td><td>16.00 – 13.50</td><td>2.35 – 1.90</td><td>1.50 – 1.00</td><td>0.50 – 0.10</td><td>&lt; 2</td><td>&lt; 1.0</td><td>&lt; 0.35</td><td>&lt; 0.08</td><td>&lt; 0.03</td><td>&lt; 0.01</td></tr></tbody></table>

The samples are submitted to a specific thermomechanical path to ensure that the microstructure in the state previous to the compression is homogeneous and fully recrystallized (Fig. 1). This is achieved by means of an initial 5 minutes heat treatment in the Gleeble machine at a temperature above the testing temperature ($T_{test}$), followed by a sudden temperature descent for a 1-minute-long holding at $T_{test}$. The tests are carried out at high strain rates $\dot{\epsilon}_{nominal} \in [20, 80] s^{-1}$ and different strain levels $\epsilon_{nominal} \in [0.25, 0.8]$.

## Metallographic preparation and microstructural characterization

After hot compression, samples were cut into identical quarters, aiming to save material and to be able to analyze the microstructure both in the compressed-state and after a final annealing process. The cut was made across the compression direction, i.e. the longitudinal direction of the deformed cylinder. Microstructure analyses were made as near as possible to the center of the sample (at around 150 µm distance to the center in the compression direction) in both cylindrical axes considering that a quarter of the original specimen was utilized. Metallographic surface preparation was carried out by means of an initial grinding using SiC abrasive papers down to a SiC granulometry of 5 µm, and then a mechanical polishing step using 3 µm diamond suspension. For optical microscopy observations, the preparation is completed by an electrochemical etching during 100 s at 1 V using $HNO_3$ (67%) to reveal grain boundaries. In order to obtain an adequate surface quality for scanning electron microscopy (SEM) and electron backscattered diffraction (EBSD) analyses, a final step after mechanical polishing consisting on an electropolishing for 4 s with a A3 Struers electrolyte and operating voltage of 25 V at 25 °C was considered.

EBSD maps were acquired using a Carl Zeiss Supra 40 field emission gun scanning electron microscope (FEGSEM) equipped with a Bruker Quantax system comprising an EBSD eF-FlashHR detector and the Esprit 2.1 software package. Step size was selected considering the dimensions of the scanned area and microstructure characteristic sizes, between 1.2 µm to 2.6 µm per pixel. The acceleration voltage was 20 kV, and the working distance was 15 mm. The open-source MTEX toolbox version 5.1.1 was used for the post-processing of EBSD data [8], using Matlab version R2018a. The high indexing rate, above 98%, indicated that no data cleaning was necessary. Grains must be composed by more than 3 pixels, with a lower threshold of 10° misorientation angle to define grain boundaries. Twins are identified by a 60° misorientation along the axis <111> with a 5° tolerance. All EBSD data were filtered using the half-quadratic filter developed by Bergmann et al. [9] to reduce measurement noise. The half-quadratic filter effectively removes the spatially independent noise from the orientation measurement data while maintaining the sharp gradients at the grain boundaries and subgrain boundaries [10]. Scanning electron micrographs were obtained using the backscattered electron detector with 12 kV voltage at a working distance of 10 mm. The orientation data is represented
* $Z_3(\dot{\epsilon}_f, T_i)$ differs from $Z_2(\dot{\epsilon}_n, T_i)$ in the strain rate value considered. In this case, local strain rate at the end of the deformation ($\dot{\epsilon}_f = \epsilon_{SH}/t_{compression}$) is used.

## Material and experimental procedure

All experiments were conducted on cylindrical specimens (ø10 mm x L12 mm) sectioned from a 10mm diameter hot rolled bar of A-286 alloy, whose chemical composition is given in Table 1.

<table><caption>Table 1. A-286 superalloy chemical composition.</caption><thead><tr><th>Elem.</th><th>Fe</th><th>Ni</th><th>Cr</th><th>Ti</th><th>Mo</th><th>V</th><th>Mn</th><th>Si</th><th>Al</th><th>C</th><th>S</th><th>B</th></tr></thead><tbody><tr><th>% Wt.</th><td>Base</td><td>27.00 – 24.00</td><td>16.00 – 13.50</td><td>2.35 – 1.90</td><td>1.50 – 1.00</td><td>0.50 – 0.10</td><td>&lt; 2</td><td>&lt; 1.0</td><td>&lt; 0.35</td><td>&lt; 0.08</td><td>&lt; 0.03</td><td>&lt; 0.01</td></tr></tbody></table>

The samples are submitted to a specific thermomechanical path to ensure that the microstructure in the state previous to the compression is homogeneous and fully recrystallized (Fig. 1). This is achieved by means of an initial 5 minutes heat treatment in the Gleeble machine at a temperature above the testing temperature ($T_{test}$), followed by a sudden temperature descent for a 1-minute-long holding at $T_{test}$. The tests are carried out at high strain rates $\dot{\epsilon}_{nominal} \in [20, 80] s^{-1}$ and different strain levels $\epsilon_{nominal} \in [0.25, 0.8]$.

## Metallographic preparation and microstructural characterization

After hot compression, samples were cut into identical quarters, aiming to save material and to be able to analyze the microstructure both in the compressed-state and after a final annealing process. The cut was made across the compression direction, i.e. the longitudinal direction of the deformed cylinder. Microstructure analyses were made as near as possible to the center of the sample (at around 150 µm distance to the center in the compression direction) in both cylindrical axes considering that a quarter of the original specimen was utilized. Metallographic surface preparation was carried out by means of an initial grinding using SiC abrasive papers down to a SiC granulometry of 5 µm, and then a mechanical polishing step using 3 µm diamond suspension. For optical microscopy observations, the preparation is completed by an electrochemical etching during 100 s at 1 V using $HNO_3$ (67%) to reveal grain boundaries. In order to obtain an adequate surface quality for scanning electron microscopy (SEM) and electron backscattered diffraction (EBSD) analyses, a final step after mechanical polishing consisting on an electropolishing for 4 s with a A3 Struers electrolyte and operating voltage of 25 V at 25 °C was considered.

EBSD maps were acquired using a Carl Zeiss Supra 40 field emission gun scanning electron microscope (FEGSEM) equipped with a Bruker Quantax system comprising an EBSD e⁻ FlashHR detector and the Esprit 2.1 software package. Step size was selected considering the dimensions of the scanned area and microstructure characteristic sizes, between 1.2 µm to 2.6 µm per pixel. The acceleration voltage was 20 kV, and the working distance was 15 mm. The open-source MTEX toolbox version 5.1.1 was used for the post-processing of EBSD data [8], using Matlab version R2018a. The high indexing rate, above 98%, indicated that no data cleaning was necessary. Grains must be composed by more than 3 pixels, with a lower threshold of 10° misorientation angle to define grain boundaries. Twins are identified by a 60° misorientation along the axis <111> with a 5° tolerance. All EBSD data were filtered using the half-quadratic filter developed by Bergmann et al. [9] to reduce measurement noise. The half-quadratic filter effectively removes the spatially independent noise from the orientation measurement data while maintaining the sharp gradients at the grain boundaries and subgrain boundaries [10]. Scanning electron micrographs were obtained using the backscattered electron detector with 12 kV voltage at a working distance of 10 mm. The orientation data is represented
by the Kernel Average Misorientation (KAM) parameter. The KAM parameter is a value assigned to each pixel/point in the EBSD map that indicates the average misorientation angle between that pixel and its neighbors. It can be expressed mathematically by the following formula:

$$KAM(p_i) = \frac{1}{v_i} \sum_{j=1}^{v_i} \theta_{i,j}, \qquad (3)$$

where $\theta_{i,j}$ is the misorientation angle between pixel $p_i$ and pixel $p_j$, and $v_i$ is the number of neighboring pixels. For the calculation of KAM, only first and second order neighbors will be considered, and misorientations above 10°, that correspond to grain boundaries, are ignored.

<figure><img src="image_1123.png" /><figcaption>Fig. 1. a), c) and d) Microstructure in the state previous to hot compression, after an initial 5-minutes-long heat treatment and a 1-minute-long heat treatment at the test temperature, in the Gleeble. a) 2nd order KAM map with grain boundaries in white (defined by a misorientation angle higher than 10°). b) Grain size surface fraction histogram. c) and d) Backscattered electrons SEM images, with TiN or TiC as the black second phase particles.</figcaption></figure>

## Initial microstructure before compression

In Fig. 1 c and Fig. 1 d, it can be seen that there are second-phase particles uniformly distributed across the microstructure, which correspond to titanium carbides or titanium nitrides. With $\bar{d}_{SPP} = 5.6$ µm and $f_{SPP} = 0.4\%$, they are not expected to be capable of significantly hindering grain boundary migration during thermomechanical processing, and thus they do not have a relevant impact on dynamic and post-dynamic microstructural evolutions with respect to other attributes such as stored energy value and distribution, as proven for similar materials [11]. It is worth mentioning that this microstructure has undergone an initial heat treatment in the Gleeble machine as indicated before. In various works, it has been reported that the Joule effect heating mode of Gleeble has an impact on the microstructure, and hence an electric current flow through
by the Kernel Average Misorientation (KAM) parameter. The KAM parameter is a value assigned to each pixel/point in the EBSD map that indicates the average misorientation angle between that pixel and its neighbors. It can be expressed mathematically by the following formula:

$$KAM(p_i) = \frac{1}{v_i} \sum_{j=1}^{v_i} \theta_{i,j}, \qquad (3)$$

where $\theta_{i,j}$ is the misorientation angle between pixel $p_i$ and pixel $p_j$, and $v_i$ is the number of neighboring pixels. For the calculation of KAM, only first and second order neighbors will be considered, and misorientations above 10°, that correspond to grain boundaries, are ignored.

<figure><img src="image_1125.png" /><figcaption>Fig. 1. a), c) and d) Microstructure in the state previous to hot compression, after an initial 5-minutes-long heat treatment and a 1-minute-long heat treatment at the test temperature, in the Gleeble. a) 2nd order KAM map with grain boundaries in white (defined by a misorientation angle higher than 10°). b) Grain size surface fraction histogram. c) and d) Backscattered electrons SEM images, with TiN or TiC as the black second phase particles.</figcaption></figure>

## Initial microstructure before compression

In Fig. 1 c and Fig. 1 d, it can be seen that there are second-phase particles uniformly distributed across the microstructure, which correspond to titanium carbides or titanium nitrides. With $\bar{d}_{SPP} = 5.6$ µm and $f_{SPP} = 0.4\%$, they are not expected to be capable of significantly hindering grain boundary migration during thermomechanical processing, and thus they do not have a relevant impact on dynamic and post-dynamic microstructural evolutions with respect to other attributes such as stored energy value and distribution, as proven for similar materials [11]. It is worth mentioning that this microstructure has undergone an initial heat treatment in the Gleeble machine as indicated before. In various works, it has been reported that the Joule effect heating mode of Gleeble has an impact on the microstructure, and hence an electric current flow through
a sample has an influence on several solid-state transformations, accelerating recrystallization and precipitation kinetics with respect to conventional radiative systems [12-13].

## Hot compression tests

Three thermocouples are spot-welded on the surface of the cylindrical sample. The temperature is controlled by the regulatory thermocouple located at the middle height of the sample (TC2 in Fig. 2), while the two remaining thermocouples are located at a distance of approximately 1 mm to each respective sample edge. These last two thermocouples allow to have a better control of the temperature gradient along the cylinder axis of the sample. The position of the thermocouples along the longitudinal axis must be measured for each sample. The module HydraWedge®, available in the Gleeble 3500 machine, is used, which allows to carry out high strain rate hot compression tests, up to 100 s⁻¹, with good force-displacement control. However, it presents the following drawbacks: strain is measured indirectly via anvil displacement; there are non-negligible friction forces between the anvils and the sample; there is an axial temperature gradient which can vary significantly, with temperature differences up to 25 °C between the mid-height thermocouple and the “edge” ones. Graphite foils of a total width of 0.25 mm are adhered to the interfaces between the end of the sample and each anvil for lubrication purposes. As mentioned in the section Introduction, Joule heating is switched off during compression. Immediately after deformation, a Helium quenching is applied, which results in a cooling rate of approximately 40 °C/s.

## FE simulations of hot compression tests

The compression tests are simulated with the aim to estimate the local thermomechanical conditions at the center of the specimen, considering self-heating. Forge® software is used [14]. Before the start of compression, there is an initial iterative step to reproduce the experimental temperature profile by adjusting the heating power. Therefore, an analytical thermal model is developed to accurately estimate sample temperature before compression by coupling the thermal and electrical analyses. The tungsten carbide anvils are also modeled, as shown in Fig. 2. A triangular mesh of size 0.2 mm is applied in the sample, except for a region comprising from sample edges up to a 1 mm distance from them, where the mesh is more refined (mesh size of 0.1 mm). The anvils are also meshed. The temperature field on the sample is defined as follows:

- Along the compression direction, at the surface of the sample, a parabolic temperature profile is approximated to the profile captured experimentally by the three aforementioned spot-welded thermocouples. The procedure to accurately reproduce the experimental temperature profile at the surface of the sample will be described next.

- Along the radial direction, a parabolic temperature profile is estimated by means of the average surface temperature of the sample (calculated from the longitudinal temperature distribution) and the thermophysical properties of the material. Knowing that the temperature axial distribution on the surface of the sample is given by the expression $T_{surface}(z) = a_0z^2 + b_0z + c_0$, where $a_0, b_0$ and $c_0$ are constants determined by the thermocouple data, the temperature at any point of the sample is given by:

$$T(r, z) = T_{surface}(z) + \frac{\sigma_{SB}\varepsilon_{R}(\bar{T}_{surface}^{4} - T_{0}^{4})}{2\lambda R_{0}} \cdot (R_{0}^{2} - r^{2}), \quad (4)$$

where $\sigma_{SB}$ is the Stefan-Boltzmann constant, $\varepsilon_{R}$ is the material emissivity, $\lambda$ stands for the thermal conductivity of the material, $R_{0}$ is the radius of the sample just before compression, $\bar{T}_{surface}$ is the average temperature on the surface of the sample and $T_{0}$ is the room temperature.
a sample has an influence on several solid-state transformations, accelerating recrystallization and precipitation kinetics with respect to conventional radiative systems [12-13].

## Hot compression tests

Three thermocouples are spot-welded on the surface of the cylindrical sample. The temperature is controlled by the regulatory thermocouple located at the middle height of the sample (TC2 in Fig. 2), while the two remaining thermocouples are located at a distance of approximately 1 mm to each respective sample edge. These last two thermocouples allow to have a better control of the temperature gradient along the cylinder axis of the sample. The position of the thermocouples along the longitudinal axis must be measured for each sample. The module HydraWedge®, available in the Gleeble 3500 machine, is used, which allows to carry out high strain rate hot compression tests, up to 100 s⁻¹, with good force-displacement control. However, it presents the following drawbacks: strain is measured indirectly via anvil displacement; there are non-negligible friction forces between the anvils and the sample; there is an axial temperature gradient which can vary significantly, with temperature differences up to 25 °C between the mid-height thermocouple and the “edge” ones. Graphite foils of a total width of 0.25 mm are adhered to the interfaces between the end of the sample and each anvil for lubrication purposes. As mentioned in the section Introduction, Joule heating is switched off during compression. Immediately after deformation, a Helium quenching is applied, which results in a cooling rate of approximately 40 °C/s.

## FE simulations of hot compression tests

The compression tests are simulated with the aim to estimate the local thermomechanical conditions at the center of the specimen, considering self-heating. Forge® software is used [14]. Before the start of compression, there is an initial iterative step to reproduce the experimental temperature profile by adjusting the heating power. Therefore, an analytical thermal model is developed to accurately estimate sample temperature before compression by coupling the thermal and electrical analyses. The tungsten carbide anvils are also modeled, as shown in Fig. 2. A triangular mesh of size 0.2 mm is applied in the sample, except for a region comprising from sample edges up to a 1 mm distance from them, where the mesh is more refined (mesh size of 0.1 mm). The anvils are also meshed. The temperature field on the sample is defined as follows:

- Along the compression direction, at the surface of the sample, a parabolic temperature profile is approximated to the profile captured experimentally by the three aforementioned spot-welded thermocouples. The procedure to accurately reproduce the experimental temperature profile at the surface of the sample will be described next.

- Along the radial direction, a parabolic temperature profile is estimated by means of the average surface temperature of the sample (calculated from the longitudinal temperature distribution) and the thermophysical properties of the material. Knowing that the temperature axial distribution on the surface of the sample is given by the expression $T_{surface}(z) = a_0z^2 + b_0z + c_0$, where $a_0, b_0$ and $c_0$ are constants determined by the thermocouple data, the temperature at any point of the sample is given by:

$$T(r, z) = T_{surface}(z) + \frac{\sigma_{SB}\varepsilon_{R}(\bar{T}_{surface}^{4}-T_{0}^{4})}{2\lambda R_{0}} \cdot (R_{0}^{2}-r^{2}), \quad (4)$$

where $\sigma_{SB}$ is the Stefan-Boltzmann constant, $\varepsilon_{R}$ is the material emissivity, $\lambda$ stands for the thermal conductivity of the material, $R_0$ is the radius of the sample just before compression, $\bar{T}_{surface}$ is the average temperature on the surface of the sample and $T_0$ is the room temperature.
<figure><img src="image_1124.png" /><figcaption>Fig. 2. In the left, temperature field of the sample and the anvils at the end of the heating stage. In the right, temperature field of the sample and the anvils at the end of the compression stage. The square in black corresponds approximately to the region where the microstructure is characterized. Only a quarter section of the axysimmetric specimen is shown in this figure.</figcaption></figure>

Moreover, during the stage of temperature set-up previous to compression, the Joule heating power $P_{Joule}$ (W) can be described as follows, in order to account for radiation and axial anvil conduction losses [15-16]:

$$P_{Joule} = - \int_{\partial \Omega} \vec{\nabla T} \cdot \vec{n} \, d(\partial \Omega). \qquad (5)$$

Using a constant approximated value of the heat transfer coefficient $h$ ($Wm^{-2}K^{-1}$) in the interface between the anvil and the specimen, and hence estimating the anvil temperature $\bar{T}_{anvil}$ via the expression $h(\bar{T}_{sample edge} - \bar{T}_{anvil}) = \pm \lambda \cdot \frac{\partial T}{\partial z}|_{z=0,l}$, the value of the power density $\dot{Q}_{Joule}^0$ ($Wm^{-3}$) is adjusted iteratively until reaching a good accordance with the experimental temperature profile described by thermocouple data (a difference of less than a 1% of the thermocouple-measured temperature between experimental and simulation data is required to validate the power density ($\dot{Q}_{Joule}^0$) value used for the heating simulation). Hence, the simulations consist of an initial heating stage to adjust the power density value to the experimental axial surface temperature profile, and a final compression stage where $\dot{Q}_{Joule}^0 = 0$ $Wm^{-3}$ (Fig. 2). For the compression stage, a Coulomb-Tresca friction model has been used for the interface between the sample and the anvils. A rheological model for the A-286 alloy under hot compression calibrated from several experiments is utilized. Concerning the estimation of sample dimensions, it is worth indicating that thermal expansion is taken into account, as described below, since the dependence of mass density $\rho(T)$ with temperature is known for the alloy [15]:

$$l(T_{test}) = l_0(T_0) \left( \frac{\rho(T_0)}{\rho(T_{test})} \right)^{\frac{1}{3}} \qquad (6)$$

The expression in Eq. 6 is used to estimate the length of the sample $l(T_{test})$ by only considering thermal expansion at test temperature. It is also used to estimate the radius of the
<figure><img src="image_1126.png" /><figcaption>Fig. 2. In the left, temperature field of the sample and the anvils at the end of the heating stage. In the right, temperature field of the sample and the anvils at the end of the compression stage. The square in black corresponds approximately to the region where the microstructure is characterized. Only a quarter section of the axysimmetric specimen is shown in this figure.</figcaption></figure>

Moreover, during the stage of temperature set-up previous to compression, the Joule heating power $P_{Joule}$ (W) can be described as follows, in order to account for radiation and axial anvil conduction losses [15-16]:

$$P_{Joule} = - \int_{\partial \Omega} \vec{\nabla T} \cdot \vec{n} \, d(\partial \Omega). \qquad (5)$$

Using a constant approximated value of the heat transfer coefficient $h$ ($Wm^{-2}K^{-1}$) in the interface between the anvil and the specimen, and hence estimating the anvil temperature $\bar{T}_{anvil}$ via the expression $h(\bar{T}_{sample edge} - \bar{T}_{anvil}) = \pm \lambda \cdot \frac{\partial T}{\partial z}|_{z=0,l}$, the value of the power density $\dot{Q}_{Joule}^0$ ($Wm^{-3}$) is adjusted iteratively until reaching a good accordance with the experimental temperature profile described by thermocouple data (a difference of less than a 1% of the thermocouple-measured temperature between experimental and simulation data is required to validate the power density ($\dot{Q}_{Joule}^0$) value used for the heating simulation). Hence, the simulations consist of an initial heating stage to adjust the power density value to the experimental axial surface temperature profile, and a final compression stage where $\dot{Q}_{Joule}^0 = 0$ $Wm^{-3}$ (Fig. 2). For the compression stage, a Coulomb-Tresca friction model has been used for the interface between the sample and the anvils. A rheological model for the A-286 alloy under hot compression calibrated from several experiments is utilized. Concerning the estimation of sample dimensions, it is worth indicating that thermal expansion is taken into account, as described below, since the dependence of mass density $\rho(T)$ with temperature is known for the alloy [15]:

$$l(T_{test}) = l_0(T_0) \left( \frac{\rho(T_0)}{\rho(T_{test})} \right)^{\frac{1}{3}} \qquad (6)$$

The expression in Eq. 6 is used to estimate the length of the sample $l(T_{test})$ by only considering thermal expansion at test temperature. It is also used to estimate the radius of the
sample $r_0(T_{test})$ at a given time, by replacing $l$ for $r$. Moreover, this expression is used to calculate the final length that the sample must reach for the compression to stop $l_f(T_f)$. The final dimensions of the compressed specimen are measured at room temperature, thus, in the simulation of the compression stage, the anvil (and hence the sample) follows the experimental jaw displacement, from the initial calculated dimensions of the sample (in Eq. 7) to the final thermally corrected length of the sample, to compensate for the elastic correction. Sample dimensions at the end of the heating stage, just before compression, are estimated considering both thermal expansion and creep, so the formula in Eq. 6 is combined with the displacements captured by the LVDT gauge as defined in Fig. 3 associated to creep, $\Delta l_1^{creep}$ and $\Delta l_2^{creep}$, as follows:

$$l_0 = \left( \frac{\rho(T_{plateau})}{\rho(T_{compression})} \right)^{\frac{1}{3}} \cdot \left( \left( \frac{\rho(T_0)}{\rho(T_{plateau})} \right)^{\frac{1}{3}} l_0(T_0) - \Delta l_1^{creep} \right) - \Delta l_2^{creep}.$$

<figure><img src="image_1125.png" /></figure>

(7)

<caption>Fig. 3. Temperature evolution and length variation (captured by the LVDT gauge) of a sample during the thermal path followed before compression, indicating the effect of creep.</caption>

## Results and discussion

After Gleeble hot compression, all the deformed-state microstructures analyzed can be classified in three types according to their recrystallized fraction ($f_{RX}$), as shown in Fig. 4:

<figure><img src="image_1126.png" /><figcaption>Fig. 4. Optical microscopy images (after an electrochemical etching to reveal grain boundaries as described in the section Metallographic preparation and microstructural characterization) of three different microstructures representatives of each class: Non-recrystallized microstructures (NonRX), with $f_{RX} \le 10\%$; partially recrystallized microstructures (PRX), with $10\% < f_{RX} < 80\%$; recrystallized microstructures (RX), with $f_{RX} \ge 80\%$.</figcaption></figure>
sample $r_0(T_{test})$ at a given time, by replacing $l$ for $r$. Moreover, this expression is used to calculate the final length that the sample must reach for the compression to stop $l_f(T_f)$. The final dimensions of the compressed specimen are measured at room temperature, thus, in the simulation of the compression stage, the anvil (and hence the sample) follows the experimental jaw displacement, from the initial calculated dimensions of the sample (in Eq. 7) to the final thermally corrected length of the sample, to compensate for the elastic correction. Sample dimensions at the end of the heating stage, just before compression, are estimated considering both thermal expansion and creep, so the formula in Eq. 6 is combined with the displacements captured by the LVDT gauge as defined in Fig. 3 associated to creep, $\Delta l_1^{creep}$ and $\Delta l_2^{creep}$, as follows:

$$l_0 = \left( \frac{\rho(T_{plateau})}{\rho(T_{compression})} \right)^{\frac{1}{3}} \cdot \left( \left( \frac{\rho(T_0)}{\rho(T_{plateau})} \right)^{\frac{1}{3}} l_0(T_0) - \Delta l_1^{creep} \right) - \Delta l_2^{creep}$$

<figure><img src="image_1127.png" /></figure>

(7)

<caption>Fig. 3. Temperature evolution and length variation (captured by the LVDT gauge) of a sample during the thermal path followed before compression, indicating the effect of creep.</caption>

## Results and discussion

After Gleeble hot compression, all the deformed-state microstructures analyzed can be classified in three types according to their recrystallized fraction ($f_{RX}$), as shown in Fig. 4:

<figure><img src="image_1128.png" /><figcaption>Fig. 4. Optical microscopy images (after an electrochemical etching to reveal grain boundaries as described in the section Metallographic preparation and microstructural characterization) of three different microstructures representatives of each class: Non-recrystallized microstructures (NonRX), with $f_{RX} \le 10\%$; partially recrystallized microstructures (PRX), with $10\% < f_{RX} < 80\%$; recrystallized microstructures (RX), with $f_{RX} \ge 80\%$.</figcaption></figure>
In all the optical microscopy images of the compressed microstructures, non-recrystallized grains and hence $f_{RX}$ are identified by imaging processing software by implementing specific criteria of high grain size and non-equiaxiality aspect ratio for each microstructure (Fig. 4). The size and density of recrystallized grains in the state after compression, which will have an impact on static microstructural evolutions, depend on the thermomechanical parameters of deformation. Fig. 5 shows the relationship between thermomechanical parameters of deformation, such as the local strain level ($\epsilon_{SH}$) and the different Zener-Hollomon parameter estimations, and the microstructure via the $f_{RX}$ range at which it belongs (according to the color code of the data points). In addition to this, the relationship between $\ln(Z)$ and $\epsilon_{SH}$ allows to evaluate the accuracy of the different estimations of the Zener-Hollomon parameter.

<figure><img src="image_1127.png" /><figcaption>Fig. 5. Local strain level at the center of the sample ($\epsilon_{SH}$) vs. Zener-Hollomon parameter calculated from (see section Introduction for the detailed description of the different estimations of the Zener-Hollomon parameter) a) nominal global values b) local values after compression c) local values before compression, with no self-heating phenomena considered and nominal strain rate d) local temperature before compression and local strain rate after compression. Circle mark points refer to the tests performed at $\dot{\epsilon}_n = 80 \text{ s}^{-1}$ while square marks stand for the tests carried out at $\dot{\epsilon}_n = 20 \text{ s}^{-1}$. Additionally, the colors (green, yellow and red) are consistent with the criteria defined in Fig. 4 to identify different microstructure types according to $f_{RX}$ (recrystallized, partially recrystallized and non-recrystallized microstructures, respectively).</figcaption></figure>

The use of local parameters instead of nominal estimations in the calculation of the Zener-Hollomon parameter results in a better distribution of points according to their microstructure type ($f_{RX}$ range) in Fig. 5 b with respect to Fig. 5 a. The use of nominal parameters in $Z_0$ calculation is not as predictive of the compressed-state microstructure as the use of local temperature estimations in $Z_1$, $Z_2$ and $Z_3$ calculation. This is shown in Fig. 5 a, where identical $Z_0$ values are shared by conditions with very different $f_{RX}$ values (according to data point color). However, the consideration of self-heating (Fig. 5 b) or not (Fig. 5 c and Fig. 5 d) in Zener-Hollomon parameter calculation has less impact on the arrangement of data points according to their color, as it can be seen from the few differences among these graphs. For both the tests performed at $\dot{\epsilon}_n = 80 \text{ s}^{-1}$ (circle data points) and the tests performed at $\dot{\epsilon}_n = 20 \text{ s}^{-1}$ (square data points), it can be seen that the arrangement of points according to their $f_{RX}$ value is slightly better defined in Fig. 5 b: high strain and low Zener-Hollomon parameter tests yield a fully recrystallized microstructure, while low strain and high Zener-Hollomon parameter tests yield a non-recrystallized microstructure.
In all the optical microscopy images of the compressed microstructures, non-recrystallized grains and hence $f_{RX}$ are identified by imaging processing software by implementing specific criteria of high grain size and non-equiaxiality aspect ratio for each microstructure (Fig. 4). The size and density of recrystallized grains in the state after compression, which will have an impact on static microstructural evolutions, depend on the thermomechanical parameters of deformation. Fig. 5 shows the relationship between thermomechanical parameters of deformation, such as the local strain level ($\epsilon_{SH}$) and the different Zener-Hollomon parameter estimations, and the microstructure via the $f_{RX}$ range at which it belongs (according to the color code of the data points). In addition to this, the relationship between $\ln(Z)$ and $\epsilon_{SH}$ allows to evaluate the accuracy of the different estimations of the Zener-Hollomon parameter.

<figure><img src="image_1129.png" /><figcaption>Fig. 5. Local strain level at the center of the sample ($\epsilon_{SH}$) vs. Zener-Hollomon parameter calculated from (see section Introduction for the detailed description of the different estimations of the Zener-Hollomon parameter) a) nominal global values b) local values after compression c) local values before compression, with no self-heating phenomena considered and nominal strain rate d) local temperature before compression and local strain rate after compression. Circle mark points refer to the tests performed at $\dot{\epsilon}_n = 80 \text{ s}^{-1}$ while square marks stand for the tests carried out at $\dot{\epsilon}_n = 20 \text{ s}^{-1}$. Additionally, the colors (green, yellow and red) are consistent with the criteria defined in Fig. 4 to identify different microstructure types according to $f_{RX}$ (recrystallized, partially recrystallized and non-recrystallized microstructures, respectively).</figcaption></figure>

The use of local parameters instead of nominal estimations in the calculation of the Zener-Hollomon parameter results in a better distribution of points according to their microstructure type ($f_{RX}$ range) in Fig. 5 b with respect to Fig. 5 a. The use of nominal parameters in $Z_0$ calculation is not as predictive of the compressed-state microstructure as the use of local temperature estimations in $Z_1$, $Z_2$ and $Z_3$ calculation. This is shown in Fig. 5 a, where identical $Z_0$ values are shared by conditions with very different $f_{RX}$ values (according to data point color). However, the consideration of self-heating (Fig. 5 b) or not (Fig. 5 c and Fig. 5 d) in Zener-Hollomon parameter calculation has less impact on the arrangement of data points according to their color, as it can be seen from the few differences among these graphs. For both the tests performed at $\dot{\epsilon}_n = 80 \text{ s}^{-1}$ (circle data points) and the tests performed at $\dot{\epsilon}_n = 20 \text{ s}^{-1}$ (square data points), it can be seen that the arrangement of points according to their $f_{RX}$ value is slightly better defined in Fig. 5 b: high strain and low Zener-Hollomon parameter tests yield a fully recrystallized microstructure, while low strain and high Zener-Hollomon parameter tests yield a non-recrystallized microstructure.
<figure><img src="image_1128.png" /><figcaption>Fig. 6. Logarithmic relation between the recrystallized grain size (D<sub>RX</sub>) and the different estimations of the Zener-Hollomon parameter (Z). A least squares linear regression line (dotted line) and the coefficient of regression R<sup>2</sup> is shown for every set of points corresponding to a different Zener-Hollomon parameter estimation.</figcaption></figure>

The relationship between recrystallized grain size and Zener-Hollomon parameter can be found in Fig. 6. It can be observed that there is no correlation for any of the different Zener-Hollomon parameter estimations analyzed, as the R² coefficients indicate. For the hot-deformed A-286 alloy at ἕ<sub>nominal</sub> ∈ [20,80]s<sup>-1</sup>, Derby's law fails to establish a relationship between the thermomechanical parameters of deformation and recrystallized grain size. The consideration of local self-heating at the end of deformation in the calculation of Z₁ results in a higher dispersion of the results with respect to recrystallized grain size, which suggests that the estimated self-heating phenomena does not have an impact on dynamic and post-dynamic grain growth, in contrast with the results found in Fig. 5 regarding recrystallization with respect to local strain level.

## Conclusion

Self-heating during hot deformation, especially at high strain rate, must be taken into account to assess plastic stored energy value and distribution in order to correctly predict static recrystallization and grain growth mechanisms. The mastering of static microstructural evolutions in the A-286 alloy would allow to predict the formation of heterogeneous microstructures in static conditions. The power-law relationship between recrystallized grain size (a consequence of stored energy) and Zener-Hollomon parameter as defined in Derby's law is sensitive to the use of local thermomechanical values instead of the nominal ones. However, local temperature and strain rate at the end of deformation do not share a correlation with the recrystallized grain size obtained after hot compression tests at high strain rate (Zener-Hollomon parameter (Z) is found to have a strong negative correlation with the recrystallized grain size). The most accurate estimation of the thermomechanical parameters of hot deformation to better predict dynamic and post-dynamic microstructural evolutions is still open to discussion, and a question now arises about which is the most microstructure-impactful moment in the thermomechanical path during the hot compression process, since the thermomechanical conditions at the end of deformation are not representative enough of the resulting compressed-state microstructure. The microstructures obtained from high strain rate Gleeble testing differ from those obtained from testing at lower strain rates due to a lower dislocation density and different intragranular dislocation distributions, so the link between the thermomechanical parameters of deformation and the deformed-state microstructure as stated by Derby would no longer be valid. Furthermore, the lack of a Derby relationship at these conditions could suggest
<figure><img src="image_1130.png" /><figcaption>Fig. 6. Logarithmic relation between the recrystallized grain size (D~RX~) and the different estimations of the Zener-Hollomon parameter (Z). A least squares linear regression line (dotted line) and the coefficient of regression R² is shown for every set of points corresponding to a different Zener-Hollomon parameter estimation.</figcaption></figure>

The relationship between recrystallized grain size and Zener-Hollomon parameter can be found in Fig. 6. It can be observed that there is no correlation for any of the different Zener-Hollomon parameter estimations analyzed, as the R² coefficients indicate. For the hot-deformed A-286 alloy at ε~nominal~ ∈ [20,80]s⁻¹, Derby's law fails to establish a relationship between the thermomechanical parameters of deformation and recrystallized grain size. The consideration of local self-heating at the end of deformation in the calculation of Z₁ results in a higher dispersion of the results with respect to recrystallized grain size, which suggests that the estimated self-heating phenomena does not have an impact on dynamic and post-dynamic grain growth, in contrast with the results found in Fig. 5 regarding recrystallization with respect to local strain level.

## Conclusion

Self-heating during hot deformation, especially at high strain rate, must be taken into account to assess plastic stored energy value and distribution in order to correctly predict static recrystallization and grain growth mechanisms. The mastering of static microstructural evolutions in the A-286 alloy would allow to predict the formation of heterogeneous microstructures in static conditions. The power-law relationship between recrystallized grain size (a consequence of stored energy) and Zener-Hollomon parameter as defined in Derby's law is sensitive to the use of local thermomechanical values instead of the nominal ones. However, local temperature and strain rate at the end of deformation do not share a correlation with the recrystallized grain size obtained after hot compression tests at high strain rate (ε~nominal~ ≥ 20 s⁻¹). The most accurate estimation of the thermomechanical parameters of hot deformation to better predict dynamic and post-dynamic microstructural evolutions is still open to discussion, and a question now arises about which is the most microstructure-impactful moment in the thermomechanical path during the hot compression process, since the thermomechanical conditions at the end of deformation are not representative enough of the resulting compressed-state microstructure. The microstructures obtained from high strain rate Gleeble testing differ from those obtained from testing at lower strain rates due to a lower dislocation density and different intragranular dislocation distributions, so the link between the thermomechanical parameters of deformation and the deformed-state microstructure as stated by Derby would no longer be valid. Furthermore, the lack of a Derby relationship at these conditions could suggest