0.004 1/s and the associated relatively low deviation from the nominal strain rate, even in DSRC tests, makes the identification of a distinction difficult.

<figure><img src="image_883.png" /><figcaption>Fig. 9. Overview of strain rate sensitive material properties of DP600 obtained from DSRC and OSRC tensile tests</figcaption></figure>

## Summary and Outlook

To increase the prediction quality of numerical simulations and consequently improve tool and part design, precise material data is crucial. A current issue in state of the art characterization procedures is either non-constant strain rates or the unknown impact of constant strain rate testing on the determination of material properties. This research presented a novel and versatile method for a 3D-DIC strain rate controlled tensile test, to provide constant strain rate testing. The method is independent from the testing machine and the specimen-geometry, such that influences like machine stiffness are negligible and do not affect the strain rate anymore. DSRC and OSRC tensile test where conducted and the resulting properties compared. Hereby, the standard gauge length of A80 tensile test specimen geometry of 80 mm was used as input for controlling the strain rate. A significant increase in the constancy and precision of the strain rate was observed for quasi-static testing at 0.4 %/s. The RMSE of the strain rate during tensile testing could be improved from 0.012 %/s for DC05 respectively 0.011 %/s for DP600 to 2.4·10⁻⁵ %/s respectively 5.6·10⁻⁵ %/s. Also the maximum deviation to the nominal strain rate was reduced from 0.224 %/s to 0.037 %/s respectively 0.278 %/s to 0.084 %/s. Due to the low strain rate sensitivity at room temperature of steels in general and the investigated materials DC05 and DP600, nearly no influence on the measurement of the mechanical properties could be observed [8]. However, in future work the OSRC with a 3D-DIC system can be applied for characterization tests at elevated temperatures, where strain rate sensitivity is more pronounced for most metals. Because of the increase of strain rate in the region of necking, an adaptive and tailored strain rate control system, which controls the strain rate in these regions, provides the potential to improve the determination of material data up to higher strains.

## Acknowledgement

This research was supported by the Bavarian Research Foundation (BFS) within the scope of the project 1565-22 "Optical strain rate control in material characterization". The authors are grateful to the industrial partners from AUDI AG, Carl Zeiss GOM Metrology GmbH, MATFEM
0.004 1/s and the associated relatively low deviation from the nominal strain rate, even in DSRC tests, makes the identification of a distinction difficult.

<figure><img src="image_891.png" /><figcaption>Fig. 9. Overview of strain rate sensitive material properties of DP600 obtained from DSRC and OSRC tensile tests</figcaption></figure>

## Summary and Outlook

To increase the prediction quality of numerical simulations and consequently improve tool and part design, precise material data is crucial. A current issue in state of the art characterization procedures is either non-constant strain rates or the unknown impact of constant strain rate testing on the determination of material properties. This research presented a novel and versatile method for a 3D-DIC strain rate controlled tensile test, to provide constant strain rate testing. The method is independent from the testing machine and the specimen-geometry, such that influences like machine stiffness are negligible and do not affect the strain rate anymore. DSRC and OSRC tensile test where conducted and the resulting properties compared. Hereby, the standard gauge length of A80 tensile test specimen geometry of 80 mm was used as input for controlling the strain rate. A significant increase in the constancy and precision of the strain rate was observed for quasi-static testing at 0.4 %/s. The RMSE of the strain rate during tensile testing could be improved from 0.012 %/s for DC05 respectively 0.011 %/s for DP600 to 2.4·10⁻⁵ %/s respectively 5.6·10⁻⁵ %/s. Also the maximum deviation to the nominal strain rate was reduced from 0.224 %/s to 0.037 %/s respectively 0.278 %/s to 0.084 %/s. Due to the low strain rate sensitivity at room temperature of steels in general and the investigated materials DC05 and DP600, nearly no influence on the measurement of the mechanical properties could be observed [8]. However, in future work the OSRC with a 3D-DIC system can be applied for characterization tests at elevated temperatures, where strain rate sensitivity is more pronounced for most metals. Because of the increase of strain rate in the region of necking, an adaptive and tailored strain rate control system, which controls the strain rate in these regions, provides the potential to improve the determination of material data up to higher strains.

## Acknowledgement

This research was supported by the Bavarian Research Foundation (BFS) within the scope of the project 1565-22 "Optical strain rate control in material characterization". The authors are grateful to the industrial partners from AUDI AG, Carl Zeiss GOM Metrology GmbH, MATFEM
Ingenieurgesellschaft mbH, Me-go GmbH and ZwickRoell GmbH & Co. KG for their support in the project. Further thanks goes to the laboratory assistants supporting the execution of this work.

## References

[1] R. Neugebauer, A. Sterzing, and R. Müller, "New methods for more accurate material characterization," *Prod. Eng. Res. Devel.*, vol. 4, no. 4, pp. 427-432, 2010. https://doi.org/10.1007/s11740-010-0244-3

[2] Thomas Haase, "A Method for the Conversion of Experimental Flow Curves to constant Strain Rates as Input for Material Models," *13. LS-DYNA Forum, Bamberg, 2014*, 2014. [Online].

[3] A. Georgios and L. Nicoleta, *SUSTAINABLE CONSUMPTION AND PRODUCTION*, 2023. Accessed: Nov. 22 2023. [Online]. Available: https://www.europarl.europa.eu/factsheets/en/sheet/77/ sustainable-consumption-and-production

[4] P. Peters, N. Manopulo, C. Lange, and P. Hora, "A strain rate dependent anisotropic hardening model and its validation through deep drawing experiments," *Int J Mater Form*, vol. 7, no. 4, pp. 447-457, 2014. https://doi.org/10.1007/s12289-013-1140-0

[5] E. L. Bartholomew, K. J. Krystyan, D. A. J. Millar, and J. Wulff, "A Machine for Tensile Testing at Controlled Temperatures and Constant True Strain Rates," *Review of Scientific Instruments*, vol. 24, no. 3, pp. 196-202, 1953. https://doi.org/10.1063/1.1770662

[6] S. Suttner and M. Merklein, "Experimental and numerical investigation of a strain rate controlled hydraulic bulge test of sheet metal," *Journal of Materials Processing Technology*, vol. 235, pp. 121-133, 2016. https://doi.org/10.1016/j.jmatprotec.2016.04.022

[7] D. K. ddBanerjee, M. A. Iadicola, and A. A. Creuziger, "Understanding Deformation Behavior in Uniaxial Tensile Tests of Steel Specimens at Varying Strain Rates," *J. RES. NATL. INST. STAN.*, vol. 126, 2021. https://doi.org/10.6028/jres.126.050

[8] C. Zener and J. H. Hollomon, "Effect of Strain Rate Upon Plastic Flow of Steel," *Journal of Applied Physics*, vol. 15, no. 1, pp. 22-32, 1944. https://doi.org/10.1063/1.1707363
Ingenieurgesellschaft mbH, Me-go GmbH and ZwickRoell GmbH & Co. KG for their support in the project. Further thanks goes to the laboratory assistants supporting the execution of this work.

## References

[1] R. Neugebauer, A. Sterzing, and R. Müller, "New methods for more accurate material characterization," *Prod. Eng. Res. Devel.*, vol. 4, no. 4, pp. 427-432, 2010.
https://doi.org/10.1007/s11740-010-0244-3

[2] Thomas Haase, "A Method for the Conversion of Experimental Flow Curves to constant Strain Rates as Input for Material Models," *13. LS-DYNA Forum, Bamberg, 2014*, 2014. [Online].

[3] A. Georgios and L. Nicoleta, *SUSTAINABLE CONSUMPTION AND PRODUCTION*, 2023.
Accessed: Nov. 22 2023. [Online]. Available:
https://www.europarl.europa.eu/factsheets/en/sheet/77/ sustainable-consumption-and-production

[4] P. Peters, N. Manopulo, C. Lange, and P. Hora, "A strain rate dependent anisotropic hardening model and its validation through deep drawing experiments," *Int J Mater Form*, vol. 7, no. 4, pp. 447-457, 2014. https://doi.org/10.1007/s12289-013-1140-0

[5] E. L. Bartholomew, K. J. Krystyan, D. A. J. Millar, and J. Wulff, "A Machine for Tensile Testing at Controlled Temperatures and Constant True Strain Rates," *Review of Scientific Instruments*, vol. 24, no. 3, pp. 196-202, 1953. https://doi.org/10.1063/1.1770662

[6] S. Suttner and M. Merklein, "Experimental and numerical investigation of a strain rate controlled hydraulic bulge test of sheet metal," *Journal of Materials Processing Technology*, vol. 235, pp. 121-133, 2016. https://doi.org/10.1016/j.jmatprotec.2016.04.022

[7] D. K. ddBanerjee, M. A. Iadicola, and A. A. Creuziger, "Understanding Deformation Behavior in Uniaxial Tensile Tests of Steel Specimens at Varying Strain Rates," *J. RES. NATL. INST. STAN.*, vol. 126, 2021. https://doi.org/10.6028/jres.126.050

[8] C. Zener and J. H. Hollomon, "Effect of Strain Rate Upon Plastic Flow of Steel," *Journal of Applied Physics*, vol. 15, no. 1, pp. 22-32, 1944. https://doi.org/10.1063/1.1707363
# A novel method to characterize low-temperature brittle failure at various pre-strains

BECHLER Jan<fnref n="1" />,ª*, LINGBEEK Roald²,ᵇ, GRODOTZKI Joshua<fnref n="1" />,ᶜ and TEKKAYA A. Erman<fnref n="1" />,ᵈ

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

²Autoliv B.V. & Co. KG Werk Nord, Otto-Hahn-Straße 4, 25337 Elmshorn, Germany

ªJan.bechler@iul.tu-dortmund.de, ᵇRoald.Lingbeek@autoliv.com,

ᶜJoshua.Grodotzki@iul.tu-dortmund.de, ᵈErman.Tekkaya@iul.tu-dortmund.de

**Keywords:** Deep Drawing, Cold Work Embrittlement (CWE), Ductile to Brittle Transition Temperature (DBTT), Impact Toughness

**Abstract.** A new characterization method for the determination of the pre-strain dependent ductile-to-brittle-transition-temperature (DBTT) of steel is presented in this paper. From four different types of mild steel circular cups were deep drawn. Out of these cups, rings were cut, which were used as Charpy impact test specimens. The Charpy-notch was placed at two different positions along the circumference of the ring: due to the steel anisotropy (rolling direction) the side walls of the cup have variations in strain, hence, notches are placed at areas with high, or low level of forming pre-strains. Using the ring samples, Charpy impact tests have been carried out at various temperature levels and the absorbed energy was measured as a function of temperature. The test results have been processed into material-specific DBTT values, including the dependency of pre-strain. As a result: high pre-strains generally raise the DBTT and therefore increase the risk of brittle failure in a low-temperature impact load. The pre-strain dependency of the DBTT varies for different types of mild steels – this is likely due to differences in the chemical composition and microstructure. Interstitial free steel with a near zero amount carbon and no added boron is most sensitive to forming pre-strain. The addition of boron in general shifts the DBTT to lower temperatures and reduces the pre-strain sensitivity.

**Introduction**

Automotive safety components, such as deep drawn airbag containers, must function correctly under low-temperature conditions (e.g., at temperatures as low as -40 °C): Any risk of brittle failure is unacceptable and must be ruled out during the component design phase. Certain grades of steel are sensitive to impact loads at low temperatures, particularly when pre-strained in the forming process. This phenomenon is known as cold work embrittlement (CWE), secondary work embrittlement (SWE), or strain-induced embrittlement [1]. Due to the complex interaction between the material, the forming process, and the subsequent impact load, the design process for such components is challenging. Here, a material characterization method that encompasses the previously mentioned physical aspects, is of great help. This paper proposes such a method.

Mild steels for deep drawn parts are commonly body-centered cubic (BCC). BCC materials change their failure mode from ductile, at high temperature, to brittle, at low temperatures<fnref n="1" />. A low temperature increases the activation energy for dislocation movements, which results in less plastic deformation and brittle material behavior becomes dominant at failure [1]. In contrast, at higher

<figure><img src="image_884.png" /></figure>

<footnote n="1">¹ The different packing order in face-centered cubic materials (FCC) causes no change in activation energy and shows therefore no change in failure mode over the temperature.</footnote>
# A novel method to characterize low-temperature brittle failure at various pre-strains

BECHLER Jan<fnref n="1" />,ª*, LINGBEEK Roald²,ᵇ, GRODOTZKI Joshua<fnref n="1" />,ᶜ and TEKKAYA A. Erman<fnref n="1" />,ᵈ

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

²Autoliv B.V. & Co. KG Werk Nord, Otto-Hahn-Straße 4, 25337 Elmshorn, Germany

ªJan.bechler@iul.tu-dortmund.de, ᵇRoald.Lingbeek@autoliv.com,

ᶜJoshua.Grodotzki@iul.tu-dortmund.de, ᵈErman.Tekkaya@iul.tu-dortmund.de

**Keywords:** Deep Drawing, Cold Work Embrittlement (CWE), Ductile to Brittle Transition Temperature (DBTT), Impact Toughness

**Abstract.** A new characterization method for the determination of the pre-strain dependent ductile-to-brittle-transition-temperature (DBTT) of steel is presented in this paper. From four different types of mild steel circular cups were deep drawn. Out of these cups, rings were cut, which were used as Charpy impact test specimens. The Charpy-notch was placed at two different positions along the circumference of the ring: due to the steel anisotropy (rolling direction) the side walls of the cup have variations in strain, hence, notches are placed at areas with high, or low level of forming pre-strains. Using the ring samples, Charpy impact tests have been carried out at various temperature levels and the absorbed energy was measured as a function of temperature. The test results have been processed into material-specific DBTT values, including the dependency of pre-strain. As a result: high pre-strains generally raise the DBTT and therefore increase the risk of brittle failure in a low-temperature impact load. The pre-strain dependency of the DBTT varies for different types of mild steels – this is likely due to differences in the chemical composition and microstructure. Interstitial free steel with a near zero amount carbon and no added boron is most sensitive to forming pre-strain. The addition of boron in general shifts the DBTT to lower temperatures and reduces the pre-strain sensitivity.

**Introduction**

Automotive safety components, such as deep drawn airbag containers, must function correctly under low-temperature conditions (e.g., at temperatures as low as -40 °C): Any risk of brittle failure is unacceptable and must be ruled out during the component design phase. Certain grades of steel are sensitive to impact loads at low temperatures, particularly when pre-strained in the forming process. This phenomenon is known as cold work embrittlement (CWE), secondary work embrittlement (SWE), or strain-induced embrittlement [1]. Due to the complex interaction between the material, the forming process, and the subsequent impact load, the design process for such components is challenging. Here, a material characterization method that encompasses the previously mentioned physical aspects, is of great help. This paper proposes such a method.

Mild steels for deep drawn parts are commonly body-centered cubic (BCC). BCC materials change their failure mode from ductile, at high temperature, to brittle, at low temperatures<fnref n="1" />. A low temperature increases the activation energy for dislocation movements, which results in less plastic deformation and brittle material behavior becomes dominant at failure [1]. In contrast, at higher

<figure><img src="image_892.png" /></figure>

<footnote n="1">¹ The different packing order in face-centered cubic materials (FCC) causes no change in activation energy and shows therefore no change in failure mode over the temperature.</footnote>
temperatures, the plastic deformation absorbs the majority of the impact energy. The transition from ductile to brittle failure is characterized by the ductile-to-brittle-transition-temperature (DBTT). It is used as a key parameter for performance evaluation at low temperatures.

Brittle failure can occur as intergranular (IGF) or transgranular failure (TGF) modes (Fig. 2 a)) [2]. In IGF, a crack propagates along the grain boundaries. For TGF, the crack passes through the grain. CWE is defined as material failure which always entails IGF [3]. IGF occurs if the strength between the grains is lower than the strength within the grains [3]. Specific alloy elements increase or decrease the strength between the grains due to their interaction with the host electron gas [4]. Carbon [5] and boron [6] increase grain boundary strength, Phosphorus [5], Silicon [7], and Sulfur [8] have a detrimental effect.

Commonly, plastic deformation will take preference over brittle failure, because the yield stress is lower than the critical stress for IGF. However, the yield strength $\sigma_{y,0}$ will be strongly increased due to the work hardening (from a forming process, such as deep drawing), high strain rate in a crash scenario $\sigma(\dot{\varepsilon})$, and low temperature $\sigma(T)$ (Fig. 1). It is assumed that the critical IGF-stress is independent of these effects [3]. Summing up: When a deep drawn component is impact loaded at low temperatures, brittle failure may become more likely than plastic deformation (shown schematically in Fig. 1). The limit stress for IGF $\sigma_F$ is explored in more detail in [3].

<figure><img src="image_885.png" /><figcaption>Fig. 1 Yield stress versus brittle failure limit stress and the influence of Carbon, Boron and Phosphorus on the brittle failure stress [3].</figcaption></figure>

A common method to characterize the brittle behavior of deep drawn components is the cup expansion test (CET) [9]. Deep drawn circular cups are trimmed to remove the “ears” resulting from the anisotropy. A conical impactor with a mass of 10 kg is dropped from 1 m height. Cups are impact-tested in a temperature range of $T = -60$ °C to $T = +20$ °C, in steps of $dT = 10$ K or $dT = 20$ K, each with 8 repeats. The DBTT is defined as the minimum temperature at which 2 out of 8 specimens exhibit brittle failure.

Previous research has shown, that (IGF) brittle failure cracks most likely occur in the shortest path along highly stretched grains boundaries [10] (Fig. 2 b)). CET of a rephosphorized steel grade revealed that brittle cracks predominantly develop in rolling direction (RD). This can be explained by the steel’s anisotropy [11] on the coil, as well as due to the cup drawing process [3]. In a circular cup, the strain in the vertical side walls varies strongly, the most elongated grains are along the RD or at 90° to RD (located underneath the cup “ears”), the least elongated grains are located 45° to RD [12].
temperatures, the plastic deformation absorbs the majority of the impact energy. The transition from ductile to brittle failure is characterized by the ductile-to-brittle-transition-temperature (DBTT). It is used as a key parameter for performance evaluation at low temperatures.

Brittle failure can occur as intergranular (IGF) or transgranular failure (TGF) modes (Fig. 2 a)) [2]. In IGF, a crack propagates along the grain boundaries. For TGF, the crack passes through the grain. CWE is defined as material failure which always entails IGF [3]. IGF occurs if the strength between the grains is lower than the strength within the grains [3]. Specific alloy elements increase or decrease the strength between the grains due to their interaction with the host electron gas [4]. Carbon [5] and boron [6] increase grain boundary strength, Phosphorus [5], Silicon [7], and Sulfur [8] have a detrimental effect.

Commonly, plastic deformation will take preference over brittle failure, because the yield stress is lower than the critical stress for IGF. However, the yield strength $\sigma_{y,0}$ will be strongly increased due to the work hardening (from a forming process, such as deep drawing), high strain rate in a crash scenario $\sigma(\dot{\varepsilon})$, and low temperature $\sigma(T)$ (Fig. 1). It is assumed that the critical IGF-stress is independent of these effects [3]. Summing up: When a deep drawn component is impact loaded at low temperatures, brittle failure may become more likely than plastic deformation (shown schematically in Fig. 1). The limit stress for IGF $\sigma_F$ is explored in more detail in [3].

<figure><img src="image_893.png" /><figcaption>Fig. 1 Yield stress versus brittle failure limit stress and the influence of Carbon, Boron and Phosphorus on the brittle failure stress [3].</figcaption></figure>

A common method to characterize the brittle behavior of deep drawn components is the cup expansion test (CET) [9]. Deep drawn circular cups are trimmed to remove the “ears” resulting from the anisotropy. A conical impactor with a mass of 10 kg is dropped from 1 m height. Cups are impact-tested in a temperature range of $T = -60$ °C to $T = +20$ °C, in steps of $dT = 10$ K or $dT = 20$ K, each with 8 repeats. The DBTT is defined as the minimum temperature at which 2 out of 8 specimens exhibit brittle failure.

Previous research has shown, that (IGF) brittle failure cracks most likely occur in the shortest path along highly stretched grains boundaries [10] (Fig. 2 b)). CET of a rephosphorized steel grade revealed that brittle cracks predominantly develop in rolling direction (RD). This can be explained by the steel’s anisotropy [11] on the coil, as well as due to the cup drawing process [3]. In a circular cup, the strain in the vertical side walls varies strongly, the most elongated grains are along the RD or at 90° to RD (located underneath the cup “ears”), the least elongated grains are located 45° to RD [12].
<figure><img src="image_886.png" /><figcaption>Fig. 2 Basic principle of a) Inter- and transgranular failure and b) The influence of the microstructure on the brittle fracture mode.</figcaption></figure>

As in each industrial deep drawn component, different levels and types of pre-strains can be found, the CET has only limited application, as it yields only a single DBTT value which may not apply to the actual part. A pre-strain dependent characterization of the steel may help to establish useful limits for the forming process (and geometry) of impact loaded deep drawn components.

## Methodology for acquiring strain-dependent DBTT

The new proposed testing procedure consists of five steps (Fig. 3). Firstly, similar to the CET, the material is deep drawn into a circular cup and due to anisotropy, “ears” will be present. Ring segments are cut from the cup (Fig.3 – 2. Step). The ring specimens have a width of 5 mm. In these rings, a notch is introduced with a radius of 0.25 ± 0.025 mm and a depth of 1 mm. Rings are produced with notches at two distinct positions: Position A (in RD, underneath a “ear top”) is at high pre-strain, Position B at low pre-strain (45° to RD, at an “ear valley”). The impact energy is then measured by Charpy impact tests (Fig. 3 – 3. Step), where, as in a normal Charpy specimen, the impact hammer hits the specimen at the location of the notch. A series of energy measurements at various temperatures yields a s-shaped curve which allows the determination of the DBTT. This DBTT is expected to be different for samples with notch at position A to samples with notch at position B. (Fig. 3 – 4. Step). In this paper, SEM analysis is used to confirm whether a ductile or brittle failure occurred in relation to the absorbed energy (Fig. 3 – 5. Step).

<figure><img src="image_887.png" /><figcaption>Fig. 3 Schematic sketch of new testing procedure.</figcaption></figure>

The Erichsen 142-12-20-40Mp was used to draw cups from different materials (see Table 1). The punch has a diameter of $d_p$ = 33 mm and a radius of $R_p$ = 4.5 mm. The die has a diameter of
<figure><img src="image_894.png" /><figcaption>Fig. 2 Basic principle of a) Inter- and transgranular failure and b) The influence of the microstructure on the brittle fracture mode.</figcaption></figure>

As in each industrial deep drawn component, different levels and types of pre-strains can be found, the CET has only limited application, as it yields only a single DBTT value which may not apply to the actual part. A pre-strain dependent characterization of the steel may help to establish useful limits for the forming process (and geometry) of impact loaded deep drawn components.

## Methodology for acquiring strain-dependent DBTT

The new proposed testing procedure consists of five steps (Fig. 3). Firstly, similar to the CET, the material is deep drawn into a circular cup and due to anisotropy, “ears” will be present. Ring segments are cut from the cup (Fig.3 – 2. Step). The ring specimens have a width of 5 mm. In these rings, a notch is introduced with a radius of 0.25 ± 0.025 mm and a depth of 1 mm. Rings are produced with notches at two distinct positions: Position A (in RD, underneath a “ear top”) is at high pre-strain, Position B at low pre-strain (45° to RD, at an “ear valley”). The impact energy is then measured by Charpy impact tests (Fig. 3 – 3. Step), where, as in a normal Charpy specimen, the impact hammer hits the specimen at the location of the notch. A series of energy measurements at various temperatures yields a s-shaped curve which allows the determination of the DBTT. This DBTT is expected to be different for samples with notch at position A to samples with notch at position B. (Fig. 3 – 4. Step). In this paper, SEM analysis is used to confirm whether a ductile or brittle failure occurred in relation to the absorbed energy (Fig. 3 – 5. Step).

<figure><img src="image_895.png" /><figcaption>Fig. 3 Schematic sketch of new testing procedure.</figcaption></figure>

The Erichsen 142-12-20-40Mp was used to draw cups from different materials (see Table 1). The punch has a diameter of $d_p$ = 33 mm and a radius of $R_p$ = 4.5 mm. The die has a diameter of
$d_D$ = 38 mm and a radius of $R_D$ = 7 mm (Fig. 4). The blank diameter and the blank holder force $F_{BH}$ were chosen equally for all materials at $d_0$ = 73 mm and $F_{BH}$ = 15 kN<fnref n="2" />.

<figure><img src="image_888.png" /><figcaption>Fig. 4 a) schematic sketch of the deep drawing process and b) Deep drawn cups of the analyzed materials.</figcaption></figure>

The forming pre-strains were calculated by FEM analysis performed with ABAQUS. For this, the flow curve was determined by a simplified approach [13]. Five solid elements with reduced integration points are used over the thickness. The deep drawing simulations were validated by strain measurements on the outside of the cup. Similar to the “circle grid method”, dots were etched on the blanks with a defined diameter $d_0$ and defined distance between adjacent points. With the change in diameter, the major and minor strains had been calculated (Eq. 1, Eq. 2). $d_1$ and $d_2$ are the changed diameter orthogonal to each other.

$$ \varphi_1 = \ln \left( \frac{d_1}{d_0} \right). \qquad (1) $$

$$ \varphi_2 = \ln \left( \frac{d_2}{d_0} \right). \qquad (2) $$

The absorbed energy of the specimen in the Charpy test is the difference in height of the pendulum hammer at the beginning and the end of the test (Eq. 3).

$$ E = m \cdot g \cdot (h_{\text{initial}} - h_{\text{end}}) \qquad (3) $$

From a Charpy measurement series, the DBTT was determined by plotting the absorbed energies over the different temperatures and fitting a tangent hyperbolic function through the test data, which takes the shape of an s-curve. The upper shelf energy (USE) is the energy absorption if the material fails in a ductile mode. The lower shelf energy (LSE) is the energy absorption for pure brittle failure. The turning point of the function is determined as the DBTT.

The testing temperature ranged from +20 °C to -150 °C and was discretized by $dT$ = 10 °C and Charpy tests at each temperature were repeated 5 times. The testing temperature had been controlled using pre-determined heat-up curves: To obtain the heat-up curves, a ring specimen was cooled down to -196 °C using liquid nitrogen, then the temperature at the specimen notch was measured repeatedly, while the ring regained room temperature. During the Charpy test series,

<footnote n="2">² This will lead to different levels of forming pre-strain at position A and B for the 4 tested materials. Compare the different r-values from Tab. 1</footnote>
$d_D$ = 38 mm and a radius of $R_D$ = 7 mm (Fig. 4). The blank diameter and the blank holder force $F_{BH}$ were chosen equally for all materials at $d_0$ = 73 mm and $F_{BH}$ = 15 kN<fnref n="2" />.

<figure><img src="image_896.png" /><figcaption>Fig. 4 a) schematic sketch of the deep drawing process and b) Deep drawn cups of the analyzed materials.</figcaption></figure>

The forming pre-strains were calculated by FEM analysis performed with ABAQUS. For this, the flow curve was determined by a simplified approach [13]. Five solid elements with reduced integration points are used over the thickness. The deep drawing simulations were validated by strain measurements on the outside of the cup. Similar to the “circle grid method”, dots were etched on the blanks with a defined diameter $d_0$ and defined distance between adjacent points. With the change in diameter, the major and minor strains had been calculated (Eq. 1, Eq. 2). $d_1$ and $d_2$ are the changed diameter orthogonal to each other.

$$ \varphi_1 = \ln \left( \frac{d_1}{d_0} \right). \qquad (1) $$

$$ \varphi_2 = \ln \left( \frac{d_2}{d_0} \right). \qquad (2) $$

The absorbed energy of the specimen in the Charpy test is the difference in height of the pendulum hammer at the beginning and the end of the test (Eq. 3).

$$ E = m \cdot g \cdot (h_{\text{initial}} - h_{\text{end}}) \qquad (3) $$

From a Charpy measurement series, the DBTT was determined by plotting the absorbed energies over the different temperatures and fitting a tangent hyperbolic function through the test data, which takes the shape of an s-curve. The upper shelf energy (USE) is the energy absorption if the material fails in a ductile mode. The lower shelf energy (LSE) is the energy absorption for pure brittle failure. The turning point of the function is determined as the DBTT.

The testing temperature ranged from +20 °C to -150 °C and was discretized by $dT$ = 10 °C and Charpy tests at each temperature were repeated 5 times. The testing temperature had been controlled using pre-determined heat-up curves: To obtain the heat-up curves, a ring specimen was cooled down to -196 °C using liquid nitrogen, then the temperature at the specimen notch was measured repeatedly, while the ring regained room temperature. During the Charpy test series,

<footnote n="2">² This will lead to different levels of forming pre-strain at position A and B for the 4 tested materials. Compare the different r-values from Tab. 1</footnote>
temperature at the notch was controlled by triggering the pendulum at the right time using the heat-up curve.

## Reference CET test series

As reference the CET was performed to compare the determined DBTT from the new method. Here, the cup specimens were cooled down with a low temperature refrigerator (Fig. 5). The cup temperature was controlled with an external thermal element which was placed in drill holes in RD and TD. Specimens were left in the refrigerator 24 hour to assure the desired temperature. The impactor geometry and energy was chosen as in the standard [9]. The tests were performed until 2 out of 8 specimen failed brittle.

<figure><img src="image_889.png" /><figcaption>Fig. 5 Experimental set-up of the CET</figcaption></figure>

## Tested materials

Four similar mild steels, with different chemical compositions, were used in these experiments. The influence of carbon was analyzed by comparing interstitial-free (IF) DX57D-IF with non-interstitial-free steels DX53D-NonIF and DC04D-NonIF. Due to the vacuum degassing process in steel making, IF-steels have a carbon content of nearly zero, to produce a steel with the best formability. Non-IF steels, where no vacuum degassing is applied, retain a small amount of carbon. DX53D-IF+Boron was included in the test program to assess the influence of boron on the DBTT in a zero-carbon IF-steel. The sheet materials had an initial thickness of $s_0 = 1.5$ mm. The mechanical properties are listed below (Tab. 1).

<table><caption>Tab. 1 Mechanical properties of the tested materials.</caption><thead><tr><th>Material</th><th>R<sub>p0.2</sub><br>(MPa)</th><th>R<sub>m</sub><br>(MPa)</th><th>A<sub>g</sub><br>(%)</th><th>A<sub>80</sub><br>(%)</th><th>r<sub>20</sub><br>-</th><th>n<sub>10-Ag</sub><br>-</th></tr></thead><tbody><tr><td>DX53D-NonIF</td><td>209</td><td>311</td><td>22.8</td><td>42.8</td><td>2.1</td><td>0.21</td></tr><tr><td>DC04D-NonIF</td><td>169</td><td>296</td><td>23.6</td><td>46.1</td><td>2</td><td>0.21</td></tr><tr><td>DX53D-IF+Boron</td><td>172</td><td>304</td><td>24.4</td><td>44.9</td><td>1.6</td><td>0.22</td></tr><tr><td>DX57D-IF</td><td>133</td><td>270</td><td>25.6</td><td>53</td><td>2.4</td><td>0.24</td></tr></tbody></table>

## Results

The measurements of the major and minor strains confirmed the numerical results in the region of interest with sufficient accuracy. Hence, the numerical results for pre-strains at position A and B can be used (Fig. 6). Forming cracks appeared for the material DX53D-IF+Boron with the etched blanks, which is the reason why it is not shown. One explanation could be the increased friction.
temperature at the notch was controlled by triggering the pendulum at the right time using the heat-up curve.

## Reference CET test series

As reference the CET was performed to compare the determined DBTT from the new method. Here, the cup specimens were cooled down with a low temperature refrigerator (Fig. 5). The cup temperature was controlled with an external thermal element which was placed in drill holes in RD and TD. Specimens were left in the refrigerator 24 hour to assure the desired temperature. The impactor geometry and energy was chosen as in the standard [9]. The tests were performed until 2 out of 8 specimen failed brittle.

<figure><img src="image_897.png" /><figcaption>Fig. 5 Experimental set-up of the CET</figcaption></figure>

## Tested materials

Four similar mild steels, with different chemical compositions, were used in these experiments. The influence of carbon was analyzed by comparing interstitial-free (IF) DX57D-IF with non-interstitial-free steels DX53D-NonIF and DC04D-NonIF. Due to the vacuum degassing process in steel making, IF-steels have a carbon content of nearly zero, to produce a steel with the best formability. Non-IF steels, where no vacuum degassing is applied, retain a small amount of carbon. DX53D-IF+Boron was included in the test program to assess the influence of boron on the DBTT in a zero-carbon IF-steel. The sheet materials had an initial thickness of $s_0$ = 1.5 mm. The mechanical properties are listed below (Tab. 1).

<table><caption>Tab. 1 Mechanical properties of the tested materials.</caption><thead><tr><th>Material</th><th>R<sub>p0.2</sub><br>(MPa)</th><th>R<sub>m</sub><br>(MPa)</th><th>A<sub>g</sub><br>(%)</th><th>A<sub>80</sub><br>(%)</th><th>r<sub>20</sub><br>-</th><th>n<sub>10-Ag</sub><br>-</th></tr></thead><tbody><tr><td>DX53D-NonIF</td><td>209</td><td>311</td><td>22.8</td><td>42.8</td><td>2.1</td><td>0.21</td></tr><tr><td>DC04D-NonIF</td><td>169</td><td>296</td><td>23.6</td><td>46.1</td><td>2</td><td>0.21</td></tr><tr><td>DX53D-IF+Boron</td><td>172</td><td>304</td><td>24.4</td><td>44.9</td><td>1.6</td><td>0.22</td></tr><tr><td>DX57D-IF</td><td>133</td><td>270</td><td>25.6</td><td>53</td><td>2.4</td><td>0.24</td></tr></tbody></table>

## Results

The measurements of the major and minor strains confirmed the numerical results in the region of interest with sufficient accuracy. Hence, the numerical results for pre-strains at position A and B can be used (Fig. 6). Forming cracks appeared for the material DX53D-IF+Boron with the etched blanks, which is the reason why it is not shown. One explanation could be the increased friction.
<table><caption>Fig. 6 Comparison of the major strain distribution of the FEM and manually measured strains for the analyzed material.</caption><thead><tr><th></th><th>DX53D-NonIF</th><th>DC04D-NonIF</th><th>DX53D-IF+Boron</th><th>DX57D-IF</th></tr></thead><tbody><tr><th>FEM</th><td>ε<sub>1,A</sub> = 0.68<br>ε<sub>1,B</sub> = 0.36</td><td>ε<sub>1,A</sub> = 0.69<br>ε<sub>1,B</sub> = 0.3</td><td>ε<sub>1,A</sub> = 0.61<br>ε<sub>1,B</sub> = 0.43</td><td>ε<sub>1,A</sub> = 0.68<br>ε<sub>1,B</sub> = 0.38</td></tr><tr><th>Experiments</th><td>ε<sub>1,A</sub> = 0.59<br>ε<sub>1,B</sub> = 0.33</td><td>ε<sub>1,A</sub> = 0.62<br>ε<sub>1,B</sub> = 0.46</td><td></td><td>ε<sub>1,A</sub> = 0.62<br>ε<sub>1,B</sub> = 0.52</td></tr></tbody></table>

The impact energy vs. temperature curves from the Charpy tests are shown in Fig. 7 a). Notice, the energy level of LSE is always very low compared to USE: at low temperature, in the brittle failure regime, nearly no energy is absorbed by any of the samples. It can already be concluded that application of a mild steel in the brittle regime should be avoided at all costs. SEM analysis clearly identified IGF and TGF in the LSE (Fig. 8 c)), and necking in the USE (Fig. 8 a)) confirming brittle failure at low and ductile failure modes at higher temperatures.

<figure><img src="image_890.png" /><figcaption>Fig. 7 a) DBTT-curve of the different materials in the configuration of high and low forming pre-strain and b) The scattering of the DBTT over the equivalent strain distribution</figcaption></figure>
<table><caption>Fig. 6 Comparison of the major strain distribution of the FEM and manually measured strains for the analyzed material.</caption><thead><tr><th></th><th>DX53D-NonIF</th><th>DC04D-NonIF</th><th>DX53D-IF+Boron</th><th>DX57D-IF</th></tr></thead><tbody><tr><th>Major strain ε<sub>1</sub></th><td>ε<sub>1,A</sub> = 0.68<br>ε<sub>1,B</sub> = 0.36</td><td>ε<sub>1,A</sub> = 0.69<br>ε<sub>1,B</sub> = 0.3</td><td>ε<sub>1,A</sub> = 0.61<br>ε<sub>1,B</sub> = 0.43</td><td>ε<sub>1,A</sub> = 0.68<br>ε<sub>1,B</sub> = 0.38</td></tr><tr><th>Experiments</th><td>ε<sub>1,A</sub> = 0.59<br>ε<sub>1,B</sub> = 0.33</td><td>ε<sub>1,A</sub> = 0.62<br>ε<sub>1,B</sub> = 0.46</td><td></td><td>ε<sub>1,A</sub> = 0.62<br>ε<sub>1,B</sub> = 0.52</td></tr></tbody></table>

The impact energy vs. temperature curves from the Charpy tests are shown in Fig. 7 a). Notice, the energy level of LSE is always very low compared to USE: at low temperature, in the brittle failure regime, nearly no energy is absorbed by any of the samples. It can already be concluded that application of a mild steel in the brittle regime should be avoided at all costs. SEM analysis clearly identified IGF and TGF in the LSE (Fig. 8 c)), and necking in the USE (Fig. 8 a)) confirming brittle failure at low and ductile failure modes at higher temperatures.

<figure><img src="image_898.png" /><figcaption>Fig. 7 a) DBTT-curve of the different materials in the configuration of high and low forming pre-strain and b) The scattering of the DBTT over the equivalent strain distribution</figcaption></figure>
<figure><img src="image_891.png" /></figure>

<figure><img src="image_892.png" /></figure>

<figure><img src="image_893.png" /></figure>

<figure><img src="image_894.png" /></figure>

<figure><img src="image_895.png" /></figure>

<figure><img src="image_896.png" /><figcaption>Fig. 8 a) ductile fracture surface of the DX53D-IF+Boron at -70 °C, b) primary ductile failure of DC04D-NonIF at -60 °C, c) entire brittle failure of DX57D-IF at -60 °C</figcaption></figure>

The curves for DX57D-IF (shown in green in Fig. 7 a) yield two interesting observations:

1. Comparing the fitted hyperbolic function for the Position A samples (high forming pre-strain, shown as a dotted line), and for the Position B samples (low forming pre-strain, shown as a solid line): The DBTT is clearly higher for the Position A-samples.

2. For the Position-A (high pre-strain) samples, the USE is clearly lower.

The second observation is made for all 4 materials. While the LSE is not clearly affected by the forming pre-strain, Position A samples (dotted lines) all show a lower USE: Highly pre-strained material has a lower impact absorption. This is easily explained: the work hardening from the forming process has reduced the ductility of the steel, locally.

The transition zone between USE and LSE is subject to scatter. For this scatter, there are several explanations, such as slight variations of the extraction point of the forming pre-strain, the dynamics of the impact tests, and the temperature determination method. To quantify this, the quality of the fit by the tangent-hyperbolic function was determined by the correlation coefficient $R^2$ (Tab. 2). Fig. 7 b) visualized the scatter around the DBTT as boxplots, plotted against the (FEM-calculated) pre-strain value at the notch. Looking again at the outcome for DX57D-IF (in green), it can be concluded that the DBTT is significantly different at the two levels of pre-strain.

In sharp contrast to DX57D-IF, the Fig. 7 a) curves for DX53D-IF+Boron show hardly any difference between Position A and Position B, neither in USE magnitude nor in DBTT. In fact, the boxplots in Fig. 7 b) lead to the conclusion that the DBTT cannot be differentiated for high and low forming pre-strain at all. It must be noted that the difference in strain magnitude is comparatively smaller for this material, which behaved less anisotropic than the other steels (this is directly visible in the small “ears” in Fig. 4 b). Notice also, that the DBTT for DX53D-IF+Boron is the lowest, compared to the other materials.

Tab. 2 sums up the results for all materials. For the DX57D-IF the DBTT is clearly above the -30 °C temperature, where automotive safety equipment must operate. On top of that, this steel is the only grade in the test program that shows a significant difference in DBTT with respect to the Position A and Position B pre-strain levels. Both Non-IF steel grades (DX53D-nonIF and DC04-nonIF) show less pre-strain dependency, but they do show a significant drop in the USE-
<figure><img src="image_899.png" /></figure>

<figure><img src="image_900.png" /></figure>

<figure><img src="image_901.png" /></figure>

<figure><img src="image_902.png" /></figure>

<figure><img src="image_903.png" /><figcaption>Fig. 8 a) ductile fracture surface of the DX53D-IF+Boron at -70 °C, b) primary ductile failure of DC04D-NonIF at -60 °C, c) entire brittle failure of DX57D-IF at -60 °C</figcaption></figure>

The curves for DX57D-IF (shown in green in Fig. 7 a) yield two interesting observations:

1. Comparing the fitted hyperbolic function for the Position A samples (high forming pre-strain, shown as a dotted line), and for the Position B samples (low forming pre-strain, shown as a solid line): The DBTT is clearly higher for the Position A-samples.

2. For the Position-A (high pre-strain) samples, the USE is clearly lower.

The second observation is made for all 4 materials. While the LSE is not clearly affected by the forming pre-strain, Position A samples (dotted lines) all show a lower USE: Highly pre-strained material has a lower impact absorption. This is easily explained: the work hardening from the forming process has reduced the ductility of the steel, locally.

The transition zone between USE and LSE is subject to scatter. For this scatter, there are several explanations, such as slight variations of the extraction point of the forming pre-strain, the dynamics of the impact tests, and the temperature determination method. To quantify this, the quality of the fit by the tangent-hyperbolic function was determined by the correlation coefficient $R^2$ (Tab. 2). Fig. 7 b) visualized the scatter around the DBTT as boxplots, plotted against the (FEM-calculated) pre-strain value at the notch. Looking again at the outcome for DX57D-IF (in green), it can be concluded that the DBTT is significantly different at the two levels of pre-strain.

In sharp contrast to DX57D-IF, the Fig. 7 a) curves for DX53D-IF+Boron show hardly any difference between Position A and Position B, neither in USE magnitude nor in DBTT. In fact, the boxplots in Fig. 7 b) lead to the conclusion that the DBTT cannot be differentiated for high and low forming pre-strain at all. It must be noted that the difference in strain magnitude is comparatively smaller for this material, which behaved less anisotropic than the other steels (this is directly visible in the small “ears” in Fig. 4 b). Notice also, that the DBTT for DX53D-IF+Boron is the lowest, compared to the other materials.

Tab. 2 sums up the results for all materials. For the DX57D-IF the DBTT is clearly above the -30 °C temperature, where automotive safety equipment must operate. On top of that, this steel is the only grade in the test program that shows a significant difference in DBTT with respect to the Position A and Position B pre-strain levels. Both Non-IF steel grades (DX53D-nonIF and DC04-nonIF) show less pre-strain dependency, but they do show a significant drop in the USE-
level when pre-strained. Whereas the DC04-NonIF DBTT is low, the DX53D-nonIF is close to typical automotive application temperatures.

<table><caption>Tab. 2 DBTT and energy absorption of the tested materials with different forming pre-strains.</caption><thead><tr><th>Material</th><th>Notch</th><th>R<sup>2</sup></th><th>ε<sub>1</sub></th><th>ε<sub>2</sub></th><th>DBTT<br>(°C)</th><th>DBTT<br>(J)</th><th>LSE<br>(J)</th><th>USE<br>(J)</th><th>USE<br>(J/mm<sup>2</sup>)</th></tr></thead><tbody><tr><td rowspan="2">DX53D-<br>NonIF</td><td>A</td><td>0.863</td><td>0.68</td><td>-0.57</td><td>-31.3</td><td>2.9</td><td>0.3</td><td>6.1</td><td>1.0</td></tr><tr><td>B</td><td>0.879</td><td>0.36</td><td>-0.42</td><td>-34.7</td><td>5.8</td><td>0.3</td><td>11.9</td><td>1.8</td></tr><tr><td rowspan="2">DC04D+ZE-<br>NonIF</td><td>A</td><td>0.889</td><td>0.69</td><td>-0.55</td><td>-65.7</td><td>4.7</td><td>0.3</td><td>9.7</td><td>1.6</td></tr><tr><td>B</td><td>0.829</td><td>0.30</td><td>-0.40</td><td>-69.9</td><td>7.1</td><td>1.0</td><td>15.2</td><td>2.2</td></tr><tr><td rowspan="2">DX53D-<br>IF+Boron</td><td>A</td><td>0.919</td><td>0.61</td><td>-0.51</td><td>-97.4</td><td>6.0</td><td>1.2</td><td>13.2</td><td>2.2</td></tr><tr><td>B</td><td>0.843</td><td>0.43</td><td>-0.45</td><td>-91.2</td><td>6.7</td><td>0.7</td><td>14.1</td><td>2.2</td></tr><tr><td rowspan="2">DX57D-IF</td><td>A</td><td>0.775</td><td>0.68</td><td>-0.58</td><td>-17.3</td><td>6.0</td><td>0.3</td><td>12.2</td><td>2.0</td></tr><tr><td>B</td><td>0.714</td><td>0.38</td><td>-0.40</td><td>-46.8</td><td>7.9</td><td>0.2</td><td>16.0</td><td>2.4</td></tr></tbody></table>

The CET was performed with the material DC04-NonIF. From the Charpy test the DBTT for the DC04-NonIF was known to be -70 °C. Hence, the tests started at -70 °C. At -70 °C, where no specimen failed brittle, only ductile failure occurs (Fig. 9). According to the standard, the temperature was decreased to -90 °C were 1 specimen out of 8 failed in a brittle mode. Subsequently, the temperature was decreased to -100 °C were 2 out of 8 cups failed brittle. According to the standard the DBTT was defined at -100 °C by the CET. All brittle fracture occurred in RD.

<figure><img src="image_897.png" /><figcaption>Fig. 9 Results of the CET at -70 °C, -90 °C and -100 °C.</figcaption></figure>

## Discussion and Conclusions

As a main finding, the contrast between DX57D-IF and DX53D-IF+Boron must be further discussed. In the ductile regime, the DX57D-IF has the highest USE-level, which can be explained by the highest level of failure strain A<sub>80</sub> – it simply is the most ductile of the 4 tested materials. For an impact-loaded component this makes it the material of preference, at first sight. However, when this is a deep drawn component, the energy absorption will be locally highly reduced due to the work hardening. Now, when the impact occurs in cold condition, the DX57D-IF may transform to the brittle regime even when the temperature is not very low, and energy absorption may be
level when pre-strained. Whereas the DC04-NonIF DBTT is low, the DX53D-nonIF is close to typical automotive application temperatures.

<table><caption>Tab. 2 DBTT and energy absorption of the tested materials with different forming pre-strains.</caption><thead><tr><th>Material</th><th>Notch</th><th>R<sup>2</sup></th><th>ε<sub>1</sub></th><th>ε<sub>2</sub></th><th>DBTT<br>(°C)</th><th>DBTT<br>(J)</th><th>LSE<br>(J)</th><th>USE<br>(J)</th><th>USE<br>(J/mm<sup>2</sup>)</th></tr></thead><tbody><tr><td rowspan="2">DX53D-<br>NonIF</td><td>A</td><td>0.863</td><td>0.68</td><td>-0.57</td><td>-31.3</td><td>2.9</td><td>0.3</td><td>6.1</td><td>1.0</td></tr><tr><td>B</td><td>0.879</td><td>0.36</td><td>-0.42</td><td>-34.7</td><td>5.8</td><td>0.3</td><td>11.9</td><td>1.8</td></tr><tr><td rowspan="2">DC04D+ZE-<br>NonIF</td><td>A</td><td>0.889</td><td>0.69</td><td>-0.55</td><td>-65.7</td><td>4.7</td><td>0.3</td><td>9.7</td><td>1.6</td></tr><tr><td>B</td><td>0.829</td><td>0.30</td><td>-0.40</td><td>-69.9</td><td>7.1</td><td>1.0</td><td>15.2</td><td>2.2</td></tr><tr><td rowspan="2">DX53D-<br>IF+Boron</td><td>A</td><td>0.919</td><td>0.61</td><td>-0.51</td><td>-97.4</td><td>6.0</td><td>1.2</td><td>13.2</td><td>2.2</td></tr><tr><td>B</td><td>0.843</td><td>0.43</td><td>-0.45</td><td>-91.2</td><td>6.7</td><td>0.7</td><td>14.1</td><td>2.2</td></tr><tr><td rowspan="2">DX57D-IF</td><td>A</td><td>0.775</td><td>0.68</td><td>-0.58</td><td>-17.3</td><td>6.0</td><td>0.3</td><td>12.2</td><td>2.0</td></tr><tr><td>B</td><td>0.714</td><td>0.38</td><td>-0.40</td><td>-46.8</td><td>7.9</td><td>0.2</td><td>16.0</td><td>2.4</td></tr></tbody></table>

The CET was performed with the material DC04-NonIF. From the Charpy test the DBTT for the DC04-NonIF was known to be -70 °C. Hence, the tests started at -70 °C. At -70 °C, where no specimen failed brittle, only ductile failure occurs (Fig. 9). According to the standard, the temperature was decreased to -90 °C were 1 specimen out of 8 failed in a brittle mode. Subsequently, the temperature was decreased to -100 °C were 2 out of 8 cups failed brittle. According to the standard the DBTT was defined at -100 °C by the CET. All brittle fracture occurred in RD.

<figure><img src="image_904.png" /><figcaption>Fig. 9 Results of the CET at -70 °C, -90 °C and -100 °C.</figcaption></figure>

## Discussion and Conclusions

As a main finding, the contrast between DX57D-IF and DX53D-IF+Boron must be further discussed. In the ductile regime, the DX57D-IF has the highest USE-level, which can be explained by the highest level of failure strain A<sub>80</sub> – it simply is the most ductile of the 4 tested materials. For an impact-loaded component this makes it the material of preference, at first sight. However, when this is a deep drawn component, the energy absorption will be locally highly reduced due to the work hardening. Now, when the impact occurs in cold condition, the DX57D-IF may transform to the brittle regime even when the temperature is not very low, and energy absorption may be
(locally) reduced to a minimum. The DX53D-IF+Boron steel shows both a low DBTT and a small influence of the pre-strain. Both factors simplify the engineering of an impact loaded component. The reason behind this difference must be explored in more detail than possible in this paper, but the authors explain the difference from the chemical composition of the material. Boron [6] and carbon [5] increase the strength between the grain boundaries and therefore decreases the average DBTT. Even when the DX53D-IF+Boron had a near-zero carbon content, the addition of boron seems to have increased the grain boundary strength, making a brittle failure less likely to occur.

The tested batch of material showed a lower anisotropy, hence, the grain-shape change was comparatively less: Loosely said, the deep drawing deformation was distributed better. Irrespective of this, the authors think that for steels with a higher intergranular strength, the influence of the microstructure (e.g. the straightness of a potential crack path) is less important.

The high grain boundary strength of DC04D-NonIF comes from the material's carbon content, and this resulted in a DBTT value much lower than required, but not as low as the boron-alloyed steel. A similar finding is reported in literature [6]. While the USE-level is significantly reduced for the pre-strained test specimens, the DBTT temperature is fairly insensitive to the forming pre-strain. In contrast, the DX53D-NonIF, which also contains carbon to strengthen grain boundaries, shows high DBTT values, too close to the required -30 °C operating temperature. The 2.8 times higher carbon and 2 times lower phosphorus content of the DC04D-NonIF could explain the lower DBTT compared to the DX53D-NonIF. For reasons of disclosure the chemical composition of the materials cannot be revealed.

Comparing conventional cup expansion drop test (CET) to the new ring-Charpy tests, the CET finds DBTT at -100 °C for DC04-NonIF. This is lower than the -65.7 °C found for Position A in the Charpy ring test, making the new test more conservative. At this time, this difference is hard to explain. The authors want to point out the stress state at impact is very different for CET and the ring-Charpy-tests – this needs more exploration in tests and FEM-simulation context.

## Summary

The new testing method of deep drawing and cutting out Charpy specimens at position A (low $\bar{\varepsilon}$) and B (high $\bar{\varepsilon}$) allowed the analysis of the influence of the forming pre-strain on the DBTT. This is an improvement over the cup-expansion test (CET).

The forming pre-strain influenced the DBTT for interstitial-free materials with no additional boron in this investigation. The forming pre-strain influences the absorbed energy for the ductile failure of the material. The increased forming pre-strain decreased the remaining ductility of the material and therefore decreased the energy absorption. Nonetheless, the chemical composition had a higher influence on the DBTT than the forming pre-strain. Carbon and boron decrease the DBTT whereby boron decreased the DBTT significantly more than carbon. The CET determined lower DBTTs than the Charpy tests. However, for IF steels a remaining question is how the different DBTT from the forming pre-strain influences the DBTT in comparison to the Charpy test and should be analyzed in the future.

## References

[1] W. Weißbach, Werkstoffkunde: Strukturen, Eigenschaften, Prüfung, Springer-Verlag, 2010.

[2] A. G. Atkins and Y.-W. Mai, Elastic and plastic fracture: metals, polymers, ceramics, composites, biological materials, 1985.

[3] K. P. Boyle, Cold work embrittlement of interstitial-free sheet steel, 2001.

[4] A. H. Cottrell, Strength of grain boundaries in impure metals, Materials science and technology 6 (4) (1990) pp. 325-329. https://doi.org/10.1179/mst.1990.6.4.325
(locally) reduced to a minimum. The DX53D-IF+Boron steel shows both a low DBTT and a small influence of the pre-strain. Both factors simplify the engineering of an impact loaded component. The reason behind this difference must be explored in more detail than possible in this paper, but the authors explain the difference from the chemical composition of the material. Boron [6] and carbon [5] increase the strength between the grain boundaries and therefore decreases the average DBTT. Even when the DX53D-IF+Boron had a near-zero carbon content, the addition of boron seems to have increased the grain boundary strength, making a brittle failure less likely to occur.

The tested batch of material showed a lower anisotropy, hence, the grain-shape change was comparatively less: Loosely said, the deep drawing deformation was distributed better. Irrespective of this, the authors think that for steels with a higher intergranular strength, the influence of the microstructure (e.g. the straightness of a potential crack path) is less important.

The high grain boundary strength of DC04D-NonIF comes from the material's carbon content, and this resulted in a DBTT value much lower than required, but not as low as the boron-alloyed steel. A similar finding is reported in literature [6]. While the USE-level is significantly reduced for the pre-strained test specimens, the DBTT temperature is fairly insensitive to the forming pre-strain. In contrast, the DX53D-NonIF, which also contains carbon to strengthen grain boundaries, shows high DBTT values, too close to the required -30 °C operating temperature. The 2.8 times higher carbon and 2 times lower phosphorus content of the DC04D-NonIF could explain the lower DBTT compared to the DX53D-NonIF. For reasons of disclosure the chemical composition of the materials cannot be revealed.

Comparing conventional cup expansion drop test (CET) to the new ring-Charpy tests, the CET finds DBTT at -100 °C for DC04-NonIF. This is lower than the -65.7 °C found for Position A in the Charpy ring test, making the new test more conservative. At this time, this difference is hard to explain. The authors want to point out the stress state at impact is very different for CET and the ring-Charpy-tests – this needs more exploration in tests and FEM-simulation context.

## Summary

The new testing method of deep drawing and cutting out Charpy specimens at position A (low $\bar{\varepsilon}$) and B (high $\bar{\varepsilon}$) allowed the analysis of the influence of the forming pre-strain on the DBTT. This is an improvement over the cup-expansion test (CET).

The forming pre-strain influenced the DBTT for interstitial-free materials with no additional boron in this investigation. The forming pre-strain influences the absorbed energy for the ductile failure of the material. The increased forming pre-strain decreased the remaining ductility of the material and therefore decreased the energy absorption. Nonetheless, the chemical composition had a higher influence on the DBTT than the forming pre-strain. Carbon and boron decrease the DBTT whereby boron decreased the DBTT significantly more than carbon. The CET determined lower DBTTs than the Charpy tests. However, for IF steels a remaining question is how the different DBTT from the forming pre-strain influences the DBTT in comparison to the Charpy test and should be analyzed in the future.

## References

[1] W. Weißbach, Werkstoffkunde: Strukturen, Eigenschaften, Prüfung, Springer-Verlag, 2010.

[2] A. G. Atkins and Y.-W. Mai, Elastic and plastic fracture: metals, polymers, ceramics, composites, biological materials, 1985.

[3] K. P. Boyle, Cold work embrittlement of interstitial-free sheet steel, 2001.

[4] A. H. Cottrell, Strength of grain boundaries in impure metals, Materials science and technology 6 (4) (1990) pp. 325-329. https://doi.org/10.1179/mst.1990.6.4.325
[5] S. Suzuki, M. Obata, K. Abiko, and H. KIMURA, Role of carbon in preventing the intergranular fracture in iron-phosphorus alloys, International Journal of Scientific and Research Publications 25 (1) 25, no. 1 pp. 62–68 (1985). https://doi.org/10.2355/isijinternational1966.25.62

[6] E. Yasuhara, K. Sakata, T. Kato, and O. Hashimoto, Effect of boron on the resistance to secondary working embrittlement in extra-low-C cold-rolled steel sheet, ISIJ international 24 (1) (1994) pp. 99–107. https://doi.org/10.2355/isijinternational.34.99

[7] B. J. Schulz and C. J. McMahon Jr, Temper embrittlement of alloy steels, ASTM STP 499 (1972) p. 104. https://doi.org/10.1520/STP38788S

[8] S. Suzuki, S. Tanii, K. Abiko, and H. Kimura, Site competition between sulfur and carbon at grain boundaries and their effects on the grain boundary cohesion in iron, Metallurgical Transactions A 18 (6) (1991) pp. 1109–1115. https://doi.org/10.1007/BF03325721

[9] Evaluation for Susceptibility to Secondary Cold Work Embrittlement and Low Temperature Impact Fractures of Ultra Low Carbon and Phosphorus Bearing Sheet Steels: General Motors Worldwide Enginnering Stantards, 16086, GMW 16086 (2014).

[10] S. Q. Cao, J. X. Zhang, J. S. Wu, and J. G. Chen, Effects of GBCD on cold work embrittlement of high strength interstitial free steels, Materials & Design 27 (1) (2006) pp. 53–57. https://doi.org/10.1016/j.matdes.2004.09.014

[11] J. Hu, Z. Marciniak, and J. Duncan, Mechanics of sheet metal forming, Elsevier, 2002.

[12] D. M. Neto, M. C. Oliveira, J. L. Alves, and L. F. Menezes, Influence of the plastic anisotropy modelling in the reverse deep drawing process simulation, Materials & Design 60 (2014) pp. 368–379. https://doi.org/10.1016/j.matdes.2014.04.008

[13] R. Lingbeek, B. Aldemir, A. Soni, S. Jagalur, and L. Yao, Material Spread and Local Failure in Breakage Modeling for Steel Safety Components, Procedia Manufacturing 47 (2020) pp. 1449–1455. https://doi.org/10.1016/j.promfg.2020.04.316
[5] S. Suzuki, M. Obata, K. Abiko, and H. KIMURA, Role of carbon in preventing the intergranular fracture in iron-phosphorus alloys, International Journal of Scientific and Research Publications 25 (1) 25, no. 1 pp. 62–68 (1985). https://doi.org/10.2355/isijinternational1966.25.62

[6] E. Yasuhara, K. Sakata, T. Kato, and O. Hashimoto, Effect of boron on the resistance to secondary working embrittlement in extra-low-C cold-rolled steel sheet, ISIJ international 24 (1) (1994) pp. 99–107. https://doi.org/10.2355/isijinternational.34.99

[7] B. J. Schulz and C. J. McMahon Jr, Temper embrittlement of alloy steels, ASTM STP 499 (1972) p. 104. https://doi.org/10.1520/STP38788S

[8] S. Suzuki, S. Tanii, K. Abiko, and H. Kimura, Site competition between sulfur and carbon at grain boundaries and their effects on the grain boundary cohesion in iron, Metallurgical Transactions A 18 (6) (1991) pp. 1109–1115. https://doi.org/10.1007/BF03325721

[9] Evaluation for Susceptibility to Secondary Cold Work Embrittlement and Low Temperature Impact Fractures of Ultra Low Carbon and Phosphorus Bearing Sheet Steels: General Motors Worldwide Enginnering Stantards, 16086, GMW 16086 (2014).

[10] S. Q. Cao, J. X. Zhang, J. S. Wu, and J. G. Chen, Effects of GBCD on cold work embrittlement of high strength interstitial free steels, Materials & Design 27 (1) (2006) pp. 53–57. https://doi.org/10.1016/j.matdes.2004.09.014

[11] J. Hu, Z. Marciniak, and J. Duncan, Mechanics of sheet metal forming, Elsevier, 2002.

[12] D. M. Neto, M. C. Oliveira, J. L. Alves, and L. F. Menezes, Influence of the plastic anisotropy modelling in the reverse deep drawing process simulation, Materials & Design 60 (2014) pp. 368–379. https://doi.org/10.1016/j.matdes.2014.04.008

[13] R. Lingbeek, B. Aldemir, A. Soni, S. Jagalur, and L. Yao, Material Spread and Local Failure in Breakage Modeling for Steel Safety Components, Procedia Manufacturing 47 (2020) pp. 1449–1455. https://doi.org/10.1016/j.promfg.2020.04.316
# Influence of the rolling direction on ductile damage during deep drawing of dual-phase steel

MÜLLER Martina¹, a*, WOLLENWEBER Maximilian A.³, b,
MEDGHALCHI Setareh³, c, HERRIG Tim¹, d and BERGS Thomas¹, 2, e

¹ Manufacturing Technology Institute MTI of RWTH Aachen University, Campus-Boulevard 30,
52074 Aachen, Germany

² Fraunhofer Institute for Production Technology IPT, Steinbachstraße 17, 52074 Aachen,
Germany

³ Institute for Physical Metallurgy and Materials Physics IMM, Kopernikusstrasse 14, 52074
Aachen, Germany

**Keywords:** Damage, Deep Drawing, Dual Phase Steel

**Abstract.** The spatial distribution as well as the amount of the accumulated damage within a component significantly affects its performance in terms of fatigue and crash behavior. One effective approach to control the amount of the accumulated damage is through precise adjustments of process parameters and setup. This paper explored the impact of sheet metal alignment with respect to the forming tools relative to the rolling direction of cold rolling on damage accumulation in sheet metal during deep drawing of a rectangular cup and a U-profile. Firstly, numerical analysis was performed to evaluate the load paths in form of stress and strain states that occur during deep drawing. Subsequently, experimental investigations were conducted to examine the effect of different alignments of the sheet metal with respect to the rolling direction on ductile damage. Therefore, the damage accumulation in the form of void area fraction in workpiece areas that are critical to performance was quantified and compared. Overall, the results have shown that the damage accumulation in form of void area fractions is dependent on the alignment of the sheet metal with respect to the rolling direction. The void area fraction of the rectangular cup could be reduced by 22.4% when considering a sheet metal that is aligned with the rolling direction parallel to the longest, straight sides of the geometry instead of a perpendicular orientation. For the U-profile it was demonstrated that a 45°-orientation of the rolling direction to the bending radius leads to the lowest damage accumulation reducing it by 50.38% compared to a 0°-orientation.

**Introduction**

The automotive industry is currently undergoing a transformative shift, strategically incorporating lightweight measures into the product design process. This shift is primarily fueled by the industry's commitment as main contributor to CO₂ emissions [1] to meet the ambitious climate targets set by the European Union [2]. Utilizing lighter vehicles has been acknowledged as an effective strategy to reduce CO₂ emissions in transportation [3]. Consequently, the adoption of high-strength materials has become a critical element of the design process, given the additional advantages of better vehicle performance and cost savings. Dual-phase steel is a frequently used material in the domain of lightweight design and the automotive industry. It constitutes a type of high-strength, low-alloy steel that comprises a soft ferrite matrix with a harder martensitic secondary phase in the form of islands. The plastic behavior of DP steels displays a low ratio between yield strength and tensile strength, pronounced work hardening, and exceptional overall

<figure><img src="image_898.png" /></figure>

<footnote>a.m.mueller@mti.rwth-aachen.de, b.wollenweber@imm.rwth-aachen.de,</footnote>

<footnote>c.medghalchi@imm.rwth-aachen.de, d.t.herrig@mti.rwth-aachen.de, e.t.bergs@mti.rwth-aachen.de</footnote>
# Influence of the rolling direction on ductile damage during deep drawing of dual-phase steel

MÜLLER Martina¹, a*, WOLLENWEBER Maximilian A.³, b,
MEDGHALCHI Setareh³, c, HERRIG Tim¹, d and BERGS Thomas¹, 2, e

¹ Manufacturing Technology Institute MTI of RWTH Aachen University, Campus-Boulevard 30,
52074 Aachen, Germany

² Fraunhofer Institute for Production Technology IPT, Steinbachstraße 17, 52074 Aachen,
Germany

³ Institute for Physical Metallurgy and Materials Physics IMM, Kopernikusstrasse 14, 52074
Aachen, Germany

**Keywords:** Damage, Deep Drawing, Dual Phase Steel

**Abstract.** The spatial distribution as well as the amount of the accumulated damage within a component significantly affects its performance in terms of fatigue and crash behavior. One effective approach to control the amount of the accumulated damage is through precise adjustments of process parameters and setup. This paper explored the impact of sheet metal alignment with respect to the forming tools relative to the rolling direction of cold rolling on damage accumulation in sheet metal during deep drawing of a rectangular cup and a U-profile. Firstly, numerical analysis was performed to evaluate the load paths in form of stress and strain states that occur during deep drawing. Subsequently, experimental investigations were conducted to examine the effect of different alignments of the sheet metal with respect to the rolling direction on ductile damage. Therefore, the damage accumulation in the form of void area fraction in workpiece areas that are critical to performance was quantified and compared. Overall, the results have shown that the damage accumulation in form of void area fractions is dependent on the alignment of the sheet metal with respect to the rolling direction. The void area fraction of the rectangular cup could be reduced by 22.4% when considering a sheet metal that is aligned with the rolling direction parallel to the longest, straight sides of the geometry instead of a perpendicular orientation. For the U-profile it was demonstrated that a 45°-orientation of the rolling direction to the bending radius leads to the lowest damage accumulation reducing it by 50.38% compared to a 0°-orientation.

**Introduction**

The automotive industry is currently undergoing a transformative shift, strategically incorporating lightweight measures into the product design process. This shift is primarily fueled by the industry's commitment as main contributor to CO₂ emissions [1] to meet the ambitious climate targets set by the European Union [2]. Utilizing lighter vehicles has been acknowledged as an effective strategy to reduce CO₂ emissions in transportation [3]. Consequently, the adoption of high-strength materials has become a critical element of the design process, given the additional advantages of better vehicle performance and cost savings. Dual-phase steel is a frequently used material in the domain of lightweight design and the automotive industry. It constitutes a type of high-strength, low-alloy steel that comprises a soft ferrite matrix with a harder martensitic secondary phase in the form of islands. The plastic behavior of DP steels displays a low ratio between yield strength and tensile strength, pronounced work hardening, and exceptional overall

<figure><img src="image_905.png" /></figure>

<footnote>a.m.mueller@mti.rwth-aachen.de, b.wollenweber@imm.rwth-aachen.de,</footnote>

<footnote>c.medghalchi@imm.rwth-aachen.de, d.t.herrig@mti.rwth-aachen.de, e.t.bergs@mti.rwth-aachen.de</footnote>
formability reflecting the distinct mechanical properties of the two phases present. These characteristics allow for a considerable array of options for microstructural design due to the distribution of stress and strain between those phases [4]. Today, DP steels are primarily used for the production of body panels and frames by deep drawing.

Prior to the forming process, the sheets employed in deep drawing typically undergo a sequence of procedures, such as steelmaking, continuous casting, hot and cold rolling, and heat treatment. These procedures as well as especially the deep drawing process itself all have an influence on the evolution of damage in the material based on the premise that ductile damage means the formation, growth and coalescence of voids in the microstructure. This damage causes degradation of the performance of the corresponding component [5]. Based on this definition of ductile damage, dual phase steels differ from other steels in its evolution. While inclusions play a primary role in void initiation in various steels, different damage mechanisms take precedence in DP steels [5]. These mechanisms encompass martensite fracture, phase boundary detachment, and grain boundary decohesion, and their prevalence is notably influenced by the microstructure [6]. However, the dominant damage mechanisms in DP steels are primarily influenced by the microstructure, while the evolution of damage and its extent in the microstructure are contingent on the stress-strain state experienced by the material throughout the processing route. In general, the higher the hydrostatic tensile stress reflected in a positive triaxiality, the larger the voids and the more damage is caused by plastic deformation in the material [6].

Considering deep drawing, various investigations have explored the influence of different key factors that impact the process in the past. These factors include friction, blank holder force, blank shape, and punch velocity [7]. The primary focus of these studies has been on achieving defect-free components at the macroscopic level, such as avoiding wrinkling, illustrated by Chen et al. [9], minimizing sheet thinning, examined by Preedawiphat et al. [10], and controlling spring back, explored in the works of Durmaz et al. [11] and Nanu et al. [12]. Only a few, however, focused on ductile damage on a microscale as defined above. And these studies focused in particular on the influence of process parameters on ductile damage, as in [13-14]. Therefore, there are knowledge gaps in the context of this understanding of ductile damage, including its potential in manufacturing processes to enhance a components performance. This study investigates the influence of the sheet metal alignment with respect to the forming tools, concerning the rolling direction of cold rolling, on the accumulation of damage in sheet metal during deep drawing of a rectangular cup and a U-profile. The analysis involves performing numerical simulations to evaluate the load paths in terms of stress and strain states that occur during deep drawing as well as the numerical predicted damage state after forming considering an isotropic damage model. Subsequently, an experimental investigation is carried out to analyze the impact on the damage accumulation of various alignments of the sheet metal with respect to the forming tools in relation to the rolling direction. Consequently, the formation of damage in terms of void area fractions in performance critical areas of the workpiece are quantified and compared. The results are used to draw conclusions regarding the impact of the rolling direction on the damage state. Additionally, they provide insides into the optimal orientation of sheet metal taking into account the rolling direction and the alignment with respect to the forming tools to minimize damage accumulation. Furthermore, an evaluation is made of the extent to which the rolling direction should be considered in future damage models for numerical analyses.

## Materials and Methods

*Experimental deep drawing setup.* For the study, a rectangular cup and a U-profile were considered. Workpieces with a similar geometry made of DP800 with various alignments of the sheet metal ($\phi = \{0°, 45°, 90°\}$) in relation to the rolling direction were compared shortly before failure (Fig. 1a). Since the time of failure and the maximum possible punch strokes $s_{St}$ without failure varied for each variant, the analysis time relied on the variant that failed first and was
formability reflecting the distinct mechanical properties of the two phases present. These characteristics allow for a considerable array of options for microstructural design due to the distribution of stress and strain between those phases [4]. Today, DP steels are primarily used for the production of body panels and frames by deep drawing.

Prior to the forming process, the sheets employed in deep drawing typically undergo a sequence of procedures, such as steelmaking, continuous casting, hot and cold rolling, and heat treatment. These procedures as well as especially the deep drawing process itself all have an influence on the evolution of damage in the material based on the premise that ductile damage means the formation, growth and coalescence of voids in the microstructure. This damage causes degradation of the performance of the corresponding component [5]. Based on this definition of ductile damage, dual phase steels differ from other steels in its evolution. While inclusions play a primary role in void initiation in various steels, different damage mechanisms take precedence in DP steels [5]. These mechanisms encompass martensite fracture, phase boundary detachment, and grain boundary decohesion, and their prevalence is notably influenced by the microstructure [6]. However, the dominant damage mechanisms in DP steels are primarily influenced by the microstructure, while the evolution of damage and its extent in the microstructure are contingent on the stress-strain state experienced by the material throughout the processing route. In general, the higher the hydrostatic tensile stress reflected in a positive triaxiality, the larger the voids and the more damage is caused by plastic deformation in the material [6].

Considering deep drawing, various investigations have explored the influence of different key factors that impact the process in the past. These factors include friction, blank holder force, blank shape, and punch velocity [7]. The primary focus of these studies has been on achieving defect-free components at the macroscopic level, such as avoiding wrinkling, illustrated by Chen et al. [9], minimizing sheet thinning, examined by Preedawiphat et al. [10], and controlling spring back, explored in the works of Durmaz et al. [11] and Nanu et al. [12]. Only a few, however, focused on ductile damage on a microscale as defined above. And these studies focused in particular on the influence of process parameters on ductile damage, as in [13-14]. Therefore, there are knowledge gaps in the context of this understanding of ductile damage, including its potential in manufacturing processes to enhance a components performance. This study investigates the influence of the sheet metal alignment with respect to the forming tools, concerning the rolling direction of cold rolling, on the accumulation of damage in sheet metal during deep drawing of a rectangular cup and a U-profile. The analysis involves performing numerical simulations to evaluate the load paths in terms of stress and strain states that occur during deep drawing as well as the numerical predicted damage state after forming considering an isotropic damage model. Subsequently, an experimental investigation is carried out to analyze the impact on the damage accumulation of various alignments of the sheet metal with respect to the forming tools in relation to the rolling direction. Consequently, the formation of damage in terms of void area fractions in performance critical areas of the workpiece are quantified and compared. The results are used to draw conclusions regarding the impact of the rolling direction on the damage state. Additionally, they provide insides into the optimal orientation of sheet metal taking into account the rolling direction and the alignment with respect to the forming tools to minimize damage accumulation. Furthermore, an evaluation is made of the extent to which the rolling direction should be considered in future damage models for numerical analyses.

## Materials and Methods

*Experimental deep drawing setup.* For the study, a rectangular cup and a U-profile were considered. Workpieces with a similar geometry made of DP800 with various alignments of the sheet metal ($\phi$ = {0°, 45°, 90°}) in relation to the rolling direction were compared shortly before failure (Fig. 1a). Since the time of failure and the maximum possible punch strokes $s_{St}$ without failure varied for each variant, the analysis time relied on the variant that failed first and was
indicated by the punch stroke $s_{\text{St}}$. The deep drawing process is schematically depicted in Fig. 1b. The sheet thickness $s_{\text{SH}}$ was set to $s_{\text{SH}} = 1.5$ mm and the punch speed $v_p$ to $v_p = 5$ mm/s. The punch had a length $l_p$ of $l_p = 100$ mm, a width $w_p$ of $w_p = 50$ mm and a corner radius $r_{\text{pc}}$ of $r_{\text{pc}} = 10$ mm (Fig. 1c). The drawing clearance $c$ was $c = 1.8$ mm. The punch radius $r_p$ was fixed at $r_p = 3$ mm and the reference die radius $r_D$ at $r_D = 3$ mm. The sheet had a length $l_{\text{s,rec.}}$ of $l_{\text{s,rec.}} = 142$ mm, a width $w_s$ of $w_s = 92$ mm and a corner radius $r_s$ of $r_s = 10$ mm.

<figure><img src="image_899.png" /><figcaption>Fig. 1: Schematic illustration of a) sheet metal alignment considering rolling direction, b) deep drawing process and c) sheet metal</figcaption></figure>

The analysis of the damage accumulation concentrated on a reference point in the cross section (Fig. 1b) chosen at the outer bend of the bottom-to-wall transition, aligning with potential future applications for the workpieces. The damage states after forming in the reference point were examined in the form of void area fractions considering an area of ≈ 1.5*0.3 mm². Corresponding scanning electron microscopy (SEM) images were analyzed using the machine learning algorithm developed in Kusche et al. [15] and Medghalchi et al. [16], which was specifically trained for the DP800 used in this study. The images were taken at an acceleration voltage of 20kV with image resolution of 3072 pixels per 100 µm.

*Numerical deep drawing setup and process model.* The influence of the deep drawing process on load paths in form of triaxiality $\eta$, lode angle parameter $\bar{\theta}$, von Mises equivalent stress $\sigma_{\text{vm}}$ and plastic equivalent strain $\phi_{\text{pl}}$ as well as on the damage state was investigated numerically by using Abaqus/Explicit. The analysis focused on the reference point, which was also utilized for determining the void area fractions (Fig. 1b). A volume measuring 1.5x1.5x0.3 mm³ was
indicated by the punch stroke $s_{\text{St}}$. The deep drawing process is schematically depicted in Fig. 1b. The sheet thickness $s_{\text{SH}}$ was set to $s_{\text{SH}} = 1.5$ mm and the punch speed $v_p$ to $v_p = 5$ mm/s. The punch had a length $l_p$ of $l_p = 100$ mm, a width $w_p$ of $w_p = 50$ mm and a corner radius $r_{\text{pc}}$ of $r_{\text{pc}} = 10$ mm (Fig. 1c). The drawing clearance $c$ was $c = 1.8$ mm. The punch radius $r_p$ was fixed at $r_p = 3$ mm and the reference die radius $r_D$ at $r_D = 3$ mm. The sheet had a length $l_{\text{s,rec.}}$ of $l_{\text{s,rec.}} = 142$ mm, a width $w_s$ of $w_s = 92$ mm and a corner radius $r_s$ of $r_s = 10$ mm.

<figure><img src="image_906.png" /><figcaption>Fig. 1: Schematic illustration of a) sheet metal alignment considering rolling direction, b) deep drawing process and c) sheet metal</figcaption></figure>

The analysis of the damage accumulation concentrated on a reference point in the cross section (Fig. 1b) chosen at the outer bend of the bottom-to-wall transition, aligning with potential future applications for the workpieces. The damage states after forming in the reference point were examined in the form of void area fractions considering an area of ≈ 1.5*0.3 mm². Corresponding scanning electron microscopy (SEM) images were analyzed using the machine learning algorithm developed in Kusche et al. [15] and Medghalchi et al. [16], which was specifically trained for the DP800 used in this study. The images were taken at an acceleration voltage of 20kV with image resolution of 3072 pixels per 100 µm.

*Numerical deep drawing setup and process model.* The influence of the deep drawing process on load paths in form of triaxiality $\eta$, lode angle parameter $\bar{\theta}$, von Mises equivalent stress $\sigma_{\text{vm}}$ and plastic equivalent strain $\phi_{\text{pl}}$ as well as on the damage state was investigated numerically by using Abaqus/Explicit. The analysis focused on the reference point, which was also utilized for determining the void area fractions (Fig. 1b). A volume measuring 1.5x1.5x0.3 mm³ was
employed for this purpose. To numerically predict the damage state as the damage parameter *D*, an isotropic coupled damage model based on Lemaitre's approach was applied [17]. The model was calibrated in rolling direction for the DP800 used in this study. The model's parameters are listed in detail in Sprave et al. [18].

To analyze the damage distribution, a parameter *x* was introduced which passes through the corner of the component in the case of the rectangular geometry, or through the exact center of the component in the case of U-profiles (Fig. 1b). The geometry was reduced to a quarter using symmetry. Table 1 summarizes the mesh parameters and body types. The sheet was depicted by a dense mesh of volume elements (C3D8R). Structured shell elements were employed to mesh the punch, blank holder and die (R3D4). While the planar regions of the active parts were represented by a few large elements, the curvatures of the active parts were represented by many small elements. This ensured an effective balance between precise geometric representation of the parts and computational efficiency. The blank holder's movement, specifically when it first came into contact with the sheet, followed a velocity profile. To prevent the sheet's natural oscillations resulting from the blank holder's impact, the blank holder's displacement was defined as a sigmoid function. This function reduced the blank holder speed during contact buildup, thus minimizing its impact effect. The blank holder was forcefully and strategically moved with a hard contact and interaction properties set to general contact to ensure force control.

<table><caption>Table 1: Mesh parameters and body types</caption><thead><tr><td>Name</td><td>Mesh</td><td>Type of body</td><td>Global seed size</td><td>Max. deviation factor</td><td>Min. size control</td></tr></thead><tbody><tr><td>Sheet</td><td>C3D8R</td><td>Solid deformable body</td><td>0.5</td><td>0.005</td><td>0.01</td></tr><tr><td>Punch, blank holder, die</td><td>R3D4</td><td>Discrete rigid</td><td>10</td><td>0.005</td><td>0.01</td></tr></tbody></table>

## Results

*Analysis of load paths.* The load paths in form of triaxiality $\eta$, lode angle parameter $\bar{\theta}$, von Mises equivalent stress $\sigma_{vm}$ and plastic equivalent strain $\phi_{pl}$ of the deep-drawn U-profile and rectangular cup are shown in Fig. 2. The triaxiality $\eta$ curve of the U-profile initially increases degressively up to an equivalent plastic strain $\phi_{pl}$ of about $\phi_{pl} = 0.025$. The value then oscillates around a level of $\eta = 0.55$ until its end. The lode angle parameter $\bar{\theta}$ course of the U-profile decreases exponentially up to an equivalent plastic strain $\phi_{pl}$ of $\phi_{pl} = 0.05$. The von Mises equivalent stress $\sigma_{vm}$ of the U-profile, on the other hand, shows a degressively increasing curve. However, just before the maximum plastic equivalent stress $\sigma_{vm}$ is reached, the value of $\sigma_{vm}$ drops again.

The course of the triaxiality $\eta$ of the rectangular cup rises steeply up to an equivalent plastic strain $\phi_{pl}$ of $\phi_{pl} = 0.02$ and reaches a positive extreme value. After a slight drop in the triaxiality $\eta$ up to an equivalent plastic strain $\phi_{pl}$ of $\phi_{pl} = 0.14$, the curve of the triaxiality $\eta$ then increases almost linearly until the end of the forming process. Up to an equivalent plastic strain $\phi_{pl}$ of $\phi_{pl} = 0.02$, the curve of the lode angle parameter $\bar{\theta}$ decreases and reaches a negative extreme value. The curve of the lode angle parameter $\bar{\theta}$ of the rectangular cup then increases to an equivalent plastic strain $\phi_{pl}$ of $\phi_{pl} = 0.22$. After the respective increase in the lode angle parameter $\bar{\theta}$, the curve of the lode angle parameter $\bar{\theta}$ decreases and reaches a second negative extreme value at the end of the forming process. The von Mises equivalent stress $\sigma_{vm}$ of the rectangular cup initially increases degressively as a function of the equivalent plastic strain $\phi_{pl}$
employed for this purpose. To numerically predict the damage state as the damage parameter *D*, an isotropic coupled damage model based on Lemaitre's approach was applied [17]. The model was calibrated in rolling direction for the DP800 used in this study. The model's parameters are listed in detail in Sprave et al. [18].

To analyze the damage distribution, a parameter *x* was introduced which passes through the corner of the component in the case of the rectangular geometry, or through the exact center of the component in the case of U-profiles (Fig. 1b). The geometry was reduced to a quarter using symmetry. Table 1 summarizes the mesh parameters and body types. The sheet was depicted by a dense mesh of volume elements (C3D8R). Structured shell elements were employed to mesh the punch, blank holder and die (R3D4). While the planar regions of the active parts were represented by a few large elements, the curvatures of the active parts were represented by many small elements. This ensured an effective balance between precise geometric representation of the parts and computational efficiency. The blank holder's movement, specifically when it first came into contact with the sheet, followed a velocity profile. To prevent the sheet's natural oscillations resulting from the blank holder's impact, the blank holder's displacement was defined as a sigmoid function. This function reduced the blank holder speed during contact buildup, thus minimizing its impact effect. The blank holder was forcefully and strategically moved with a hard contact and interaction properties set to general contact to ensure force control.

<table><caption>Table 1: Mesh parameters and body types</caption><thead><tr><td>Name</td><td>Mesh</td><td>Type of body</td><td>Global seed size</td><td>Max. deviation factor</td><td>Min. size control</td></tr></thead><tbody><tr><td>Sheet</td><td>C3D8R</td><td>Solid deformable body</td><td>0.5</td><td>0.005</td><td>0.01</td></tr><tr><td>Punch, blank holder, die</td><td>R3D4</td><td>Discrete rigid</td><td>10</td><td>0.005</td><td>0.01</td></tr></tbody></table>

## Results

Analysis of load paths. The load paths in form of triaxiality $\eta$, lode angle parameter $\bar{\theta}$, von Mises equivalent stress $\sigma_{vm}$ and plastic equivalent strain $\phi_{pl}$ of the deep-drawn U-profile and rectangular cup are shown in Fig. 2. The triaxiality $\eta$ curve of the U-profile initially increases degressively up to an equivalent plastic strain $\phi_{pl}$ of about $\phi_{pl} = 0.025$. The value then oscillates around a level of $\eta = 0.55$ until its end. The lode angle parameter $\bar{\theta}$ course of the U-profile decreases exponentially up to an equivalent plastic strain $\phi_{pl}$ of $\phi_{pl} = 0.05$. The von Mises equivalent stress $\sigma_{vm}$ of the U-profile, on the other hand, shows a degressively increasing curve. However, just before the maximum plastic equivalent stress $\sigma_{vm}$ is reached, the value of $\sigma_{vm}$ drops again.

The course of the triaxiality $\eta$ of the rectangular cup rises steeply up to an equivalent plastic strain $\phi_{pl}$ of $\phi_{pl} = 0.02$ and reaches a positive extreme value. After a slight drop in the triaxiality $\eta$ up to an equivalent plastic strain $\phi_{pl}$ of $\phi_{pl} = 0.14$, the curve of the triaxiality $\eta$ then increases almost linearly until the end of the forming process. Up to an equivalent plastic strain $\phi_{pl}$ of $\phi_{pl} = 0.02$, the curve of the lode angle parameter $\bar{\theta}$ decreases and reaches a negative extreme value. The curve of the lode angle parameter $\bar{\theta}$ of the rectangular cup then increases to an equivalent plastic strain $\phi_{pl}$ of $\phi_{pl} = 0.22$. After the respective increase in the lode angle parameter $\bar{\theta}$, the curve of the lode angle parameter $\bar{\theta}$ decreases and reaches a second negative extreme value at the end of the forming process. The von Mises equivalent stress $\sigma_{vm}$ of the rectangular cup initially increases degressively as a function of the equivalent plastic strain $\phi_{pl}$
until an equivalent plastic strain $\varphi_{pl}$ of $\varphi_{pl} = 0.22$ is reached. The von Mises equivalent stress $\sigma_{vm}$ then drops off abruptly towards the end of the forming process.

<figure><img src="image_900.png" /><figcaption>Fig. 2: Load paths during forming of the U-profile and the rectangular cup at reference point</figcaption></figure>

Analysis of numerical predicted damage state after forming. Fig. 3 presents the damage distribution resulting from deep drawing of the U-profile, where the bottom, transition, and wall areas are taken into account. The bottom area indicates no numerical predicted damage correlation with deep drawing. However, the transition area, from the floor to the wall, displays a continuous numerical predicted damage area. This numerical predicted damage area consists of a local maximum of the damage parameter **D** at the beginning of the transition area and a successive more uniform damage distribution. The global maximum of the damage parameter **D**<sub>max</sub> arises in the lower end direction of the wall, towards the flange. The damage parameter **D** at the reference point is predicted as 0.04.

<figure><img src="image_901.png" /><figcaption>Fig. 3: Numerical predicted damage distribution in U-profile</figcaption></figure>

Fig. 4 demonstrates the progression of the damage parameter **D** throughout the rectangular cup over distance **x**. It is noteworthy that the bottom of the cup is not impacted by deep drawing considering the numerical predicted damage state. As the transition area is approached, the damage parameter **D** progressively increases and reaches its maximum **D**<sub>max</sub> before it decreases. The predicted damage parameter **D** is **D** = 0.19 at the reference point.
until an equivalent plastic strain $\varphi_{pl}$ of $\varphi_{pl} = 0.22$ is reached. The von Mises equivalent stress $\sigma_{vm}$ then drops off abruptly towards the end of the forming process.

<figure><img src="image_907.png" /><figcaption>Fig. 2: Load paths during forming of the U-profile and the rectangular cup at reference point</figcaption></figure>

Analysis of numerical predicted damage state after forming. Fig. 3 presents the damage distribution resulting from deep drawing of the U-profile, where the bottom, transition, and wall areas are taken into account. The bottom area indicates no numerical predicted damage correlation with deep drawing. However, the transition area, from the floor to the wall, displays a continuous numerical predicted damage area. This numerical predicted damage area consists of a local maximum of the damage parameter **D** at the beginning of the transition area and a successive more uniform damage distribution. The global maximum of the damage parameter **D**<sub>max</sub> arises in the lower end direction of the wall, towards the flange. The damage parameter **D** at the reference point is predicted as 0.04.

<figure><img src="image_908.png" /><figcaption>Fig. 3: Numerical predicted damage distribution in U-profile</figcaption></figure>

Fig. 4 demonstrates the progression of the damage parameter **D** throughout the rectangular cup over distance **x**. It is noteworthy that the bottom of the cup is not impacted by deep drawing considering the numerical predicted damage state. As the transition area is approached, the damage parameter **D** progressively increases and reaches its maximum **D**<sub>max</sub> before it decreases. The predicted damage parameter **D** is **D** = 0.19 at the reference point.
<figure><img src="image_902.png" /><figcaption>Fig. 4: Numerical predicted damage distribution in rectangular cup</figcaption></figure>

Analysis of accumulated damage after forming. Fig. 5 displays the void area fraction $\delta$ at the reference point in dependence of the alignments of the sheet metal in relation to the rolling direction. In the case of the rectangular cup, it is apparent that the void area fraction $\delta$ decreases as the angle of the alignments of the sheet metal increases. The values reach a maximum of $\delta_{\text{rec.cup,0}} = 0.058 \%$, $\delta_{\text{rec.cup,45}} = 0.051 \%$ and $\delta_{\text{rec.cup,90}} = 0.045 \%$. Considering the U-profile, the values reach a maximum of $\delta_{\text{U-profile,0}} = 0.026 \%$ and $\delta_{\text{U-profile,45}} = 0.0129 \%$. The sheet metal workpieces in the 90°-orientation were not analyzed, as these failed despite being deep-drawn with the same deep drawing setup.

<figure><img src="image_903.png" /><figcaption>Fig. 5: Void area fraction $\delta$ at reference point of a) rectangular cup and b) U-profile after deep drawing in dependence of the alignments of the sheet metal in relation to the rolling direction</figcaption></figure>

## Conclusion

When comparing the load paths in the reference point of the two considered geometries, noticeable disparities can be observed although the same punch edge and corner radius cause the two geometries to have a similar curvature. Firstly, the two geometries differ significantly in the required equivalent plastic strain $\phi_{pl}$ at the reference point which is considerably higher for the rectangular cup than for the U-profile. Furthermore, the stress state in form of triaxiality $\eta$ and lode angle parameter $\bar{\theta}$ of the rectangular cup indicate that non-proportional loading occurs during forming, shifting between the plain strain tension region and the biaxial tension region. This makes a corresponding numerical prediction of the damage state much more complex, as, in principle, a larger number of different loadings should be used for accurate calibration. In contrast, the stress values of the U-section are almost exclusively in the plain strain tension range.

The different stress and strain states are also reflected in the numerically predicted damage state at the reference point, which differs significantly in its value. The rectangular cup yields a value almost five times greater than the U-profile. This difference is also reflected in the determined void
<figure><img src="image_909.png" /><figcaption>Fig. 4: Numerical predicted damage distribution in rectangular cup</figcaption></figure>

Analysis of accumulated damage after forming. Fig. 5 displays the void area fraction $\delta$ at the reference point in dependence of the alignments of the sheet metal in relation to the rolling direction. In the case of the rectangular cup, it is apparent that the void area fraction $\delta$ decreases as the angle of the alignments of the sheet metal increases. The values reach a maximum of $\delta_{\text{rec.cup,0}} = 0.058 \%$, $\delta_{\text{rec.cup,45}} = 0.051 \%$ and $\delta_{\text{rec.cup,90}} = 0.045 \%$. Considering the U-profile, the values reach a maximum of $\delta_{\text{U-profile,0}} = 0.026 \%$ and $\delta_{\text{U-profile,45}} = 0.0129 \%$. The sheet metal workpieces in the 90°-orientation were not analyzed, as these failed despite being deep-drawn with the same deep drawing setup.

<figure><img src="image_910.png" /><figcaption>Fig. 5: Void area fraction $\delta$ at reference point of a) rectangular cup and b) U-profile after deep drawing in dependence of the alignments of the sheet metal in relation to the rolling direction</figcaption></figure>

## Conclusion

When comparing the load paths in the reference point of the two considered geometries, noticeable disparities can be observed although the same punch edge and corner radius cause the two geometries to have a similar curvature. Firstly, the two geometries differ significantly in the required equivalent plastic strain $\phi_{pl}$ at the reference point which is considerably higher for the rectangular cup than for the U-profile. Furthermore, the stress state in form of triaxiality $\eta$ and lode angle parameter $\bar{\theta}$ of the rectangular cup indicate that non-proportional loading occurs during forming, shifting between the plain strain tension region and the biaxial tension region. This makes a corresponding numerical prediction of the damage state much more complex, as, in principle, a larger number of different loadings should be used for accurate calibration. In contrast, the stress values of the U-section are almost exclusively in the plain strain tension range.

The different stress and strain states are also reflected in the numerically predicted damage state at the reference point, which differs significantly in its value. The rectangular cup yields a value almost five times greater than the U-profile. This difference is also reflected in the determined void
area fractions, with the rectangular cup having a void area fraction at least three times higher than that of the U-profile, regardless of the rolling direction. Accordingly, the numerical results are in qualitative agreement with the experimental results.

Considering the measured void area fractions of the rectangular cup as a function of the alignments of the sheet metal with respect to the forming tools relative to the rolling direction, there are differences in the values obtained. For the rectangular cup, it was shown that the damage accumulation is lower if the sheet metal is aligned with the rolling direction parallel to the longest, straight sides considering a 90°-orientation to the rolling direction. This enabled the void area fraction to be reduced by 22.4% from $δ_{rec.cup,0°} = 0.058\%$ to $δ_{rec.cup,90°} = 0.045\%$. For the U-profile, on the other hand, it was demonstrated that a 45°-orientation leads to the lowest damage accumulation reducing it by 50.38% from $δ_{U-profile,0°} = 0.026\%$ to $δ_{U-profile,45°} = 0.0129\%$. Therefore, it can be deduced that geometry features with a convex radius should have a 90°-orientation to the rolling direction with respect to the forming tools, while long, straight sides, as they occur with the U-profile, should be orientated at 45° to the rolling direction with respect to the forming tools in order to achieve the lowest possible damage accumulation. However, the interaction between performance, rolling direction and damage accumulation still needs to be evaluated in order to be able to draw a general conclusion.

## Outlook

Based on the results of this study, there is an endeavour in forthcoming research to refine the damage model to address material anisotropy. This entails assuring that the model comprehends the divergent behaviour of materials in distinct directions and thus taking into account a tensorial damage parameter. Moreover, non-proportional load paths need to be considered so that the model can handle different types of loads more accurately, especially when forces aren't applied uniformly. Both approaches are expected to improve the model's ability to predict the damage state, since especially with complex geometries, the geometry features, such as corners or straight sides, reflected in the rectangular cup and U-profile, may not always be aligned parallel or orthogonal relative to the rolling direction with respect to the forming tools. In addition, further investigations are necessary to understand the microstructural reasons that lead to damage as well as to determine the performance of the components depending on the alignment of the sheet metal to the rolling direction with respect to the forming tools.

## Acknowledgements

This research was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation; Project number 278868966 - TRR 188, projects A06 and B02; Damage Controlled Forming Processes). The authors would also like to thank A. Erman Tekkaya, Till Clausmeyer, Alexander Schowtjak and Jan Gerlach from the Institute of Forming Technology and Lightweight Components of TU Dortmund, sub-project S01 of TRR 188, for the usage of their implementation of the Lemaitre damage model.

## References

[1] Information on https://www.europarl.europa.eu/.

[2] Information on https://climate.ec.europa.eu/eu-action/european-green-deal_en.

[3] Information on https://www.massiverleichtbau.de

[4] T. R. Bieler, P. Eisenlohr, F. Roters, D. Kumar, D. E. Mason, M. A. Crimp, and D. Raabe, The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals. *International Journal of Plasticity*, 9 (2009) 1655-1683.
https://doi.org/10.1016/j.ijplas.2008.09.002
area fractions, with the rectangular cup having a void area fraction at least three times higher than that of the U-profile, regardless of the rolling direction. Accordingly, the numerical results are in qualitative agreement with the experimental results.

Considering the measured void area fractions of the rectangular cup as a function of the alignments of the sheet metal with respect to the forming tools relative to the rolling direction, there are differences in the values obtained. For the rectangular cup, it was shown that the damage accumulation is lower if the sheet metal is aligned with the rolling direction parallel to the longest, straight sides considering a 90°-orientation to the rolling direction. This enabled the void area fraction to be reduced by 22.4% from $δ_{rec.cup,0°} = 0.058\%$ to $δ_{rec.cup,90°} = 0.045\%$. For the U-profile, on the other hand, it was demonstrated that a 45°-orientation leads to the lowest damage accumulation reducing it by 50.38% from $δ_{U-profile,0°} = 0.026\%$ to $δ_{U-profile,45°} = 0.0129\%$. Therefore, it can be deduced that geometry features with a convex radius should have a 90°-orientation to the rolling direction with respect to the forming tools, while long, straight sides, as they occur with the U-profile, should be orientated at 45° to the rolling direction with respect to the forming tools in order to achieve the lowest possible damage accumulation. However, the interaction between performance, rolling direction and damage accumulation still needs to be evaluated in order to be able to draw a general conclusion.

## Outlook

Based on the results of this study, there is an endeavour in forthcoming research to refine the damage model to address material anisotropy. This entails assuring that the model comprehends the divergent behaviour of materials in distinct directions and thus taking into account a tensorial damage parameter. Moreover, non-proportional load paths need to be considered so that the model can handle different types of loads more accurately, especially when forces aren't applied uniformly. Both approaches are expected to improve the model's ability to predict the damage state, since especially with complex geometries, the geometry features, such as corners or straight sides, reflected in the rectangular cup and U-profile, may not always be aligned parallel or orthogonal relative to the rolling direction with respect to the forming tools. In addition, further investigations are necessary to understand the microstructural reasons that lead to damage as well as to determine the performance of the components depending on the alignment of the sheet metal to the rolling direction with respect to the forming tools.

## Acknowledgements

This research was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation; Project number 278868966 – TRR 188, projects A06 and B02; Damage Controlled Forming Processes). The authors would also like to thank A. Erman Tekkaya, Till Clausmeyer, Alexander Schowtjak and Jan Gerlach from the Institute of Forming Technology and Lightweight Components of TU Dortmund, sub-project S01 of TRR 188, for the usage of their implementation of the Lemaitre damage model.

## References

[1] Information on https://www.europarl.europa.eu/

[2] Information on https://climate.ec.europa.eu/eu-action/european-green-deal_en.

[3] Information on https://www.massiverleichtbau.de

[4] T. R. Bieler, P. Eisenlohr, F. Roters, D. Kumar, D. E. Mason, M. A. Crimp, and D. Raabe, The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals. *International Journal of Plasticity*, 9 (2009) 1655–1683.
https://doi.org/10.1016/j.ijplas.2008.09.002
[5] M. Müller, I. F. Weiser, T. Herrig, and T. Bergs, Numerical Prediction of the Influence of Process Parameters and Process Set-Up on Damage Evolution during Deep Drawing of Rectangular Cups. Engineering Proceedings, 6 (2022) 1-10.
https://doi.org/10.3390/engproc2022026006

[6] A. E. Tekkaya, P.-O. Bouchard, S. Bruschi, and C. C. Tasan, Damage in metal forming. CIRP Annals, 2 (2020) 600-623. https://doi.org/10.1016/j.cirp.2020.05.005

[7] F. A. McClintock, A Criterion for Ductile Fracture by the Growth of Holes. Journal of Applied Mechanics, 2 (1968) 363-371. https://doi.org/10.1115/1.3601204

[8] P. R. Tiwari, A. Rathore, and M. G. Bodkhe, Factors affecting the deep drawing process – A review. Materials Today: Proceedings (2022) 2902-2908.
https://doi.org/10.1016/j.matpr.2021.10.189

[9] K. Chen, A. J. Carter, and Y. P. Korkolis, Flange Wrinkling in Deep-Drawing: Experiments, Simulations and a Reduced-Order Model. JMMP, 4 (2022). https://doi.org/10.3390/jmmp6040076

[10] P. Preedawiphat, P. Koowattanasuchat, N. Mahayotsanun, and S. Mahabunphachai, Sheet thinning prediction method based on localized friction effect in deep-drawing. Advances in Mechanical Engineering, 9 (2020). https://doi.org/10.1177/1687814020953941

[11] U. Durmaz, S. Heibel, T. Schweiker, A. Prabhakar, and M. Merklein, Influence of the forming process on springback. IOP Conf. Ser.: Mater. Sci. Eng., 1 (2022). https://doi.org/10.1088/1757-899X/1238/1/012074

[12] N. Nanu and G. Brabie, Analytical model for prediction of springback parameters in the case of U stretch-bending process as a function of stresses distribution in the sheet thickness. International Journal of Mechanical Sciences, 1 (2012) 11-21.
https://doi.org/10.1016/j.ijmecsci.2012.08.007

[13] M. Nick, A. Feuerhack, T. Bergs and T. Clausmeyer, Numerical Investigation of Damage in Single-step, Two-step, and Reverse Deep Drawing of Rotationally Symmetric Cups from DP800 Dual Phase Steel. Procedia Manufacturing, 47 (2020) 636-642.
https://doi.org/10.1016/j.promfg.2020.04.195

[14] T. Bergs. M. Nick, D. Trauth, and F. Klocke, Damage Evolution in Nakajima Tests of DP800 Dual Phase Steel. Materials and Science Engineering, 418 (2018). https://doi.org/10.1088/1757-899X/418/1/012048

[15] C. Kusche, T. Reclik, M. Freund, T. Al-Samman, U. Kerzel, and S. Korte-Kerzel, Large-area, high-resolution characterisation and classification of damage mechanisms in dual-phase steel using deep learning. PloS one, 5 (2019). https://doi.org/10.1371/journal.pone.0216493

[16] S. Medghalchi, C. F. Kusche, E. Karimi, U. Kerzel, and S. Korte-Kerzel, Damage Analysis in Dual-Phase Steel Using Deep Learning: Transfer from Uniaxial to Biaxial Straining Conditions by Image Data Augmentation. JOM, 12 (2020) 4420-4430. https://doi.org/10.1007/s11837-020-04404-0

[17] J. Lemaitre, A Continuous Damage Mechanics Model for Ductile Fracture. Journal of Engineering Materials and Technology, 1 (1985) 83-89. https://doi.org/10.1115/1.3225775

[18] L. Sprave, A. Schowtjak, R. Meya, T. Clausmeyer, A. E. Tekkaya, and A. Menzel, On mesh dependencies in finite-element-based damage prediction: application to sheet metal bending. Prod. Eng. Res. Devel., 1 (2020) 123-134. https://doi.org/10.1007/s11740-019-00937-9
[5] M. Müller, I. F. Weiser, T. Herrig, and T. Bergs, Numerical Prediction of the Influence of Process Parameters and Process Set-Up on Damage Evolution during Deep Drawing of Rectangular Cups. Engineering Proceedings, 6 (2022) 1-10.
https://doi.org/10.3390/engproc2022026006

[6] A. E. Tekkaya, P.-O. Bouchard, S. Bruschi, and C. C. Tasan, Damage in metal forming. CIRP Annals, 2 (2020) 600-623. https://doi.org/10.1016/j.cirp.2020.05.005

[7] F. A. McClintock, A Criterion for Ductile Fracture by the Growth of Holes. Journal of Applied Mechanics, 2 (1968) 363-371. https://doi.org/10.1115/1.3601204

[8] P. R. Tiwari, A. Rathore, and M. G. Bodkhe, Factors affecting the deep drawing process – A review. Materials Today: Proceedings (2022) 2902-2908.
https://doi.org/10.1016/j.matpr.2021.10.189

[9] K. Chen, A. J. Carter, and Y. P. Korkolis, Flange Wrinkling in Deep-Drawing: Experiments, Simulations and a Reduced-Order Model. JMMP, 4 (2022). https://doi.org/10.3390/jmmp6040076

[10] P. Preedawiphat, P. Koowattanasuchat, N. Mahayotsanun, and S. Mahabunphachai, Sheet thinning prediction method based on localized friction effect in deep-drawing. Advances in Mechanical Engineering, 9 (2020). https://doi.org/10.1177/1687814020953941

[11] U. Durmaz, S. Heibel, T. Schweiker, A. Prabhakar, and M. Merklein, Influence of the forming process on springback. IOP Conf. Ser.: Mater. Sci. Eng., 1 (2022). https://doi.org/10.1088/1757-899X/1238/1/012074

[12] N. Nanu and G. Brabie, Analytical model for prediction of springback parameters in the case of U stretch-bending process as a function of stresses distribution in the sheet thickness. International Journal of Mechanical Sciences, 1 (2012) 11-21.
https://doi.org/10.1016/j.ijmecsci.2012.08.007

[13] M. Nick, A. Feuerhack, T. Bergs and T. Clausmeyer, Numerical Investigation of Damage in Single-step, Two-step, and Reverse Deep Drawing of Rotationally Symmetric Cups from DP800 Dual Phase Steel. Procedia Manufacturing, 47 (2020) 636-642.
https://doi.org/10.1016/j.promfg.2020.04.195

[14] T. Bergs. M. Nick, D. Trauth, and F. Klocke, Damage Evolution in Nakajima Tests of DP800 Dual Phase Steel. Materials and Science Engineering, 418 (2018). https://doi.org/10.1088/1757-899X/418/1/012048

[15] C. Kusche, T. Reclik, M. Freund, T. Al-Samman, U. Kerzel, and S. Korte-Kerzel, Large-area, high-resolution characterisation and classification of damage mechanisms in dual-phase steel using deep learning. PloS one, 5 (2019). https://doi.org/10.1371/journal.pone.0216493

[16] S. Medghalchi, C. F. Kusche, E. Karimi, U. Kerzel, and S. Korte-Kerzel, Damage Analysis in Dual-Phase Steel Using Deep Learning: Transfer from Uniaxial to Biaxial Straining Conditions by Image Data Augmentation. JOM, 12 (2020) 4420-4430. https://doi.org/10.1007/s11837-020-04404-0

[17] J. Lemaitre, A Continuous Damage Mechanics Model for Ductile Fracture. Journal of Engineering Materials and Technology, 1 (1985) 83-89. https://doi.org/10.1115/1.3225775

[18] L. Sprave, A. Schowtjak, R. Meya, T. Clausmeyer, A. E. Tekkaya, and A. Menzel, On mesh dependencies in finite-element-based damage prediction: application to sheet metal bending. Prod. Eng. Res. Devel., 1 (2020) 123-134. https://doi.org/10.1007/s11740-019-00937-9
# Pre-strain correction of forming limit curves

ARETZ Holger

Speira GmbH, Research & Development, Georg-von-Boeselager-Str. 21, 53117 Bonn,
Germany

holger.aretz@speira.com

**Keywords:** Sheet Metal Forming, Forming Limit Curve, Strain-Path Dependence

**Abstract.** Inherent features of the Nakajima test to determine forming limit curves (FLC) are inevitable bending strains and pre-strains originating from evolving punch contact in the early stage. Both effects are very annoying and bias the resulting FLC considerably, so that a direct comparison of materials with different gauges is impossible. In the present work, a simple computational procedure is developed to remove both effects from FLC data. Application examples demonstrate its capabilities.

## Introduction

The forming limit curve (FLC) represents the transition from safe forming to failure and is thus an indispensable tool in sheet metal forming to ensure defect-free products. The most frequently used methodology in laboratory FLC-determination is the *Nakajima* approach. It involves stretch-forming of circular sheet samples possessing different symmetric cut-outs over a hemispherical punch to failure, so that a broad range of different strain modes can be realized, thereby providing a discrete picture of the FLC. In the present work, the standard FLC representation in terms of minor and major in-plane *true* strains at failure (denoted as $\varepsilon_2$ and $\varepsilon_1$, respectively, with the notational convention $\varepsilon_1 \ge \varepsilon_2$) will be considered, i.e. $\text{FLC} = f(\varepsilon_1, \varepsilon_2)$. Furthermore, the material of which the samples are made is approximated as *rigid-plastic*, i.e. elastic strains are neglected.

Despite its popularity the Nakajima method introduces two undesired effects: equibiaxial pre-strains caused by the evolving punch contact in the early stage of the test (see e.g. Noder & Butcher [9]), and strains associated with bending of the samples (see e.g. Affronti & Merklein [1]). Both effects bias the resulting FLC data in different ways:

1. Due to the presence of pre-strains, the realized strain-paths are essentially bi-linear (with the only exception of the equibiaxial strain mode, which is “immune” to equibiaxial pre-straining), thus violating an important prerequisite of FLC determination, viz. linearity of the imposed strain-paths. As a consequence, the FLC’s minimum point, commonly called “FLC$_{min}$”, is displaced to the *lower right* in strain space, i.e. $\text{FLC}_{\min}$ does not coincide with plane-strain deformation (as it should be the case) and its major strain value is usually lowered (see e.g. Müschenborn & Sonne [8]), provided that the bending strains do not dominate.

2. Bending strains act in the opposite direction and *lift* the major strain values. This is because strains measured on the convex side of a bent sample are always larger than, say, in the sample’s midplane. This is a pure geometric effect and thus strongly dependent of the sample’s thickness. Accordingly, if a (maybe material dependent) sample thickness is exceeded, the lowering effect of the pre-strains on the major strain of $\text{FLC}_{\min}$ can be overcompensated by the bending strains, in which case the $\text{FLC}_{\min}$-point may be only slightly displaced in positive minor strain direction, but without altering or even elevating its major strain value compared to linear strain paths.

While the effect of bending strains does solely depend on the material’s gauge, the pre-strain effect seems to be material dependent, at least according to the author’s experience. Therefore, it is not

<figure><img src="image_904.png" /></figure>
# Pre-strain correction of forming limit curves

ARETZ Holger

Speira GmbH, Research & Development, Georg-von-Boeselager-Str. 21, 53117 Bonn,
Germany

holger.aretz@speira.com

**Keywords:** Sheet Metal Forming, Forming Limit Curve, Strain-Path Dependence

**Abstract.** Inherent features of the Nakajima test to determine forming limit curves (FLC) are inevitable bending strains and pre-strains originating from evolving punch contact in the early stage. Both effects are very annoying and bias the resulting FLC considerably, so that a direct comparison of materials with different gauges is impossible. In the present work, a simple computational procedure is developed to remove both effects from FLC data. Application examples demonstrate its capabilities.

## Introduction

The forming limit curve (FLC) represents the transition from safe forming to failure and is thus an indispensable tool in sheet metal forming to ensure defect-free products. The most frequently used methodology in laboratory FLC-determination is the *Nakajima* approach. It involves stretch-forming of circular sheet samples possessing different symmetric cut-outs over a hemispherical punch to failure, so that a broad range of different strain modes can be realized, thereby providing a discrete picture of the FLC. In the present work, the standard FLC representation in terms of minor and major in-plane *true* strains at failure (denoted as $\varepsilon_2$ and $\varepsilon_1$, respectively, with the notational convention $\varepsilon_1 \ge \varepsilon_2$) will be considered, i.e. $\text{FLC} = f(\varepsilon_1, \varepsilon_2)$. Furthermore, the material of which the samples are made is approximated as *rigid-plastic*, i.e. elastic strains are neglected.

Despite its popularity the Nakajima method introduces two undesired effects: equibiaxial pre-strains caused by the evolving punch contact in the early stage of the test (see e.g. Noder & Butcher [9]), and strains associated with bending of the samples (see e.g. Affronti & Merklein [1]). Both effects bias the resulting FLC data in different ways:

1. Due to the presence of pre-strains, the realized strain-paths are essentially bi-linear (with the only exception of the equibiaxial strain mode, which is “immune” to equibiaxial pre-straining), thus violating an important prerequisite of FLC determination, viz. linearity of the imposed strain-paths. As a consequence, the FLC’s minimum point, commonly called “FLC$_{min}$”, is displaced to the *lower right* in strain space, i.e. $\text{FLC}_{\min}$ does not coincide with plane-strain deformation (as it should be the case) and its major strain value is usually lowered (see e.g. Müschenborn & Sonne [8]), provided that the bending strains do not dominate.

2. Bending strains act in the opposite direction and *lift* the major strain values. This is because strains measured on the convex side of a bent sample are always larger than, say, in the sample’s midplane. This is a pure geometric effect and thus strongly dependent of the sample’s thickness. Accordingly, if a (maybe material dependent) sample thickness is exceeded, the lowering effect of the pre-strains on the major strain of $\text{FLC}_{\min}$ can be overcompensated by the bending strains, in which case the $\text{FLC}_{\min}$-point may be only slightly displaced in positive minor strain direction, but without altering or even elevating its major strain value compared to linear strain paths.

While the effect of bending strains does solely depend on the material’s gauge, the pre-strain effect seems to be material dependent, at least according to the author’s experience. Therefore, it is not

<figure><img src="image_911.png" /></figure>
possible to compare materials with different gauges, which is very annoying in practice. This problem is addressed in the present work, and as a solution a simple computational method to remove both the pre-strains and the bending strains from the FLC data is proposed.

For notational convenience, throughout this paper the following 2 × 1 column matrices

$$ \mathbf{FLC} := (\mathbf{FLC}_1, \mathbf{FLC}_2)^T, \mathbf{FLC}^* := (\mathbf{FLC}_1^*, \mathbf{FLC}_2^*)^T \text{ and } \boldsymbol{\varepsilon}^* := (\varepsilon_1^*, \varepsilon_2^*)^T \quad (1) $$

will be used to store a strain tuple pertaining to a single point of the FLC, associated with zero and non-zero pre-strains $\boldsymbol{\varepsilon}^*$, respectively.

## Forward Retro Analysis

The present work started with the development of a straightforward approach to reproduce given experimental or predicted FLCs using an elastic-plastic constitutive model: a given linear strain-path is incrementally followed until the target FLC is reached. Here, the key point is the formulation of a suitable termination criterion (cf. Eq. 3 below), which can handle pre-strains properly. This approach enables the acquisition of experimentally inaccessible data, such as stresses and work-conjugate equivalent plastic strains corresponding to the utilized constitutive model. This approach is called *forward retro analysis*, because a given FLC is *retrospectively* analyzed. Using concise notation, the forward retro analysis is written as

$$ \mathbf{FLC}^* = \text{retro}(\mathbf{FLC}, \boldsymbol{\varepsilon}^*) \quad (2) $$

In all cases, the following equation is to be solved (in an incremental approach this serves as a termination criterion to stop the accumulation of strain increments):

$$ \|\boldsymbol{\varepsilon}^*\| + \|\mathbf{FLC}^* - \boldsymbol{\varepsilon}^*\| = \|\mathbf{FLC}\| \quad (3) $$

As a length measure, we define the following strain norm, which is a homogeneous function of degree one in terms of the strain components $\varepsilon_1$ and $\varepsilon_2$:

$$ \eta(\varepsilon_1, \varepsilon_2) := \|\boldsymbol{\varepsilon}\| = \sqrt{\frac{2}{3} \cdot [\varepsilon_1^2 + \varepsilon_2^2 + \varepsilon_3^2]}, \quad \varepsilon_3 = -(\varepsilon_1 + \varepsilon_2) \quad (4) $$

Despite the fact that Eq. 4 is identical to the von Mises equivalent plastic strain definition, no isotropic material behavior is implied here, and Eq. 4 should be considered as a material independent and pure kinematic quantity.

To proceed, the strain difference

$$ \Delta\mathbf{FLC}^* = (\Delta\mathbf{FLC}_1^*, \Delta\mathbf{FLC}_2^*)^T := \mathbf{FLC}^* - \boldsymbol{\varepsilon}^* \quad (5) $$

and the following strain ratio

$$ \beta := \frac{\mathbf{FLC}_2}{\mathbf{FLC}_1} = \frac{\Delta\mathbf{FLC}_2^*}{\Delta\mathbf{FLC}_1^*} \quad (6) $$

are defined. In Eq. 6 two equivalent definitions of $\beta$ are provided, and the one to be used depends on which input data are *known*, as will be become clear from the derivations that follow.

Using the strain norm Eq. 4, we may write:

$$ \|\Delta\mathbf{FLC}^*\| = \Delta\mathbf{FLC}_1^* \cdot \eta(1, \beta) \quad \text{with} \quad \beta = \frac{\mathbf{FLC}_2}{\mathbf{FLC}_1} \quad (7) $$
possible to compare materials with different gauges, which is very annoying in practice. This problem is addressed in the present work, and as a solution a simple computational method to remove both the pre-strains and the bending strains from the FLC data is proposed.

For notational convenience, throughout this paper the following 2 × 1 column matrices

$$ \mathbf{FLC} := (\mathbf{FLC}_1, \mathbf{FLC}_2)^T, \quad \mathbf{FLC}^* := (\mathbf{FLC}_1^*, \mathbf{FLC}_2^*)^T \quad \text{and} \quad \boldsymbol{\varepsilon}^* := (\boldsymbol{\varepsilon}_1^*, \boldsymbol{\varepsilon}_2^*)^T \qquad (1) $$

will be used to store a strain tuple pertaining to a single point of the FLC, associated with zero and non-zero pre-strains $\boldsymbol{\varepsilon}^*$, respectively.

## Forward Retro Analysis

The present work started with the development of a straightforward approach to reproduce given experimental or predicted FLCs using an elastic-plastic constitutive model: a given linear strain-path is incrementally followed until the target FLC is reached. Here, the key point is the formulation of a suitable termination criterion (cf. Eq. 3 below), which can handle pre-strains properly. This approach enables the acquisition of experimentally inaccessible data, such as stresses and work-conjugate equivalent plastic strains corresponding to the utilized constitutive model. This approach is called *forward retro analysis*, because a given FLC is *retrospectively* analyzed. Using concise notation, the forward retro analysis is written as

$$ \mathbf{FLC}^* = \text{retro}(\mathbf{FLC}, \boldsymbol{\varepsilon}^*) \qquad (2) $$

In all cases, the following equation is to be solved (in an incremental approach this serves as a termination criterion to stop the accumulation of strain increments):

$$ \|\boldsymbol{\varepsilon}^*\| + \|\mathbf{FLC}^* - \boldsymbol{\varepsilon}^*\| = \|\mathbf{FLC}\| \qquad (3) $$

As a length measure, we define the following strain norm, which is a homogeneous function of degree one in terms of the strain components $\varepsilon_1$ and $\varepsilon_2$:

$$ \eta(\varepsilon_1, \varepsilon_2) := \|\boldsymbol{\varepsilon}\| = \sqrt{\frac{2}{3} \cdot [\varepsilon_1^2 + \varepsilon_2^2 + \varepsilon_3^2]}, \quad \varepsilon_3 = -(\varepsilon_1 + \varepsilon_2) \qquad (4) $$

Despite the fact that Eq. 4 is identical to the von Mises equivalent plastic strain definition, no isotropic material behavior is implied here, and Eq. 4 should be considered as a material independent and pure kinematic quantity.

To proceed, the strain difference

$$ \Delta\mathbf{FLC}^* = (\Delta\mathbf{FLC}_1^*, \Delta\mathbf{FLC}_2^*)^T := \mathbf{FLC}^* - \boldsymbol{\varepsilon}^* \qquad (5) $$

and the following strain ratio

$$ \beta := \frac{\mathbf{FLC}_2}{\mathbf{FLC}_1} = \frac{\Delta\mathbf{FLC}_2^*}{\Delta\mathbf{FLC}_1^*} \qquad (6) $$

are defined. In Eq. 6 two equivalent definitions of $\beta$ are provided, and the one to be used depends on which input data are *known*, as will be become clear from the derivations that follow.

Using the strain norm Eq. 4, we may write:

$$ \|\Delta\mathbf{FLC}^*\| = \Delta\mathbf{FLC}_1^* \cdot \eta(1, \beta) \quad \text{with} \quad \beta = \frac{\mathbf{FLC}_2}{\mathbf{FLC}_1} \qquad (7) $$
Inserting Eq. 7 in Eq. 3 yields the strain differences

$$ \Delta \mathrm{FLC}_1^* = \frac{\|\mathrm{FLC}\| - \|\boldsymbol{\varepsilon}^*\|}{\eta(1, \beta)}, \quad \Delta \mathrm{FLC}_2^* = \beta \cdot \Delta \mathrm{FLC}_1^* \qquad (8) $$

Finally, inserting Eq. 8 into Eq. 5 gives the desired strain solutions

$$ \mathrm{FLC}_i^* = \varepsilon_i^* + \Delta \mathrm{FLC}_i^*, \quad i = 1, 2 \qquad (9) $$

## Inverse Retro Analysis

As shown before, the forward retro analysis can be used to reproduce **FLC**\* using **FLC** and **ε**\* as input data, i.e. **FLC**\* = retro(**FLC**, **ε**\*). While this sounds not very useful at first glance, the true value of this procedure is revealed by inverting the process, called inverse retro analysis:

$$ \mathrm{FLC} = \mathrm{retro}^{-1}(\mathrm{FLC}^*, \boldsymbol{\varepsilon}^*) \qquad (10) $$

Using the strain norm Eq. 4, the right-hand side of Eq. 3 becomes

$$ \|\mathrm{FLC}\| = \mathrm{FLC}_1 \cdot \eta(1, \beta) \quad \text{with} \quad \beta = \frac{\Delta \mathrm{FLC}_2^*}{\Delta \mathrm{FLC}_1^*} \qquad (11) $$

Inserting this expression in Eq. 3 yields the final strain solutions

$$ \mathrm{FLC}_1 = \frac{\|\boldsymbol{\varepsilon}^*\| + \|\mathrm{FLC}^* - \boldsymbol{\varepsilon}^*\|}{\eta(1, \beta)}, \quad \mathrm{FLC}_2 = \beta \cdot \mathrm{FLC}_1 \qquad (12) $$

Most importantly, the inverse retro analysis enables the removal of pre-strain effects from a given FLC, which is very attractive to the sheet metal forming community, because it allows to correct experimental data accordingly, see below.

**Remark.** It should be mentioned that a similar strain-path correction approach was developed by Leppin et al. [6], who incrementally followed the nonlinear strain-paths recorded during their Nakajima tests. Basically, their incremental procedure is the most consistent approach, but it has some important disadvantages: (i) it presumes a measurement system equipped with the necessary data acquisition, which is not available in all laboratories; (ii) the strain measurement errors accumulate; (iii) for the sake of consistency, the bending strain compensation must be incrementally employed, and any approximations associated with the underlying bending correction theory accumulate. Obviously, the latter contributions will bias the integrated strains to some extent. As a final remark, the approach of Leppin et al. [6] accounts implicitly for pre-strains, whereas the approach developed in the present work takes pre-strains explicitly into account, thereby enabling both a forward and an inverse retro analysis.

## Bending Strain Analysis

*Introduction.* The overall objective is to calculate the midplane strains of the deformed Nakajima sample from known (measured) surface strains. For this purpose, the strain theory for shell bending derived in Timoshenko & Gere [10] is employed.

*General equations.* According to Timoshenko & Gere [10, pp. 440-441], the surface strains of a bent shell can be calculated from

$$ e_j(z) = \frac{e_{j,\mathrm{mid}}}{1-z/R_j} - \frac{z}{1-z/R_j} \cdot \left[ \frac{1}{(1-e_{j,\mathrm{mid}}) \cdot \hat{R}_j} - \frac{1}{R_j} \right], \quad j = 1, 2 \qquad (13) $$
Inserting Eq. 7 in Eq. 3 yields the strain differences

$$ \Delta\text{FLC}_1^* = \frac{\|\text{FLC}\| - \|\varepsilon^*\|}{\eta(1,\beta)}, \quad \Delta\text{FLC}_2^* = \beta \cdot \Delta\text{FLC}_1^* \qquad (8) $$

Finally, inserting Eq. 8 into Eq. 5 gives the desired strain solutions

$$ \text{FLC}_i^* = \varepsilon_i^* + \Delta\text{FLC}_i^*, \quad i = 1,2 \qquad (9) $$

## Inverse Retro Analysis

As shown before, the forward retro analysis can be used to reproduce **FLC**\* using **FLC** and **ε**\* as input data, i.e. **FLC**\* = retro(**FLC**, **ε**\*). While this sounds not very useful at first glance, the true value of this procedure is revealed by *inverting* the process, called *inverse retro analysis*:

$$ \text{FLC} = \text{retro}^{-1}(\text{FLC}^*, \varepsilon^*) \qquad (10) $$

Using the strain norm Eq. 4, the right-hand side of Eq. 3 becomes

$$ \|\text{FLC}\| = \text{FLC}_1 \cdot \eta(1, \beta) \quad \text{with} \quad \beta = \frac{\Delta\text{FLC}_2^*}{\Delta\text{FLC}_1^*} \qquad (11) $$

Inserting this expression in Eq. 3 yields the final strain solutions

$$ \text{FLC}_1 = \frac{\|\varepsilon^*\| + \|\text{FLC}^* - \varepsilon^*\|}{\eta(1,\beta)}, \quad \text{FLC}_2 = \beta \cdot \text{FLC}_1 \qquad (12) $$

Most importantly, the *inverse retro analysis* enables the *removal of pre-strain effects* from a given FLC, which is very attractive to the sheet metal forming community, because it allows to correct experimental data accordingly, see below.

**Remark.** It should be mentioned that a similar strain-path correction approach was developed by Leppin et al. [6], who incrementally followed the nonlinear strain-paths recorded during their Nakajima tests. Basically, their incremental procedure is the most consistent approach, but it has some important disadvantages: (i) it presumes a measurement system equipped with the necessary data acquisition, which is not available in all laboratories; (ii) the strain measurement errors accumulate; (iii) for the sake of consistency, the bending strain compensation must be *incrementally* employed, and any approximations associated with the underlying bending correction theory accumulate. Obviously, the latter contributions will bias the integrated strains to some extent. As a final remark, the approach of Leppin et al. [6] accounts *implicitly* for pre-strains, whereas the approach developed in the present work takes pre-strains *explicitly* into account, thereby enabling both a forward and an inverse retro analysis.

## Bending Strain Analysis

*Introduction.* The overall objective is to calculate the *midplane* strains of the deformed Nakajima sample from known (measured) surface strains. For this purpose, the strain theory for shell bending derived in Timoshenko & Gere [10] is employed.

*General equations.* According to Timoshenko & Gere [10, pp. 440-441], the surface strains of a bent shell can be calculated from

$$ e_j(z) = \frac{e_{j,\text{mid}}}{1-z/R_j} - \frac{z}{1-z/R_j} \cdot \left[ \frac{1}{(1-e_{j,\text{mid}}) \cdot \hat{R}_j} - \frac{1}{R_j} \right], \quad j = 1,2 \qquad (13) $$
with $R_j$ and $\hat{R}_j$ denoting the radius of curvature in the undeformed and deformed configuration, respectively. The coordinate $z$ points towards the center point of curvature and is zero in the shell's midplane. Accordingly, at the convex side of the shell $z = -t/2$ and at the concave side $z = +t/2$, with $t$ being the shell's thickness. The respective strain in the midplane is $e_{j,\text{mid}}$. Note that all strains in Eq. 13 are engineering strains. In the undeformed configuration the blank is flat, i.e. $R_j \to \infty$. Accordingly, Eq. 13 simplifies to

$$e_j(z) = e_{j,\text{mid}} - \frac{z}{(1-e_{j,\text{mid}})\cdot\hat{R}_j}, \quad j = 1,2 \qquad (14)$$

We consider the strains at the convex surface of the deformed sample, i.e. at $z = -t/2$, which are assumed of being known from measurements:

$$e_{j,\text{surf}} := e_j(z = -t/2) = e_{j,\text{mid}} + \frac{t/2}{(1-e_{j,\text{mid}})\cdot\hat{R}_j}, \quad j = 1,2 \qquad (15)$$

This may be rearranged to give the following quadratic equation in terms of $e_{j,\text{mid}}$:

$$(e_{j,\text{mid}})^2 + (-1 - e_{j,\text{surf}}) \cdot e_{j,\text{mid}} + \left(e_{j,\text{surf}} - \frac{t/2}{\hat{R}_j}\right) = 0, \quad j = 1,2 \qquad (16)$$

This equation possesses two possible solutions, but only the following one yields physically plausible results:

$$e_{j,\text{mid}} = \frac{1+e_{j,\text{surf}}}{2} - \sqrt{\left(\frac{1+e_{j,\text{surf}}}{2}\right)^2 - \left(e_{j,\text{surf}} - \frac{t/2}{\hat{R}_j}\right)}, \quad j = 1,2 \qquad (17)$$

The engineering midplane strains can be converted to logarithmic strains, and vice versa, according to

$$\varepsilon_j = \ln(1 + e_j) \Leftrightarrow e_j = \exp(\varepsilon_j) - 1, \quad j = 1,2 \qquad (18)$$

*Application to the Nakajima test.* The strain theory may be transferred to the Nakajima test as follows. Regarding the midplane's radius of curvature, we assume

$$\hat{R}_1 = \hat{R}_2 = R_p + t/2 \qquad (19)$$

with the punch radius $R_p$ and the current sheet thickness $t$. The engineering midplane strains $e_{j,\text{mid}}$ can be computed from the measured surface strains at failure (constituting the given Nakajima FLC) using Eq. 17 as follows:

$$e_{j,\text{mid}} = \frac{1+e_{j,\text{surf}}}{2} - \sqrt{\left(\frac{1+e_{j,\text{surf}}}{2}\right)^2 - \left(e_{j,\text{surf}} - \frac{t/2}{R_p+t/2}\right)}, \quad j = 1,2 \qquad (20)$$

Before entering Eq. 20, the logarithmic surface strains must be converted to engineering strains using Eq. 18, and after evaluation of Eq. 20 the engineering strains $e_{j,\text{mid}}$ can be converted back to logarithmic strains.

Using concise notation, we may summarize the outlined computation as follows:
with $R_j$ and $\hat{R}_j$ denoting the radius of curvature in the undeformed and deformed configuration, respectively. The coordinate $z$ points towards the center point of curvature and is zero in the shell's midplane. Accordingly, at the convex side of the shell $z = -t/2$ and at the concave side $z = +t/2$, with $t$ being the shell's thickness. The respective strain in the midplane is $e_{j,\text{mid}}$. Note that all strains in Eq. 13 are engineering strains. In the undeformed configuration the blank is flat, i.e. $R_j \to \infty$. Accordingly, Eq. 13 simplifies to

$$e_j(z) = e_{j,\text{mid}} - \frac{z}{(1-e_{j,\text{mid}})\cdot\hat{R}_j}, \quad j = 1,2 \qquad (14)$$

We consider the strains at the convex surface of the deformed sample, i.e. at $z = -t/2$, which are assumed of being known from measurements:

$$e_{j,\text{surf}} := e_j(z = -t/2) = e_{j,\text{mid}} + \frac{t/2}{(1-e_{j,\text{mid}})\cdot\hat{R}_j}, \quad j = 1,2 \qquad (15)$$

This may be rearranged to give the following quadratic equation in terms of $e_{j,\text{mid}}$:

$$(e_{j,\text{mid}})^2 + (-1 - e_{j,\text{surf}}) \cdot e_{j,\text{mid}} + \left(e_{j,\text{surf}} - \frac{t/2}{\hat{R}_j}\right) = 0, \quad j = 1,2 \qquad (16)$$

This equation possesses two possible solutions, but only the following one yields physically plausible results:

$$e_{j,\text{mid}} = \frac{1+e_{j,\text{surf}}}{2} - \sqrt{\left(\frac{1+e_{j,\text{surf}}}{2}\right)^2 - \left(e_{j,\text{surf}} - \frac{t/2}{\hat{R}_j}\right)}, \quad j = 1,2 \qquad (17)$$

The engineering midplane strains can be converted to logarithmic strains, and vice versa, according to

$$\varepsilon_j = \ln(1 + e_j) \Leftrightarrow e_j = \exp(\varepsilon_j) - 1, \quad j = 1,2 \qquad (18)$$

*Application to the Nakajima test.* The strain theory may be transferred to the Nakajima test as follows. Regarding the midplane's radius of curvature, we assume

$$\hat{R}_1 = \hat{R}_2 = R_p + t/2 \qquad (19)$$

with the punch radius $R_p$ and the current sheet thickness $t$. The engineering midplane strains $e_{j,\text{mid}}$ can be computed from the measured surface strains at failure (constituting the given Nakajima FLC) using Eq. 17 as follows:

$$e_{j,\text{mid}} = \frac{1+e_{j,\text{surf}}}{2} - \sqrt{\left(\frac{1+e_{j,\text{surf}}}{2}\right)^2 - \left(e_{j,\text{surf}} - \frac{t/2}{R_p+t/2}\right)}, \quad j = 1,2 \qquad (20)$$

Before entering Eq. 20, the logarithmic surface strains must be converted to engineering strains using Eq. 18, and after evaluation of Eq. 20 the engineering strains $e_{j,\text{mid}}$ can be converted back to logarithmic strains.

Using concise notation, we may summarize the outlined computation as follows:
1. Input: punch radius $R_p$; initial blank thickness $t_0$; measured surface strains $\varepsilon_{j,surf}$, $j = 1,2$

2. Calculate current blank thickness: $t = t_0 \cdot \exp(-(\varepsilon_{1,surf} + \varepsilon_{2,surf}))$

3. Convert to engineering strains: $e_{j,surf} = \exp(\varepsilon_{j,surf}) - 1, j = 1,2$

4. Calculate midplane strains: $e_{j,\text{mid}} = \text{surface2midplane}(e_{j,surf}, R_p, t), j = 1,2$

5. Convert to logarithmic strains (output): $\varepsilon_{j,\text{mid}} = \ln(1 + e_{j,\text{mid}}), j = 1,2$

The logarithmic midplane strains are regarded as to constitute a bending compensated FLC. In general, the bending strains must be subtracted from the FLC data *before* the pre-strains are removed.

*Remark.* Hill [5, p. 287] rates

$$ \text{neutral plane} \approx \text{central plane throughout bending} \qquad (21) $$

as a good approximation for plane-strain bending, provided that the radius of curvature is larger than four to five times the sheet's thickness. Furthermore, in a detailed analysis, assuming *ideal-plastic* material behavior, Hill [5, p. 293] and Lippmann [7, p. 94] derived the radius of the neutral surface in plane-strain *bending under tension* of sheets as $\hat{R} = \sqrt{R_i \cdot R_o} \cdot \exp(-p/(4k))$, with $R_i$ and $R_o$ being the inner and outer radius of the bent sheet, respectively, *p* the contact pressure caused by the bending tool and *k* the material's shear yield stress. Even this result suggests that the approximation in Eq. 21 can be justified under usual conditions if *p* is not too large compared to the yield stress.

## Application Example I

The intention of the considered application example is to demonstrate the capabilities of the *forward retro analysis*. In the following scenario we consider two user-supplied reference FLCs, which are assumed of being given:

1. A user-supplied FLC based on linear strain-paths, called FLC.

2. A user-supplied FLC based on bi-linear strain-paths, called FLC*.

Both FLCs were calculated using the elastic-plastic CST/M-K necking model, as described in detail in the paper of Aretz [2], because in a simulation model the pre-strains can be prescribed *exactly*, which is impossible in experiments. Bending strains are *absent* in all FLC simulations.

The material considered in all FLC simulations is a rolled sheet made of the aluminium alloy AA2008-T4, whose material data were taken from the paper of Lege et al. [4]. In accordance with the said reference, the yield surface was described by means of the orthotropic non-quadratic “Yld89” yield function (cf. Barlat & Lian [3]), whose material dependent parameters are $a = 0.119316E+01$, $h = 0.115889E+01$, $p = 0.102435E+01$, along with the yield function’s exponent $M = 8$. Isotropic hardening was assumed, and the hardening curve was described using the Voce ansatz function of the form $Y_{\text{ref}}(\bar{\varepsilon}) = A - B \cdot \exp(-C \cdot \bar{\varepsilon})$ with the parameters $A = 408$, $B = 233$ and $C = 6.14$.

<table><caption>Table 1. Definition of the considered pre-straining scenarios.</caption><thead><tr><td>Scenario No.</td><td>Pre-strains {ε<sub>2</sub><sup>*</sup>, ε<sub>1</sub><sup>*</sup>}</td></tr></thead><tbody><tr><td>1</td><td>{-0.04, 0.1}</td></tr><tr><td>2</td><td>{0, 0.1}</td></tr><tr><td>3</td><td>{0.05, 0.05}</td></tr></tbody></table>

In all FLC simulations the *same* material model was used, and only the pre-straining according to Table 1 was varied. The FLC simulation results pertaining to zero and non-zero pre-strains are shown in Fig. 1, labelled as “FLC (sim.)” and “FLC* (sim.)”, respectively. The objective is to use
1. Input: punch radius $R_p$; initial blank thickness $t_0$; measured surface strains $\varepsilon_{j,surf}$, $j = 1,2$

2. Calculate current blank thickness: $t = t_0 \cdot \exp(-(\varepsilon_{1,surf} + \varepsilon_{2,surf}))$

3. Convert to engineering strains: $e_{j,surf} = \exp(\varepsilon_{j,surf}) - 1, j = 1,2$

4. Calculate midplane strains: $e_{j,\text{mid}} = \text{surface2midplane}(e_{j,surf}, R_p, t), j = 1,2$

5. Convert to logarithmic strains (output): $\varepsilon_{j,\text{mid}} = \ln(1 + e_{j,\text{mid}}), j = 1,2$

The logarithmic midplane strains are regarded as to constitute a bending compensated FLC. In general, the bending strains must be subtracted from the FLC data *before* the pre-strains are removed.

*Remark.* Hill [5, p. 287] rates

$$ \text{neutral plane} \approx \text{central plane throughout bending} \qquad (21) $$

as a good approximation for plane-strain bending, provided that the radius of curvature is larger than four to five times the sheet's thickness. Furthermore, in a detailed analysis, assuming *ideal-plastic* material behavior, Hill [5, p. 293] and Lippmann [7, p. 94] derived the radius of the neutral surface in plane-strain *bending under tension* of sheets as $\hat{R} = \sqrt{R_i \cdot R_o} \cdot \exp(-p/(4k))$, with $R_i$ and $R_o$ being the inner and outer radius of the bent sheet, respectively, *p* the contact pressure caused by the bending tool and *k* the material's shear yield stress. Even this result suggests that the approximation in Eq. 21 can be justified under usual conditions if *p* is not too large compared to the yield stress.

## Application Example I

The intention of the considered application example is to demonstrate the capabilities of the *forward retro analysis*. In the following scenario we consider two user-supplied reference FLCs, which are assumed of being given:

1. A user-supplied FLC based on linear strain-paths, called FLC.

2. A user-supplied FLC based on bi-linear strain-paths, called FLC*.

Both FLCs were calculated using the elastic-plastic CST/M-K necking model, as described in detail in the paper of Aretz [2], because in a simulation model the pre-strains can be prescribed *exactly*, which is impossible in experiments. Bending strains are *absent* in all FLC simulations.

The material considered in all FLC simulations is a rolled sheet made of the aluminium alloy AA2008-T4, whose material data were taken from the paper of Lege et al. [4]. In accordance with the said reference, the yield surface was described by means of the orthotropic non-quadratic “Yld89” yield function (cf. Barlat & Lian [3]), whose material dependent parameters are $a = 0.119316E+01$, $h = 0.115889E+01$, $p = 0.102435E+01$, along with the yield function’s exponent $M = 8$. Isotropic hardening was assumed, and the hardening curve was described using the Voce ansatz function of the form $Y_{\text{ref}}(\bar{\varepsilon}) = A - B \cdot \exp(-C \cdot \bar{\varepsilon})$ with the parameters $A = 408$, $B = 233$ and $C = 6.14$.

<table><caption>Table 1. Definition of the considered pre-straining scenarios.</caption><thead><tr><td>Scenario No.</td><td>Pre-strains {$\varepsilon_2^*, \varepsilon_1^*$}</td></tr></thead><tbody><tr><td>1</td><td>{-0.04, 0.1}</td></tr><tr><td>2</td><td>{0, 0.1}</td></tr><tr><td>3</td><td>{0.05, 0.05}</td></tr></tbody></table>

In all FLC simulations the *same* material model was used, and only the pre-straining according to Table 1 was varied. The FLC simulation results pertaining to zero and non-zero pre-strains are shown in Fig. 1, labelled as “FLC (sim.)” and “FLC* (sim.)”, respectively. The objective is to use
the forward retro analysis to calculate FLC* from FLC for the pre-straining scenarios defined in Table 1. The results of these analyses are labelled as “FLC* (forward retro)” in Fig. 1, and reveal that FLC* ≈ retro(FLC, ε*) holds with very good accuracy (compare grey and green curves) in each scenario.

<figure><img src="image_905.png" /><figcaption>Figure 1. Forward retro analysis results for different pre-straining scenarios (cf. Table 1); from left to right: {ε₂*, ε₁*} = {-0.04, 0.1}, {ε₂*, ε₁*} = {0, 0.1} and {ε₂*, ε₁*} = {0.05, 0.05}.</figcaption></figure>

## Application Example II

The intention of the second application example is to demonstrate the capabilities of the inverse retro analysis. Here, the pre-straining scenario no. 1 defined in Table 1 is considered. Figure 2 summarizes the associated results.

The reference FLC shown in Fig. 2 (left), labelled as “Reference (input)”, pertains to zero pre-strains and is identical to the previous application example. Application of the forward retro analysis to the FLC “Reference (input)”, along with the pre-strains {ε₂*, ε₁*} = {-0.04, 0.1}, yields the FLC “Forward retro (output)” in Fig. 2 (left).

In Fig. 2 (right) the process is reversed: the results of the previous forward retro analysis, now labelled as “Forward retro (input)” in Fig. 2 (right), are fed to the inverse retro analysis, yielding the FLC “Inverse retro (output)” in Fig. 2 (right), which is identical to the FLC “Reference (input)” in Fig. 2 (left). This confirms the important theoretical prerequisite

$$ \text{FLC} = \text{retro}^{-1}(\text{retro}(\text{FLC}, \varepsilon^*), \varepsilon^*) \qquad (22) $$

which means that the forward and inverse retro analysis neutralize each other.
the forward retro analysis to calculate FLC* from FLC for the pre-straining scenarios defined in Table 1. The results of these analyses are labelled as “FLC* (forward retro)” in Fig. 1, and reveal that FLC* ≈ retro(FLC, ε*) holds with very good accuracy (compare grey and green curves) in each scenario.

<figure><img src="image_912.png" /><figcaption>Figure 1. Forward retro analysis results for different pre-straining scenarios (cf. Table 1); from left to right: {$ε_2^*, ε_1^*$} = {-0.04, 0.1}, {$ε_2^*, ε_1^*$} = {0, 0.1} and {$ε_2^*, ε_1^*$} = {0.05, 0.05}.</figcaption></figure>

## Application Example II

The intention of the second application example is to demonstrate the capabilities of the inverse retro analysis. Here, the pre-straining scenario no. 1 defined in Table 1 is considered. Figure 2 summarizes the associated results.

The reference FLC shown in Fig. 2 (left), labelled as “Reference (input)”, pertains to zero pre-strains and is identical to the previous application example. Application of the forward retro analysis to the FLC “Reference (input)”, along with the pre-strains {$ε_2^*, ε_1^*$} = {-0.04, 0.1}, yields the FLC “Forward retro (output)” in Fig. 2 (left).

In Fig. 2 (right) the process is reversed: the results of the previous forward retro analysis, now labelled as “Forward retro (input)” in Fig. 2 (right), are fed to the inverse retro analysis, yielding the FLC “Inverse retro (output)” in Fig. 2 (right), which is identical to the FLC “Reference (input)” in Fig. 2 (left). This confirms the important theoretical prerequisite

$$ \text{FLC} = \text{retro}^{-1}(\text{retro}(\text{FLC}, \varepsilon^*), \varepsilon^*) \quad (22) $$

which means that the forward and inverse retro analysis neutralize each other.
<figure><img src="image_906.png" /><figcaption>Figure 2. Inverse retro analysis results for the pre-straining scenario no. 1 (cf. Table 1).</figcaption></figure>

## Application Example III

In the last application example, measured FLCs acquired by Nakajima tests are considered, which are intrinsically biased by pre-strains. Therefore, these FLCs will be labelled as FLC*. Both pre-strains and bending strains will be removed from the FLC* data as described above. In what follows, it is assumed that pre-straining occurred under equibiaxial tension, and that the minor strain value of the pre-strains equals the minor strain of FLC\*<sub>min</sub>, which reads in concise notation $\epsilon_{1}^{*} = \epsilon_{2}^{*} = (\mathbf{FLC}_{\min}^{*})_{2}$. The punch radius used in the Nakajima experiments was $R_{p}$ = 50 mm, which is required for the bending strain calculation. It is important that the bending strain correction is carried out *before* the pre-strain correction.

The considered materials were made of two aluminium alloys of the 5000 class with different gauges (1.5 mm and 2.5 mm, respectively), which leads to different contributions of the respective bending correction. For both materials FLCs were determined according to the DIN EN ISO 12004-2 standard, with 3 repetitions for each sample geometry. The averaged results are shown in Fig. 3 (labelled as “Input”). In the first step, the bending strains were corrected, i.e. the midplane strains were calculated from the measured surface strains, yielding the intermediate results labelled as “Bending correction” in Fig. 3. Thereafter, these midplane strains are fed to the inverse retro analysis, along with the pre-strains $\epsilon_{1}^{*} = \epsilon_{2}^{*} = (\mathbf{FLC}_{\min}^{*})_{2}$, yielding the final results labelled as “Output” in Fig. 3. It is very interesting to observe how the FLCs changed after the final correction step: while the FLC<sub>min</sub>-point of the 1.5 mm material was significantly lifted in major strain direction, this is not the case for the 2.5 mm material, where the FLC<sub>min</sub>-point experiences almost no change in terms of its major strain. The primary reason for this observation is that the thinner material’s FLC\*<sub>min</sub>-point is originally much more displaced along the minor strain axis, which causes a correspondingly large correction by the inverse retro analysis towards a larger major strain value. Furthermore, the bending correction is larger for the thicker material, i.e. the thicker material “looses” more major strain after bending correction than the thinner material, as substantiated below. As expected, after the final correction step FLC<sub>min</sub> coincides with plane strain deformation in both cases, i.e. FLC<sub>min</sub> has moved to the major strain axis.
<figure><img src="image_913.png" /><figcaption>Figure 2. Inverse retro analysis results for the pre-straining scenario no. 1 (cf. Table 1).</figcaption></figure>

## Application Example III

In the last application example, measured FLCs acquired by Nakajima tests are considered, which are intrinsically biased by pre-strains. Therefore, these FLCs will be labelled as FLC*. Both pre-strains and bending strains will be removed from the FLC* data as described above. In what follows, it is assumed that pre-straining occurred under equibiaxial tension, and that the minor strain value of the pre-strains equals the minor strain of FLC\*<sub>min</sub>, which reads in concise notation $\epsilon_{1}^{*} = \epsilon_{2}^{*} = (\mathbf{FLC}_{\min}^{*})_{2}$. The punch radius used in the Nakajima experiments was $R_{p}$ = 50 mm, which is required for the bending strain calculation. It is important that the bending strain correction is carried out *before* the pre-strain correction.

The considered materials were made of two aluminium alloys of the 5000 class with different gauges (1.5 mm and 2.5 mm, respectively), which leads to different contributions of the respective bending correction. For both materials FLCs were determined according to the DIN EN ISO 12004-2 standard, with 3 repetitions for each sample geometry. The averaged results are shown in Fig. 3 (labelled as “Input”). In the first step, the bending strains were corrected, i.e. the midplane strains were calculated from the measured surface strains, yielding the intermediate results labelled as “Bending correction” in Fig. 3. Thereafter, these midplane strains are fed to the inverse retro analysis, along with the pre-strains $\epsilon_{1}^{*} = \epsilon_{2}^{*} = (\mathbf{FLC}_{\min}^{*})_{2}$, yielding the final results labelled as “Output” in Fig. 3. It is very interesting to observe how the FLCs changed after the final correction step: while the FLC<sub>min</sub>-point of the 1.5 mm material was significantly lifted in major strain direction, this is not the case for the 2.5 mm material, where the FLC<sub>min</sub>-point experiences almost no change in terms of its major strain. The primary reason for this observation is that the thinner material’s FLC\*<sub>min</sub>-point is originally much more displaced along the minor strain axis, which causes a correspondingly large correction by the inverse retro analysis towards a larger major strain value. Furthermore, the bending correction is larger for the thicker material, i.e. the thicker material “looses” more major strain after bending correction than the thinner material, as substantiated below. As expected, after the final correction step FLC<sub>min</sub> coincides with plane strain deformation in both cases, i.e. FLC<sub>min</sub> has moved to the major strain axis.
<figure><img src="image_907.png" /><figcaption>Figure 3. Bending correction followed by pre-strain correction using the inverse retro analysis for two materials made of 5xxx aluminium alloys with different compositions and gauges.</figcaption></figure>

<figure><img src="image_908.png" /><figcaption>Figure 4. Direct comparison of the corrected FLC data shown in Fig. 3 (left) and calculated bending strains (right).</figcaption></figure>

In Fig. 4 a direct comparison of both FLCs after the final correction step is shown. In accordance with the main motivation for the present work, it is now possible to compare different materials made of different gauges in terms of their respective FLC. In addition, Fig. 4 does also show the bending contribution in terms of the major and minor strains, calculated as the respective difference between the uncorrected FLC data and the corresponding data after bending correction, in accordance with Affronti & Merklein [1]. As expected, the bending strains are much larger in the case of the thicker material. Furthermore, one may see that the bending strain depends almost linearly on the strain ratio $ε_2/ε_1$: according to the results in Fig. 4, the smallest bending strain occurs in equibiaxial stretching and the largest one in the narrowest samples.
<figure><img src="image_914.png" /><figcaption>Figure 3. Bending correction followed by pre-strain correction using the inverse retro analysis for two materials made of 5xxx aluminium alloys with different compositions and gauges.</figcaption></figure>

<figure><img src="image_915.png" /><figcaption>Figure 4. Direct comparison of the corrected FLC data shown in Fig. 3 (left) and calculated bending strains (right).</figcaption></figure>

In Fig. 4 a direct comparison of both FLCs after the final correction step is shown. In accordance with the main motivation for the present work, it is now possible to compare different materials made of different gauges in terms of their respective FLC. In addition, Fig. 4 does also show the bending contribution in terms of the major and minor strains, calculated as the respective difference between the uncorrected FLC data and the corresponding data after bending correction, in accordance with Affronti & Merklein [1]. As expected, the bending strains are much larger in the case of the thicker material. Furthermore, one may see that the bending strain depends almost linearly on the strain ratio $ε_2/ε_1$: according to the results in Fig. 4, the smallest bending strain occurs in equibiaxial stretching and the largest one in the narrowest samples.
**Summary**

In the present work, a method to comprehensively correct FLC data acquired by means of the Nakajima test in terms of bending strains and pre-strains is proposed, which enables a direct comparison of FLCs pertaining to different materials made of different gauges.

It was demonstrated that the pre-strain correction reproduces theoretical results very well, and that it is self-consistent in the sense that it possesses the ability to neutralize itself, which is an important theoretical prerequisite.

The application of both the pre-strain and the bending correction to experimental data yields plausible results, but there are aspects of the proposed correction method which should be addressed in more detail in future work:

1. Regarding pre-strain correction, the following assumptions were implied: (i) the pre-strains can be identified as $\varepsilon_1^* = \varepsilon_2^* = (\text{FLC}_{\min})_2$ and (ii) these pre-strains can be *universally* assigned to all FLC samples. The question is whether this rather simple approach is sufficiently accurate or needs to be refined.

2. The proposed bending strain correction is based on the shell bending kinematics found in the textbook of Timoshenko & Gere [10, pp. 440-441]. The associated assumptions introduced in the present work (e.g. Eq. 19) should be carefully verified and, if necessary, revised in future work.

**Acknowledgements**

The present work is dedicated to Profs. Dorel Banabic and Frédéric Barlat on the occasion of their retirement. Both have been an invaluable source of inspiration to me.

**References**

[1] E. Affronti, M. Merklein: Analysis of the bending effects and the biaxial pre-straining in sheet metal stretch forming processes for the determination of the forming limits, Int. J. of Mechanical Sciences 138-139 (2018) 295-309. https://doi.org/10.1016/j.ijmecsci.2018.02.024

[2] H. Aretz: An extension of Hill's localized necking model, Int. J. of Engineering Science 48 (2010) 312-331. https://doi.org/10.1016/j.ijengsci.2009.09.007

[3] F. Barlat, J. Lian: Plastic behaviour and stretchability of sheet metals – Part I: A yield function for orthotropic sheets under plane stress condition, Int. J. of Plasticity 5 (1989) 51-66. https://doi.org/10.1016/0749-6419(89)90019-3

[4] D. J. Lege, F. Barlat, J. C. Brem: Characterization and modeling of the mechanical behavior and formability of a 2008-T4 sheet sample, Int. J. of Mechanical Sciences 31 (1989) 549-563. https://doi.org/10.1016/0020-7403(89)90104-5

[5] R. Hill: *The Mathematical Theory of Plasticity*, Oxford University Press, 1950 (Reprinted 2009)

[6] C. Leppin, J. Li, D. Daniel: Application of a method to correct the effect of non-proportional strain paths on Nakajima test based forming limit curves, Proc. Numisheet 2008 Conference, Interlaken, Switzerland

[7] H. Lippmann: *Mechanik des plastischen Fließens*, Springer-Verlag, 1981. https://doi.org/10.1007/978-3-642-52209-3

[8] W. Müschenborn, H.-M. Sonne: Einfluß des Formänderungsweges auf die Grenzformänderungen des Feinblechs, Archiv des Eisenhüttenwesens 46 (1975) 597-602. https://doi.org/10.1002/srin.197503686
**Summary**

In the present work, a method to comprehensively correct FLC data acquired by means of the Nakajima test in terms of bending strains and pre-strains is proposed, which enables a direct comparison of FLCs pertaining to different materials made of different gauges.

It was demonstrated that the pre-strain correction reproduces theoretical results very well, and that it is self-consistent in the sense that it possesses the ability to neutralize itself, which is an important theoretical prerequisite.

The application of both the pre-strain and the bending correction to experimental data yields plausible results, but there are aspects of the proposed correction method which should be addressed in more detail in future work:

1. Regarding pre-strain correction, the following assumptions were implied: (i) the pre-strains can be identified as $\epsilon_1^* = \epsilon_2^* = (\text{FLC}_{\text{min}})^2$ and (ii) these pre-strains can be *universally* assigned to all FLC samples. The question is whether this rather simple approach is sufficiently accurate or needs to be refined.

2. The proposed bending strain correction is based on the shell bending kinematics found in the textbook of Timoshenko & Gere [10, pp. 440-441]. The associated assumptions introduced in the present work (e.g. Eq. 19) should be carefully verified and, if necessary, revised in future work.

**Acknowledgements**

The present work is dedicated to Profs. Dorel Banabic and Frédéric Barlat on the occasion of their retirement. Both have been an invaluable source of inspiration to me.

**References**

[1] E. Affronti, M. Merklein: Analysis of the bending effects and the biaxial pre-straining in sheet metal stretch forming processes for the determination of the forming limits, Int. J. of Mechanical Sciences 138-139 (2018) 295-309. https://doi.org/10.1016/j.ijmecsci.2018.02.024

[2] H. Aretz: An extension of Hill's localized necking model, Int. J. of Engineering Science 48 (2010) 312-331. https://doi.org/10.1016/j.ijengsci.2009.09.007

[3] F. Barlat, J. Lian: Plastic behaviour and stretchability of sheet metals – Part I: A yield function for orthotropic sheets under plane stress condition, Int. J. of Plasticity 5 (1989) 51-66. https://doi.org/10.1016/0749-6419(89)90019-3

[4] D. J. Lege, F. Barlat, J. C. Brem: Characterization and modeling of the mechanical behavior and formability of a 2008-T4 sheet sample, Int. J. of Mechanical Sciences 31 (1989) 549-563. https://doi.org/10.1016/0020-7403(89)90104-5

[5] R. Hill: *The Mathematical Theory of Plasticity*, Oxford University Press, 1950 (Reprinted 2009)

[6] C. Leppin, J. Li, D. Daniel: Application of a method to correct the effect of non-proportional strain paths on Nakajima test based forming limit curves, Proc. Numisheet 2008 Conference, Interlaken, Switzerland

[7] H. Lippmann: *Mechanik des plastischen Fließens*, Springer-Verlag, 1981. https://doi.org/10.1007/978-3-642-52209-3

[8] W. Müschenborn, H.-M. Sonne: Einfluß des Formänderungsweges auf die Grenzformänderungen des Feinblechs, Archiv des Eisenhüttenwesens 46 (1975) 597-602. https://doi.org/10.1002/srin.197503686
[9] J. Noder, C. Butcher: A comparative investigation into the influence of the constitutive model on the prediction of in-plane formability for Nakajima and Marciniak tests, Int. J. of Mechanical Sciences 163 (2019) 105138. https://doi.org/10.1016/j.ijmecsci.2019.105138

[10] S. Timoshenko. J. M. Gere: *Theory of Elastic Stability*, 2nd ed., Dover Publication, 2009
[9] J. Noder, C. Butcher: A comparative investigation into the influence of the constitutive model on the prediction of in-plane formability for Nakajima and Marciniak tests, Int. J. of Mechanical Sciences 163 (2019) 105138. https://doi.org/10.1016/j.ijmecsci.2019.105138

[10] S. Timoshenko. J. M. Gere: *Theory of Elastic Stability*, 2nd ed., Dover Publication, 2009
# Effect of different oxide layer shares on the upsetting of titanium aluminide specimens

DÖRING Sebastian¹,ᵃ*, PEDDINGHAUS Julius¹,ᵇ, BRUNOTTE Kai¹,ᶜ and BEHRENS Bernd-Arno¹,ᵈ

¹Leibniz University Hannover, Institute of Forming Technology and Machines, An der Universität 2, 30823 Garbsen, Germany

ᵃs.doering@ifum.uni-hannover.de, ᵇpeddinghaus@ifum.uni-hannover.de,

ᶜbrunotte@ifum.uni-hannover.de, ᵈbehrens@ifum.uni-hannover.de

**Keywords:** Powder Metallurgy, Lightweight Materials, Hot Forming

**Abstract.** By ball milling in a low-oxygen atmosphere, it was possible to show that titanium aluminides (TiAl) can be processed into components by pressing and sintering in the same atmosphere. The properties (e.g. hardness and density) that can be realised with established processes such as field-assisted sintering (FAST) or hot isostatic pressing (HIP) were not achieved. Pores in the component are closed by forming processes, which improves the mechanical properties. In this work, powder-metallurgically processed TiAl was hot-formed in a low-oxygen atmosphere. The forging parameters and pre-consolidation were characterised with regard to their effect on the component properties. Force, hardness and porosity measurements as well as metallographic analyses were used to evaluate the process and the resulting specimens. It was found that a pre-consolidation and a higher degree of deformation lead to a lower porosity and a higher hardness.

**Introduction**

Titanium aluminides (TiAl) are intermetallic materials that have great application potential in the aerospace industry [1]. These materials are characterised by high temperature resistance combined with a low density of 3.9 - 4.2 g/cm³. As a result, titanium aluminides can replace dense nickel-based superalloys (density approx. 8 g/cm³) in engines and thus reduce fuel consumption [2]. The properties of titanium aluminides are mainly determined by the microstructure, which in turn is determined by the manufacturing and processing methods [3].

In order to achieve high chemical and microstructural homogeneity, TiAl alloys are processed by powder metallurgical methods such as field-assisted sintering (FAST) or hot isostatic pressing (HIP) [4]. When processing TiAl powder using the conventional process route, consisting of pressing and sintering, the low ductility and the oxide layers present a challenge. Titanium aluminides have a high affinity for oxygen, which influences the microstructure development and thus the technological properties [5]. In the previously established FAST and HIP processes, the oxide layers are of secondary importance, as the oxide layers are broken up during the process [6, 7].

By ball milling in an oxygen-reduced atmosphere beforehand, oxide-free particle surfaces can be produced and sustained on the TiAl powder. Prior work has shown that TiAl powder can be conventionally pressed and sintered after such a ball-milling process. To sustain the oxide-free particle surfaces, these subsequent steps must however also be carried out under oxygen-free atmosphere. However, these samples exhibited a certain porosity, and the samples also had a lower hardness than comparable material after casting [8]. In principle, the pores in a component can be closed in a forging process, which increases the strength of the component and thus increases the load-bearing capacity of a component. As the strength correlates with the hardness, the hardness also increases as a result of the forming process. This could increase the component properties of

<figure><img src="image_909.png" /></figure>
# Effect of different oxide layer shares on the upsetting of titanium aluminide specimens

DÖRING Sebastian¹,ᵃ*, PEDDINGHAUS Julius¹,ᵇ, BRUNOTTE Kai¹,ᶜ and BEHRENS Bernd-Arno¹,ᵈ

¹Leibniz University Hannover, Institute of Forming Technology and Machines, An der Universität 2, 30823 Garbsen, Germany

ᵃs.doering@ifum.uni-hannover.de, ᵇpeddinghaus@ifum.uni-hannover.de,

ᶜbrunotte@ifum.uni-hannover.de, ᵈbehrens@ifum.uni-hannover.de

**Keywords:** Powder Metallurgy, Lightweight Materials, Hot Forming

**Abstract.** By ball milling in a low-oxygen atmosphere, it was possible to show that titanium aluminides (TiAl) can be processed into components by pressing and sintering in the same atmosphere. The properties (e.g. hardness and density) that can be realised with established processes such as field-assisted sintering (FAST) or hot isostatic pressing (HIP) were not achieved. Pores in the component are closed by forming processes, which improves the mechanical properties. In this work, powder-metallurgically processed TiAl was hot-formed in a low-oxygen atmosphere. The forging parameters and pre-consolidation were characterised with regard to their effect on the component properties. Force, hardness and porosity measurements as well as metallographic analyses were used to evaluate the process and the resulting specimens. It was found that a pre-consolidation and a higher degree of deformation lead to a lower porosity and a higher hardness.

**Introduction**

Titanium aluminides (TiAl) are intermetallic materials that have great application potential in the aerospace industry [1]. These materials are characterised by high temperature resistance combined with a low density of 3.9 - 4.2 g/cm³. As a result, titanium aluminides can replace dense nickel-based superalloys (density approx. 8 g/cm³) in engines and thus reduce fuel consumption [2]. The properties of titanium aluminides are mainly determined by the microstructure, which in turn is determined by the manufacturing and processing methods [3].

In order to achieve high chemical and microstructural homogeneity, TiAl alloys are processed by powder metallurgical methods such as field-assisted sintering (FAST) or hot isostatic pressing (HIP) [4]. When processing TiAl powder using the conventional process route, consisting of pressing and sintering, the low ductility and the oxide layers present a challenge. Titanium aluminides have a high affinity for oxygen, which influences the microstructure development and thus the technological properties [5]. In the previously established FAST and HIP processes, the oxide layers are of secondary importance, as the oxide layers are broken up during the process [6, 7].

By ball milling in an oxygen-reduced atmosphere beforehand, oxide-free particle surfaces can be produced and sustained on the TiAl powder. Prior work has shown that TiAl powder can be conventionally pressed and sintered after such a ball-milling process. To sustain the oxide-free particle surfaces, these subsequent steps must however also be carried out under oxygen-free atmosphere. However, these samples exhibited a certain porosity, and the samples also had a lower hardness than comparable material after casting [8]. In principle, the pores in a component can be closed in a forging process, which increases the strength of the component and thus increases the load-bearing capacity of a component. As the strength correlates with the hardness, the hardness also increases as a result of the forming process. This could increase the component properties of

<figure><img src="image_916.png" /></figure>
the TiAl sintered parts. However, oxygen also poses a challenge during forming and can lead to material loss [9, 10].

In the present study, the potential of enhancing the properties of TiAl samples produced by pressing and sintering in an oxygen-reduced atmosphere through a subsequent hot forming process in an extremely low-oxygen atmosphere was analysed. In order to analyse the potential of this sinter forging route for TiAl materials, the forging parameters and the prior consolidation are characterised in terms of the effects on the resulting component properties and the process.

## Material and methods

The tests were carried out with the titanium aluminide alloy GE48 (composition in % by weight: 59.60 Ti, 33.00 Al, 2.60 Cr and 4.80 Nb). The TiAl powder was produced by inert gas atomisation with argon; the powder had a spherical shape. The oxygen concentration of the powder is 909 ppmw. 90 % of the powder particles of the initial powder had a particle size below 134 µm. In the cast state according to supplier information, the density of this material is 3.97 g/cm³ and the hardness is 285 HV10 (Gesellschaft für Elektrometallurgie, Nuremberg, Germany).

The study was based on prior work on pressing and sintering of TiAl powder ball-milled under an oxygen-reduced atmosphere, the TiAl powder was processed inside a glovebox (GS Glovebox Systemtechnik GmbH, Malsch, Germany) with an extremely low-oxygen atmosphere. First, it is pre-flushing with argon 5.0 (purity ≥ 99.99 %). Due to the residual oxygen in argon, it is then rinsed with an argon/monosilane mixture (99 % Ar/ 1 % SiH₄). This causes the residual oxygen in the argon 5.0 to react with the monosilane to form silicon dioxide, resulting in a major decrease in the oxygen concentration [11]. The oxygen partial pressure is measured using a sensor box with a lambda probe (L-Probe EM2020, Mesa GmbH, Schmalkalden, Germany). All experiments were carried out at an oxygen partial pressure below 10⁻¹⁸ ppmv [8].

In the glovebox with the low-oxygen atmosphere, the GE48 powder was filled into 2 milling containers (FRITSCH GmbH, Idar-Oberstein, Germany, 250 ml, hardened steel DIN EN 10027-1 X105CrMo17). Each container was filled with 40 g TiAl powder and 345 g steel balls (X105CrMo17) with a diameter of 10 mm. In prior work, toluene ($C_7H_8$) was added to the powder [8]. It was shown that the toluene reacts with the TiAl and the MAX-phase $Ti_2AlC$ is formed. In order to eliminate the influence of the MAX phase on the forming process within these investigations, no toluene was added to the TiAl powder. The ball milling containers were sealed gas-tight inside the box, so that the oxygen partial pressure was below 10⁻¹⁸ ppmv during the entire ball milling process. The ball milling process was carried out with the planetary ball mill Pulverisette 5/4 (FRITSCH GmbH, Idar-Oberstein, Germany). In prior work the active ball milling time was 4 min with a milling speed of 300 1/min. This was followed by an interruption of 4 min. In contrast to prior work with toluene, this cycle was carried out a total of four times to ensure that the TiAl powder does not agglomerate [8, 12].

After the ball milling process, the ball milling containers were reopened inside the glovebox with the low-oxygen atmosphere. Green compacts were then produced from the TiAl powder. Compaction was carried out by coaxial pressing on the manually driven hydraulic press (MSE Teknoloji. Osb/Gebze/Kocaeli, Turkey) inside the glovebox. This hydraulic press is located inside the glovebox in a separate gas-lockable box. In each case, 20 g of TiAl powder was filled into the die and compacted at 800 MPa, the samples had a diameter of 20 mm. These green compacts were then filled into capsules inside the glovebox and sealed gas-tight. The capsules consist of three parts, with 2 lids and a hollow cylinder, the inner diameter of the hollow cylinder and the outer diameter of the lids were 20 mm. A press fit of H6/n6 according to DIN ISO 2768 between hollow cylinder and lid ensures a gas-tight seal. To seal the capsules with the TiAl, the separate box with the hydraulic press (MSE Teknoloji) was closed and a negative pressure of less than 50 mbar was created inside using a vacuum pump. This is necessary to ensure that there are no gas pockets in the capsule which can cause porosity. The lids were then pressed into place using the hydraulic
the TiAl sintered parts. However, oxygen also poses a challenge during forming and can lead to material loss [9, 10].

In the present study, the potential of enhancing the properties of TiAl samples produced by pressing and sintering in an oxygen-reduced atmosphere through a subsequent hot forming process in an extremely low-oxygen atmosphere was analysed. In order to analyse the potential of this sinter forging route for TiAl materials, the forging parameters and the prior consolidation are characterised in terms of the effects on the resulting component properties and the process.

## Material and methods

The tests were carried out with the titanium aluminide alloy GE48 (composition in % by weight: 59.60 Ti, 33.00 Al, 2.60 Cr and 4.80 Nb). The TiAl powder was produced by inert gas atomisation with argon; the powder had a spherical shape. The oxygen concentration of the powder is 909 ppmw. 90 % of the powder particles of the initial powder had a particle size below 134 µm. In the cast state according to supplier information, the density of this material is 3.97 g/cm³ and the hardness is 285 HV10 (Gesellschaft für Elektrometallurgie, Nuremberg, Germany).

The study was based on prior work on pressing and sintering of TiAl powder ball-milled under an oxygen-reduced atmosphere, the TiAl powder was processed inside a glovebox (GS Glovebox Systemtechnik GmbH, Malsch, Germany) with an extremely low-oxygen atmosphere. First, it is pre-flushing with argon 5.0 (purity ≥ 99.99 %). Due to the residual oxygen in argon, it is then rinsed with an argon/monosilane mixture (99 % Ar/ 1 % SiH₄). This causes the residual oxygen in the argon 5.0 to react with the monosilane to form silicon dioxide, resulting in a major decrease in the oxygen concentration [11]. The oxygen partial pressure is measured using a sensor box with a lambda probe (L-Probe EM2020, Mesa GmbH, Schmalkalden, Germany). All experiments were carried out at an oxygen partial pressure below 10⁻¹⁸ ppmv [8].

In the glovebox with the low-oxygen atmosphere, the GE48 powder was filled into 2 milling containers (FRITSCH GmbH, Idar-Oberstein, Germany, 250 ml, hardened steel DIN EN 10027-1 X105CrMo17). Each container was filled with 40 g TiAl powder and 345 g steel balls (X105CrMo17) with a diameter of 10 mm. In prior work, toluene ($C_7H_8$) was added to the powder [8]. It was shown that the toluene reacts with the TiAl and the MAX-phase $Ti_2AlC$ is formed. In order to eliminate the influence of the MAX phase on the forming process within these investigations, no toluene was added to the TiAl powder. The ball milling containers were sealed gas-tight inside the box, so that the oxygen partial pressure was below 10⁻¹⁸ ppmv during the entire ball milling process. The ball milling process was carried out with the planetary ball mill Pulverisette 5/4 (FRITSCH GmbH, Idar-Oberstein, Germany). In prior work the active ball milling time was 4 min with a milling speed of 300 1/min. This was followed by an interruption of 4 min. In contrast to prior work with toluene, this cycle was carried out a total of four times to ensure that the TiAl powder does not agglomerate [8, 12].

After the ball milling process, the ball milling containers were reopened inside the glovebox with the low-oxygen atmosphere. Green compacts were then produced from the TiAl powder. Compaction was carried out by coaxial pressing on the manually driven hydraulic press (MSE Teknoloji. Osb/Gebze/Kocaeli, Turkey) inside the glovebox. This hydraulic press is located inside the glovebox in a separate gas-lockable box. In each case, 20 g of TiAl powder was filled into the die and compacted at 800 MPa, the samples had a diameter of 20 mm. These green compacts were then filled into capsules inside the glovebox and sealed gas-tight. The capsules consist of three parts, with 2 lids and a hollow cylinder, the inner diameter of the hollow cylinder and the outer diameter of the lids were 20 mm. A press fit of H6/n6 according to DIN ISO 2768 between hollow cylinder and lid ensures a gas-tight seal. To seal the capsules with the TiAl, the separate box with the hydraulic press (MSE Teknoloji) was closed and a negative pressure of less than 50 mbar was created inside using a vacuum pump. This is necessary to ensure that there are no gas pockets in the capsule which can cause porosity. The lids were then pressed into place using the hydraulic
press. The closure ensures that the following forming outside of the glovebox can also take place in an oxygen-reduced atmosphere. With regard to the choice of material and wall thicknesses, the design of the capsules was based on Wan et al. and Li et al. who have carried out investigations on the forming of cast and encapsulated semi-finished TiAl products [13, 14]. All components were made of 1.4301 steel and the capsules had a length of 32 mm and an outer diameter of 30 mm. To investigate the effects of the oxide-free surfaces during forming, 20 g of unground and therefore oxidized GE48 powder was filled into capsules as a reference. As it is not possible to produce green compacts from untreated TiAl powder by die pressing due to the oxide layers, the powder was filled into the capsules without compaction [8].

Half of the capsules with ground TiAl powder and half of the capsules with unground TiAl powder were then subjected to a sintering process to identify the effect of prior consolidation and the interaction with possible further consolidation through the hot forming process. This results in 4 different process routes for this study. This sinter process was carried out in a sintering tube furnace (Thermconcept ROC, Bremen, Germany) at 1150 °C for 60 min at a heating rate of 5 K/min. The temperature selected for the sintering process is based on previous FAST experiments in which a desired near γ microstructure was successfully achieved at this temperature [15]. The furnace is directly connected to the glovebox via a cooled flange to guarantee a transfer under the required low-oxygen atmosphere. The sintering furnace is supplied with the low-oxygen atmosphere of the glovebox by a fan, so that the oxygen partial pressure during sintering was also below 10⁻¹⁸ ppmv. The other half of the samples without sintering are analysed in terms of the potential for a direct powder forging process, investigated by Jiang et al. for nickel-based superalloys [16].

For the hot forming tests, the corresponding samples were heated in an oven at the selected temperature for 60 minutes so that the sample had a homogeneous heat distribution. During the forming process, the force was recorded with the force measuring device (see Figure 1) based on strain gauges. The maximum force was recorded in order to analyse the effect of the different process routes and the forming parameters on the deformation behaviour and mechanisms. The maximum correlates with the highest degree of deformation towards the end of the forming operation and is therefore most significant to characterise forming.

<figure><img src="image_910.png" /><figcaption>Figure 1. Setup of the forming tests</figcaption></figure>
press. The closure ensures that the following forming outside of the glovebox can also take place in an oxygen-reduced atmosphere. With regard to the choice of material and wall thicknesses, the design of the capsules was based on Wan et al. and Li et al. who have carried out investigations on the forming of cast and encapsulated semi-finished TiAl products [13, 14]. All components were made of 1.4301 steel and the capsules had a length of 32 mm and an outer diameter of 30 mm. To investigate the effects of the oxide-free surfaces during forming, 20 g of unground and therefore oxidized GE48 powder was filled into capsules as a reference. As it is not possible to produce green compacts from untreated TiAl powder by die pressing due to the oxide layers, the powder was filled into the capsules without compaction [8].

Half of the capsules with ground TiAl powder and half of the capsules with unground TiAl powder were then subjected to a sintering process to identify the effect of prior consolidation and the interaction with possible further consolidation through the hot forming process. This results in 4 different process routes for this study. This sinter process was carried out in a sintering tube furnace (Thermconcept ROC, Bremen, Germany) at 1150 °C for 60 min at a heating rate of 5 K/min. The temperature selected for the sintering process is based on previous FAST experiments in which a desired near γ microstructure was successfully achieved at this temperature [15]. The furnace is directly connected to the glovebox via a cooled flange to guarantee a transfer under the required low-oxygen atmosphere. The sintering furnace is supplied with the low-oxygen atmosphere of the glovebox by a fan, so that the oxygen partial pressure during sintering was also below 10⁻¹⁸ ppmv. The other half of the samples without sintering are analysed in terms of the potential for a direct powder forging process, investigated by Jiang et al. for nickel-based superalloys [16].

For the hot forming tests, the corresponding samples were heated in an oven at the selected temperature for 60 minutes so that the sample had a homogeneous heat distribution. During the forming process, the force was recorded with the force measuring device (see Figure 1) based on strain gauges. The maximum force was recorded in order to analyse the effect of the different process routes and the forming parameters on the deformation behaviour and mechanisms. The maximum correlates with the highest degree of deformation towards the end of the forming operation and is therefore most significant to characterise forming.

<figure><img src="image_917.png" /><figcaption>Figure 1. Setup of the forming tests</figcaption></figure>
All samples were hot-formed on a Dunkes HD 250 (S. Dunkes GmbH Maschinenfabrik, Kirchheim unter Teck, Germany) hydraulic press. The forming parameters were varied individually in order to investigate the influence of the forming temperature, pressing speed and the degree of forming on the process and the properties of the samples. To determine the degree of deformation from the initial height h₀ before upsetting and the final height h after upsetting, the following formula is used:

$$ \varphi = \ln(h/h_0) \qquad (1) $$

The parameters in Table 1 were used in the tests, which were carried out according to a partial factorial test plan. This results in 7 forming processes for each specimen preparation state, resulting in a total of 28 tests for this study. The parameters used are based on previous hot forging tests in which recrystallisation was observed at higher forming speeds and degrees [17]. Due to the gas-tight closure of the capsules, each forming process was carried out in an oxygen-reduced atmosphere.

<table><caption>Table 1. Applied forming parameters</caption><thead><tr><td>Degree of forming φ</td><td>Pressing speed v</td><td>Forming temperature U<sub>T</sub></td></tr></thead><tbody><tr><td>-1.16 [-]</td><td>10 [mm/s]</td><td>1050 [°C]</td></tr><tr><td>-0.76 [-]</td><td>30 [mm/s]</td><td>1150 [°C]</td></tr><tr><td>-0.47 [-]</td><td>50 [mm/s]</td><td>1250 [°C]</td></tr></tbody></table>

After the forming tests, cross-sectional samples were produced to determine the properties of the titanium aluminide. In order to be able to evaluate the mechanical properties despite the limited specimen dimensions, hardness measurements according to Vickers HV1 were carried out using a Qness Q10A hardness [10]. In the centre of each specimen, 15 hardness measurements in accordance with ISO 6507 were carried out and averaged. A confocal LED microscope (Smartproof 5, ZEISS, Oberkochen, Germany) was used to quantify the porosity in the centre of the sample. The cross-section surface was scanned in an area of 550 µm x 480 µm. To differentiate between roughness due to specimen preparation and depressions in the surface caused by porosity, a tolerance plane of 0.25 µm below the highest peak was applied in accordance to [18]. Any cavity below this plane is defined as porosity. The areas with the indentations were compared with the scanned cross-sectional area. Evaluating the surface porosity ratio of the specimens allows a comparative analysis of the densification behavior of the applied forming processes. In addition, global stitched images of the entire cross-section of the sample were taken with a Leica DM6 light microscope (Leica, Wetzlar, Germany) in order to make statements about the overall microstructural homogeneity of the formed specimens. By applying these analytical methods, to establish a deeper understanding of the correlation between the pre-consolidation of low-oxygen processed powder and the resulting properties.

## Results and discussion

### Process forces

For a qualitative statement on the influence of pre-consolidation on the process force, the maximum force was measured for each test. The maximum force was reached at the end of the forming process. The results of the maximum force is listed in Table 2 for each test with different specimen conditions and forming parameters. The results show that the influence of the forming parameters on the maximum force during the forming process is greater than the influence of the process route of the TiAl powder. As expected, the process forces increase with lower forming temperature, a higher degree of forming and a higher pressing speed. There is no clear influence of milling and sintering before hot forming on the process forces. The oxide layers, which inhibit particle bonding and thus minimise the strength, do not result in a significant reduction in the
All samples were hot-formed on a Dunkes HD 250 (S. Dunkes GmbH Maschinenfabrik, Kirchheim unter Teck, Germany) hydraulic press. The forming parameters were varied individually in order to investigate the influence of the forming temperature, pressing speed and the degree of forming on the process and the properties of the samples. To determine the degree of deformation from the initial height h₀ before upsetting and the final height h after upsetting, the following formula is used:

$$ \varphi = \ln(h/h_0) \qquad (1) $$

The parameters in Table 1 were used in the tests, which were carried out according to a partial factorial test plan. This results in 7 forming processes for each specimen preparation state, resulting in a total of 28 tests for this study. The parameters used are based on previous hot forging tests in which recrystallisation was observed at higher forming speeds and degrees [17]. Due to the gas-tight closure of the capsules, each forming process was carried out in an oxygen-reduced atmosphere.

<table><caption>Table 1. Applied forming parameters</caption><thead><tr><td>Degree of forming φ</td><td>Pressing speed v</td><td>Forming temperature U<sub>T</sub></td></tr></thead><tbody><tr><td>-1.16 [-]</td><td>10 [mm/s]</td><td>1050 [°C]</td></tr><tr><td>-0.76 [-]</td><td>30 [mm/s]</td><td>1150 [°C]</td></tr><tr><td>-0.47 [-]</td><td>50 [mm/s]</td><td>1250 [°C]</td></tr></tbody></table>

After the forming tests, cross-sectional samples were produced to determine the properties of the titanium aluminide. In order to be able to evaluate the mechanical properties despite the limited specimen dimensions, hardness measurements according to Vickers HV1 were carried out using a Qness Q10A hardness [10]. In the centre of each specimen, 15 hardness measurements in accordance with ISO 6507 were carried out and averaged. A confocal LED microscope (Smartproof 5, ZEISS, Oberkochen, Germany) was used to quantify the porosity in the centre of the sample. The cross-section surface was scanned in an area of 550 µm x 480 µm. To differentiate between roughness due to specimen preparation and depressions in the surface caused by porosity, a tolerance plane of 0.25 µm below the highest peak was applied in accordance to [18]. Any cavity below this plane is defined as porosity. The areas with the indentations were compared with the scanned cross-sectional area. Evaluating the surface porosity ratio of the specimens allows a comparative analysis of the densification behavior of the applied forming processes. In addition, global stitched images of the entire cross-section of the sample were taken with a Leica DM6 light microscope (Leica, Wetzlar, Germany) in order to make statements about the overall microstructural homogeneity of the formed specimens. By applying these analytical methods, to establish a deeper understanding of the correlation between the pre-consolidation of low-oxygen processed powder and the resulting properties.

## Results and discussion

### Process forces

For a qualitative statement on the influence of pre-consolidation on the process force, the maximum force was measured for each test. The maximum force was reached at the end of the forming process. The results of the maximum force is listed in Table 2 for each test with different specimen conditions and forming parameters. The results show that the influence of the forming parameters on the maximum force during the forming process is greater than the influence of the process route of the TiAl powder. As expected, the process forces increase with lower forming temperature, a higher degree of forming and a higher pressing speed. There is no clear influence of milling and sintering before hot forming on the process forces. The oxide layers, which inhibit particle bonding and thus minimise the strength, do not result in a significant reduction in the
forming forces. Overall, the forces for each forming parameter setup were highest for the ground and sintered variants. This can possibly be caused by the stronger particle cohesion after pressing and sintering and therefore stronger overall material strength which must be overcome in forming.

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

Porosity

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