that post-dynamic recrystallization and grain growth have a bigger impact in the resulting microstructure than dynamic evolutions.

## Acknowledgements

This study was carried out with Aperam funding and support, in the frame of the DIGIMU consortium, handled by MINES Paris and Armines.

## References

[1] X. Huang, J. Li, B.S. Amirkhiz, P. Liu, "Effect of water density on the oxidation behavior of alloy A-286 at 625 °C - A TEM study", Journal of Nuclear Materials, 2015, vol. 467, pp. 758-769. https://doi.org/10.1016/j.jnucmat.2015.10.023

[2] O. Takakuwa, Y. Ogawa, J. Yamabe, H. Matsunaga, "Hydrogen-induced ductility loss of precipitation-strengthened Fe-Ni-Cr based Superalloy", Materials Science & Engineering A, 2019, vol. 739, pp. 335-342. https://doi.org/10.1016/j.msea.2018.10.040

[3] A. Nicolaÿ, J.M. Franchet, J Cormier, H. Mansour, M. de Graef, A. Seret, N. Bozzolo, "Discrimination of dynamically and post-dynamically recrystallized grains based on EBSD data: application to Inconel 718.", Journal of Microscopy, (2018), Vol. 00, pp. 1-13. https://doi.org/10.1111/jmi.12769

[4] Sela, A., Ortiz-de-Zarate, G., Soler, D., Germain, G., Gallegos, L., & Arrazola, P. J. "Adiabatic self-heating determination for Ti6Al4V at different temperatures.", International Journal of Heat and Mass Transfer, 2023, 204, 123747. https://doi.org/10.1016/j.ijheatmasstransfer.2022.123747

[5] Zener, Clarence, and John Herbert Hollomon. "Effect of strain rate upon plastic flow of steel." Journal of Applied physics 15.1 (1944): 22-32. https://doi.org/10.1063/1.1707363

[6] Rollett, Anthony, Gregory S. Rohrer, and John Humphreys, Recrystallization and related annealing phenomena, Newnes, 2017.

[7] B. Derby, The dependence of grain size on stress during dynamic recrystallisation. Acta metallurgica et materialia, 39(5), (1991) 955-962. https://doi.org/10.1016/0956-7151(91)90295-C

[8] Bachmann, F., Hielscher, R., & Schaeben, H. (2011). "Grain detection from 2d and 3d EBSD data—Specification of the MTEX algorithm", Ultramicroscopy, 2011, 111(12), 1720-1733. https://doi.org/10.1016/j.ultramic.2011.08.002

[9] Bergmann R, Chan RH, Hielscher R, Persch J, Steidl G "Restoration of manifold-valued images by half-quadratic minimization", Inverse Probl. Imaging, 2016, 10:281–304. https://doi.org/10.3934/ipi.2016001

[10] Lehto, P., & Remes, "EBSD characterisation of grain size distribution and grain sub-structures for ferritic steel weld metals" Welding in the World, 2022, 66(2), 363-377. https://doi.org/10.1007/s40194-021-01225-w

[11] A. Agnoli, M. Bernacki, R. Logé, J.-M. Franchet, J. Laigo, N. Bozzolo, "Selective growth of low stored energy grains during δ sub-solvus annealing in the Inconel 718 nickel-based superalloy", Metallurgical and Materials Transactions A, 2015, vol. 46, pp. 4405-4421. https://doi.org/10.1007/s11661-015-3035-9

[12] A. Nicolaÿ, J.M. Franchet, J Cormier, R. E. Logé, G. Fiorucci, J. Fausty, M. Van der Meer, N. Bozzolo, "Influence of Joule effect heating on recrystallization phenomena in inconel 718.",
that post-dynamic recrystallization and grain growth have a bigger impact in the resulting microstructure than dynamic evolutions.

## Acknowledgements

This study was carried out with Aperam funding and support, in the frame of the DIGIMU consortium, handled by MINES Paris and Armines.

## References

[1] X. Huang, J. Li, B.S. Amirkhiz, P. Liu, "Effect of water density on the oxidation behavior of alloy A-286 at 625 °C - A TEM study", Journal of Nuclear Materials, 2015, vol. 467, pp. 758-769. https://doi.org/10.1016/j.jnucmat.2015.10.023

[2] O. Takakuwa, Y. Ogawa, J. Yamabe, H. Matsunaga, "Hydrogen-induced ductility loss of precipitation-strengthened Fe-Ni-Cr based Superalloy", Materials Science & Engineering A, 2019, vol. 739, pp. 335-342. https://doi.org/10.1016/j.msea.2018.10.040

[3] A. Nicolaÿ, J.M. Franchet, J Cormier, H. Mansour, M. de Graef, A. Seret, N. Bozzolo, "Discrimination of dynamically and post-dynamically recrystallized grains based on EBSD data: application to Inconel 718.", Journal of Microscopy, (2018), Vol. 00, pp. 1-13. https://doi.org/10.1111/jmi.12769

[4] Sela, A., Ortiz-de-Zarate, G., Soler, D., Germain, G., Gallegos, L., & Arrazola, P. J. "Adiabatic self-heating determination for Ti6Al4V at different temperatures.", International Journal of Heat and Mass Transfer, 2023, 204, 123747. https://doi.org/10.1016/j.ijheatmasstransfer.2022.123747

[5] Zener, Clarence, and John Herbert Hollomon. "Effect of strain rate upon plastic flow of steel." Journal of Applied physics 15.1 (1944): 22-32. https://doi.org/10.1063/1.1707363

[6] Rollett, Anthony, Gregory S. Rohrer, and John Humphreys, Recrystallization and related annealing phenomena, Newnes, 2017.

[7] B. Derby, The dependence of grain size on stress during dynamic recrystallisation. Acta metallurgica et materialia, 39(5), (1991) 955-962. https://doi.org/10.1016/0956-7151(91)90295-C

[8] Bachmann, F., Hielscher, R., & Schaeben, H. (2011). "Grain detection from 2d and 3d EBSD data—Specification of the MTEX algorithm", Ultramicroscopy, 2011, 111(12), 1720-1733. https://doi.org/10.1016/j.ultramic.2011.08.002

[9] Bergmann R, Chan RH, Hielscher R, Persch J, Steidl G "Restoration of manifold-valued images by half-quadratic minimization", Inverse Probl. Imaging, 2016, 10:281–304. https://doi.org/10.3934/ipi.2016001

[10] Lehto, P., & Remes, "EBSD characterisation of grain size distribution and grain sub-structures for ferritic steel weld metals" Welding in the World, 2022, 66(2), 363-377. https://doi.org/10.1007/s40194-021-01225-w

[11] A. Agnoli, M. Bernacki, R. Logé, J.-M. Franchet, J. Laigo, N. Bozzolo, "Selective growth of low stored energy grains during δ sub-solvus annealing in the Inconel 718 nickel-based superalloy", Metallurgical and Materials Transactions A, 2015, vol. 46, pp. 4405–4421. https://doi.org/10.1007/s11661-015-3035-9

[12] A. Nicolaÿ, J.M. Franchet, J Cormier, R. E. Logé, G. Fiorucci, J. Fausty, M. Van der Meer, N. Bozzolo, "Influence of Joule effect heating on recrystallization phenomena in inconel 718.",
Metallurgical and Materials Transactions, A 52.10 (2021), 4572-4596.
https://doi.org/10.1007/s11661-021-06411-5

[13] H. Conrad, "Effects of electric current on solid state phase transformations in metals." Materials Science and Engineering, A 287.2 (2000), 227-237.

[14] P. de Micheli, A. Settefrati, S. Marie, J. Barlier, P. Lasne, et al. "Towards the simulation of the whole manufacturing chain processes with FORGE," in Proc. Int. Conf. on "New development in forging technology," 2015, pp. 1-25.

[15] Carlslaw, H. S., and J. C. Jaeger. "Conduction of Heat in Solids." Oxford University Press, Oxford (1959).

[16] Evans, R. W., and P. J. Scharning. "Axisymmetric compression test and hot working properties of alloys." Materials science and technology 17.8 (2001), 995-1004.
https://doi.org/10.1179/026708301101510843
Metallurgical and Materials Transactions, A 52.10 (2021), 4572-4596.
https://doi.org/10.1007/s11661-021-06411-5

[13] H. Conrad, "Effects of electric current on solid state phase transformations in metals." Materials Science and Engineering, A 287.2 (2000), 227-237.

[14] P. de Micheli, A. Settefrati, S. Marie, J. Barlier, P. Lasne, et al. "Towards the simulation of the whole manufacturing chain processes with FORGE," in Proc. Int. Conf. on "New development in forging technology," 2015, pp. 1-25.

[15] Carlslaw, H. S., and J. C. Jaeger. "Conduction of Heat in Solids." Oxford University Press, Oxford (1959).

[16] Evans, R. W., and P. J. Scharning. "Axisymmetric compression test and hot working properties of alloys." Materials science and technology 17.8 (2001), 995-1004.
https://doi.org/10.1179/026708301101510843
# The effect of friction compensation on the slope of flow curves obtained by stack compression tests

KÖLÜS Martin L.¹,ᵃ, BORBÉLY Richárd¹,ᵇ and BÉRES Gábor J.¹,ᶜ*

¹Research Group of Mechanical- and Laser Beam Technologies, John von Neumann University, 6000 Kecskemét, Hungary

ᵃkolus.martin@nje.hu, ᵇborbely.richard@nje.hu, ᶜberes.gabor@nje.hu

**Keywords:** Disk Compression Tests, Stack Compression Tests, Friction Compensation, Pressure Dependent Friction

**Abstract.** Knowing the large strain flow curve is essential for many simulation processes in the sheet metal forming technology. A prominent example is the clinching, when the strains are one magnitude higher than in tension, but a simple deep-drawing process could also lead to appropriately large strains. To obtain the flow stress - plastic strain curves beyond the range of tension, several processes are known. In the most multi-axial cases, a conversion factor based on the work equivalence is usually applied on the measurement results to obtain the equivalent quantities. However, at compressive experiments, the friction and the geometrical issues may distort the curve shape even further, hiding the real material behavior. In this study we investigate the effect of the friction determination method and the value of the friction coefficient on the curve shape. Pressure dependent and constant friction coefficients were inversely specified by disk compression tests and related finite element modeling. In possession of the friction values, the conversions of the physically measured compressive stress to equivalent flow stress during disk and stack compression tests with different height-diameter ratios are discussed here.

**Introduction**

Flow curves are essential input parameters of the metal forming simulations (as well as manual calculations in the earlier times) next to the yield and the failure criteria. Accordingly, there are several procedures to obtain the flow stress – plastic strain point pairs. At bulk forming, the problematic of large strain flow curves’ measurement is perhaps simpler in the sense that the workpiece dimensions allow more in terms of buckling. In the sheet metal forming, the shell-like behavior of the sheet prevents the simple compression of specimen. As a result, testing methods with different nature are spread, such as in-plane torsion test, hydraulic bulge test, plane strain compression test, stack compression test and post-necking tensile tests [1]. Of these cases, we discuss the effect of the friction compensation at stack compression test in this paper.

The stack compression test was introduced by Merklein and Kuppert [2] in 2009, among the first pioneers. The issue of the contact friction and the flow stress were already discussed by Coppieters et al. [3], one year later. They used the modified two specimen method for the Coulomb friction coefficient evaluation, and they obtained a constant parameter. Also, a constant friction coefficient was proposed and used by Coppieters et al. [4] and An and Vegter [5] to investigate the effect of the hydrostatic pressure and the frictional behavior for the stacked specimens.

Recently, Kuwabara et al. [6] came up with an interesting study also about the effect of the hydrostatic pressure on the flow stress (strength differential effect). They reported that the lubricant may deteriorate due to the thinning and extension of the oil film. To mitigate the uncertainty resulting from the friction conditions, they modified the experimental setup. Kraus et al. [7] also proposed to consider the friction coefficient as a variable, instead of a constant value. They measured and introduced the changing of both the Coulomb friction coefficient and the friction factor. This kind of changing was mathematically described in the function of the normal

<figure><img src="image_1129.png" /></figure>
# The effect of friction compensation on the slope of flow curves obtained by stack compression tests

KÖLÜS Martin L.¹,ᵃ, BORBÉLY Richárd¹,ᵇ and BÉRES Gábor J.¹,ᶜ*

¹Research Group of Mechanical- and Laser Beam Technologies, John von Neumann University, 6000 Kecskemét, Hungary

ᵃkolus.martin@nje.hu, ᵇborbely.richard@nje.hu, ᶜberes.gabor@nje.hu

**Keywords:** Disk Compression Tests, Stack Compression Tests, Friction Compensation, Pressure Dependent Friction

**Abstract.** Knowing the large strain flow curve is essential for many simulation processes in the sheet metal forming technology. A prominent example is the clinching, when the strains are one magnitude higher than in tension, but a simple deep-drawing process could also lead to appropriately large strains. To obtain the flow stress - plastic strain curves beyond the range of tension, several processes are known. In the most multi-axial cases, a conversion factor based on the work equivalence is usually applied on the measurement results to obtain the equivalent quantities. However, at compressive experiments, the friction and the geometrical issues may distort the curve shape even further, hiding the real material behavior. In this study we investigate the effect of the friction determination method and the value of the friction coefficient on the curve shape. Pressure dependent and constant friction coefficients were inversely specified by disk compression tests and related finite element modeling. In possession of the friction values, the conversions of the physically measured compressive stress to equivalent flow stress during disk and stack compression tests with different height-diameter ratios are discussed here.

**Introduction**

Flow curves are essential input parameters of the metal forming simulations (as well as manual calculations in the earlier times) next to the yield and the failure criteria. Accordingly, there are several procedures to obtain the flow stress – plastic strain point pairs. At bulk forming, the problematic of large strain flow curves’ measurement is perhaps simpler in the sense that the workpiece dimensions allow more in terms of buckling. In the sheet metal forming, the shell-like behavior of the sheet prevents the simple compression of specimen. As a result, testing methods with different nature are spread, such as in-plane torsion test, hydraulic bulge test, plane strain compression test, stack compression test and post-necking tensile tests [1]. Of these cases, we discuss the effect of the friction compensation at stack compression test in this paper.

The stack compression test was introduced by Merklein and Kuppert [2] in 2009, among the first pioneers. The issue of the contact friction and the flow stress were already discussed by Coppieters et al. [3], one year later. They used the modified two specimen method for the Coulomb friction coefficient evaluation, and they obtained a constant parameter. Also, a constant friction coefficient was proposed and used by Coppieters et al. [4] and An and Vegter [5] to investigate the effect of the hydrostatic pressure and the frictional behavior for the stacked specimens.

Recently, Kuwabara et al. [6] came up with an interesting study also about the effect of the hydrostatic pressure on the flow stress (strength differential effect). They reported that the lubricant may deteriorate due to the thinning and extension of the oil film. To mitigate the uncertainty resulting from the friction conditions, they modified the experimental setup. Kraus et al. [7] also proposed to consider the friction coefficient as a variable, instead of a constant value. They measured and introduced the changing of both the Coulomb friction coefficient and the friction factor. This kind of changing was mathematically described in the function of the normal

<figure><img src="image_1131.png" /></figure>
pressure by Gil et al. [8]. They used a non-linear equation to calculate the changing tendency of the lubricant conditions, which relationship will be used in this manuscript later, too.

## The compression tests' method

The compression tests were carried out on an Instron 4482 universal material testing machine (max. load 100 kN), at room temperature. The compressive dies were produced by the institutional cutting workshop using Böhler K110 (X153CrMoV12) commercial tool steel material. The heat treating after the turning process was also done in-house, following the merchant's recommendation [9]. The dies were polished in the final step (Fig. 1).

<figure><img src="image_1130.png" /><figcaption>Fig. 1 The compressive dies after the complete manufacturing process</figcaption></figure>

## Workpiece setups

Three workpiece setups are investigated in this study with the aim to consider the effect of the aspect ratio on the friction conversation. The disks' nominal diameters are uniformly Ø10 mm, or exactly 9.73...9.88 mm, so as a statistical mean of 9.81 mm. The nominal sheet thickness is 1.0 mm, or the exact mean of them is 1.05 mm based on fifty thickness measurements. (The average diameter is also a statistical result of fifty disks.)

Setups by stacking a single (1x10), five (5x10) and ten (10x10) disks on each other were formed, which resulted in 1/10, 1/2 and 1/1 aspect (height-diameter) ratios. In order to compile the 'towers' as accurate as possible before the compression, a self-developed positioning device was manufactured by 3D printing. The rolling direction was indicated and considered during the compilation of the laser cut disks.

<figure><img src="image_1131.png" /><figcaption>Fig. 2 A 10x10 stack with the positioning device (left) and the initial 'towers' of the 1x10 and the 10x10 configurations</figcaption></figure>

The workpiece material is a commercial mild steel, DC04 cold rolled sheet. The basic material parameters were obtained by standard tensile tests (ISO 6892-1:2016), at room temperature and the average values are listed in Table 1. The testing equipment and the applied $1.7 \times 10^{-3} \text{ s}^{-1}$ strain rate were the same with which we performed the compression tests. Both the strength and deformation properties were monitored in parallel, perpendicular and 45° to the rolling direction, but for some characteristics (e.g. YS – yield strength; UTS – ultimate tensile strength; $A_{80}$ – total elongation; and $A_g$ - uniform elongation), only the averages are indicated.

As lubricant, Luba 21® oil for high pressures was applied on the top and bottom surfaces. Between the disks, the surfaces remained untreated.
pressure by Gil et al. [8]. They used a non-linear equation to calculate the changing tendency of the lubricant conditions, which relationship will be used in this manuscript later, too.

## The compression tests' method

The compression tests were carried out on an Instron 4482 universal material testing machine (max. load 100 kN), at room temperature. The compressive dies were produced by the institutional cutting workshop using Böhler K110 (X153CrMoV12) commercial tool steel material. The heat treating after the turning process was also done in-house, following the merchant's recommendation [9]. The dies were polished in the final step (Fig. 1).

<figure><img src="image_1132.png" /><figcaption>Fig. 1 The compressive dies after the complete manufacturing process</figcaption></figure>

## Workpiece setups

Three workpiece setups are investigated in this study with the aim to consider the effect of the aspect ratio on the friction conversation. The disks' nominal diameters are uniformly Ø10 mm, or exactly 9.73...9.88 mm, so as a statistical mean of 9.81 mm. The nominal sheet thickness is 1.0 mm, or the exact mean of them is 1.05 mm based on fifty thickness measurements. (The average diameter is also a statistical result of fifty disks.)

Setups by stacking a single (1x10), five (5x10) and ten (10x10) disks on each other were formed, which resulted in 1/10, 1/2 and 1/1 aspect (height-diameter) ratios. In order to compile the 'towers' as accurate as possible before the compression, a self-developed positioning device was manufactured by 3D printing. The rolling direction was indicated and considered during the compilation of the laser cut disks.

<figure><img src="image_1133.png" /><figcaption>Fig. 2 A 10x10 stack with the positioning device (left) and the initial 'towers' of the 1x10 and the 10x10 configurations</figcaption></figure>

The workpiece material is a commercial mild steel, DC04 cold rolled sheet. The basic material parameters were obtained by standard tensile tests (ISO 6892-1:2016), at room temperature and the average values are listed in Table 1. The testing equipment and the applied $1.7 \times 10^{-3} \text{ s}^{-1}$ strain rate were the same with which we performed the compression tests. Both the strength and deformation properties were monitored in parallel, perpendicular and 45° to the rolling direction, but for some characteristics (e.g. YS – yield strength; UTS – ultimate tensile strength; $A_{80}$ – total elongation; and $A_g$ - uniform elongation), only the averages are indicated.

As lubricant, Luba 21® oil for high pressures was applied on the top and bottom surfaces. Between the disks, the surfaces remained untreated.
<table><caption>Table 1 Basic, average material parameters of the DC04 steel</caption><thead><tr><th>YS<br>[N/mm²]</th><th>UTS<br>[N/mm²]</th><th>A<sub>80</sub><br>[%]</th><th>A<sub>g</sub><br>[%]</th><th>r<sub>0</sub><br>[-]</th><th>r<sub>45</sub><br>[-]</th><th>r<sub>90</sub><br>[-]</th><th>r̄<br>[-]</th></tr></thead><tbody><tr><td>229</td><td>336</td><td>37.9</td><td>20.4</td><td>1.823</td><td>1.311</td><td>2.380</td><td>1.706</td></tr></tbody></table>

## Determination of the machine error

It is an old observation already that the machine stroke cannot provide adequate data for the specimen strain. This is especially true for high loads, mainly if the load occurs during small displacements. As the aspect ratio decreases, the process moves more and more in this direction at our investigations. To obtain the accurate stack heights, pre-tests were carried out for all setups with the object of determining the relationship between the true stack height and the machine stroke. The true specimen heights (i.e. the distance of the dies) were measured using gauge bars, after interrupting the pre-tests at different displacements. Based on our observations, linear approximations fit well the relationship between the true stack height and the machine cross head displacement (Fig. 3), i.e. the machine stroke values were suitable to convert the geometrical data to average strains.

<figure><img src="image_1132.png" /><figcaption>Fig. 3 Determination of the machine error through comparing the true specimen height reduction with the machine crosshead stroke</figcaption></figure>

Knowing the exact specimen height at every moment, volume constancy was used to calculate the current, average compression true strain (ε<sub>c</sub>) and true stress (σ<sub>c</sub>) (Fig. 4).
<table><caption>Table 1 Basic, average material parameters of the DC04 steel</caption><thead><tr><th>YS<br>[N/mm²]</th><th>UTS<br>[N/mm²]</th><th>A<sub>80</sub><br>[%]</th><th>A<sub>g</sub><br>[%]</th><th>r<sub>0</sub><br>[-]</th><th>r<sub>45</sub><br>[-]</th><th>r<sub>90</sub><br>[-]</th><th>r̄<br>[-]</th></tr></thead><tbody><tr><td>229</td><td>336</td><td>37.9</td><td>20.4</td><td>1.823</td><td>1.311</td><td>2.380</td><td>1.706</td></tr></tbody></table>

## Determination of the machine error

It is an old observation already that the machine stroke cannot provide adequate data for the specimen strain. This is especially true for high loads, mainly if the load occurs during small displacements. As the aspect ratio decreases, the process moves more and more in this direction at our investigations. To obtain the accurate stack heights, pre-tests were carried out for all setups with the object of determining the relationship between the true stack height and the machine stroke. The true specimen heights (i.e. the distance of the dies) were measured using gauge bars, after interrupting the pre-tests at different displacements. Based on our observations, linear approximations fit well the relationship between the true stack height and the machine cross head displacement (Fig. 3), i.e. the machine stroke values were suitable to convert the geometrical data to average strains.

<figure><img src="image_1134.png" /><figcaption>Fig. 3 Determination of the machine error through comparing the true specimen height reduction with the machine crosshead stroke</figcaption></figure>

Knowing the exact specimen height at every moment, volume constancy was used to calculate the current, average compression true strain (ε<sub>c</sub>) and true stress (σ<sub>c</sub>) (Fig. 4).
<figure><img src="image_1133.png" /><figcaption>Fig. 4 Force – height reduction (left) and compressive stress – compressive strain (right) curves</figcaption></figure>

Determination of friction coefficients

Inverse friction coefficient determination was performed on the basis of the comparison of experimental and modeled characteristics. Both the barreling effect and the force-stroke (force-height reduction) curves of individual disks were monitored in the Simufact Forming® software to find the best matches while the friction coefficient was changed. Other input parameters of the
<figure><img src="image_1135.png" /><figcaption>Fig. 4 Force – height reduction (left) and compressive stress – compressive strain (right) curves</figcaption></figure>

Determination of friction coefficients

Inverse friction coefficient determination was performed on the basis of the comparison of experimental and modeled characteristics. Both the barreling effect and the force-stroke (force-height reduction) curves of individual disks were monitored in the Simufact Forming® software to find the best matches while the friction coefficient was changed. Other input parameters of the
finite element modeling, such as the flow curve and the yield criterion (Hill' 48) were originated from the tensile tests results.

Our former results [10] showed that a constant 0.07 Coulomb friction coefficient closely approximates reality among these lubricant conditions. At the same time, it is also known (and partially evident from our former measurements, too) that the friction coefficient changes during the forming processes (see [6,7,8]). By continuously fitting the modeled barreling of the disks to the experimental results at different pressure values, the parameters of the non-linear pressure dependent friction coefficient (μPD) equation supposed by [8] have become available. In Eq. 1, p refers to the current pressure as well as p0 and μ0 are reference parameters and n is the pressure exponent.

$$ \mu_{PD} = \mu_0 \cdot \left(\frac{p}{p_0}\right)^{n-1}. \qquad (1) $$

According to our experiences these parameters fluctuate depending on some circumstances, e.g. on which direction (parallel or perpendicular to the rolling direction) is monitored, but a possible set of parameters could be the following: p0 = 618; μ0 = 0.077; n = -0.187 (though n is supposed to be between 0...1). The deeper question of these parameters is still under consideration, although readers can find more detailed discussion in [11].

## Flow curves' calculation

At compressive experiments, primarily the friction and the geometrical effects deflect the biaxial tension stress state existing in the sample's cross section, resulting in triaxial. Thus, to obtain equivalent quantities, the Siebel-type and the friction-hill analyses are usually used. Here we calculated the necessary compensations based on the latter model, as it describes:

$$ \sigma_f = \frac{2 \cdot \sigma_c}{\left(\frac{2h}{\mu d}\right)^2 \cdot \left(e^{\frac{d\mu}{h}} - \frac{d\mu}{h} - 1\right)}. \qquad (2) $$

In Eq. 2, h and d are the instantaneous height and diameter of the stack, as well as μ represents the friction coefficient.

Fig. 5 depicts the compensated compressive true stress - average compressive true strain curves as well as the flow curves (σf - εpl curves). The friction and geometry-based conversion is the same for both cases, but the diagrams on the left contain the elastic stages, too. The importance of this lies in the fact that the yield point falls somewhere lower if using the pressure dependent friction coefficient in Eq. 2. At the same time, the degree of the separation of the two curves are more pronounced, the smaller the aspect ratio. These results show that slopes of the high strain flow curves are about the same between 1/2...1/1 aspect ratios, regardless the friction compensation method. However, both the position and the slope of the flow curves deviate significantly if pressure dependent friction coefficient is considered, at a small aspect ratio. As a reference, the flow curve obtained from the tensile tests (T) is also indicated in the diagrams of Fig. 5.
finite element modeling, such as the flow curve and the yield criterion (Hill' 48) were originated from the tensile tests results.

Our former results [10] showed that a constant 0.07 Coulomb friction coefficient closely approximates reality among these lubricant conditions. At the same time, it is also known (and partially evident from our former measurements, too) that the friction coefficient changes during the forming processes (see [6,7,8]). By continuously fitting the modeled barreling of the disks to the experimental results at different pressure values, the parameters of the non-linear pressure dependent friction coefficient (μPD) equation supposed by [8] have become available. In Eq. 1, p refers to the current pressure as well as p0 and μ0 are reference parameters and n is the pressure exponent.

$$ \mu_{PD} = \mu_0 \cdot \left(\frac{p}{p_0}\right)^{n-1}. \qquad (1) $$

According to our experiences these parameters fluctuate depending on some circumstances, e.g. on which direction (parallel or perpendicular to the rolling direction) is monitored, but a possible set of parameters could be the following: p0 = 618; μ0 = 0.077; n = -0.187 (though n is supposed to be between 0...1). The deeper question of these parameters is still under consideration, although readers can find more detailed discussion in [11].

## Flow curves' calculation

At compressive experiments, primarily the friction and the geometrical effects deflect the biaxial tension stress state existing in the sample's cross section, resulting in triaxial. Thus, to obtain equivalent quantities, the Siebel-type and the friction-hill analyses are usually used. Here we calculated the necessary compensations based on the latter model, as it describes:

$$ \sigma_f = \frac{2 \cdot \sigma_c}{\left(\frac{2h}{\mu d}\right)^2 \cdot \left(e^{\frac{d\mu}{h}} - \frac{d\mu}{h} - 1\right)}. \qquad (2) $$

In Eq. 2, h and d are the instantaneous height and diameter of the stack, as well as μ represents the friction coefficient.

Fig. 5 depicts the compensated compressive true stress - average compressive true strain curves as well as the flow curves (σf - εpl curves). The friction and geometry-based conversion is the same for both cases, but the diagrams on the left contain the elastic stages, too. The importance of this lies in the fact that the yield point falls somewhere lower if using the pressure dependent friction coefficient in Eq. 2. At the same time, the degree of the separation of the two curves are more pronounced, the smaller the aspect ratio. These results show that slopes of the high strain flow curves are about the same between 1/2...1/1 aspect ratios, regardless the friction compensation method. However, both the position and the slope of the flow curves deviate significantly if pressure dependent friction coefficient is considered, at a small aspect ratio. As a reference, the flow curve obtained from the tensile tests (T) is also indicated in the diagrams of Fig. 5.
<figure><img src="image_1134.png" /><figcaption>Fig. 5 Compressive true stress after friction compensation – compressive strain curves (left) and flow curves (right)</figcaption></figure>

Important to mention that all the measurement results are suitable for determining the flow curves at high strain ranges using the work equivalence principle [12]. But as discussed above, the slope of the curves could deviate from each other if different friction interpretation is applied, especially for small aspect ratios. It is illustrated in Fig. 6.
<figure><img src="image_1136.png" /><figcaption>Fig. 5 Compressive true stress after friction compensation – compressive strain curves (left) and flow curves (right)</figcaption></figure>

Important to mention that all the measurement results are suitable for determining the flow curves at high strain ranges using the work equivalence principle [12]. But as discussed above, the slope of the curves could deviate from each other if different friction interpretation is applied, especially for small aspect ratios. It is illustrated in Fig. 6.
To calculate the work equivalence, the tensile test results were considered as reference ($W_{ref}^T$), and the plastic work at the final point of the reference curve was considered as the basis of the calculations:

$$W_{ref}^T = W^{pl}, \qquad (3)$$

$$\sigma_{ref}^T \cdot \varepsilon_{ref}^T = \sigma_f \cdot \varepsilon^{pl}. \qquad (4)$$

<figure><img src="image_1135.png" /><figcaption>Fig. 6 Flow stress – plastic work curves after the work equivalence compensation</figcaption></figure>

## Summary

This paper deals with the friction compensation methodology at stack compression tests, for three setups with different aspect (height-diameter) ratios. Constant Coulomb friction coefficient obtained from former research of the authors and currently calculated pressure dependent coefficient were factored into the friction-hill analysis to draw the flow stress curves. The results show that there is no significant deviation between the flow stresses obtained by constant and pressure dependent coefficients at 1/2...1/1 aspect ratios. However, under the aspect ratio of 1/2, the flow stress curves strongly separate from each other depending on the friction evaluation, causing relevant differences in the slopes of the curves. This can be misleading if the stack compression test is applied for expanding a reference (e.g. a tensile test originated) flow curve for higher strains. The equation parameters and the methodology to obtain the pressure dependent function of the friction coefficient is still under consideration.
To calculate the work equivalence, the tensile test results were considered as reference ($W_{ref}^T$), and the plastic work at the final point of the reference curve was considered as the basis of the calculations:

$$W_{ref}^T = W^{pl}, \qquad (3)$$

$$\sigma_{ref}^T \cdot \varepsilon_{ref}^T = \sigma_f \cdot \varepsilon^{pl}. \qquad (4)$$

<figure><img src="image_1137.png" /><figcaption>Fig. 6 Flow stress – plastic work curves after the work equivalence compensation</figcaption></figure>

## Summary

This paper deals with the friction compensation methodology at stack compression tests, for three setups with different aspect (height-diameter) ratios. Constant Coulomb friction coefficient obtained from former research of the authors and currently calculated pressure dependent coefficient were factored into the friction-hill analysis to draw the flow stress curves. The results show that there is no significant deviation between the flow stresses obtained by constant and pressure dependent coefficients at 1/2...1/1 aspect ratios. However, under the aspect ratio of 1/2, the flow stress curves strongly separate from each other depending on the friction evaluation, causing relevant differences in the slopes of the curves. This can be misleading if the stack compression test is applied for expanding a reference (e.g. a tensile test originated) flow curve for higher strains. The equation parameters and the methodology to obtain the pressure dependent function of the friction coefficient is still under consideration.
## References

[1] S. Coppieters, H. Traphöner, F. Stiebert, T. Balan, T. Kuwabara, A.E. Tekkaya, Large strain flow curve identification for sheet metal, J. Mat. Pro. Tec. 308 (2022) 117725. https://doi.org/10.1016/j.jmatprotec.2022.117725

[2] M. Merklein, A. Kuppert, A method for the layer compression test considering the anisotropic material behavior, Int. J. Mater. Form. Vol. 2 (2009) 483-486. doi:10.1007/s12289-009-0592-8

[3] S. Coppieters, P. Lava, H. Sol, A. Van Bael, P. Van Houtte, D. Debruyne, Determination of the flow stress and contact friction of sheet metal in a multi-layered upsetting test, J. Mat. Pro. Tec. 210 (2010) 1290-1296. https://doi.org/10.1016/j.jmatprotec.2010.03.017

[4] S. Coppieters, M. Jackel, C. Kraus, T. Kuwabara, F. Barlat, Influence of hydrostatic pressure shift on the flow stress in sheet metal, Proc. Manuf. 47 (2020) 1245-1249. https://doi.org/10.1016/j.promfg.2020.04.196

[5] Y.G. An, H. Vegter, Analytical and experimental study of frictional behavior in through-thickness compression test, J. Mat. Pro. Tec. 160 (2005) 148-155. https://doi.org/10.1016/j.jmatprotec.2004.05.026

[6] T. Kuwabara, R. Tachibana, Y. Takada, T. Koizumi, S. Coppieters, F. Barlat, Effect of hydrostatic stress on the strength differential effect in low-carbon steel sheet, Int. J. Mater. Form. 15:13 (2022) https://doi.org/10.1007/s12289-022-01650-2

[7] M. Kraus, M. Lenzen, M. Merklein, Contact pressure-dependent friction characterization by using a single sheet metal compression test, Wear 476 (2021) 203679. https://doi.org/10.1016/j.wear.2021.203679

[8] I. Gil, J. Mendiguren, L. Galdos, E. Magurra, E.S. de Argandona, Influence of the pressure dependent coefficient of friction on deep drawing springback prediction, Tribology International 103 (2016) 266-273. http://dx.doi.org/10.1016/j.triboint.2016.07.004

[9] voestalpine BÖHLER Edelstahl GmbH & Co KG, products/k110/product description

[10] G.J. Beres, Z. Weltsch, R. Borbely, M.L. Kolus, Zs. Lukacs, M. Tisza, An extended stress-based forming limit diagram focusing on the wrinkling phenomenon and the effect of the normal pressure on clamped surfaces, J. Mat. Pro. Tec. 322 (2023) 118196. https://doi.org/10.1016/j.jmatprotec.2023.118196

[11] M.L. Kolus, R. Borbely, G.J. Beres, Friction coefficient validation by finite element method for disk compression test, in Conference Book of 38th International Colloquium on Advanced Manufacturing and Repair Technologies in Vehicle Industry (2023) 17-22.

[12] C. Chermette, K. Unruh, I. Peshekhodov, J. Chottin, T. Balan, A new analytical method for determination of the flow curve for high-strength sheet steels using the plane strain compression test, Int. J. Mater. Form. 13 (2020) 269-292. https://doi.org/10.1007/s12289-019-01485-4
## References

[1] S. Coppieters, H. Traphöner, F. Stiebert, T. Balan, T. Kuwabara, A.E. Tekkaya, Large strain flow curve identification for sheet metal, J. Mat. Pro. Tec. 308 (2022) 117725. https://doi.org/10.1016/j.jmatprotec.2022.117725

[2] M. Merklein, A. Kuppert, A method for the layer compression test considering the anisotropic material behavior, Int. J. Mater. Form. Vol. 2 (2009) 483-486. doi:10.1007/s12289-009-0592-8

[3] S. Coppieters, P. Lava, H. Sol, A. Van Bael, P. Van Houtte, D. Debruyne, Determination of the flow stress and contact friction of sheet metal in a multi-layered upsetting test, J. Mat. Pro. Tec. 210 (2010) 1290-1296. https://doi.org/10.1016/j.jmatprotec.2010.03.017

[4] S. Coppieters, M. Jackel, C. Kraus, T. Kuwabara, F. Barlat, Influence of hydrostatic pressure shift on the flow stress in sheet metal, Proc. Manuf. 47 (2020) 1245-1249. https://doi.org/10.1016/j.promfg.2020.04.196

[5] Y.G. An, H. Vegter, Analytical and experimental study of frictional behavior in through-thickness compression test, J. Mat. Pro. Tec. 160 (2005) 148-155. https://doi.org/10.1016/j.jmatprotec.2004.05.026

[6] T. Kuwabara, R. Tachibana, Y. Takada, T. Koizumi, S. Coppieters, F. Barlat, Effect of hydrostatic stress on the strength differential effect in low-carbon steel sheet, Int. J. Mater. Form. 15:13 (2022) https://doi.org/10.1007/s12289-022-01650-2

[7] M. Kraus, M. Lenzen, M. Merklein, Contact pressure-dependent friction characterization by using a single sheet metal compression test, Wear 476 (2021) 203679. https://doi.org/10.1016/j.wear.2021.203679

[8] I. Gil, J. Mendiguren, L. Galdos, E. Magurra, E.S. de Argandona, Influence of the pressure dependent coefficient of friction on deep drawing springback prediction, Tribology International 103 (2016) 266-273. http://dx.doi.org/10.1016/j.triboint.2016.07.004

[9] voestalpine BÖHLER Edelstahl GmbH & Co KG, products/k110/product description

[10] G.J. Beres, Z. Weltsch, R. Borbely, M.L. Kolus, Zs. Lukacs, M. Tisza, An extended stress-based forming limit diagram focusing on the wrinkling phenomenon and the effect of the normal pressure on clamped surfaces, J. Mat. Pro. Tec. 322 (2023) 118196. https://doi.org/10.1016/j.jmatprotec.2023.118196

[11] M.L. Kolus, R. Borbely, G.J. Beres, Friction coefficient validation by finite element method for disk compression test, in Conference Book of 38th International Colloquium on Advanced Manufacturing and Repair Technologies in Vehicle Industry (2023) 17-22.

[12] C. Chermette, K. Unruh, I. Peshekhodov, J. Chottin, T. Balan, A new analytical method for determination of the flow curve for high-strength sheet steels using the plane strain compression test, Int. J. Mater. Form. 13 (2020) 269-292. https://doi.org/10.1007/s12289-019-01485-4
# Inverse estimation of material model parameters using Bayesian data assimilation

YAMANAKA Akinori¹,ᵃ*, SUDA Michihiko¹,ᵇ, SUEKI Sae¹,ᶜ, SASAKI Kengo¹,ᵈ, and FUNAMOTO Ryuki¹,ᵉ

¹Tokyo University of Agriculture and Technology, 2-24-16, Naka-cho, Koganei, Tokyo, 184-8588, Japan

ᵃa-yamana@cc.tuat.ac.jp, ᵇs226270r@st.go.tuat.ac.jp, ᶜs227904y@st.go.tuat.ac.jp, ᵈs234450w@st.go.tuat.ac.jp, ᵉs222036v@st.go.tuat.ac.jp

**Keywords:** Bayesian Data Assimilation, Digital Image Correlation, Material Model

**Abstract.** This study proposes a new method for the inverse estimation of the parameters included in material models from full-field measurement data that are obtained using the digital image correlation method. This approach is based on data assimilation according to the Bayes' theorem (Bayesian data assimilation). In this study, we demonstrate the assimilation of experimental data obtained from uniaxial tensile, forming, and fracture tests of aluminum alloys into elastoplastic finite element and phase-field crack propagation simulations. The proposed method allows the simultaneous estimation of multiple material model parameters. The Bayesian data assimilation is a promising methodology for estimating the parameters of different material models and constructing digital twins of material deformation.

**Introduction**

The sustainability in the development of newer products may be achieved by reducing the overall material consumption during the R&D of new devices. An effective approach to satisfy this requirement is constructing numerical simulation models that can quantitatively reproduce real phenomena in a virtual space, i.e., digital twins. However, the construction of digital twins requires accurate identification of the physical properties of target materials and the parameters included in material models that describe material behavior. The conventional methods for identifying material model parameters from experimental data include the virtual field method (VFM) [1] and the finite element model updating (FEMU) method [2, 3]. In recent years, inverse parameter estimation methods based on data assimilation (DA) and Bayes' theorem (Bayesian DA) have garnered considerable attention. The Bayesian DA algorithms can be classified into sequential and nonsequential. A representative example of the application of sequential DA in materials engineering is the ensemble Kalman filter, which has been applied to a phase-field model describing the evolution of microstructures in materials [4-6]. Furthermore, nonsequential DA has been applied to elastoplastic finite element (EPFE) simulations of the deformation and forming of metallic materials. In this study, we suggest the use of nonsequential DA to inversely estimate the material model parameters (i.e., parameters included in the constitutive equations, work-hardening laws, and yield functions) according to the full-field measurement data acquired using the digital image correlation (DIC) method. This paper presents the results of three case studies wherein nonsequential DA was applied to EPFE and phase-field crack (PFC) propagation simulations.

<figure><img src="image_1136.png" /></figure>
# Inverse estimation of material model parameters using Bayesian data assimilation

YAMANAKA Akinori¹,ᵃ*, SUDA Michihiko¹,ᵇ, SUEKI Sae¹,ᶜ, SASAKI Kengo¹,ᵈ, and FUNAMOTO Ryuki¹,ᵉ

¹Tokyo University of Agriculture and Technology, 2-24-16, Naka-cho, Koganei, Tokyo, 184-8588, Japan

ᵃa-yamana@cc.tuat.ac.jp, ᵇs226270r@st.go.tuat.ac.jp, ᶜs227904y@st.go.tuat.ac.jp, ᵈs234450w@st.go.tuat.ac.jp, ᵉs222036v@st.go.tuat.ac.jp

**Keywords:** Bayesian Data Assimilation, Digital Image Correlation, Material Model

**Abstract.** This study proposes a new method for the inverse estimation of the parameters included in material models from full-field measurement data that are obtained using the digital image correlation method. This approach is based on data assimilation according to the Bayes' theorem (Bayesian data assimilation). In this study, we demonstrate the assimilation of experimental data obtained from uniaxial tensile, forming, and fracture tests of aluminum alloys into elastoplastic finite element and phase-field crack propagation simulations. The proposed method allows the simultaneous estimation of multiple material model parameters. The Bayesian data assimilation is a promising methodology for estimating the parameters of different material models and constructing digital twins of material deformation.

**Introduction**

The sustainability in the development of newer products may be achieved by reducing the overall material consumption during the R&D of new devices. An effective approach to satisfy this requirement is constructing numerical simulation models that can quantitatively reproduce real phenomena in a virtual space, i.e., digital twins. However, the construction of digital twins requires accurate identification of the physical properties of target materials and the parameters included in material models that describe material behavior. The conventional methods for identifying material model parameters from experimental data include the virtual field method (VFM) [1] and the finite element model updating (FEMU) method [2, 3]. In recent years, inverse parameter estimation methods based on data assimilation (DA) and Bayes' theorem (Bayesian DA) have garnered considerable attention. The Bayesian DA algorithms can be classified into sequential and nonsequential. A representative example of the application of sequential DA in materials engineering is the ensemble Kalman filter, which has been applied to a phase-field model describing the evolution of microstructures in materials [4-6]. Furthermore, nonsequential DA has been applied to elastoplastic finite element (EPFE) simulations of the deformation and forming of metallic materials. In this study, we suggest the use of nonsequential DA to inversely estimate the material model parameters (i.e., parameters included in the constitutive equations, work-hardening laws, and yield functions) according to the full-field measurement data acquired using the digital image correlation (DIC) method. This paper presents the results of three case studies wherein nonsequential DA was applied to EPFE and phase-field crack (PFC) propagation simulations.

<figure><img src="image_1138.png" /></figure>
## Bayesian DA

DA is based on the state-space model, which consists of a system model and an observation model expressed using the following equations [7]:

$$x_t = M(x_{t-1}) + v_t, \qquad (1)$$

where $x_t$ is a state vector at time $t = n\Delta t$ ($n = 1, 2, 3, \dots, N_{end}$), which includes the state variables of the system and the material parameters to be estimated, $M$ expresses the simulation model, and $v_t$ is the system noise at time $t$, which corresponds to imperfections in the simulation model and numerical errors. The observational model is expressed as follows:

$$y_t = H(x_t) + w_t, \qquad (2)$$

Here, $y_t$ is an observation vector that includes the experimental data obtained at time $t$. In this study, $y_t$ contains the displacement and strain measured using the DIC method, $H$ is an observation operator that extracts the components of $x_t$ for comparison with those of $y_t$, and $w_t$ is the observational noise at time $t$, which shows the experimental errors.

DA tracks the evolution of the probability density function (PDF) $x_t$ according to Eq. (1), and corrects the PDF based on Bayes' theorem, which is given using the following equation:

$$p(x_t|y_{1:t}) = p(y_t|x_t)p(x_t|y_{1:t-1}). \qquad (3)$$

Here, $p(x_t|y_{1:t-1})$, $p(y_t|x_t)$, and $p(x_t|y_{1:t})$ are the conditional PDFs referred to as the prior, likelihood, and posterior distributions, respectively. For example, $y_{1:t-1}$ describes the observation data from time $t=1$ to $t=(n-1)\Delta t$, where $n$ is the number of time steps. It should be noted that all the PDFs in this study are assumed to be multivariate Gaussian functions.

The nonsequential DA is based on the four-dimensional variational method (4DVar). As shown in Fig. 1, the 4DVar searches the optimal state vector at time $t=0$, $x_0^{opt}$, which minimizes the cost function given by the following equation:

$$J(x_0) = \frac{1}{2}(x_0 - x_0^b)^T B^{-1}(x_0 - x_0^b) + \sum_{t=0}^{t_{end}} \frac{1}{2}(H(x_t) - y_t)^T R_t^{-1}(H(x_t) - y_t) \qquad (4)$$

The second term on the right-hand sides of Eq. (4) is the time integration of the misfit between the simulation and observation data (mathematically given by $H(x_t) - y_t$): $x_0^b$ is a vector that contains the initially estimated system state and initial values of the parameters to be estimated. Further, $B$ and $R_t$ are the covariance matrices of $p(x_0)$ and $p(w_t)$, respectively.

<figure><img src="image_1137.png" /><figcaption>Figure 1 Schematic of 4DVar-based DA</figcaption></figure>

To search for optimal state vector $x_0^{opt}$, the conventional 4DVar calculates the gradient of the cost function, which is given by

$$\nabla J(x_0) = B^{-1}(x_0 - x_0^b) + \sum_{t=0}^{t_{end}} \hat{H}_t^T R_t^{-1} (\hat{H}_t(x_0) - y_t), \qquad (5)$$
## Bayesian DA

DA is based on the state-space model, which consists of a system model and an observation model expressed using the following equations [7]:

$$x_t = M(x_{t-1}) + v_t, \qquad (1)$$

where $x_t$ is a state vector at time $t = n\Delta t$ ($n = 1, 2, 3, \dots, N_{end}$), which includes the state variables of the system and the material parameters to be estimated, $M$ expresses the simulation model, and $v_t$ is the system noise at time $t$, which corresponds to imperfections in the simulation model and numerical errors. The observational model is expressed as follows:

$$y_t = H(x_t) + w_t, \qquad (2)$$

Here, $y_t$ is an observation vector that includes the experimental data obtained at time $t$. In this study, $y_t$ contains the displacement and strain measured using the DIC method, $H$ is an observation operator that extracts the components of $x_t$ for comparison with those of $y_t$, and $w_t$ is the observational noise at time $t$, which shows the experimental errors.

DA tracks the evolution of the probability density function (PDF) $x_t$ according to Eq. (1), and corrects the PDF based on Bayes' theorem, which is given using the following equation:

$$p(x_t|y_{1:t}) = p(y_t|x_t)p(x_t|y_{1:t-1}). \qquad (3)$$

Here, $p(x_t|y_{1:t-1})$, $p(y_t|x_t)$, and $p(x_t|y_{1:t})$ are the conditional PDFs referred to as the prior, likelihood, and posterior distributions, respectively. For example, $y_{1:t-1}$ describes the observation data from time $t=1$ to $t=(n-1)\Delta t$, where $n$ is the number of time steps. It should be noted that all the PDFs in this study are assumed to be multivariate Gaussian functions.

The nonsequential DA is based on the four-dimensional variational method (4DVar). As shown in Fig. 1, the 4DVar searches the optimal state vector at time $t=0$, $x_0^{opt}$, which minimizes the cost function given by the following equation:

$$J(x_0) = \frac{1}{2}(x_0 - x_0^b)^T B^{-1}(x_0 - x_0^b) + \sum_{t=0}^{t_{end}} \frac{1}{2}(H(x_t) - y_t)^T R_t^{-1}(H(x_t) - y_t) \qquad (4)$$

The second term on the right-hand sides of Eq. (4) is the time integration of the misfit between the simulation and observation data (mathematically given by $H(x_t) - y_t$): $x_0^b$ is a vector that contains the initially estimated system state and initial values of the parameters to be estimated. Further, $B$ and $R_t$ are the covariance matrices of $p(x_0)$ and $p(w_t)$, respectively.

<figure><img src="image_1139.png" /><figcaption>Figure 1 Schematic of 4DVar-based DA</figcaption></figure>

To search for optimal state vector $x_0^{opt}$, the conventional 4DVar calculates the gradient of the cost function, which is given by

$$\nabla J(x_0) = B^{-1}(x_0 - x_0^b) + \sum_{t=0}^{t_{end}} \hat{H}_t^T R_t^{-1} (\hat{H}_t(x_0) - y_t), \qquad (5)$$
where

$$ \hat{H}_t(\mathbf{x}_0) = H(M'(\mathbf{x}_0)) = H_t M_{t-1} M_{t-2} \cdots M_0. \qquad (6) $$

To calculate Eq. (6), adjoint model $M_t$ is required. $M_t$ can be expressed as follows:

$$ M_t = \frac{\partial M(\mathbf{x}_t)}{\partial \mathbf{x}_t}. \qquad (7) $$

However, the analytical derivation of $M_t$ is difficult for nonlinear simulation models (e.g., the EPFE and PFC simulation models). Therefore, the ensemble 4DVar (En4DVar), which helps calculate the gradient of the cost function using the ensemble approximation of the PDF, has been used in previous studies [9, 10]. Furthermore, the DMC-TPE method [11] has been recently suggested for use as an adjoint model-free approach for 4DVar DA. The flowchart for estimating optimal state vector $\mathbf{x}_0^{opt}$ using the DMC-TPE method has been explained in a previous study [11].

## Applications of Bayesian DA

This section presents the results of three case studies (I–III) using the En4DVar- and DMC-TPE-based DA methods. In case study I, En4DVar-based DA was used to estimate the parameters of the strain-hardening law according to the full-field measurement data of the uniaxial tensile test [12]. In this case study, a round bar specimen of the A5052-O aluminum alloy was used. MultiDIC [13] was used for full-field measurements of the displacement and strain fields at the specimen surface. The Hill'48 yield function and Swift's strain-hardening equation ($\sigma = C (\epsilon^p + \epsilon_0)^n$) were used for the EPFE simulation because this yield function and strain-hardening equation are widely adopted. The material models were implemented in ABAQUS/standard using UMMDp [14]. Parameters $C$, $\epsilon_0$, and $n$ in Swift's strain-hardening equation were targeted for estimation using En4DVar-based DA. The initial parameter estimates were assumed to follow a Gaussian PDF. The mean values of the initial estimates were $C = 650$ MPa, $\epsilon_0 = 0.0042$, and $n = 0.3$, and the standard deviation was 10 % of the mean.

Figure 2(a) shows the variation in the cost function according to the number of iterative minimization calculations. Because the En4DVar minimizes the cost function based on the gradient of the cost function, the value of the cost function decreases monotonically. The optimal estimated values of the parameters that yielded the minimum value of the cost function were $C = 320$ MPa, $\epsilon_0 = 0.0053$, and $n = 0.113$. By integrating the estimated parameters, the EPFE simulation reproduced the experimentally observed strain evolution, as shown in Fig. 2(b) and (c). A recent study demonstrated that En4DVar-based DA helps estimate not only the strain-hardening parameters but also the yield function parameters [15]. In future works, the estimated parameters can be compared with those identified using a classical analytical calibration method.

In case study II, DMC-TPE-based DA was used to estimate the parameters of the anisotropic yield function. To obtain the observational data, hole expansion tests and DIC measurements were performed on an A5052-O aluminum alloy sheet with a nominal thickness of 1.0 mm. In this study, we used the Yld2000-2d yield function, which describes the anisotropic plastic deformation behavior of the material. The parameters of the Yld2000-2d yield function were estimated from the full-field strain field data that were obtained using the DIC method. Furthermore, by integrating the estimated parameters, a deep-drawing test and an EPFE simulation were performed to verify the feasibility of the parameters. Figure 3(a) and (b) show the results of the deep-drawing simulation based on the estimated parameters and the experimentally obtained displacement distribution, respectively. It can be seen that the simulation qualitatively reproduces the experimentally measured displacement field. Figure 3(c) shows the absolute difference between the experimentally measured displacement field and simulation results. Although the simulation results slightly differ from the experimental results, the accuracy of the simulation may be improved by simultaneously estimating not only the yield function parameters but also the process parameters (e.g., the friction coefficient between the blank and punch).
where

$$ \hat{H}_t(\mathbf{x}_0) = H(M'(\mathbf{x}_0)) = H_t M_{t-1} M_{t-2} \cdots M_0. \qquad (6) $$

To calculate Eq. (6), adjoint model $M_t$ is required. $M_t$ can be expressed as follows:

$$ M_t = \frac{\partial M(\mathbf{x}_t)}{\partial \mathbf{x}_t}. \qquad (7) $$

However, the analytical derivation of $M_t$ is difficult for nonlinear simulation models (e.g., the EPFE and PFC simulation models). Therefore, the ensemble 4DVar (En4DVar), which helps calculate the gradient of the cost function using the ensemble approximation of the PDF, has been used in previous studies [9, 10]. Furthermore, the DMC-TPE method [11] has been recently suggested for use as an adjoint model-free approach for 4DVar DA. The flowchart for estimating optimal state vector $\mathbf{x}_0^{opt}$ using the DMC-TPE method has been explained in a previous study [11].

## Applications of Bayesian DA

This section presents the results of three case studies (I–III) using the En4DVar- and DMC-TPE-based DA methods. In case study I, En4DVar-based DA was used to estimate the parameters of the strain-hardening law according to the full-field measurement data of the uniaxial tensile test [12]. In this case study, a round bar specimen of the A5052-O aluminum alloy was used. MultiDIC [13] was used for full-field measurements of the displacement and strain fields at the specimen surface. The Hill'48 yield function and Swift's strain-hardening equation ($\sigma = C (\epsilon^p + \epsilon_0)^n$) were used for the EPFE simulation because this yield function and strain-hardening equation are widely adopted. The material models were implemented in ABAQUS/standard using UMMDp [14]. Parameters $C$, $\epsilon_0$, and $n$ in Swift's strain-hardening equation were targeted for estimation using En4DVar-based DA. The initial parameter estimates were assumed to follow a Gaussian PDF. The mean values of the initial estimates were $C = 650$ MPa, $\epsilon_0 = 0.0042$, and $n = 0.3$, and the standard deviation was 10 % of the mean.

Figure 2(a) shows the variation in the cost function according to the number of iterative minimization calculations. Because the En4DVar minimizes the cost function based on the gradient of the cost function, the value of the cost function decreases monotonically. The optimal estimated values of the parameters that yielded the minimum value of the cost function were $C = 320$ MPa, $\epsilon_0 = 0.0053$, and $n = 0.113$. By integrating the estimated parameters, the EPFE simulation reproduced the experimentally observed strain evolution, as shown in Fig. 2(b) and (c). A recent study demonstrated that En4DVar-based DA helps estimate not only the strain-hardening parameters but also the yield function parameters [15]. In future works, the estimated parameters can be compared with those identified using a classical analytical calibration method.

In case study II, DMC-TPE-based DA was used to estimate the parameters of the anisotropic yield function. To obtain the observational data, hole expansion tests and DIC measurements were performed on an A5052-O aluminum alloy sheet with a nominal thickness of 1.0 mm. In this study, we used the Yld2000-2d yield function, which describes the anisotropic plastic deformation behavior of the material. The parameters of the Yld2000-2d yield function were estimated from the full-field strain field data that were obtained using the DIC method. Furthermore, by integrating the estimated parameters, a deep-drawing test and an EPFE simulation were performed to verify the feasibility of the parameters. Figure 3(a) and (b) show the results of the deep-drawing simulation based on the estimated parameters and the experimentally obtained displacement distribution, respectively. It can be seen that the simulation qualitatively reproduces the experimentally measured displacement field. Figure 3(c) shows the absolute difference between the experimentally measured displacement field and simulation results. Although the simulation results slightly differ from the experimental results, the accuracy of the simulation may be improved by simultaneously estimating not only the yield function parameters but also the process parameters (e.g., the friction coefficient between the blank and punch).
<figure><img src="image_1138.png" /><figcaption>Figure 2 (a) Transition of cost function $J(x_0)$. Evolution of strain $\epsilon_{zz}$ during uniaxial tensile deformation of the A5052-O aluminum alloy round bar specimen (b) measured by the DIC method and (c) simulated by the EPFE simulation with optimally estimated strain-hardening parameters.</figcaption></figure>

<figure><img src="image_1139.png" /><figcaption>Figure 3 Distribution of displacement along x-direction ($U_x$) around the top of punch at a punch height of 60 mm (a) calculated using deep-drawing simulation with optimally estimated yield function parameters ($U_x^{sim}$) and (b) measured by the DIC method ($U_x^{obs}$). Absolute difference between simulated and measured displacements is shown in (c).</figcaption></figure>

The PFC model is a numerical model that is suitable for simulating complex crack propagation behaviors, including crack branching and coarsening. In case study III, the parameters included in the PFC model, which represent the crack propagation behavior in sheet metals, were estimated using DA according to the DMC-TPE algorithm. A uniaxial tensile specimen was prepared from the same aluminum alloy sheet used in case study II. An asymmetrically notched specimen was fabricated to promote crack propagation. The crack propagation tests were performed at a tensile deformation rate ($\dot{U}$) of 2.0 mm/min.

The PFC model proposed by Khalil et al. [16] was used in this study. The model consists of the stress equilibrium equation and the governing equation of order parameter $\phi$, as shown below:
$$ (1 - \phi)^2 \nabla \cdot \sigma = 0, \qquad (8) $$

where $\phi$ is the order parameter, which is defined as $\phi = 1$ inside a crack, and $\phi = 0$ inside an uncracked material. Here, $\phi$ changed smoothly from 0 to 1 at the diffusive interface between the cracked and uncracked domains. Further, $\sigma$ denotes the Cauchy stress tensor. The governing equation for $\phi$ is as follows:

$$ K \left( \frac{\phi}{l} - l \nabla^2 \phi \right) - 2(1-\phi) g_{\text{strain}}(\mathbf{u}) = 0 \qquad (9) $$

Here, $K$ is a parameter related to the fracture toughness, $l$ is a parameter that controls the thickness of the diffusive interface, $g_{\text{strain}}(\mathbf{u})$ is the strain energy, and $\mathbf{u}$ is the displacement vector.
<figure><img src="image_1140.png" /><figcaption>Figure 2 (a) Transition of cost function $J(x_0)$. Evolution of strain $\epsilon_{zz}$ during uniaxial tensile deformation of the A5052-O aluminum alloy round bar specimen (b) measured by the DIC method and (c) simulated by the EPFE simulation with optimally estimated strain-hardening parameters.</figcaption></figure>

<figure><img src="image_1141.png" /><figcaption>Figure 3 Distribution of displacement along x-direction ($U_x$) around the top of punch at a punch height of 60 mm (a) calculated using deep-drawing simulation with optimally estimated yield function parameters ($U_x^{sim}$) and (b) measured by the DIC method ($U_x^{obs}$). Absolute difference between simulated and measured displacements is shown in (c).</figcaption></figure>

The PFC model is a numerical model that is suitable for simulating complex crack propagation behaviors, including crack branching and coarsening. In case study III, the parameters included in the PFC model, which represent the crack propagation behavior in sheet metals, were estimated using DA according to the DMC-TPE algorithm. A uniaxial tensile specimen was prepared from the same aluminum alloy sheet used in case study II. An asymmetrically notched specimen was fabricated to promote crack propagation. The crack propagation tests were performed at a tensile deformation rate ($\dot{U}$) of 2.0 mm/min.

The PFC model proposed by Khalil et al. [16] was used in this study. The model consists of the stress equilibrium equation and the governing equation of order parameter $\phi$, as shown below:
$$ (1 - \phi)^2 \nabla \cdot \sigma = 0, \qquad (8) $$

where $\phi$ is the order parameter, which is defined as $\phi = 1$ inside a crack, and $\phi = 0$ inside an uncracked material. Here, $\phi$ changed smoothly from 0 to 1 at the diffusive interface between the cracked and uncracked domains. Further, $\sigma$ denotes the Cauchy stress tensor. The governing equation for $\phi$ is as follows:

$$ K \left( \frac{\phi}{l} - l \nabla^2 \phi \right) - 2(1-\phi) g_{\text{strain}}(\mathbf{u}) = 0 \qquad (9) $$

Here, $K$ is a parameter related to the fracture toughness, $l$ is a parameter that controls the thickness of the diffusive interface, $g_{\text{strain}}(\mathbf{u})$ is the strain energy, and $\mathbf{u}$ is the displacement vector.
For the crack propagation simulation using the PFC model, the weak forms of Eqs. (8) and (9) are solved using the FEM. In this study, the PFC model was implemented according to ABAQUS/Standard based on the user-defined subroutine UELMAT [16]. The plastic deformation of the specimens was analyzed using the von Mises yield function and Voce's strain-hardening equation:

$$ \sigma = \sigma_0 + Q_{\infty} \left\{ 1 - \exp(-b\epsilon^p) \right\}. \qquad (10) $$

Here, $\sigma_0$, $Q_\infty$, and $b$ are the strain-hardening parameters. In case study III, the four parameters (i.e., $K$, $\sigma_0$, $Q_\infty$ and $b$) were estimated from the full-field strain field data by DMC-TPE-based DA. The initial estimates of the parameters were $K = 500$ kJ/mm², $\sigma_0 = 87.2$ MPa, $Q_\infty = 165.6$ MPa, and $b = 15.5$.

After the minimization of the cost function, the optimally estimated parameters included in vector $x_0^{opt}$ were $K = 472.9$ kJ/mm², $\sigma_0 = 94.37$ MPa, $Q_\infty = 295.85$ MPa, and $b = 9.68$. The computation time required to obtain the optimally estimated parameters was approximately 35 h without parallel computing. Figure 4(a) shows the evolution of true strain $\epsilon_{yy}$ measured using the DIC method as the cracks propagated. It can be observed that the strain concentration occurs near the crack tip as forced displacement $U$ increased along the $y$ direction. Figure 4(b) shows the crack propagation behavior (i.e., time evolution of $\phi$) obtained from the PFC simulation using the optimally estimated parameters. It is evident that the PFC simulation reproduces a crack propagation behavior similar to that observed in the experiment.

<figure><img src="image_1140.png" /><figcaption>Figure 4 Evolution of (a) true strain $\epsilon_{yy}$ during the crack propagation measured using the DIC method and (b) order parameter $\phi$ describing the crack propagation behavior simulated by the PFC model with optimally estimated parameters.</figcaption></figure>

## Summary

This paper presents the results of three case studies wherein Bayesian DA methods were applied to the deformation (case study I), formation (case study II), and fracture (case study III) of the A5052-O aluminum alloy. The results show that Bayesian DA is an effective tool for estimating the parameters of material models by utilizing full-field measurement data from the DIC method, which contributes to the digital twin construction of each phenomenon.

## Acknowledgement

This research was financially supported by the Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (KAKENHI) (B) (Grant Numbers: 20H02476 and 23H01297) and the AMADA Foundation (Grant Number: AF-2021002-A2).

## References

[1] M. Grédiac, E. Toussaint, F. Pierron, Special virtual fields for the direct determination of
For the crack propagation simulation using the PFC model, the weak forms of Eqs. (8) and (9) are solved using the FEM. In this study, the PFC model was implemented according to ABAQUS/Standard based on the user-defined subroutine UELMAT [16]. The plastic deformation of the specimens was analyzed using the von Mises yield function and Voce's strain-hardening equation:

$$ \sigma = \sigma_0 + Q_{\infty} \left\{ 1 - \exp(-b\epsilon^p) \right\}. \qquad (10) $$

Here, $\sigma_0$, $Q_\infty$, and $b$ are the strain-hardening parameters. In case study III, the four parameters (i.e., $K$, $\sigma_0$, $Q_\infty$ and $b$) were estimated from the full-field strain field data by DMC-TPE-based DA. The initial estimates of the parameters were $K = 500$ kJ/mm², $\sigma_0 = 87.2$ MPa, $Q_\infty = 165.6$ MPa, and $b = 15.5$.

After the minimization of the cost function, the optimally estimated parameters included in vector $x_0^{opt}$ were $K = 472.9$ kJ/mm², $\sigma_0 = 94.37$ MPa, $Q_\infty = 295.85$ MPa, and $b = 9.68$. The computation time required to obtain the optimally estimated parameters was approximately 35 h without parallel computing. Figure 4(a) shows the evolution of true strain $\epsilon_{yy}$ measured using the DIC method as the cracks propagated. It can be observed that the strain concentration occurs near the crack tip as forced displacement $U$ increased along the $y$ direction. Figure 4(b) shows the crack propagation behavior (i.e., time evolution of $\phi$) obtained from the PFC simulation using the optimally estimated parameters. It is evident that the PFC simulation reproduces a crack propagation behavior similar to that observed in the experiment.

<figure><img src="image_1142.png" /><figcaption>Figure 4 Evolution of (a) true strain $\epsilon_{yy}$ during the crack propagation measured using the DIC method and (b) order parameter $\phi$ describing the crack propagation behavior simulated by the PFC model with optimally estimated parameters.</figcaption></figure>

## Summary

This paper presents the results of three case studies wherein Bayesian DA methods were applied to the deformation (case study I), formation (case study II), and fracture (case study III) of the A5052-O aluminum alloy. The results show that Bayesian DA is an effective tool for estimating the parameters of material models by utilizing full-field measurement data from the DIC method, which contributes to the digital twin construction of each phenomenon.

## Acknowledgement

This research was financially supported by the Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (KAKENHI) (B) (Grant Numbers: 20H02476 and 23H01297) and the AMADA Foundation (Grant Number: AF-2021002-A2).

## References

[1] M. Grédiac, E. Toussaint, F. Pierron, Special virtual fields for the direct determination of
material parameters with the virtual fields method. 2—Application to in-plane properties, Int. J. Solids Struct. 39 (2002) 2707-2730. https://doi.org/10.1016/S0020-7683(02)00128-2

[2] K.T. Kavanagh, R.W. Clough, Finite element applications in the characterization of elastic solids, Int. J. Solids Struct. 7 (1971) 11-23. https://doi.org/10.1016/0020-7683(71)90015-1

[3] Y. Zhang, A. Yamanaka, S. Cooreman, T. Kuwabara, S. Coppieters, Inverse identification of plastic anisotropy through multiple non-conventional mechanical experiments, International Journal of Solids and Structures, Vol. 285 (2023) 112534. https://doi.org/10.1016/j.ijsolstr.2023.112534

[4] K. Sasaki, A. Yamanaka, S. Ito, H. Nagao, Data Assimilation for Phase-field Models based on the Ensemble Kalman Filter, Comput. Mater. Sci., 141 (2018) 141-152. https://doi.org/10.1016/j.commatsci.2017.09.025

[5] A. Yamanaka, Y. Maeda, K. Sasaki, Ensemble Kalman filter-based data assimilation for three-dimensional multi-phase-field model: Estimation of anisotropic grain boundary properties, Mater. Design, 165, (2019), 107577. https://doi.org/10.1016/j.matdes.2018.107577

[6] K. Takahashi, A. Yamanaka, Quantitative three-dimensional phase-field modeling of dendritic solidification coupled with local ensemble transform Kalman filter, Comput. Mater. Sci., 190 (2021) 110296. https://doi.org/10.1016/j.commatsci.2021.110296

[7] W. Lahoz, B. Khattatov, R. Ménard, Data Assimilation: Making Sense of Observations, Springer, Germany, 2010. https://doi.org/10.1007/978-3-540-74703-1

[8] G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res. Oceans, 99 (1994) 10143-10162. https://doi.org/10.1029/94JC00572

[9] A. Ishii, A. Yamanaka, Y. Okada, A. Yamamoto, Estimation of solid-state sintering and material parameters using phase-field modeling and ensemble four-dimensional variational method, Model. Simul. Mater. Sci. Eng., 29 (2021) 065012. https://doi.org/10.1088/1361-651X/ac13cd

[10] S. Sueki, A. Ishii, S. Coppieters, A. Yamanaka, Inverse characterization of a material model using an ensemble-based four-dimensional variational method, Int. J. Solids Struct., 279 (2023) 112350. https://doi.org/10.1016/j.ijsolstr.2023.112350

[11] A. Ishii, A. Yamamoto, A. Yamanaka, DMC-TPE: Tree-Structured Parzen Estimator-Based Efficient Data Assimilation Method for Phase-Field Simulation of Solid-State Sintering, Sci. Technol. Adv. Mater. Methods, 3 (2023) 2239133 https://doi.org/10.1080/27660400.2023.2239133

[12] S. Sueki, A. Ishii and A. Yamanaka, Bayesian data assimilation for inverse material modelling using 3D-digital image correlation measurement, Proceedings of JSME International Conference on Materials and Processing 2022, Paper #Tu-3B-3.

[13] D. Solav, K.M. Moerman, A.M. Jaeger, K. Genovese, H.M. Herr, IEEE Access, 6 (2018) 30520-30535. https://doi.org/10.1109/ACCESS.2018.2843725

[14] H. Takizawa, T. Kuwabara, K. Oide, J. Yoshida, Development of the subroutine library 'UMMDp' for anisotropic yield function commonly applicable to commercial FEM codes, J. Phys. Conf. Ser., 734 (2016), 032028. https://doi.org/10.1088/1742-6596/734/3/032028

[15] S. Sueki, A. Ishii, A. Yamanaka, Inverse Estimation of Material Model Parameters Using Digital Image Correlation and Ensemble-based Four-dimensional Variational Methods, J. Jpn. Soc. Technol. Plast., 64 (2023) 195-201. https://doi.org/10.9773/sosei.64.195

[16] Z. Khalil, A.Y. Elghazouli, E. Martínez-Pañeda, A generalised phase field model for fatigue crack growth in elastic-plastic solids with an efficient monolithic solver, Comput. Methods Appl. Mech. Eng., 388 (2022) 114286. http://dx.doi.org/10.1016/j.cma.2021.114286
material parameters with the virtual fields method. 2—Application to in-plane properties, Int. J. Solids Struct. 39 (2002) 2707-2730. https://doi.org/10.1016/S0020-7683(02)00128-2

[2] K.T. Kavanagh, R.W. Clough, Finite element applications in the characterization of elastic solids, Int. J. Solids Struct. 7 (1971) 11-23. https://doi.org/10.1016/0020-7683(71)90015-1

[3] Y. Zhang, A. Yamanaka, S. Cooreman, T. Kuwabara, S. Coppieters, Inverse identification of plastic anisotropy through multiple non-conventional mechanical experiments, International Journal of Solids and Structures, Vol. 285 (2023) 112534. https://doi.org/10.1016/j.ijsolstr.2023.112534

[4] K. Sasaki, A. Yamanaka, S. Ito, H. Nagao, Data Assimilation for Phase-field Models based on the Ensemble Kalman Filter, Comput. Mater. Sci., 141 (2018) 141-152. https://doi.org/10.1016/j.commatsci.2017.09.025

[5] A. Yamanaka, Y. Maeda, K. Sasaki, Ensemble Kalman filter-based data assimilation for three-dimensional multi-phase-field model: Estimation of anisotropic grain boundary properties, Mater. Design, 165, (2019), 107577. https://doi.org/10.1016/j.matdes.2018.107577

[6] K. Takahashi, A. Yamanaka, Quantitative three-dimensional phase-field modeling of dendritic solidification coupled with local ensemble transform Kalman filter, Comput. Mater. Sci., 190 (2021) 110296. https://doi.org/10.1016/j.commatsci.2021.110296

[7] W. Lahoz, B. Khattatov, R. Ménard, Data Assimilation: Making Sense of Observations, Springer, Germany, 2010. https://doi.org/10.1007/978-3-540-74703-1

[8] G. Evensen, Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics, J. Geophys. Res. Oceans, 99 (1994) 10143-10162. https://doi.org/10.1029/94JC00572

[9] A. Ishii, A. Yamanaka, Y. Okada, A. Yamamoto, Estimation of solid-state sintering and material parameters using phase-field modeling and ensemble four-dimensional variational method, Model. Simul. Mater. Sci. Eng., 29 (2021) 065012. https://doi.org/10.1088/1361-651X/ac13cd

[10] S. Sueki, A. Ishii, S. Coppieters, A. Yamanaka, Inverse characterization of a material model using an ensemble-based four-dimensional variational method, Int. J. Solids Struct., 279 (2023) 112350. https://doi.org/10.1016/j.ijsolstr.2023.112350

[11] A. Ishii, A. Yamamoto, A. Yamanaka, DMC-TPE: Tree-Structured Parzen Estimator-Based Efficient Data Assimilation Method for Phase-Field Simulation of Solid-State Sintering, Sci. Technol. Adv. Mater. Methods, 3 (2023) 2239133 https://doi.org/10.1080/27660400.2023.2239133

[12] S. Sueki, A. Ishii and A. Yamanaka, Bayesian data assimilation for inverse material modelling using 3D-digital image correlation measurement, Proceedings of JSME International Conference on Materials and Processing 2022, Paper #Tu-3B-3.

[13] D. Solav, K.M. Moerman, A.M. Jaeger, K. Genovese, H.M. Herr, IEEE Access, 6 (2018) 30520-30535. https://doi.org/10.1109/ACCESS.2018.2843725

[14] H. Takizawa, T. Kuwabara, K. Oide, J. Yoshida, Development of the subroutine library 'UMMDp' for anisotropic yield function commonly applicable to commercial FEM codes, J. Phys. Conf. Ser., 734 (2016), 032028. https://doi.org/10.1088/1742-6596/734/3/032028

[15] S. Sueki, A. Ishii, A. Yamanaka, Inverse Estimation of Material Model Parameters Using Digital Image Correlation and Ensemble-based Four-dimensional Variational Methods, J. Jpn. Soc. Technol. Plast., 64 (2023) 195-201. https://doi.org/10.9773/sosei.64.195

[16] Z. Khalil, A.Y. Elghazouli, E. Martínez-Pañeda, A generalised phase field model for fatigue crack growth in elastic-plastic solids with an efficient monolithic solver, Comput. Methods Appl. Mech. Eng., 388 (2022) 114286. http://dx.doi.org/10.1016/j.cma.2021.114286
# Bending of third generation steel:
Experimental and numerical approach

ESPANHOL PEREIRA Catarina Sofia¹,²,ª, VINCZE Gabriela¹,²,ᵇ*, DIAS PRATES Pedro André¹,²,ᶜ and BUTUC Marilena Carmen¹,²,ᵈ

¹Centre of Mechanical Technology and Automation (TEMA), Department of Mechanical Engineering, University of Aveiro, Portugal

²LASI – Intelligent Systems Associate Laboratory, Portugal

ªcsep@ua.pt, ᵇgvincze@ua.pt, ᶜprates@ua.pt, ᵈcbutuc@ua.pt

**Keywords:** Bending, Advanced High-Strength Steel, Springback/Forward

**Abstract.** The development of a new class of material, such as third-generation advanced high-strength steel, requires a deep understanding of mechanical behaviour before its use in industry. One of the critical issues in sheet metal forming processes is dimensional accuracy, which is strongly dependent on the elastic recovery that can manifest itself as springback or spring forward [1]. This undesired defect results from the materials parameters and/or tools geometries [2]. One common way to evaluate the springback/forward is through bending tests. This work aims to investigate the existence of springback/forward on the selected material, its magnitude and dependence on the tool geometry and test the performance of the commercial finite element software Abaqus 2017 to predict the observed behaviour. The 3rd GEN steel, CR980XG3™ produced by US Steel is subjected to V-bending to evaluate the existence of springback or spring-forward. Furthermore, to evaluate the dependence of springback/forward on the tool's geometry, the V-bending tools with different bending angles (namely, 60º and 90º) and different punch radii (namely, 5º and 10º) are used. The load is applied by a Shimadzu AG-X plus 100kN, and the deformation is measured by digital image correlation (DIC) using the GOM ARAMIS 3D 5M system. Numerical analysis of the bending tests is made using Abaqus 2017 software. The experimental results show that the material exhibits springback and spring forward depending on the tool geometry. The strain distribution through the thickness is correlated with the springback/forward.

**Introduction**

The automotive industry has increased the production of lightweight and safe materials for vehicle components such as the body shell, brakes, steering system, and others [3]. High-tensile strength sheet metals are engineered to enhance component strength and reduce weight by creating a robust metallic structure with minimal thickness [4]. Since the new European regulations on CO2 emission, lightweight materials such as advanced high-strength steels (AHSS) are much valorised, given the vehicle's weight reduction [5]. There are three AHSS generations and this classification is based on ductility and strength [6,7]. The 3rd generation combines the low-cost and easy production of the 1st generation and the high strength and elongation of the 2nd. It also has the additional goal of minimizing the industry's environmental footprint, given the automotive production's emissions and the vehicle's fuel consumption [6,7].

In sheet metal forming processes, the V-bending process is commonly used in the industry [2,8,9], and springback often occurs, since in the bending process, the metal elasticity limit can be exceeded [3,10]. Springback results in a modification of the part's shape, posing challenges to the accuracy of the dimensions and quality of the bent components [1,11]. The change in geometry is due to elastic recovery in the sheet metal after the unloading [2,8]. In the bending process, tensile

<figure><img src="image_1141.png" /></figure>
# Bending of third generation steel:
Experimental and numerical approach

ESPANHOL PEREIRA Catarina Sofia¹,²,ª, VINCZE Gabriela¹,²,ᵇ*, DIAS PRATES Pedro André¹,²,ᶜ and BUTUC Marilena Carmen¹,²,ᵈ

¹Centre of Mechanical Technology and Automation (TEMA), Department of Mechanical Engineering, University of Aveiro, Portugal

²LASI – Intelligent Systems Associate Laboratory, Portugal

ªcsep@ua.pt, ᵇgvincze@ua.pt, ᶜprates@ua.pt, ᵈcbutuc@ua.pt

**Keywords:** Bending, Advanced High-Strength Steel, Springback/Forward

**Abstract.** The development of a new class of material, such as third-generation advanced high-strength steel, requires a deep understanding of mechanical behaviour before its use in industry. One of the critical issues in sheet metal forming processes is dimensional accuracy, which is strongly dependent on the elastic recovery that can manifest itself as springback or spring forward [1]. This undesired defect results from the materials parameters and/or tools geometries [2]. One common way to evaluate the springback/forward is through bending tests. This work aims to investigate the existence of springback/forward on the selected material, its magnitude and dependence on the tool geometry and test the performance of the commercial finite element software Abaqus 2017 to predict the observed behaviour. The 3rd GEN steel, CR980XG3™ produced by US Steel is subjected to V-bending to evaluate the existence of springback or spring-forward. Furthermore, to evaluate the dependence of springback/forward on the tool's geometry, the V-bending tools with different bending angles (namely, 60º and 90º) and different punch radii (namely, 5º and 10º) are used. The load is applied by a Shimadzu AG-X plus 100kN, and the deformation is measured by digital image correlation (DIC) using the GOM ARAMIS 3D 5M system. Numerical analysis of the bending tests is made using Abaqus 2017 software. The experimental results show that the material exhibits springback and spring forward depending on the tool geometry. The strain distribution through the thickness is correlated with the springback/forward.

**Introduction**

The automotive industry has increased the production of lightweight and safe materials for vehicle components such as the body shell, brakes, steering system, and others [3]. High-tensile strength sheet metals are engineered to enhance component strength and reduce weight by creating a robust metallic structure with minimal thickness [4]. Since the new European regulations on CO2 emission, lightweight materials such as advanced high-strength steels (AHSS) are much valorised, given the vehicle's weight reduction [5]. There are three AHSS generations and this classification is based on ductility and strength [6,7]. The 3rd generation combines the low-cost and easy production of the 1st generation and the high strength and elongation of the 2nd. It also has the additional goal of minimizing the industry's environmental footprint, given the automotive production's emissions and the vehicle's fuel consumption [6,7].

In sheet metal forming processes, the V-bending process is commonly used in the industry [2,8,9], and springback often occurs, since in the bending process, the metal elasticity limit can be exceeded [3,10]. Springback results in a modification of the part's shape, posing challenges to the accuracy of the dimensions and quality of the bent components [1,11]. The change in geometry is due to elastic recovery in the sheet metal after the unloading [2,8]. In the bending process, tensile

<figure><img src="image_1143.png" /></figure>
and compressive stress occurs in the outside (pressed against the die) and inside (pressed against the punch) surfaces of the bend, respectively [11–13]. Therefore, the inner surface of the metal sheet tends to contract, while the outer side has a propensity to expand, resulting in the occurrence of springback [11,13]. Based on the bend angle achieved after the unloading, springback is called positive if this angle surpasses the final bending angle before the punch removal. In the opposite situation, springback can be called negative or spring forward [11,13]. Springback's value can be obtained through the difference between the abovementioned angles.

Several factors influence the springback occurrence in sheet metal, mainly sheet metal geometrical and mechanical properties, tools' geometrical properties, and the bending process characteristics. The goal of the present study is to investigate the existence of springback in the 3rd generation steel CR980XG3TM, its magnitude and the tool geometry influence, through bending tests. Furthermore, the Abaqus 2017 software is used to predict the springback phenomenon in the same bending conditions. Experimental and numerical studies (using the finite element method) were conducted to evaluate springback's influencing factors. Lems [14] examined the elastic constants during plastic deformation and demonstrated they suffer alteration throughout the process. Cleveland and Gosh's [15] research indicates a reduction in Young's modulus as plastic strain increases. Using the Abaqus 2017 software, Fu [16] performed a study on springback factors in air-bending tests. Fei and Hodgson [17] studied the influence of the material's (TRIP steels) mechanical properties on springback and revealed the friction's influence on the springback phenomenon. Ramezani et al. [18] also designed a model using finite element modelling to establish a relationship between springback and kinetic friction during V-bending tests, for ultra-high-tensile-strength steels. With the aid of an analytical model, and very much like the present study, Leu and Zhuang [19] studied the correlation between several factors, such as punch radius, and sheet metal thickness with the springback phenomenon in a high-strength material, through experimental tests and numerical simulation. On a more experimental procedure, Garcia-Romeu et al. [20] conducted a study with aluminium and stainless steel, concluding that an increase in die width and the bending angle increases springback in the metal sheet.

As previously noted, the present study assesses the occurrence, extent, and contributing factors of springback in third-generation steel. This objective is achieved through an experimental procedure involving bending tests and numerical simulations conducted using the Abaqus 2017 software. The use of these computational tools offers several advantages, both from an economic and ecological standpoint (eliminating the need for material acquisition, cutting, machining, and measurement) and in terms of time efficiency (negating the necessity for machinery calibration or setup, as well as the acquisition or fabrication of the bending tools).

## Experimental procedure

The experimental procedure was carried out to study the springback in material USS CR980XG3TM steel. Its mechanical properties were determined in another study and the data is available in the article of Santos et al. [21]. The material exhibits a Young's modulus (E) of 195 ± 5 GPa and a Poisson's ratio (ν) of 0.289 ± 0.003. Fig. 1 presents the material's true stress-strain curves in rolling direction (RD).
and compressive stress occurs in the outside (pressed against the die) and inside (pressed against the punch) surfaces of the bend, respectively [11–13]. Therefore, the inner surface of the metal sheet tends to contract, while the outer side has a propensity to expand, resulting in the occurrence of springback [11,13]. Based on the bend angle achieved after the unloading, springback is called positive if this angle surpasses the final bending angle before the punch removal. In the opposite situation, springback can be called negative or spring forward [11,13]. Springback's value can be obtained through the difference between the abovementioned angles.

Several factors influence the springback occurrence in sheet metal, mainly sheet metal geometrical and mechanical properties, tools' geometrical properties, and the bending process characteristics. The goal of the present study is to investigate the existence of springback in the 3rd generation steel CR980XG3TM, its magnitude and the tool geometry influence, through bending tests. Furthermore, the Abaqus 2017 software is used to predict the springback phenomenon in the same bending conditions. Experimental and numerical studies (using the finite element method) were conducted to evaluate springback's influencing factors. Lems [14] examined the elastic constants during plastic deformation and demonstrated they suffer alteration throughout the process. Cleveland and Gosh's [15] research indicates a reduction in Young's modulus as plastic strain increases. Using the Abaqus 2017 software, Fu [16] performed a study on springback factors in air-bending tests. Fei and Hodgson [17] studied the influence of the material's (TRIP steels) mechanical properties on springback and revealed the friction's influence on the springback phenomenon. Ramezani et al. [18] also designed a model using finite element modelling to establish a relationship between springback and kinetic friction during V-bending tests, for ultra-high-tensile-strength steels. With the aid of an analytical model, and very much like the present study, Leu and Zhuang [19] studied the correlation between several factors, such as punch radius, and sheet metal thickness with the springback phenomenon in a high-strength material, through experimental tests and numerical simulation. On a more experimental procedure, Garcia-Romeu et al. [20] conducted a study with aluminium and stainless steel, concluding that an increase in die width and the bending angle increases springback in the metal sheet.

As previously noted, the present study assesses the occurrence, extent, and contributing factors of springback in third-generation steel. This objective is achieved through an experimental procedure involving bending tests and numerical simulations conducted using the Abaqus 2017 software. The use of these computational tools offers several advantages, both from an economic and ecological standpoint (eliminating the need for material acquisition, cutting, machining, and measurement) and in terms of time efficiency (negating the necessity for machinery calibration or setup, as well as the acquisition or fabrication of the bending tools).

## Experimental procedure

The experimental procedure was carried out to study the springback in material USS CR980XG3TM steel. Its mechanical properties were determined in another study and the data is available in the article of Santos et al. [21]. The material exhibits a Young's modulus (E) of 195 ± 5 GPa and a Poisson's ratio (ν) of 0.289 ± 0.003. Fig. 1 presents the material's true stress-strain curves in rolling direction (RD).
<figure><img src="image_1142.png" /><figcaption>Fig. 1 - True stress-strain curves of the studied material.</figcaption></figure>

The specimens' dimensions consist of 160×20×1.6 mm (approx.), obtained using a CNC machining centre. V-shaped bending tools featuring different radii and bend angles were employed. It was determined that four tests would be carried out for each bending tool. Four V-shaped punches were used, with a 60° or 90° punch angle, and a 5 or 10 mm bending radius. The tests performed with each punch and the determined displacement for each type are indicated in Table 1, along with the corresponding test specimens' thickness.

<table><caption>Table 1 - Punches' displacement and test specimen's average thickness.</caption><thead><tr><td>Test</td><td>Punch displacement [mm]</td><td>Test specimen's thickness [mm]</td></tr></thead><tbody><tr><td>V-P60R10</td><td>46,65</td><td>≈1,57</td></tr><tr><td>V-P60R5</td><td>51,75</td><td>≈1,57</td></tr><tr><td>V-P90R10</td><td>50,0</td><td>≈1,56</td></tr><tr><td>V-P90R5</td><td>52,0</td><td>≈1,56</td></tr></tbody></table>

The Shimadzu AG-X plus testing machine, controlled by the TRAPEZIUM X software, performed the bending tests, and the GOM ARAMIS 3D 5M system measures the deformation through the thickness by Digital Image Correlation (DIC) using the Aramis software. Fig. 2 exhibits the bending tests' setup.

<figure><img src="image_1143.png" /><figcaption>Fig. 2 - Shimadzu and GOM ARAMIS setup, with die, punch, and test specimen in place</figcaption></figure>

Strain calculations along a chosen area were conducted using the “epsilon X” option in the Aramis software. Subsequently, images capturing the maximum flexion and punch withdrawal of the specimen were saved for later springback assessment, using the SOLIDWORKS software, as observed in Fig. 3. The TRAPEZIUM software oversees the bending test. It provides information,
<figure><img src="image_1144.png" /><figcaption>Fig. 1 - True stress-strain curves of the studied material.</figcaption></figure>

The specimens' dimensions consist of 160×20×1.6 mm (approx.), obtained using a CNC machining centre. V-shaped bending tools featuring different radii and bend angles were employed. It was determined that four tests would be carried out for each bending tool. Four V-shaped punches were used, with a 60° or 90° punch angle, and a 5 or 10 mm bending radius. The tests performed with each punch and the determined displacement for each type are indicated in Table 1, along with the corresponding test specimens' thickness.

<table><caption>Table 1 - Punches' displacement and test specimen's average thickness.</caption><thead><tr><td>Test</td><td>Punch displacement [mm]</td><td>Test specimen's thickness [mm]</td></tr></thead><tbody><tr><td>V-P60R10</td><td>46,65</td><td>≈1,57</td></tr><tr><td>V-P60R5</td><td>51,75</td><td>≈1,57</td></tr><tr><td>V-P90R10</td><td>50,0</td><td>≈1,56</td></tr><tr><td>V-P90R5</td><td>52,0</td><td>≈1,56</td></tr></tbody></table>

The Shimadzu AG-X plus testing machine, controlled by the TRAPEZIUM X software, performed the bending tests, and the GOM ARAMIS 3D 5M system measures the deformation through the thickness by Digital Image Correlation (DIC) using the Aramis software. Fig. 2 exhibits the bending tests' setup.

<figure><img src="image_1145.png" /><figcaption>Fig. 2 - Shimadzu and GOM ARAMIS setup, with die, punch, and test specimen in place</figcaption></figure>

Strain calculations along a chosen area were conducted using the “epsilon X” option in the Aramis software. Subsequently, images capturing the maximum flexion and punch withdrawal of the specimen were saved for later springback assessment, using the SOLIDWORKS software, as observed in Fig. 3. The TRAPEZIUM software oversees the bending test. It provides information,
including force and displacement data. These values are essential for analyzing the material and tool behavior, and for visualizing force vs. displacement curves.

<figure><img src="image_1144.png" /><figcaption>Fig. 3 - Angle measurement in SOLIDWORKS, with pictures taken by the Aramis software.</figcaption></figure>

## Results

Springback is estimated through angle measurements using the SOLIDWORKS software. Two images taken by the Aramis system, the first taken in the moment of maximum flexion and the second in the unloading moment, are loaded in SOLIDWORKS for angle measurement, as represented in Fig. 3. Fig. 4 illustrates the obtained mean springback value, in percentage, in the test specimens. Springback is calculated through the subtraction of the final angle value to the angle in the moment of maximum flexion, which is the following: $springback (\%) = \frac{\theta_d-\theta_{maxFlex}}{\theta_{maxFlex}} * 100$. $\theta_d$ represents the angle after the unloading moment and $\theta_{maxFlex}$ represents the angle in the moment of maximum flexion.

<figure><img src="image_1145.png" /><figcaption>Fig. 4 - Springback results (%) in V-bending tests: P60 and P90 refer to the punch bending angle, and R5 and R10 refer to the punch radius.</figcaption></figure>

The data analysis indicates that specimens subjected to punches with a smaller bending radius (5 mm) experienced less springback than those subjected to punches with a larger radius (10 mm). This behaviour is in agreement to the literature [22–24], namely: a lower bending radius causes a reduction in the specimen's springback when compared to a higher radius. The lowest springback mean value was achieved by specimens subjected to the V-P60R5 punch, with a mean value of -2.34%; the V-P60R10 punch was responsible for causing greater springback in the specimens. It is noteworthy that punches with a smaller bending radius of 5 mm caused negative springback, i.e., spring-forward, whereas punches with a larger radius caused positive springback values.

In addition to springback analysis, it was decided to study the specimens' strain after the bending tests. The procedure involves analyzing the images captured by the Aramis system, represented in Fig. 5, where a color gradient indicates the strain in that area. The outermost zone is marked in red, representing positive strain; the inner zone in blue represents negative strain. In between, the deformation has a neutral value, represented in green (neutral zone).
including force and displacement data. These values are essential for analyzing the material and tool behavior, and for visualizing force vs. displacement curves.

<figure><img src="image_1146.png" /><figcaption>Fig. 3 - Angle measurement in SOLIDWORKS, with pictures taken by the Aramis software.</figcaption></figure>

## Results

Springback is estimated through angle measurements using the SOLIDWORKS software. Two images taken by the Aramis system, the first taken in the moment of maximum flexion and the second in the unloading moment, are loaded in SOLIDWORKS for angle measurement, as represented in Fig. 3. Fig. 4 illustrates the obtained mean springback value, in percentage, in the test specimens. Springback is calculated through the subtraction of the final angle value to the angle in the moment of maximum flexion, which is the following: $springback (\%) = \frac{\theta_d - \theta_{maxFlex}}{\theta_{maxFlex}} * 100$. $\theta_d$ represents the angle after the unloading moment and $\theta_{maxFlex}$ represents the angle in the moment of maximum flexion.

<figure><img src="image_1147.png" /><figcaption>Fig. 4 - Springback results (%) in V-bending tests: P60 and P90 refer to the punch bending angle, and R5 and R10 refer to the punch radius.</figcaption></figure>

The data analysis indicates that specimens subjected to punches with a smaller bending radius (5 mm) experienced less springback than those subjected to punches with a larger radius (10 mm). This behaviour is in agreement to the literature [22–24], namely: a lower bending radius causes a reduction in the specimen’s springback when compared to a higher radius. The lowest springback mean value was achieved by specimens subjected to the V-P60R5 punch, with a mean value of -2.34%; the V-P60R10 punch was responsible for causing greater springback in the specimens. It is noteworthy that punches with a smaller bending radius of 5 mm caused negative springback, i.e., spring-forward, whereas punches with a larger radius caused positive springback values.

In addition to springback analysis, it was decided to study the specimens’ strain after the bending tests. The procedure involves analyzing the images captured by the Aramis system, represented in Fig. 5, where a color gradient indicates the strain in that area. The outermost zone is marked in red, representing positive strain; the inner zone in blue represents negative strain. In between, the deformation has a neutral value, represented in green (neutral zone).
<figure><img src="image_1146.png" /><figcaption>Fig. 5 - Strain distribution through the thickness observed in specimen tested with the V-P60R10 punch (image from Aramis); the red value (10,8%) is the maximum strain value and the blue value (-5,1%) is the lowest.</figcaption></figure>

For each punch, the mean of both the maximum positive and negative deformation values was computed across all specimens subjected to that specific punch. The results are illustrated in Fig. 6.

<figure><img src="image_1147.png" /><figcaption>Fig. 6 - Mean strain in the test specimens.</figcaption></figure>

Concerning the average positive strain, the V-P60R5 punch achieved the highest value, registering 23.07% of strain. Subsequently, the V-P90R5 punch followed with an average positive strain of 18.55%. In contrast, the V-P60R10 punch exhibited the lowest mean positive strain at 10.18%. The 5 mm radius punches induced a greater positive strain in the test specimens when compared with the 10 mm radius punches. This behaviour agrees with the literature [22–24]. The V-P60R10 punch yielded the highest mean negative strain at -9.89%, while the remaining punches exhibited comparable negative strain values. A relationship can be established between springback and strain, as specimens subjected to punches with smaller bending radii exhibit less springback and higher strain. The opposite is observed in specimens subjected to the 10 mm radius punch. It is noted that the occurrence of spring-forward is consistent with the use of the 5 mm bending radius, which may be related to a higher positive strain value.

The analysis of the maximum force exerted for each test specimen follows, where the data were recorded by the TRAPEZIUM software and are illustrated in Fig. 7.
<figure><img src="image_1148.png" /><figcaption>Fig. 5 - Strain distribution through the thickness observed in specimen tested with the V-P60R10 punch (image from Aramis); the red value (10,8%) is the maximum strain value and the blue value (-5,1%) is the lowest.</figcaption></figure>

For each punch, the mean of both the maximum positive and negative deformation values was computed across all specimens subjected to that specific punch. The results are illustrated in Fig. 6.

<figure><img src="image_1149.png" /><figcaption>Fig. 6 - Mean strain in the test specimens.</figcaption></figure>

Concerning the average positive strain, the V-P60R5 punch achieved the highest value, registering 23.07% of strain. Subsequently, the V-P90R5 punch followed with an average positive strain of 18.55%. In contrast, the V-P60R10 punch exhibited the lowest mean positive strain at 10.18%. The 5 mm radius punches induced a greater positive strain in the test specimens when compared with the 10 mm radius punches. This behaviour agrees with the literature [22–24]. The V-P60R10 punch yielded the highest mean negative strain at -9.89%, while the remaining punches exhibited comparable negative strain values. A relationship can be established between springback and strain, as specimens subjected to punches with smaller bending radii exhibit less springback and higher strain. The opposite is observed in specimens subjected to the 10 mm radius punch. It is noted that the occurrence of spring-forward is consistent with the use of the 5 mm bending radius, which may be related to a higher positive strain value.

The analysis of the maximum force exerted for each test specimen follows, where the data were recorded by the TRAPEZIUM software and are illustrated in Fig. 7.
<figure><img src="image_1148.png" /><figcaption>Fig. 7 - Maximum load applied in the test specimens in each punch.</figcaption></figure>

Examining the results, it is observed that the test specimens subjected to the 5 mm radius punches experienced a higher maximum load when compared to those subjected to the remaining punches. The highest and lowest values of the maximum load were achieved by the V-P60R5 and V-P60R10 punches, respectively, in the order of 10 kN and 5 kN. In the previously conducted analysis, it was found that the V-P60R5 punch induced less springback and higher positive strain in the specimens. Ergo, a relationship can be established between the maximum load, springback, and strain: an increase in the maximum load applied to the specimens results in a higher positive strain, and consequently, reduced springback. More localized deformation in the specimen, in the contact zone between the punch and the sheet, leads to a high asymmetry between tension and compression deformations, causing less springback.

Finite element modelling

The simulation models in the Abaqus 2017 software were based on the experimental procedure conducted for the bending tests. The test piece was modelled with a total of 3200 linear hexahedral elements of type C3D8I, using a Structured meshing technique. The choice of the meshing element's geometry is due to the test piece's geometry since it has rectangular faces. To note, four layers of elements through the specimen thickness of the specimen was used. The dies and punches were modelled as analytical rigid 3D surfaces drawn with the respective bending angle, and bending radius, like in the experimental procedure. For the simulations, the material definition was carried out with the material's stress-strain curve, assuming the isotropic hardening. The initial yield locus shape is described by the Von Mises yield function.

The Abaqus 2017 simulations contained two models: one for simulating the bending test with tools and the specimen, and a second one solely responsible for simulating springback. To simulate bending, it was necessary to create a step and four boundary conditions. To streamline the simulation, the specimen was designed at only a quarter of its actual size, with two symmetry boundary conditions (along X and Z axes) assigned. The matrix was given a fixed boundary condition, and the punch had a linear and downward movement along the Y-axis. The displacement value for the corresponding bending test was determined experimentally. The algorithm used for contact interactions was the surface-to-surface type. After completing the simulation for bending, a model was created to simulate springback, which included all the information from the previous model except for the tools and their boundary conditions, as well as the interactions between the tool-specimen assembly. The bending step was replaced with the springback step, and a new boundary condition was introduced to fix the specimen. To maintain the information from the previous model, a Predefined Field was established.

The springback calculation was carried out in the same manner as the experimental procedure. Fig. 8 shows a comparison of the experimental results with those from simulations.
<figure><img src="image_1150.png" /><figcaption>Fig. 7 - Maximum load applied in the test specimens in each punch.</figcaption></figure>

Examining the results, it is observed that the test specimens subjected to the 5 mm radius punches experienced a higher maximum load when compared to those subjected to the remaining punches. The highest and lowest values of the maximum load were achieved by the V-P60R5 and V-P60R10 punches, respectively, in the order of 10 kN and 5 kN. In the previously conducted analysis, it was found that the V-P60R5 punch induced less springback and higher positive strain in the specimens. Ergo, a relationship can be established between the maximum load, springback, and strain: an increase in the maximum load applied to the specimens results in a higher positive strain, and consequently, reduced springback. More localized deformation in the specimen, in the contact zone between the punch and the sheet, leads to a high asymmetry between tension and compression deformations, causing less springback.

Finite element modelling

The simulation models in the Abaqus 2017 software were based on the experimental procedure conducted for the bending tests. The test piece was modelled with a total of 3200 linear hexahedral elements of type C3D8I, using a Structured meshing technique. The choice of the meshing element's geometry is due to the test piece's geometry since it has rectangular faces. To note, four layers of elements through the specimen thickness of the specimen was used. The dies and punches were modelled as analytical rigid 3D surfaces drawn with the respective bending angle, and bending radius, like in the experimental procedure. For the simulations, the material definition was carried out with the material's stress-strain curve, assuming the isotropic hardening. The initial yield locus shape is described by the Von Mises yield function.

The Abaqus 2017 simulations contained two models: one for simulating the bending test with tools and the specimen, and a second one solely responsible for simulating springback. To simulate bending, it was necessary to create a step and four boundary conditions. To streamline the simulation, the specimen was designed at only a quarter of its actual size, with two symmetry boundary conditions (along X and Z axes) assigned. The matrix was given a fixed boundary condition, and the punch had a linear and downward movement along the Y-axis. The displacement value for the corresponding bending test was determined experimentally. The algorithm used for contact interactions was the surface-to-surface type. After completing the simulation for bending, a model was created to simulate springback, which included all the information from the previous model except for the tools and their boundary conditions, as well as the interactions between the tool-specimen assembly. The bending step was replaced with the springback step, and a new boundary condition was introduced to fix the specimen. To maintain the information from the previous model, a Predefined Field was established.

The springback calculation was carried out in the same manner as the experimental procedure. Fig. 8 shows a comparison of the experimental results with those from simulations.
<figure><img src="image_1149.png" /><figcaption>Fig. 8 - Springback results comparison between experimental and numerical results.</figcaption></figure>

The occurrence of springback/forward is related to the punch radius in the experimental results, as mentioned earlier. Despite some variations between the experimental and numerical results, a consistent trend is observed, suggesting the presence of the punch geometry's influence on the material's springback in numerical simulations.

In the numerical simulations, the value of springback was higher for the radius 5 punches, which is contrary to what was observed in the experimental results. Among all the punches, the V-P60R5 punch caused the highest springback value at -8.98%, while the V-P90R10 punch showed the lowest at -2.08%. Except for the V-P60R10 punch, spring-forward was observed in all punches. However, this phenomenon may be related to the disparity between numerical and experimental results mentioned earlier. It is also noticeable that the springback value for the V-P60R10 punch shows the greatest deviation from the experimental value in the numerical simulations. An investigation was conducted to determine the cause of the deviation between experimental and numerical results. To test the influence of the friction coefficient value on the springback phenomenon, simulations were conducted with values of 0.05 and 0.15, in addition to the initial value of 0.1. However, the analysis showed that the difference in springback of the test specimen was negligible, so the value of 0.1 was retained. The steps' definition in the Abaqus 2017 software was also studied and it was concluded that changing the increment values (minimum, maximum, etc.) affects the springback of the test specimen, reducing it. Furthermore, modifying the boundary condition of the test specimen can also be explored to determine if it has any effect on the springback.

Besides springback analysis, the maximum load applied to each specimen in the numerical simulations was also studied. The charts were plotted with the Abaqus 2017' data and are shown in Fig. 9 where a comparison can be seen between the numerical outcomes and the experimental results.
<figure><img src="image_1151.png" /><figcaption>Fig. 8 - Springback results comparison between experimental and numerical results.</figcaption></figure>

The occurrence of springback/forward is related to the punch radius in the experimental results, as mentioned earlier. Despite some variations between the experimental and numerical results, a consistent trend is observed, suggesting the presence of the punch geometry's influence on the material's springback in numerical simulations.

In the numerical simulations, the value of springback was higher for the radius 5 punches, which is contrary to what was observed in the experimental results. Among all the punches, the V-P60R5 punch caused the highest springback value at -8.98%, while the V-P90R10 punch showed the lowest at -2.08%. Except for the V-P60R10 punch, spring-forward was observed in all punches. However, this phenomenon may be related to the disparity between numerical and experimental results mentioned earlier. It is also noticeable that the springback value for the V-P60R10 punch shows the greatest deviation from the experimental value in the numerical simulations. An investigation was conducted to determine the cause of the deviation between experimental and numerical results. To test the influence of the friction coefficient value on the springback phenomenon, simulations were conducted with values of 0.05 and 0.15, in addition to the initial value of 0.1. However, the analysis showed that the difference in springback of the test specimen was negligible, so the value of 0.1 was retained. The steps' definition in the Abaqus 2017 software was also studied and it was concluded that changing the increment values (minimum, maximum, etc.) affects the springback of the test specimen, reducing it. Furthermore, modifying the boundary condition of the test specimen can also be explored to determine if it has any effect on the springback.

Besides springback analysis, the maximum load applied to each specimen in the numerical simulations was also studied. The charts were plotted with the Abaqus 2017' data and are shown in Fig. 9 where a comparison can be seen between the numerical outcomes and the experimental results.
<figure><img src="image_1150.png" /><figcaption>Fig. 9 - Comparison between the maximum load applied results of the Isotropic simulation and the experimental procedure.</figcaption></figure>

It has been observed that there is a significant difference between the numerical and experimental outcomes, similar to the springback analysis. However, there is a noticeable pattern concerning the impact of the punch radius on springback, and its correlation with the maximum load. The test results indicate that 5 mm radius punches applied a higher maximum load on the samples, resulting in more springback than the 10 mm radius punches, which caused less springback. Hence, the numerical simulations were in line with the experimental results, as expected.

## Conclusions

This study focuses on exploring the springback and spring-forward characteristics of third-generation steel, specifically USS CR980XG3TM, through V-bending tests and numerical simulations. Four V-punches with varying bending angles and radii were used to conduct the tests. The study reveals that tool geometry has a significant impact on the springback of third-generation steel specimens during bending tests. The research also analyzes the effects of punch angle and radius on springback, strain, and maximum load. Experimental results show that a smaller punch radius leads to decreased springback due to more localized deformation in the contact zone between the punch top and the sheet. Furthermore, the experimental study indicates that the 60-degree punch induces higher springback than the 90-degree punch, though this difference is less pronounced with a 5 mm radius. The procedure has some limitations, such as reduced specimen thickness, which affects strain data capture by the Aramis system and may cause occasional inaccuracies.

Regarding the numerical study, the Abaqus 2017 software simulations were based on experimental bending tests, with careful consideration given to modelling parameters, meshing techniques, and material properties. Two distinct models were employed, one for simulating the bending test and another for assessing springback, each with specific boundary conditions and steps.

Despite observed some differences between numerical simulations and experimental results, a consistent trend emerged, highlighting the influence of punch geometry on material springback in the simulations. An investigation into the discrepancy of the results revealed negligible impact from variations in the friction coefficient, leading to the retention of the initial value of 0.1. Additionally, an exploration of step definitions in the Abaqus 2017 software indicated that modifying increment values and boundary conditions could potentially affect the springback phenomenon, providing avenues for further study.

In summary, while some differences persist between numerical simulations and experimental outcomes, this study sheds light on the significance of punch geometry in influencing material
<figure><img src="image_1152.png" /><figcaption>Fig. 9 - Comparison between the maximum load applied results of the Isotropic simulation and the experimental procedure.</figcaption></figure>

It has been observed that there is a significant difference between the numerical and experimental outcomes, similar to the springback analysis. However, there is a noticeable pattern concerning the impact of the punch radius on springback, and its correlation with the maximum load. The test results indicate that 5 mm radius punches applied a higher maximum load on the samples, resulting in more springback than the 10 mm radius punches, which caused less springback. Hence, the numerical simulations were in line with the experimental results, as expected.

## Conclusions

This study focuses on exploring the springback and spring-forward characteristics of third-generation steel, specifically USS CR980XG3TM, through V-bending tests and numerical simulations. Four V-punches with varying bending angles and radii were used to conduct the tests. The study reveals that tool geometry has a significant impact on the springback of third-generation steel specimens during bending tests. The research also analyzes the effects of punch angle and radius on springback, strain, and maximum load. Experimental results show that a smaller punch radius leads to decreased springback due to more localized deformation in the contact zone between the punch top and the sheet. Furthermore, the experimental study indicates that the 60-degree punch induces higher springback than the 90-degree punch, though this difference is less pronounced with a 5 mm radius. The procedure has some limitations, such as reduced specimen thickness, which affects strain data capture by the Aramis system and may cause occasional inaccuracies.

Regarding the numerical study, the Abaqus 2017 software simulations were based on experimental bending tests, with careful consideration given to modelling parameters, meshing techniques, and material properties. Two distinct models were employed, one for simulating the bending test and another for assessing springback, each with specific boundary conditions and steps.

Despite observed some differences between numerical simulations and experimental results, a consistent trend emerged, highlighting the influence of punch geometry on material springback in the simulations. An investigation into the discrepancy of the results revealed negligible impact from variations in the friction coefficient, leading to the retention of the initial value of 0.1. Additionally, an exploration of step definitions in the Abaqus 2017 software indicated that modifying increment values and boundary conditions could potentially affect the springback phenomenon, providing avenues for further study.

In summary, while some differences persist between numerical simulations and experimental outcomes, this study sheds light on the significance of punch geometry in influencing material
springback. Further research into refining simulation parameters and boundary conditions may
contribute to a more accurate representation of the observed phenomena in practical bending tests.

Acknowledgments

The authors acknowledge support from the Portuguese Foundation of Science and Technology (FCT), in its State Budget component (OE) through project 2022.05783.PTDC (https://doi.org/10.54499/2022.05783.PTDC). This article was also supported by the projects UIDB/00481/2020 and UIDP/00481/2020 - FCT, (https://doi.org/10.54499/UIDB/00481/2020) and (https://doi.org/10.54499/UIDP/00481/2020) and CENTRO-01-0145-FEDER-022083 Centro Portugal Regional Programme (Centro2020), under the PORTUGAL 2020 Partnership Agreement, through the European Regional Development Fund.

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[11] S. Thipprakmas, S. Rojananan, Investigation of spring-go phenomenon using finite element method, Mater Des 29 (2008) 1526–1532. https://doi.org/10.1016/j.matdes.2008.02.002.

[12] H. Tschaetsch, Metal Forming Practise, Springer, 2006.

[13] M. Wasif, A. Fatima, A. Ahmed, S.A. Iqbal, Investigation and Optimization of Parameters for the Reduced Springback in JSC-590 Sheet Metals Occurred During the V-Bending Process, Transactions of the Indian Institute of Metals 74 (2021) 2751–2760. https://doi.org/10.1007/s12666-021-02357-9
springback. Further research into refining simulation parameters and boundary conditions may
contribute to a more accurate representation of the observed phenomena in practical bending tests.

Acknowledgments

The authors acknowledge support from the Portuguese Foundation of Science and Technology (FCT), in its State Budget component (OE) through project 2022.05783.PTDC (https://doi.org/10.54499/2022.05783.PTDC). This article was also supported by the projects UIDB/00481/2020 and UIDP/00481/2020 - FCT, (https://doi.org/10.54499/UIDB/00481/2020) and (https://doi.org/10.54499/UIDP/00481/2020) and CENTRO-01-0145-FEDER-022083 Centro Portugal Regional Programme (Centro2020), under the PORTUGAL 2020 Partnership Agreement, through the European Regional Development Fund.

References

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[2] D.K. Leu, Relationship between mechanical properties and geometric parameters to limitation condition of springback based on springback-radius concept in V-die bending process, International Journal of Advanced Manufacturing Technology. 101 (2019) 913–926. https://doi.org/10.1007/s00170-018-2970-1

[3] W. Zhang, J. Xu, Advanced lightweight materials for Automobiles: A review, Mater Des. 221 (2022). https://doi.org/10.1016/j.matdes.2022.110994

[4] S. 'Kalpakjian, S. 'Schmid, Manufacturing Engineering & Technology, 7th ed., Pearson Higher Education, 2013.

[5] J.H. Schmitt, T. Iung, New developments of advanced high-strength steels for automotive applications, C R Phys 19 (2018) 641–656. https://doi.org/10.1016/j.crhy.2018.11.004

[6] T.K. 'Roy, B. 'Bhattacharya, C. 'Ghosh, S.K.'Ajmani, Advanced High Strength Steel: Processing and Applications, Springer Singapore, 2018.

[7] M.Y.'Demeri, Advanced High-Strength Steels: Science, Technology, and Applications, ASM International, 2013.

[8] K. 'Lange, Handbook of Metal Forming, McGraw-Hill, 1985.

[9] S. Thipprakmas, W. Phanitwong, Process parameter design of spring-back and spring-go in V-bending process using Taguchi technique, Mater Des 32 (2011) 4430–4436. https://doi.org/10.1016/j.matdes.2011.03.069

[10] M. Özdemir, H. Gökmeşe, Microstructural characterization and deformation of X10CrAlSi24 sheet material applied V-bending process, Tehnicki Vjesnik 25 (2018) 846–854. https://doi.org/10.17559/TV-20170425022545

[11] S. Thipprakmas, S. Rojananan, Investigation of spring-go phenomenon using finite element method, Mater Des 29 (2008) 1526–1532. https://doi.org/10.1016/j.matdes.2008.02.002.

[12] H. Tschaetsch, Metal Forming Practise, Springer, 2006.

[13] M. Wasif, A. Fatima, A. Ahmed, S.A. Iqbal, Investigation and Optimization of Parameters for the Reduced Springback in JSC-590 Sheet Metals Occurred During the V-Bending Process, Transactions of the Indian Institute of Metals 74 (2021) 2751–2760. https://doi.org/10.1007/s12666-021-02357-9
[14] W. Lems, The change of Young's modulus of copper and silver after deformation at low temperature and its recovery, Physica 28 (1962) 445-452. https://doi.org/10.1016/0031-8914(62)90022-8

[15] R.M. Cleveland, A.K. Ghosh, Inelastic effects on springback in metals, Int J Plast 18 (2002) 769-785. https://doi.org/10.1016/S0749-6419(01)00054-7

[16] Z.M. Fu, Numerical Simulation of Springback in Air-Bending Forming of Sheet Metal, Applied Mechanics and Materials 121-126 (2011) 3602-3606. https://doi.org/10.4028/www.scientific.net/AMM.121-126.3602.

[17] D. Fei, P. Hodgson, Experimental and numerical studies of springback in air v-bending process for cold rolled TRIP steels, Nuclear Engineering and Design 236 (2006) 1847-1851. https://doi.org/10.1016/j.nucengdes.2006.01.016

[18] M. Ramezani, Z.M. Ripin, A friction model for dry contacts during metal-forming processes, The International Journal of Advanced Manufacturing Technology 51 (2010) 93-102. https://doi.org/10.1007/s00170-010-2608-4

[19] D.-K. Leu, Z.-W. Zhuang, Springback prediction of the vee bending process for high-strength steel sheets, Journal of Mechanical Science and Technology 30 (2016) 1077-1084. https://doi.org/10.1007/s12206-016-0212-8

[20] M.L. Garcia-Romeu, J. Ciurana, I. Ferrer, Springback determination of sheet metals in an air bending process based on an experimental work, J Mater Process Technol 191 (2007) 174-177. https://doi.org/10.1016/j.jmatprotec.2007.03.019

[21] R.O. Santos, L.P. Moreira, M.C. Butuc, G. Vincze, A.B. Pereira, Damage Analysis of Third-Generation Advanced High-Strength Steel Based on the Gurson-Tvergaard-Needleman (GTN) Model, Metals (Basel) 12 (2022) 214. https://doi.org/10.3390/met12020214

[22] Ö. Tekaslan, N. Gerger, U. Şeker, Determination of spring-back of stainless steel sheet metal in "V" bending dies, Mater Des 29 (2008) 1043-1050. https://doi.org/10.1016/j.matdes.2007.04.004

[23] Springback/Springforward Behaviour of DP Steels Used in the Automotive Industry, Tehnicki Vjesnik - Technical Gazette 27 (2020). https://doi.org/10.17559/TV-20181004144024

[24] M.L. Garcia-Romeu, J. Ciurana, I. Ferrer, Springback determination of sheet metals in an air bending process based on an experimental work, J Mater Process Technol 191 (2007) 174-177. https://doi.org/10.1016/j.jmatprotec.2007.03.019
[14] W. Lems, The change of Young's modulus of copper and silver after deformation at low temperature and its recovery, Physica 28 (1962) 445-452. https://doi.org/10.1016/0031-8914(62)90022-8

[15] R.M. Cleveland, A.K. Ghosh, Inelastic effects on springback in metals, Int J Plast 18 (2002) 769-785. https://doi.org/10.1016/S0749-6419(01)00054-7

[16] Z.M. Fu, Numerical Simulation of Springback in Air-Bending Forming of Sheet Metal, Applied Mechanics and Materials 121-126 (2011) 3602-3606. https://doi.org/10.4028/www.scientific.net/AMM.121-126.3602.

[17] D. Fei, P. Hodgson, Experimental and numerical studies of springback in air v-bending process for cold rolled TRIP steels, Nuclear Engineering and Design 236 (2006) 1847-1851. https://doi.org/10.1016/j.nucengdes.2006.01.016

[18] M. Ramezani, Z.M. Ripin, A friction model for dry contacts during metal-forming processes, The International Journal of Advanced Manufacturing Technology 51 (2010) 93-102. https://doi.org/10.1007/s00170-010-2608-4

[19] D.-K. Leu, Z.-W. Zhuang, Springback prediction of the vee bending process for high-strength steel sheets, Journal of Mechanical Science and Technology 30 (2016) 1077-1084. https://doi.org/10.1007/s12206-016-0212-8

[20] M.L. Garcia-Romeu, J. Ciurana, I. Ferrer, Springback determination of sheet metals in an air bending process based on an experimental work, J Mater Process Technol 191 (2007) 174-177. https://doi.org/10.1016/j.jmatprotec.2007.03.019

[21] R.O. Santos, L.P. Moreira, M.C. Butuc, G. Vincze, A.B. Pereira, Damage Analysis of Third-Generation Advanced High-Strength Steel Based on the Gurson-Tvergaard-Needleman (GTN) Model, Metals (Basel) 12 (2022) 214. https://doi.org/10.3390/met12020214

[22] Ö. Tekaslan, N. Gerger, U. Şeker, Determination of spring-back of stainless steel sheet metal in "V" bending dies, Mater Des 29 (2008) 1043-1050. https://doi.org/10.1016/j.matdes.2007.04.004

[23] Springback/Springforward Behaviour of DP Steels Used in the Automotive Industry, Tehnicki Vjesnik - Technical Gazette 27 (2020). https://doi.org/10.17559/TV-20181004144024

[24] M.L. Garcia-Romeu, J. Ciurana, I. Ferrer, Springback determination of sheet metals in an air bending process based on an experimental work, J Mater Process Technol 191 (2007) 174-177. https://doi.org/10.1016/j.jmatprotec.2007.03.019
# Influence of pre-strain on fracture toughness of 3rd generation advanced high strength steels

GRIFÉ Laura¹,ª*, FRÓMETA David¹,ᵇ, PAYÀ Anna¹,ᶜ and CASELLAS Daniel¹,²,ᵈ

¹Eurecat, Centre Tecnològic de Catalunya, Unit of Metallic and Ceramic Materials, Plaça de la Ciència, 2, Manresa 08243, Spain

²Division of Mechanics of Solid Materials, Luleå University of Technology, 971 87 Luleå, Sweden

ªlaura.grife@eurecat.org, ᵇdavid.frometa@eurecat.org, ᵇanna.paya@eurecat.org, ᵈdaniel.casellas@eurecat.org

**Keywords:** Fracture Toughness, Pre-Deformation, 3rd Gen Advanced High Strength Steels

**Abstract.** The present work investigates the influence of pre-strain on the fracture toughness of 3rd Generation Advanced High Strength Steels (AHSS). Specifically, a Carbide Free Bainitic (CFB) and a Quenching and Partitioning (Q&P) steel have been studied, the properties of which are crucial for lightweight vehicle construction. Fracture toughness, which is a key parameter for crash performance applications, is assessed using the Essential Work of Fracture methodology. The study investigates the pre-straining states of uniaxial tension, plane strain, and equibiaxial tension in 1.5 mm Q&P and 1.4 mm CFB sheet-form steels of 1180 MPa tensile strength. Overall, Q&P steel demonstrates superior fracture toughness compared to CFB steel. Remarkably, the specific essential work of fracture (w<sub>e</sub>) remains unaffected by pre-straining across different strain states. Nevertheless, pre-straining exerts a notable influence on the non-essential plastic work (βw<sub>p</sub>) due to the plastic energy consumed during pre-deformation. These results suggest that pre-strain has little or no influence on the fracture properties of AHSS, which is relevant for the design and manufacturing of high crash-performance and safety-related components.

**Introduction**

3rd Gen Advanced High Strength Steels (AHSS) have emerged as one of the most promising materials for lightweight construction of vehicle structures. Their outstanding mechanical properties combining high strength, high fracture toughness and good formability are the result of their complex multiphase microstructures and the strain-induced transformation of retained austenite to martensite, also known as Transformation Induced Plasticity (TRIP) effect. The TRIP effect is known to be dependent on the stress state and deformation level [1] and, therefore, the strain induced during the sheet forming process may significantly affect the local properties of the final component. It is well known that the presence of martensite in a microstructure increases the yield strength and the tensile strength, while decreases the fracture elongation. Thus, the amount of transformed martensite due to the TRIP effect during cold forming will also change the tensile properties. It is affecting sheet formability, but this change does not inform about part performance as crash or fatigue [2]. Accordingly, anticipating the influence of such pre-deformation on critical in-use properties, such as crashworthiness, is crucial for the safe design of structural and crash-relevant components.

The present work aims to assess the influence of pre-strain on the fracture toughness of two 3rd Generation AHSS for automotive applications. Fracture toughness has been shown to be a good indicator of the impact fracture performance of AHSS and Press Hardened Steels [2,3]. Frómeta et al. [3] established a correlation between the fracture toughness of different AHSS and their crash

<figure><img src="image_1151.png" /></figure>
# Influence of pre-strain on fracture toughness of 3rd generation advanced high strength steels

GRIFÉ Laura¹,ª*, FRÓMETA David¹,ᵇ, PAYÀ Anna¹,ᶜ and CASELLAS Daniel¹,²,ᵈ

¹Eurecat, Centre TecnOLògic de Catalunya, Unit of Metallic and Ceramic Materials, Plaça de la Ciència, 2, Manresa 08243, Spain

²Division of Mechanics of Solid Materials, Luleå University of Technology, 971 87 Luleå, Sweden

ªlaura.grife@eurecat.org, ᵇdavid.frometa@eurecat.org, ᵇanna.paya@eurecat.org, ᵈdaniel.casellas@eurecat.org

**Keywords:** Fracture Toughness, Pre-Deformation, 3rd Gen Advanced High Strength Steels

**Abstract.** The present work investigates the influence of pre-strain on the fracture toughness of 3rd Generation Advanced High Strength Steels (AHSS). Specifically, a Carbide Free Bainitic (CFB) and a Quenching and Partitioning (Q&P) steel have been studied, the properties of which are crucial for lightweight vehicle construction. Fracture toughness, which is a key parameter for crash performance applications, is assessed using the Essential Work of Fracture methodology. The study investigates the pre-straining states of uniaxial tension, plane strain, and equibiaxial tension in 1.5 mm Q&P and 1.4 mm CFB sheet-form steels of 1180 MPa tensile strength. Overall, Q&P steel demonstrates superior fracture toughness compared to CFB steel. Remarkably, the specific essential work of fracture (w<sub>e</sub>) remains unaffected by pre-straining across different strain states. Nevertheless, pre-straining exerts a notable influence on the non-essential plastic work (βw<sub>p</sub>) due to the plastic energy consumed during pre-deformation. These results suggest that pre-strain has little or no influence on the fracture properties of AHSS, which is relevant for the design and manufacturing of high crash-performance and safety-related components.

**Introduction**

3rd Gen Advanced High Strength Steels (AHSS) have emerged as one of the most promising materials for lightweight construction of vehicle structures. Their outstanding mechanical properties combining high strength, high fracture toughness and good formability are the result of their complex multiphase microstructures and the strain-induced transformation of retained austenite to martensite, also known as Transformation Induced Plasticity (TRIP) effect. The TRIP effect is known to be dependent on the stress state and deformation level [1] and, therefore, the strain induced during the sheet forming process may significantly affect the local properties of the final component. It is well known that the presence of martensite in a microstructure increases the yield strength and the tensile strength, while decreases the fracture elongation. Thus, the amount of transformed martensite due to the TRIP effect during cold forming will also change the tensile properties. It is affecting sheet formability, but this change does not inform about part performance as crash or fatigue [2]. Accordingly, anticipating the influence of such pre-deformation on critical in-use properties, such as crashworthiness, is crucial for the safe design of structural and crash-relevant components.

The present work aims to assess the influence of pre-strain on the fracture toughness of two 3rd Generation AHSS for automotive applications. Fracture toughness has been shown to be a good indicator of the impact fracture performance of AHSS and Press Hardened Steels [2,3]. Frómeta et al. [3] established a correlation between the fracture toughness of different AHSS and their crash

<figure><img src="image_1153.png" /></figure>
failure behavior, thus being an invaluable material parameter to estimate crashworthiness at the lab scale. The study investigates the effect of different pre-straining states, namely, uniaxial tension, plane strain, and equibiaxial tension, on the fracture properties of a Quenching and Partitioning (Q&P) and a Carbide Free Bainitic (CFB) sheet steels of 1180 MPa tensile strength.

Fracture toughness characterization is here conducted using the Essential Work of Fracture (EWF) methodology [4]. The EWF is a simple procedure to measure the crack propagation resistance of AHSS sheets and it has been used to understand their cracking performance [2,3,5, 6, 7, 8]. The results for undeformed and pre-strained specimens are compared and discussed.

## Materials

The materials analyzed in this study are two 3rd Gen AHSS for automotive applications. Specifically, a Carbon Free Bainitic (CFB) and a Quenching and Partitioning (Q&P) steel of 1.4 and 1.5 mm thickness, respectively

The microstructure of CFB is characterized by a matrix of carbide-free bainite with globular islands of martensite/retained austenite and laths of retained austenite. The Q&P steel presents a matrix of tempered or carbon-depleted martensite with lath-like retained austenite and globular islands of retained austenite/martensite and bainite. Both steel grades have a content of retained austenite of 12-16% as presented by Frómeta et al. [3].

The tensile properties of the two TRIP-assisted steels were determined according to UNE EN ISO 6892-1 in the transverse direction with respect to the rolling direction. The obtained results are presented in Table 1. Both steels show very similar tensile properties.

<table><caption>Table 1 – Mechanical properties of the investigated TRIP-assisted steel grades at transverse direction: thickness, *t*; yield strength, *σ*<sub>YS</sub>; ultimate tensile strength, *σ*<sub>UTS</sub>; uniform elongation (elongation at *σ*<sub>UTS</sub>), Ag; elongation at fracture (initial gauge length 80 mm), A<sub>80</sub>; strain hardening exponent between 2 and 4% of deformation, n<sub>2-4</sub>.</caption><thead><tr><th>Steel denomination</th><th>t [mm]</th><th>σ<sub>YS</sub> [MPa]</th><th>σ<sub>UTS</sub> [MPa]</th><th>Ag [%]</th><th>A<sub>80</sub> [%]</th><th>n<sub>2-4</sub> [-]</th></tr></thead><tbody><tr><td>HF1180 CFB</td><td>1.4</td><td>987</td><td>1216</td><td>9.2</td><td>12.6</td><td>0.11</td></tr><tr><td>HF1180 Q&P</td><td>1.5</td><td>1034</td><td>1191</td><td>9.2</td><td>13.1</td><td>0.09</td></tr></tbody></table>

## Experimental procedure

### EWF tests

EWF tests were performed according to CWA 17793:2021 [7]. DENT specimens of 240 x 55 mm machined in the transverse direction were used (Fig. 1a), this geometry has shown to be the most suitable because the transverse stress between the notches is tensile and there is no buckling [9]. Initial notches were machined by electrical discharge machining. Then, fatigue pre-cracks were nucleated at the notch root following the recommendations of the ASTM E1820 [10]. The cracks were extended about 1-1.5 mm per side (Fig. 1a). EWF tests were performed at a 250 kN universal testing machine, equipped with a digital videoextensometer (Fig. 1b). 5 different ligament lengths (*l*<sub>0</sub>) ranging from 6 to 14 mm were used and about 3 specimens per ligament length were tested. The specimens were tested up to fracture at a constant crosshead speed of 1 mm/min. The load-line displacement was measured by means of the videoextensometer using initial extensometer marks separated 50 mm.

To check that the ligament section is fully yielded before crack initiation and ensure the validity of the EWF measurements, a full field strain analysis was performed on the surface of the ligament area.
failure behavior, thus being an invaluable material parameter to estimate crashworthiness at the lab scale. The study investigates the effect of different pre-straining states, namely, uniaxial tension, plane strain, and equibiaxial tension, on the fracture properties of a Quenching and Partitioning (Q&P) and a Carbide Free Bainitic (CFB) sheet steels of 1180 MPa tensile strength.

Fracture toughness characterization is here conducted using the Essential Work of Fracture (EWF) methodology [4]. The EWF is a simple procedure to measure the crack propagation resistance of AHSS sheets and it has been used to understand their cracking performance [2,3,5, 6, 7, 8]. The results for undeformed and pre-strained specimens are compared and discussed.

## Materials

The materials analyzed in this study are two 3rd Gen AHSS for automotive applications. Specifically, a Carbon Free Bainitic (CFB) and a Quenching and Partitioning (Q&P) steel of 1.4 and 1.5 mm thickness, respectively

The microstructure of CFB is characterized by a matrix of carbide-free bainite with globular islands of martensite/retained austenite and laths of retained austenite. The Q&P steel presents a matrix of tempered or carbon-depleted martensite with lath-like retained austenite and globular islands of retained austenite/martensite and bainite. Both steel grades have a content of retained austenite of 12-16% as presented by Frómeta et al. [3].

The tensile properties of the two TRIP-assisted steels were determined according to UNE EN ISO 6892-1 in the transverse direction with respect to the rolling direction. The obtained results are presented in Table 1. Both steels show very similar tensile properties.

<table><caption>Table 1 – Mechanical properties of the investigated TRIP-assisted steel grades at transverse direction: thickness, t; yield strength, σ<sub>YS</sub>; ultimate tensile strength, σ<sub>UTS</sub>; uniform elongation (elongation at σ<sub>UTS</sub>), Ag; elongation at fracture (initial gauge length 80 mm), A<sub>80</sub>; strain hardening exponent between 2 and 4% of deformation, n<sub>2-4</sub>.</caption><thead><tr><td>Steel denomination</td><td>t [mm]</td><td>σ<sub>YS</sub> [MPa]</td><td>σ<sub>UTS</sub> [MPa]</td><td>Ag [%]</td><td>A<sub>80</sub> [%]</td><td>n<sub>2-4</sub> [-]</td></tr></thead><tbody><tr><td>HF1180 CFB</td><td>1.4</td><td>987</td><td>1216</td><td>9.2</td><td>12.6</td><td>0.11</td></tr><tr><td>HF1180 Q&P</td><td>1.5</td><td>1034</td><td>1191</td><td>9.2</td><td>13.1</td><td>0.09</td></tr></tbody></table>

## Experimental procedure

### EWF tests

EWF tests were performed according to CWA 17793:2021 [7]. DENT specimens of 240 x 55 mm machined in the transverse direction were used (Fig. 1a), this geometry has shown to be the most suitable because the transverse stress between the notches is tensile and there is no buckling [9]. Initial notches were machined by electrical discharge machining. Then, fatigue pre-cracks were nucleated at the notch root following the recommendations of the ASTM E1820 [10]. The cracks were extended about 1-1.5 mm per side (Fig. 1a). EWF tests were performed at a 250 kN universal testing machine, equipped with a digital videoextensometer (Fig. 1b). 5 different ligament lengths (l<sub>0</sub>) ranging from 6 to 14 mm were used and about 3 specimens per ligament length were tested. The specimens were tested up to fracture at a constant crosshead speed of 1 mm/min. The load-line displacement was measured by means of the videoextensometer using initial extensometer marks separated 50 mm.

To check that the ligament section is fully yielded before crack initiation and ensure the validity of the EWF measurements, a full field strain analysis was performed on the surface of the ligament area.
<figure><img src="image_1152.png" /><figcaption>Figure 1 – a) DENT specimen geometry and detail of the fatigue pre-crack at the notch root. b) Universal testing machine used for EWF tests.</figcaption></figure>

## Pre-straining

The pre-straining process comprised two methods: uniaxial pre-straining and plane strain/biaxial pre-straining. In the uniaxial approach, rectangular specimens measuring 120 x 45 mm were uniaxially deformed in the transverse direction until the predetermined pre-strain level was achieved at a constant displacement rate of 2.5 mm/min. Subsequently, smaller DENT specimens sized at 70 x 40 mm were extracted from the pre-strained specimens (Fig. 2).

<figure><img src="image_1153.png" /><figcaption>Figure 2 – Schematization of the experimental procedure followed for EWF tests with uniaxial pre-strained specimens.</figcaption></figure>

For plane strain and biaxial pre-straining, Marciniak stretching tests were employed using two different geometries to induce a biaxial and a plane strain state. The pre-strain tests were conducted using an Erichsen machine equipped with a Digital Image Correlation (DIC) system (Fig. 3a). The materials underwent pre-straining up to the specified deformation levels at a punch speed of 67 mm/min, with 15 specimens for each condition. The pre-straining methodology involved stretching one specimen up to fracture while monitoring strain levels with the DIC equipment with a frame rate of 3 Hz. The stage number at which the selected strain level was achieved determined the punch stroke, which was then applied to the remaining specimens. Fig. 3b illustrates a specimen pre-strained in a plane pre-strain condition. After both pre-straining processes, DENT specimens measuring 70 x 40 mm were extracted from the top flat zone of the stretched specimens (Fig. 2).
<figure><img src="image_1154.png" /><figcaption>Figure 1 – a) DENT specimen geometry and detail of the fatigue pre-crack at the notch root. b) Universal testing machine used for EWF tests.</figcaption></figure>

## Pre-straining

The pre-straining process comprised two methods: uniaxial pre-straining and plane strain/biaxial pre-straining. In the uniaxial approach, rectangular specimens measuring 120 x 45 mm were uniaxially deformed in the transverse direction until the predetermined pre-strain level was achieved at a constant displacement rate of 2.5 mm/min. Subsequently, smaller DENT specimens sized at 70 x 40 mm were extracted from the pre-strained specimens (Fig. 2).

<figure><img src="image_1155.png" /><figcaption>Figure 2 – Schematization of the experimental procedure followed for EWF tests with uniaxial pre-strained specimens.</figcaption></figure>

For plane strain and biaxial pre-straining, Marciniak stretching tests were employed using two different geometries to induce a biaxial and a plane strain state. The pre-strain tests were conducted using an Erichsen machine equipped with a Digital Image Correlation (DIC) system (Fig. 3a). The materials underwent pre-straining up to the specified deformation levels at a punch speed of 67 mm/min, with 15 specimens for each condition. The pre-straining methodology involved stretching one specimen up to fracture while monitoring strain levels with the DIC equipment with a frame rate of 3 Hz. The stage number at which the selected strain level was achieved determined the punch stroke, which was then applied to the remaining specimens. Fig. 3b illustrates a specimen pre-strained in a plane pre-strain condition. After both pre-straining processes, DENT specimens measuring 70 x 40 mm were extracted from the top flat zone of the stretched specimens (Fig. 2).
From the extracted specimens, fatigue pre-cracking and EWF tests were conducted, with an adjusted load-line displacement length of 25 mm due to the smaller specimen size. The employment of different specimen sizes does not affect EWF results for it describes a local property of the material, as shown by Hilhorst et al. on [11].

<figure><img src="image_1154.png" /><figcaption>Figure 3 – a) Erichsen equipment and DIC system to perform pre-strain tests at Centro Ricerche Fiat laboratories. b) Pre-strained specimen in plane strain condition.</figcaption></figure>

<figure><img src="image_1155.png" /><figcaption>Figure 4 – Extraction of DENT specimens for EWF tests from plane strained and biaxial pre-strained specimens.</figcaption></figure>

Fig. 5 shows the deformation paths followed for specimen pre-straining. The selected pre-strain level was 7.4%, which is also indicated.
From the extracted specimens, fatigue pre-cracking and EWF tests were conducted, with an adjusted load-line displacement length of 25 mm due to the smaller specimen size. The employment of different specimen sizes does not affect EWF results for it describes a local property of the material, as shown by Hilhorst et al. on [11].

<figure><img src="image_1156.png" /><figcaption>Figure 3 – a) Erichsen equipment and DIC system to perform pre-strain tests at Centro Ricerche Fiat laboratories. b) Pre-strained specimen in plane strain condition.</figcaption></figure>

<figure><img src="image_1157.png" /><figcaption>Figure 4 – Extraction of DENT specimens for EWF tests from plane strained and biaxial pre-strained specimens.</figcaption></figure>

Fig. 5 shows the deformation paths followed for specimen pre-straining. The selected pre-strain level was 7.4%, which is also indicated.
<figure><img src="image_1156.png" /><figcaption>Figure 5 – Deformation paths and strain level selected for pre-straining the specimens.</figcaption></figure>

**Results and discussion**

Fig. 6 presents the EWF results for CFB and Q&P steels in the four pre-straining conditions studied. The pre-straining state and the final thickness of the metal sheet are indicated. In all cases, $w_f$ values escalate linearly with the ligament length. It can be observed that Q&P steel presents higher fracture toughness than CFB. The first remark that can be made is that the pre-straining mainly affects the plastic work, $\beta w_p$, obtained by the slope of the linear regression. Also, it can be noted that the linear fitting is good for both materials in the undeformed state (R² = 0.83 – 0.92). However, the correlation coefficient of the linear regression of the pre-strained states decreased due to the local effects of pre-straining that can be detected experimentally by the application of the EWF methodology; nevertheless, the trend on the linear adjustments is clear.

With the support of a DIC system and a succeeding full field strain analysis of the specimen surface, it was verified that the ligament area is fully yielded before fracture initiation in all the studied conditions, indicating that the EWF methodology is valid to characterize fracture toughness of these materials.
<figure><img src="image_1158.png" /><figcaption>Figure 5 – Deformation paths and strain level selected for pre-straining the specimens.</figcaption></figure>

## Results and discussion

Fig. 6 presents the EWF results for CFB and Q&P steels in the four pre-straining conditions studied. The pre-straining state and the final thickness of the metal sheet are indicated. In all cases, $w_f$ values escalate linearly with the ligament length. It can be observed that Q&P steel presents higher fracture toughness than CFB. The first remark that can be made is that the pre-straining mainly affects the plastic work, $\beta w_p$, obtained by the slope of the linear regression. Also, it can be noted that the linear fitting is good for both materials in the undeformed state (R² = 0.83 – 0.92). However, the correlation coefficient of the linear regression of the pre-strained states decreased due to the local effects of pre-straining that can be detected experimentally by the application of the EWF methodology; nevertheless, the trend on the linear adjustments is clear.

With the support of a DIC system and a succeeding full field strain analysis of the specimen surface, it was verified that the ligament area is fully yielded before fracture initiation in all the studied conditions, indicating that the EWF methodology is valid to characterize fracture toughness of these materials.
<figure><img src="image_1157.png" /><figcaption>Figure 6 – EWF results for the investigated TRIP-assisted steels investigated: a) CFB and b) Q&P; in the four pre-straining states tested.</figcaption></figure>

Moreover, it can be noted that fracture toughness is different for both steels in the undeformed state, despite the tensile properties being very similar. Results show that CFB presents considerably lower fracture resistance than Q&P. These results are in agreement with previous works [12] underlying that fracture toughness cannot be directly estimated from tensile parameters. Therefore, a proper description of crack initiation and propagation resistance behavior of high-strength metal sheets requires the application of a fracture mechanics approach.

A summary of the $w_e$ and $\beta w_p$ results for the two tested steels and the pre-straining conditions performed is presented in Fig. 7 and Fig. 8, respectively. It is important to note that the fracture toughness remains constant across the different pre-straining conditions in both TRIP-assisted steels investigated. The energy absorbed for the generation of new fracture surfaces is similar independently of the strain state of the steel sheet. The non-essential plastic work is notably reduced subsequent to experiencing pre-straining. This behaviour is coherent as the pre-straining
<figure><img src="image_1159.png" /><figcaption>Figure 6 – EWF results for the investigated TRIP-assisted steels investigated: a) CFB and b) Q&P; in the four pre-straining states tested.</figcaption></figure>

Moreover, it can be noted that fracture toughness is different for both steels in the undeformed state, despite the tensile properties being very similar. Results show that CFB presents considerably lower fracture resistance than Q&P. These results are in agreement with previous works [12] underlying that fracture toughness cannot be directly estimated from tensile parameters. Therefore, a proper description of crack initiation and propagation resistance behavior of high-strength metal sheets requires the application of a fracture mechanics approach.

A summary of the $w_e$ and $\beta w_p$ results for the two tested steels and the pre-straining conditions performed is presented in Fig. 7 and Fig. 8, respectively. It is important to note that the fracture toughness remains constant across the different pre-straining conditions in both TRIP-assisted steels investigated. The energy absorbed for the generation of new fracture surfaces is similar independently of the strain state of the steel sheet. The non-essential plastic work is notably reduced subsequent to experiencing pre-straining. This behaviour is coherent as the pre-straining
process consumes plastic work and the remaining plasticity available for the fracture process is reduced.

The study results provide additional detail and depth to the understanding of the parameters influencing fracture resistance mechanisms since very few works are available. Hodges et al. [13] determined by Charpy tests that the global fracture resistance of AHSS was affected by pre-straining. However, it should be considered that such impact tests give the overall fracture energy, without giving information about the energy spent in fracture resistance and the plastic work. The results provided by EWF methodology accurately separate both contributions and permit to readily assess the effect of sheet pre-strain level on the fracture toughness. Since, fracture toughness allows estimating the edge cracking resistance [3] and crashworthiness [2], the results obtained in this work through the EWF method can be used to identify material and processing challenges associated with edge cracking [14] and crash performance of AHSS.

<figure><img src="image_1158.png" /><figcaption>Figure 7 – Summary of $w_e$ values in four pre-strain conditions for materials: a) CFB and b) Q&P.</figcaption></figure>

<figure><img src="image_1159.png" /><figcaption>Figure 8 – Summary of $\beta w_p$ values in four pre-strain conditions for materials: a) CFB and b) Q&P.</figcaption></figure>

## Conclusions

The present investigation on the effect of pre-straining in the fracture resistance of two TRIP-assisted 3rd Gen AHSS steels has provided the following significant findings:

a) The EWF methodology has shown to be applicable to characterize the fracture toughness of the two AHSS investigated in four different strain states.

b) Q&P steel exhibits superior overall fracture toughness compared to CFB steel.

c) Pre-straining presents a significant effect on the non-essential plastic work, $\beta w_p$.

d) Biaxial pre-straining presents the greatest reduction on non-essential plastic work.
process consumes plastic work and the remaining plasticity available for the fracture process is reduced.

The study results provide additional detail and depth to the understanding of the parameters influencing fracture resistance mechanisms since very few works are available. Hodges et al. [13] determined by Charpy tests that the global fracture resistance of AHSS was affected by pre-straining. However, it should be considered that such impact tests give the overall fracture energy, without giving information about the energy spent in fracture resistance and the plastic work. The results provided by EWF methodology accurately separate both contributions and permit to readily assess the effect of sheet pre-strain level on the fracture toughness. Since, fracture toughness allows estimating the edge cracking resistance [3] and crashworthiness [2], the results obtained in this work through the EWF method can be used to identify material and processing challenges associated with edge cracking [14] and crash performance of AHSS.

<figure><img src="image_1160.png" /><figcaption>Figure 7 – Summary of $w_e$ values in four pre-strain conditions for materials: a) CFB and b) Q&P.</figcaption></figure>

<figure><img src="image_1161.png" /><figcaption>Figure 8 – Summary of $\beta w_p$ values in four pre-strain conditions for materials: a) CFB and b) Q&P.</figcaption></figure>

## Conclusions

The present investigation on the effect of pre-straining in the fracture resistance of two TRIP-assisted 3rd Gen AHSS steels has provided the following significant findings:

a) The EWF methodology has shown to be applicable to characterize the fracture toughness of the two AHSS investigated in four different strain states.

b) Q&P steel exhibits superior overall fracture toughness compared to CFB steel.

c) Pre-straining presents a significant effect on the non-essential plastic work, $\beta w_p$.

d) Biaxial pre-straining presents the greatest reduction on non-essential plastic work.
e) The specific essential work of fracture, $w_e$, remains unaffected by pre-straining across all investigated strain states.

f) Accordingly, crash performance and edge cracking resistance are not expected to be greatly affected by sheet pre-straining during cold forming.

## Acknowledgements

This work has been financially supported by the European Commission, Research Fund for Coal and Steel programme under Grant Agreement 800693 — CrashTough — RFCS-2017. The authors would like to acknowledge the participation of Centro Ricerche Fiat (CRF) in the pre-deformation process of plane-strain and equibiaxial strain specimens.

## References

[1] P.J. Jacques, Q. Furnémont, F. Lani, T. Pardoen, F. Delannay, Multiscale mechanics of TRIP-assisted multiphase steels: Characterization and mechanical testing. Acta Mater. 55 (2007) 3681–3693. https://doi.org/10.1016/j.actamat.2007.02.029

[2] D. Frómeta, A. Lara, S. Molas, D. Casellas, J. Rehrl, C. Suppan, P. Larour, J. Calvo, On the correlation between fracture toughness and crash resistance of advanced high strength steels, Eng. Frac. Mech. 205 (2019) 319-332. https://doi.org/10.1016/j.engfracmech.2018.10.005

[3] D. Casellas, A. Lara, D. Frómeta, D. Gutiérrez, S. Molas, Ll. Pérez, J. Rehrl, C Suppan. Fracture toughness to understand stretch-flangeability and edge cracking resistance in AHSS. Metallurgical and Materials Transactions A 48 (2017) 86-94. https://doi.org/10.1007/s11661-016-3815-x

[4] D.Frómeta, M. Tedesco, J. Calvo, A. Lara, S. Molas, D. Casellas. Assessing edge cracking resistance in AHSS automotive parts by the Essential Work of Fracture methodology. Journal of Physics: Conference Series 896 (2017) p. 012102. https://doi.org/10.1088/1742-6596/896/1/012102

[5] D. Frómeta, A. Lara, S. Parareda, D. Casellas. Evaluation of edge formability in high strength sheets through a fracture mechanics approach. AIP conference proceedings 2113 (2017).

[6] D. Frómeta, A. Lara, B Casas, D. Casellas, Fracture toughness measurements to understand local ductility of advanced high strength steels. IOP Conference Series: Materials Science and Engineering 65 (2019) p. 012071. https://doi.org/10.1088/1757-899X/651/1/012071

[7] CWA 17793:2021, Test method for determination of the essential work of fracture of thin ductile metallic sheets.

[8] D. Frómeta, S., Parareda, A. Lara, S. Molas, D. Casellas, P., Jonsén, J. Calvo. Identification of fracture toughness parameters to understand the fracture resistance of advanced high strength sheet steels. Engineering fracture mechanics 229 (2020) 106949. https://doi.org/10.1016/j.engfracmech.2020.106949

[9] Y.W. Mai and P. Powell. Essential work of fracture and J-integral measurements for ductile polymers. J. Polym. Sci. 29 (1991) 785-793. https://doi.org/10.1002/polb.1991.090290702

[10] ASTM E1820. Standard test method for measurement of fracture toughness. American Society for Testing and Materials.

[11] A. Hilhorst, T. Pardoen, P.J. Jacques, Optimization of the essential work of fracture method for characterization of the fracture resistance of metallic sheets. Engineering Fracture Mechanics 268 (2022) 108442. https://doi.org/10.1016/j.engfracmech.2022.108442
e) The specific essential work of fracture, $w_e$, remains unaffected by pre-straining across all investigated strain states.

f) Accordingly, crash performance and edge cracking resistance are not expected to be greatly affected by sheet pre-straining during cold forming.

## Acknowledgements

This work has been financially supported by the European Commission, Research Fund for Coal and Steel programme under Grant Agreement 800693 — CrashTough — RFCS-2017. The authors would like to acknowledge the participation of Centro Ricerche Fiat (CRF) in the pre-deformation process of plane-strain and equibiaxial strain specimens.

## References

[1] P.J. Jacques, Q. Furnémont, F. Lani, T. Pardoen, F. Delannay, Multiscale mechanics of TRIP-assisted multiphase steels: Characterization and mechanical testing. Acta Mater. 55 (2007) 3681–3693. https://doi.org/10.1016/j.actamat.2007.02.029

[2] D. Frómeta, A. Lara, S. Molas, D. Casellas, J. Rehrl, C. Suppan, P. Larour, J. Calvo, On the correlation between fracture toughness and crash resistance of advanced high strength steels, Eng. Frac. Mech. 205 (2019) 319-332. https://doi.org/10.1016/j.engfracmech.2018.10.005

[3] D. Casellas, A. Lara, D. Frómeta, D. Gutiérrez, S. Molas, Ll. Pérez, J. Rehrl, C Suppan. Fracture toughness to understand stretch-flangeability and edge cracking resistance in AHSS. Metallurgical and Materials Transactions A 48 (2017) 86-94. https://doi.org/10.1007/s11661-016-3815-x

[4] D.Frómeta, M. Tedesco, J. Calvo, A. Lara, S. Molas, D. Casellas. Assessing edge cracking resistance in AHSS automotive parts by the Essential Work of Fracture methodology. Journal of Physics: Conference Series 896 (2017) p. 012102. https://doi.org/10.1088/1742-6596/896/1/012102

[5] D. Frómeta, A. Lara, S. Parareda, D. Casellas. Evaluation of edge formability in high strength sheets through a fracture mechanics approach. AIP conference proceedings 2113 (2017).

[6] D. Frómeta, A. Lara, B Casas, D. Casellas, Fracture toughness measurements to understand local ductility of advanced high strength steels. IOP Conference Series: Materials Science and Engineering 65 (2019) p. 012071. https://doi.org/10.1088/1757-899X/651/1/012071

[7] CWA 17793:2021, Test method for determination of the essential work of fracture of thin ductile metallic sheets.

[8] D. Frómeta, S., Parareda, A. Lara, S. Molas, D. Casellas, P., Jonsén, J. Calvo. Identification of fracture toughness parameters to understand the fracture resistance of advanced high strength sheet steels. Engineering fracture mechanics 229 (2020) 106949. https://doi.org/10.1016/j.engfracmech.2020.106949

[9] Y.W. Mai and P. Powell. Essential work of fracture and J-integral measurements for ductile polymers. J. Polym. Sci. 29 (1991) 785-793. https://doi.org/10.1002/polb.1991.090290702

[10] ASTM E1820. Standard test method for measurement of fracture toughness. American Society for Testing and Materials.

[11] A. Hilhorst, T. Pardoen, P.J. Jacques, Optimization of the essential work of fracture method for characterization of the fracture resistance of metallic sheets. Engineering Fracture Mechanics 268 (2022) 108442. https://doi.org/10.1016/j.engfracmech.2022.108442
[12] D. Frómeta, N. Cuadrado, J. Rehrl, C. Suppan, T. Dieudonné, P. Dietsch, J. Calvo, D. Casellas, Microstructural effects on fracture toughness of ultra-high strength dual phase sheet steels, Materials Science & Engineering A 802 (2021) 140631.
https://doi.org/10.1016/j.msea.2020.140631

[13] A.D. Hodges, S. Tedesco, L. Golem, G. Huang, Strain Amount and Strain Path Effects on Instrumented Charpy Toughness of Baked Third Generation Advanced High Strength Steels. SAE Technical Paper 2021-01-0266, 2021, doi:10.4271/2021-01-0266. https://doi.org/10.4271/2021-01-0266

[14] A. Payà, A. Lara, D. Frómeta, M. Marés, I. Gil, and C. Arenillas. Effect of pre-strain on edge formability of high strength steels for chassis applications. In IOP Conference Series: Materials Science and Engineering (Vol. 1284, No. 1, p. 012025). IOP Publishing (2023).
https://doi.org/10.1088/1757-899X/1284/1/012025
[12] D. Frómeta, N. Cuadrado, J. Rehrl, C. Suppan, T. Dieudonné, P. Dietsch, J. Calvo, D. Casellas, Microstructural effects on fracture toughness of ultra-high strength dual phase sheet steels, Materials Science & Engineering A 802 (2021) 140631.
https://doi.org/10.1016/j.msea.2020.140631

[13] A.D. Hodges, S. Tedesco, L. Golem, G. Huang, Strain Amount and Strain Path Effects on Instrumented Charpy Toughness of Baked Third Generation Advanced High Strength Steels. SAE Technical Paper 2021-01-0266, 2021, doi:10.4271/2021-01-0266. https://doi.org/10.4271/2021-01-0266

[14] A. Payà, A. Lara, D. Frómeta, M. Marés, I. Gil, and C. Arenillas. Effect of pre-strain on edge formability of high strength steels for chassis applications. In IOP Conference Series: Materials Science and Engineering (Vol. 1284, No. 1, p. 012025). IOP Publishing (2023).
https://doi.org/10.1088/1757-899X/1284/1/012025
# A temperature-dependent crystal plasticity model for predicting cyclic loading behaviors of a magnesium alloy

BONG Hyuk Jong¹,ᵃ*

¹Materials Deformation Department, Korea Institute of Materials Science, Changwon 51508, Republic of Korea

ᵃhjbong@kims.re.kr

**Keywords:** Crystal Plasticity, Magnesium Alloy, Cyclic Loading, Twin-Detwin

**Abstract.** In this work, a crystal plasticity finite element (CPFE) model to predict cyclic loading behaviors at elevated temperatures of a wrought magnesium alloy, i.e., AZ31B sheet, is proposed. The temperature-dependent mechanical behavior is systematically modeled by modifying the strain-hardening model. The twinning-detwinning, a key deformation mechanism that occurs during the cyclic loadings in the magnesium sheet, is also modeled based on the well-known predominant twinning reorientation (PTR) scheme. Furthermore, to better predict the detwinning behavior, a concept of residual twin is also introduced and employed in the PTR scheme. The modified strain-hardening and enhanced-PTR model considering the twinning-detwinning are implemented in the CPFE framework. Using the developed model, mechanical responses of the AZ31B sheet under cyclic loading conditions at various testing temperatures up to 200°C are predicted and compared with the experimental data, and the prediction results are promising.

**Introduction**

Magnesium and its alloys are the lightest structural metallic material and attract more attention as a lightweight material. However, the wide application of magnesium alloys is hindered by their poor formability at room temperature (RT) due to the limited number of independent deformation mechanisms [1]. However, non-basal slip systems become more active at higher temperatures since their critical resolved shear stresses (CRSSs), known to be significant temperature dependent [2–4], decrease, and they can accommodate the plastic deformation with other slip or twin systems of magnesium alloys with its hexagonal close-packed (HCP) crystal structure.

To overcome the limited formability and asymmetric mechanical responses of magnesium alloy sheets at room temperature, temperature-assisted forming of such alloys is widely employed. Therefore, the material modeling for the magnesium alloy sheets considering complicated slip and twin mechanisms under elevated temperature conditions as well as strain-path changes is vital.

Crystal plasticity is useful for understanding underlying deformation mechanisms and predicting micro- and macro-properties as a consequence of such deformation mechanisms. To date, the crystal plasticity models consider the heterogeneous CRSSs of available slip and twin systems for the HCP-structured magnesium alloys [5–9]. Among them, several attempts have been made to model the temperature-dependent mechanical responses [7,10]. In addition, the crystal plasticity models to predict abnormal mechanical responses under the cyclic loading scenarios considering the twinning-detwinning phenomenon have been proposed [8,9]. However, a material model based on a crystal plasticity framework for cyclic loading scenarios at elevated temperatures for magnesium alloys has not been proposed yet.

The main focus of the current work is to develop a CPFE model that can reproduce the mechanical responses of HCP-structured magnesium (and its alloys) under various temperatures and cyclic loading scenarios at the same time. In the newly developed model, there are two main advancements compared to conventional CPFE model: (a) the temperature dependency of non-basal slip systems is systematically modeled by modifying the strain-hardening model, i.e.,

<figure><img src="image_1160.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.
# A temperature-dependent crystal plasticity model for predicting cyclic loading behaviors of a magnesium alloy

BONG Hyuk Jong¹,ᵃ*

¹Materials Deformation Department, Korea Institute of Materials Science, Changwon 51508, Republic of Korea

ᵃhjbong@kims.re.kr

**Keywords:** Crystal Plasticity, Magnesium Alloy, Cyclic Loading, Twin-Detwin

**Abstract.** In this work, a crystal plasticity finite element (CPFE) model to predict cyclic loading behaviors at elevated temperatures of a wrought magnesium alloy, i.e., AZ31B sheet, is proposed. The temperature-dependent mechanical behavior is systematically modeled by modifying the strain-hardening model. The twinning-detwinning, a key deformation mechanism that occurs during the cyclic loadings in the magnesium sheet, is also modeled based on the well-known predominant twinning reorientation (PTR) scheme. Furthermore, to better predict the detwinning behavior, a concept of residual twin is also introduced and employed in the PTR scheme. The modified strain-hardening and enhanced-PTR model considering the twinning-detwinning are implemented in the CPFE framework. Using the developed model, mechanical responses of the AZ31B sheet under cyclic loading conditions at various testing temperatures up to 200°C are predicted and compared with the experimental data, and the prediction results are promising.

**Introduction**

Magnesium and its alloys are the lightest structural metallic material and attract more attention as a lightweight material. However, the wide application of magnesium alloys is hindered by their poor formability at room temperature (RT) due to the limited number of independent deformation mechanisms [1]. However, non-basal slip systems become more active at higher temperatures since their critical resolved shear stresses (CRSSs), known to be significant temperature dependent [2–4], decrease, and they can accommodate the plastic deformation with other slip or twin systems of magnesium alloys with its hexagonal close-packed (HCP) crystal structure.

To overcome the limited formability and asymmetric mechanical responses of magnesium alloy sheets at room temperature, temperature-assisted forming of such alloys is widely employed. Therefore, the material modeling for the magnesium alloy sheets considering complicated slip and twin mechanisms under elevated temperature conditions as well as strain-path changes is vital.

Crystal plasticity is useful for understanding underlying deformation mechanisms and predicting micro- and macro-properties as a consequence of such deformation mechanisms. To date, the crystal plasticity models consider the heterogeneous CRSSs of available slip and twin systems for the HCP-structured magnesium alloys [5–9]. Among them, several attempts have been made to model the temperature-dependent mechanical responses [7,10]. In addition, the crystal plasticity models to predict abnormal mechanical responses under the cyclic loading scenarios considering the twinning-detwinning phenomenon have been proposed [8,9]. However, a material model based on a crystal plasticity framework for cyclic loading scenarios at elevated temperatures for magnesium alloys has not been proposed yet.

The main focus of the current work is to develop a CPFE model that can reproduce the mechanical responses of HCP-structured magnesium (and its alloys) under various temperatures and cyclic loading scenarios at the same time. In the newly developed model, there are two main advancements compared to conventional CPFE model: (a) the temperature dependency of non-basal slip systems is systematically modeled by modifying the strain-hardening model, i.e.,

<figure><img src="image_1162.png" /></figure>
extended Voce hardening, as a function of temperature, and (b) the well-known predominant twinning reorientation (PTR) scheme is enhanced by incorporating the twinning-detwinning mechanism and introducing the concept of residual twin as a criterion for detwinning termination.

## Materials

A commercially available AZ31B magnesium alloy sheet with a thickness of 1.4 mm is investigated in the current study. The chemical composition of the corresponding magnesium alloy is listed in Table 1.

<table><caption>Table 1. Chemical composition of AZ31B sheet (unit in wt%)</caption><thead><tr><th>Al</th><th>Zn</th><th>Mn</th><th>Si</th><th>Fe</th><th>Cu</th><th>Mg</th></tr></thead><tbody><tr><td>2.5-3.5</td><td>0.6-1.4</td><td>0.2-1.0</td><td>&lt;0.1</td><td>&lt;0.005</td><td>&lt;0.05</td><td>Balance</td></tr></tbody></table>

To obtain the initial crystallographic texture of the AZ31B, an X-ray diffraction test was conducted. Using the measured pole figures shown in Fig. 1, the orientation distribution function (ODF) was estimated, and 1,000 discrete Euler angles were extracted from the estimated ODF. They were then used as input for the CPFE simulation afterward.

<figure><img src="image_1161.png" /><figcaption>Fig. 1. {0002} and {1010} pole figures of as-received AZ31B sheet measured by X-ray diffraction</figcaption></figure>

## Experimental

Uniaxial tension tests of the AZ31B sheet were conducted using an ASTM E8 specimen with a gauge length of 25 mm. The samples were heated up using a heating chamber. The strain fields were measured using the ARAMIS digital image correlation (DIC) system, and recorded images were analyzed to obtain stress-strain data along with the load data from the universal testing machine and r-value defined as $r = -\epsilon_w/(\epsilon_w + \epsilon_L)$. The r-value is calculated from the recorded DIC image during the test.

The mechanical responses under the cyclic loading conditions were obtained by conducting in-plane cyclic tests. The testing apparatus and dimensions of the testing sample are shown in Fig. 2. The clamping dies were heated up using cartridge heaters. The temperature of the sample was measured using attached thermocouples and tests were initiated once the target temperature of the sample was reached. Out-of-plane buckling was prevented by applying side force to the anti-buckling plates of 2 kN for tests at RT and 100 °C and 1 kN for the tests at 150 and 200 °C. The biaxial stress induced by the side force and frictional force between the anti-buckling plates and the sample was corrected according to the procedure reported elsewhere [11]. Using the sample testing apparatus, the in-plane uniaxial tension tests were also conducted at the same temperature conditions.

The uniaxial tension, uniaxial compression, and cyclic tests were conducted along the rolling direction at a strain rate of ~0.001/s and RT, 100, 150, and 200 °C.
extended Voce hardening, as a function of temperature, and (b) the well-known predominant twinning reorientation (PTR) scheme is enhanced by incorporating the twinning-detwinning mechanism and introducing the concept of residual twin as a criterion for detwinning termination.

## Materials

A commercially available AZ31B magnesium alloy sheet with a thickness of 1.4 mm is investigated in the current study. The chemical composition of the corresponding magnesium alloy is listed in Table 1.

<table><caption>Table 1. Chemical composition of AZ31B sheet (unit in wt%)</caption><thead><tr><th>Al</th><th>Zn</th><th>Mn</th><th>Si</th><th>Fe</th><th>Cu</th><th>Mg</th></tr></thead><tbody><tr><td>2.5-3.5</td><td>0.6-1.4</td><td>0.2-1.0</td><td>&lt;0.1</td><td>&lt;0.005</td><td>&lt;0.05</td><td>Balance</td></tr></tbody></table>

To obtain the initial crystallographic texture of the AZ31B, an X-ray diffraction test was conducted. Using the measured pole figures shown in Fig. 1, the orientation distribution function (ODF) was estimated, and 1,000 discrete Euler angles were extracted from the estimated ODF. They were then used as input for the CPFE simulation afterward.

<figure><img src="image_1163.png" /><figcaption>Fig. 1. {0002} and {1010} pole figures of as-received AZ31B sheet measured by X-ray diffraction</figcaption></figure>

## Experimental

Uniaxial tension tests of the AZ31B sheet were conducted using an ASTM E8 specimen with a gauge length of 25 mm. The samples were heated up using a heating chamber. The strain fields were measured using the ARAMIS digital image correlation (DIC) system, and recorded images were analyzed to obtain stress-strain data along with the load data from the universal testing machine and r-value defined as $r = -\varepsilon_w/(\varepsilon_w + \varepsilon_L)$. The r-value is calculated from the recorded DIC image during the test.

The mechanical responses under the cyclic loading conditions were obtained by conducting in-plane cyclic tests. The testing apparatus and dimensions of the testing sample are shown in Fig. 2. The clamping dies were heated up using cartridge heaters. The temperature of the sample was measured using attached thermocouples and tests were initiated once the target temperature of the sample was reached. Out-of-plane buckling was prevented by applying side force to the anti-buckling plates of 2 kN for tests at RT and 100 °C and 1 kN for the tests at 150 and 200 °C. The biaxial stress induced by the side force and frictional force between the anti-buckling plates and the sample was corrected according to the procedure reported elsewhere [11]. Using the sample testing apparatus, the in-plane uniaxial tension tests were also conducted at the same temperature conditions.

The uniaxial tension, uniaxial compression, and cyclic tests were conducted along the rolling direction at a strain rate of ~0.001/s and RT, 100, 150, and 200 °C.
<figure><img src="image_1162.png" /><figcaption>Fig. 2. In-plane cyclic testing apparatus and dimension of testing sample</figcaption></figure>

## Crystal plasticity modeling

A CPFE model to model temperature dependency under cyclic loading conditions based on a rate-dependent crystal plasticity framework [12,13] was developed. The basic formulations of the crystal plasticity for the HCP-structured metallic materials can be found elsewhere [5], and only a summary of the CPFE model is presented in this section along with the core formulation for the temperature-dependency and twinning-detwinning model [14].

The temperature dependency of the non-basal slip systems is addressed by modifying the Voce hardening model as

$$ \tau_{crss}^{\alpha}(T, \Gamma) = \tau_{0}^{\alpha}(T) + (\tau_{1}^{\alpha}(T) + \theta_{1}^{\alpha}(T) \cdot \Gamma) \left\{ 1 - \exp \left( -\frac{\theta_{0}^{\alpha}(T) \cdot \Gamma}{\tau_{1}^{\alpha}(T)} \right) \right\} \quad (1) $$

where $\Gamma$ is the total accumulated shear strain over all available slip and twin systems, and $\tau_0^\alpha(T)$, $\tau_1^\alpha(T)$, $\theta_0^\alpha(T)$, and $\theta_1^\alpha(T)$ are temperature-dependent hardening parameters of $\alpha$-th non-basal slip systems and they are modeled as follows according to Singh et al. [15]:

$$ \begin{pmatrix} \tau_0^\alpha(T) \\ \tau_1^\alpha(T) \\ \theta_0^\alpha(T) \\ \theta_1^\alpha(T) \end{pmatrix} = k(T) \begin{pmatrix} \tau_0^\alpha(T_R) \\ \tau_1^\alpha(T_R) \\ \theta_0^\alpha(T_R) \\ \theta_1^\alpha(T_R) \end{pmatrix} \quad (2) $$

where $T_R$ is the reference temperature, and $k(T)$ is the temperature-dependent scaling factor.

Twinning-detwinning based on the PTR scheme is proposed, and the details are elsewhere [14]. Here, only a flowchart describing the activation of twinning and detwinning is presented in Fig. 3. According to the PTR model, when the volume fraction of twins in a particular grain exceeds the threshold value $f_c$, that grain will be re-orientated. $f_c$ evolves as

$$ f_c = \min(1.0, A + B \frac{F_E}{F_A}) \quad (3) $$

where $F_E (= \sum_{m=1}^{NG^{twin}} w^m)$ is the effective twin volume fraction and $F_A (= \sum_{n=1}^{NG} (w^n \sum_{\beta=1}^{N^{TW}} f^\beta))$. Here, $NG^{twin}$, $NG$, and $w$, $N^{twin}$ are the total number of fully twinned grains, total number of grains, volume fraction of the corresponding grains overall grains, and numbers of twin variants, respectively.

The basic assumption of the proposed twinning-detwinning model is that only one of two, i.e., twin and detwin, will be active during each loading step. It is worthwhile to note that slip systems
<figure><img src="image_1164.png" /><figcaption>Fig. 2. In-plane cyclic testing apparatus and dimension of testing sample</figcaption></figure>

## Crystal plasticity modeling

A CPFE model to model temperature dependency under cyclic loading conditions based on a rate-dependent crystal plasticity framework [12,13] was developed. The basic formulations of the crystal plasticity for the HCP-structured metallic materials can be found elsewhere [5], and only a summary of the CPFE model is presented in this section along with the core formulation for the temperature-dependency and twinning-detwinning model [14].

The temperature dependency of the non-basal slip systems is addressed by modifying the Voce hardening model as

$$ \tau_{crss}^{\alpha}(T, \Gamma) = \tau_{0}^{\alpha}(T) + (\tau_{1}^{\alpha}(T) + \theta_{1}^{\alpha}(T) \cdot \Gamma) \left\{ 1 - \exp \left( - \frac{\theta_{0}^{\alpha}(T) \cdot \Gamma}{\tau_{1}^{\alpha}(T)} \right) \right\} \quad (1) $$

where $\Gamma$ is the total accumulated shear strain over all available slip and twin systems, and $\tau_0^\alpha(T)$, $\tau_1^\alpha(T)$, $\theta_0^\alpha(T)$, and $\theta_1^\alpha(T)$ are temperature-dependent hardening parameters of $\alpha$-th non-basal slip systems and they are modeled as follows according to Singh et al. [15]:

$$ \begin{pmatrix} \tau_0^\alpha(T) \\ \tau_1^\alpha(T) \\ \theta_0^\alpha(T) \\ \theta_1^\alpha(T) \end{pmatrix} = k(T) \begin{pmatrix} \tau_0^\alpha(T_R) \\ \tau_1^\alpha(T_R) \\ \theta_0^\alpha(T_R) \\ \theta_1^\alpha(T_R) \end{pmatrix} \quad (2) $$

where $T_R$ is the reference temperature, and $k(T)$ is the temperature-dependent scaling factor.

Twinning-detwinning based on the PTR scheme is proposed, and the details are elsewhere [14]. Here, only a flowchart describing the activation of twinning and detwinning is presented in Fig. 3. According to the PTR model, when the volume fraction of twins in a particular grain exceeds the threshold value $f_c$, that grain will be re-orientated. $f_c$ evolves as

$$ f_c = \min(1.0, A + B \frac{F_E}{F_A}) \quad (3) $$

where $F_E ( = \sum_{m=1}^{NG^{twin}} w^m )$ is the effective twin volume fraction and $F_A ( = \sum_{n=1}^{NG} (w^n \sum_{\beta=1}^{N^{TW}} f^\beta) )$. Here, $NG^{twin}$, $NG$, and $w$, $N^{twin}$ are the total number of fully twinned grains, total number of grains, volume fraction of the corresponding grains overall grains, and numbers of twin variants, respectively.

The basic assumption of the proposed twinning-detwinning model is that only one of two, i.e., twin and detwin, will be active during each loading step. It is worthwhile to note that slip systems
are active at all time steps. During the first loading, the twin should be active. Then, at the onset of the second loading, the grains are categorized into the following three different twinning-detwinning scenarios according to the twin volume fraction. In Case I, the volume fraction exceeds the threshold at the first loading and the grain is completely re-oriented. In such a case, it can be assumed that detwinning is activated at the second loading. In Case II, the grain is not completely re-oriented at the first loading, but some amount of twin is created more than $f_r$, the residual twin fraction. The residual twin represents persisting twins, even after full reverse loading. If the fraction of twin generated at first loading is greater than $f_r$, detwin will be activated during the second loading, otherwise this Case III is assumed. In this case, the grain does not have the right orientation for the twin to take place, and the twin is activated during the second loading. During the next loading sequence, twin occurs again in the grain that is detwinned at the second loading. However, in Case III in the second loading, the three cases can be assumed again as shown in Fig. 3.

<figure><img src="image_1163.png" /><figcaption>Fig. 3. Flowchart of proposed twinning-detwinning model [14].</figcaption></figure>

## Results

The material parameters are identified by simultaneously fitting using a representative volume element (RVE) incorporated with the developed CPFE model to stress-strain curves obtained from the uniaxial tension and compression tests, and (accumulated) r-value obtained from the uniaxial tension tests at all investigated testing temperatures. The RVE model consists of 1,000 finite elements (FEs) represented by 8-node brick element with reduced integration. Each FE represents a single crystal, and thus, the RVE represents a polycrystal with 1,000 grains. Periodic boundary condition is imposed as described in previous studies by the author [16,17]. The following four slip/twin systems are assumed to be active: {0001}(1120) basal slip, {1011}(1120) prismatic slip, {1122}(1123) pyramidal <a+c> slip, and {1012}(1011) tensile twin.
are active at all time steps. During the first loading, the twin should be active. Then, at the onset of the second loading, the grains are categorized into the following three different twinning-detwinning scenarios according to the twin volume fraction. In Case I, the volume fraction exceeds the threshold at the first loading and the grain is completely re-oriented. In such a case, it can be assumed that detwinning is activated at the second loading. In Case II, the grain is not completely re-oriented at the first loading, but some amount of twin is created more than $f_r$, the residual twin fraction. The residual twin represents persisting twins, even after full reverse loading. If the fraction of twin generated at first loading is greater than $f_r$, detwin will be activated during the second loading, otherwise this Case III is assumed. In this case, the grain does not have the right orientation for the twin to take place, and the twin is activated during the second loading. During the next loading sequence, twin occurs again in the grain that is detwinned at the second loading. However, in Case III in the second loading, the three cases can be assumed again as shown in Fig. 3.

<figure><img src="image_1165.png" /><figcaption>Fig. 3. Flowchart of proposed twinning-detwinning model [14].</figcaption></figure>

## Results

The material parameters are identified by simultaneously fitting using a representative volume element (RVE) incorporated with the developed CPFE model to stress-strain curves obtained from the uniaxial tension and compression tests, and (accumulated) r-value obtained from the uniaxial tension tests at all investigated testing temperatures. The RVE model consists of 1,000 finite elements (FEs) represented by 8-node brick element with reduced integration. Each FE represents a single crystal, and thus, the RVE represents a polycrystal with 1,000 grains. Periodic boundary condition is imposed as described in previous studies by the author [16,17]. The following four slip/twin systems are assumed to be active: {0001}(1120) basal slip, {1011}(1120) prismatic slip, {1122}(1123) pyramidal <a+c> slip, and {1012}(1011) tensile twin.
The fitting results are shown in Fig. 4. The identified constitutive parameters for the extended Voce hardening model are listed in Table 2. The PTR model parameters in Eq. (3) are identified as A=0.7 and B=1 for all the testing temperatures. It is worthwhile to note that the Voce hardening parameters for the tensile detwin are identified by fitting to the cyclic loading data obtained from the in-plane cyclic tests, which will appear later on.

<figure><img src="image_1164.png" /><figcaption>Fig. 4. Measured and simulated stress-strain curves under (a) uniaxial tension and (b) uniaxial compression, and (c) r-values under uniaxial tension</figcaption></figure>

<table><caption>Table 2 Identified constitutive parameters for the extended Voce hardening model</caption><thead><tr><th>Temperature</th><th>Slip/Twin Systems</th><th>τ<sub>0</sub></th><th>τ<sub>1</sub></th><th>θ<sub>0</sub></th><th>θ<sub>1</sub></th><th>q<sub>ij</sub></th></tr></thead><tbody><tr><td rowspan="5">25</td><td>Basal</td><td>25</td><td>1</td><td>90</td><td>0</td><td>2.5</td></tr><tr><td>Prismatic</td><td>105</td><td>21</td><td>390</td><td>10</td><td>2.5</td></tr><tr><td>Pyramidal &lt;a+c&gt;</td><td>108</td><td>29</td><td>440</td><td>15</td><td>4</td></tr><tr><td>Tensile twin</td><td>28</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td>Tensile detwin</td><td>21</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td rowspan="5">100</td><td>Basal</td><td>25</td><td>1</td><td>90</td><td>0</td><td>2.5</td></tr><tr><td>Prismatic</td><td>74</td><td>16</td><td>280</td><td>7.5</td><td>2.5</td></tr><tr><td>Pyramidal &lt;a+c&gt;</td><td>76</td><td>20.5</td><td>320</td><td>11</td><td>4</td></tr><tr><td>Tensile twin</td><td>28</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td>Tensile detwin</td><td>21</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td rowspan="5">150</td><td>Basal</td><td>25</td><td>1</td><td>90</td><td>0</td><td>2.5</td></tr><tr><td>Prismatic</td><td>50</td><td>12</td><td>185</td><td>6</td><td>2.5</td></tr><tr><td>Pyramidal &lt;a+c&gt;</td><td>51</td><td>15</td><td>210</td><td>8.5</td><td>4</td></tr><tr><td>Tensile twin</td><td>28</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td>Tensile detwin</td><td>21</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td rowspan="5">200</td><td>Basal</td><td>25</td><td>1</td><td>90</td><td>0</td><td>2.5</td></tr><tr><td>Prismatic</td><td>33</td><td>7</td><td>130</td><td>3</td><td>2.5</td></tr><tr><td>Pyramidal &lt;a+c&gt;</td><td>34</td><td>8</td><td>140</td><td>5</td><td>4</td></tr><tr><td>Tensile twin</td><td>28</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr><tr><td>Tensile detwin</td><td>21</td><td>5</td><td>10</td><td>5</td><td>2.5</td></tr></tbody></table>