forming and fracture limits from experiments for neither criteria. However, with *Sc* = 6, the TTSIC criterion can be said to predict conservative results compared to the experimental fracture strains, and the shape of the predicted fracture curves seems reasonable. For the OCLFC, *Sc* = 2 was sufficient to obtain a reasonable shape for the fracture limit diagram and predict conservative results. Larger scale factors were therefore not tested for OCLFC. The magnitudes of the predicted fracture strains were clearly underestimated for strain paths in vicinity to the two plane-strain axes, while strain paths near the equibiaxial axis displayed better agreement with the experimental data. Although the introduction of thickness perturbations helped, the OCLFC was not able to recreate the shape of the fracture strain curves in a satisfying way.

## Conclusions

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

## References

[1] A. Graf, W. Hosford, Effect of Changing Strain Paths on Forming Limit Diagrams of Al 2008-T4, Metall. Trans. A. 24 (1993) 2503-2512.

[2] A. Graf, W. Hosford, Influence of strain-path changes on forming limit diagrams of Al 6111 T4, Int. J. Mech. Sci. 36(10) (1994) 897-8910.

[3] D. Vysochinskiy, T. Coudert, O.S. Hopperstad, O.-G. Lademo, A. Reyes, Experimental study on the formability of AA6016 sheets pre-strained by rolling, Int. J. Mater. Forming. 11(4) (2018). 541-557. https://10.1007/s12289-017-1363-6

[4] D. Vysochinskiy, Formability of aluminium alloy subjected to prestrain by rolling, PhD Thesis, Department of Structural Engineering, Trondheim, Norwegian University of Science and Technology, 2014.

[5] W. Muhammad, U. Ali, A. P. Brahme, J. Kang, R. K. Mishra, K. Inal, Experimental analyses and numerical modeling of texture evolution and the development of surface roughness during bending of an extruded aluminum alloy using a multiscale modeling framework, Int. J. Plast. 117 (2019) 93-121. https://doi.org/10.1016/j.ijplas.2017.09.013

[6] Y. Hu, G. Zhou, R. Liu, X. Yuan, L. Cao, B. Yang, D. Li, P. Wu, On the correlation between roping, texture, and morphology of aluminium alloy sheets, J. Mater. Res. Technol. 26 (2023) 571-586. https://doi.org/10.1016/j.jmrt.2023.07.209

[7] Y. Hu, G. Zhou, X. Yuan, D. Li, L. Cao, W. Zhang, P. Wu, An artificial neural network-based model for roping prediction in aluminum alloy sheet, Acta Mater. 245 (2023) 118605. https://doi.org/10.1016/j.actamat.2022.118605

[8] O. Engler, C. Schäfer, H. J. Brinkman, Crystal-plasticity simulation of the correlation of microtexture and roping in AA 6xxx Al-Mg-Si sheet alloys for automotive applications, Acta Mater. 60(13-14) (2012) 5217-5232. http://dx.doi.org/10.1016/j.actamat.2012.06.039
[9] O. Engler, H. Moo-Young, C. N. Tome, Crystal-plasticity analysis of ridging in ferritic stainless steel sheets, Metall. Mater. Trans. A (Physical Metallurgy and Materials Science). 36A(11) (2005) 3127-3139.

[10] L. Qin, M. Seefeldt, P. Van Houtte, Analysis of roping of aluminum sheet materials based on the meso-scale moving window approach, Acta Mater. 84 (2015) 215-228. https://doi.org/10.1016/j.actamat.2014.10.054

[11] H. Jin, Breakup of texture alignment and reduction of roping in AA6111 aluminium alloy, Mater. Sci. Technol. 33(11) (2017) 1388-1396. https://doi.org/10.1080/02670836.2017.1295518

[12] S. Tveit, Formability analysis of AA6016-T4 aluminium alloy sheets subjected to roping, MSc Thesis, Department of Civil Engineering and Energy Technology, Oslo, Oslo Metropolitan University, 2020.

[13] Z. Marciniak, K. Kuczynski, Limit strains in the process of stretch-forming sheet metal, Int. J. Mech. Sci. 9 (1967) 600-620.

[14] J. O. Hallquist, Theoretical Manual, Livermore Software Technology Corporation, Livermore, California, 1998.

[15] LS-DYNA Keyword User's Manual. Volume II Material Models, Version 971 R6.1.0, Livermore Software Technology Corporation, Livermore, California, 2012.

[16] H. Aretz, Applications of a new plane stress yield function to orthotropic steel and aluminium sheet metals, Modelling Simul. Mater. Sci. Eng. 12 (2004) 491-509.

[17] J. D. Bressan, J. A. Williams, The use of a shear instability criterion to predict local necking in sheet metal deformation, Int. J. Mech. Sci. 25(3) (1983) 155-168.

[18] M. G. Cockcroft, D. J. Latham, Ductility and the Workability of Metals, Journal of the institute of metals. 96 (1968) 33-39.

[19] P. D. Wu, D. J. Lloyd, Analysis of surface roughening in AA6111 automotive sheet, Acta Mater. 52(7) (2004) 1785-1798. https://doi.org/10.1016/j.actamat.2003.12.039

[20] H. Aretz, A comparison between geometrical and material imperfections in localized necking prediction, in: E. Cueto, F. Chinesta (Ed.), Proc. 10th ESAFORM Conference on Material Forming, AIP Conference Proceedings, 2007, pp. 287-292.

[21] MATLAB & Simulink, Sum of Sine Models - R2020a, MathWorks Nordic, [Online]. Read on https://se.mathworks.com/help/curvefit/sum-of-sine.html the 7th April 2020
[9] O. Engler, H. Moo-Young, C. N. Tome, Crystal-plasticity analysis of ridging in ferritic stainless steel sheets, Metall. Mater. Trans. A (Physical Metallurgy and Materials Science). 36A(11) (2005) 3127-3139.

[10] L. Qin, M. Seefeldt, P. Van Houtte, Analysis of roping of aluminum sheet materials based on the meso-scale moving window approach, Acta Mater. 84 (2015) 215-228. https://doi.org/10.1016/j.actamat.2014.10.054

[11] H. Jin, Breakup of texture alignment and reduction of roping in AA6111 aluminium alloy, Mater. Sci. Technol. 33(11) (2017) 1388-1396. https://doi.org/10.1080/02670836.2017.1295518

[12] S. Tveit, Formability analysis of AA6016-T4 aluminium alloy sheets subjected to roping, MSc Thesis, Department of Civil Engineering and Energy Technology, Oslo, Oslo Metropolitan University, 2020.

[13] Z. Marciniak, K. Kuczynski, Limit strains in the process of stretch-forming sheet metal, Int. J. Mech. Sci. 9 (1967) 600-620.

[14] J. O. Hallquist, Theoretical Manual, Livermore Software Technology Corporation, Livermore, California, 1998.

[15] LS-DYNA Keyword User's Manual. Volume II Material Models, Version 971 R6.1.0, Livermore Software Technology Corporation, Livermore, California, 2012.

[16] H. Aretz, Applications of a new plane stress yield function to orthotropic steel and aluminium sheet metals, Modelling Simul. Mater. Sci. Eng. 12 (2004) 491-509.

[17] J. D. Bressan, J. A. Williams, The use of a shear instability criterion to predict local necking in sheet metal deformation, Int. J. Mech. Sci. 25(3) (1983) 155-168.

[18] M. G. Cockcroft, D. J. Latham, Ductility and the Workability of Metals, Journal of the institute of metals. 96 (1968) 33-39.

[19] P. D. Wu, D. J. Lloyd, Analysis of surface roughening in AA6111 automotive sheet, Acta Mater. 52(7) (2004) 1785-1798. https://doi.org/10.1016/j.actamat.2003.12.039

[20] H. Aretz, A comparison between geometrical and material imperfections in localized necking prediction, in: E. Cueto, F. Chinesta (Ed.), Proc. 10th ESAFORM Conference on Material Forming, AIP Conference Proceedings, 2007, pp. 287-292.

[21] MATLAB & Simulink, Sum of Sine Models - R2020a, MathWorks Nordic, [Online]. Read on https://se.mathworks.com/help/curvefit/sum-of-sine.html the 7th April 2020
# Local ductility in steel sheet, towards a practical and robust measurement method

LINGBEEK Roald<fnref n="1" />,ª*, LAROUR Patrick<fnref n="2" />,ᵇ, RUCK Astrid<fnref n="1" />,ᶜ, BEIER Thorsten<fnref n="3" />,ᵈ, CORNETTE Dominique<fnref n="4" />,ᵉ, SCHADOW Thomas<fnref n="5" />,>f and WESTHÄUSER Sebastian<fnref n="6" />,⁹

**Keywords:** Local Ductility, Sheet Metal, AHSS, Measurement Robustness

**Abstract.** Local ductility has become a useful and significant mechanical property of sheet metal, particularly for advanced-high-strength-steels (AHSS) [1, 2]. The widely accepted hole expansion test (HET) [3] has two problems: It mixes up the effects of shear cutting with the intrinsic material ductility, and the variance of test results is large. As a complementary approach, a workgroup consisting of universities, steel producers, automotive suppliers and OEMs have developed VDA-guideline 238-110 [4] for industrial application of the *TTS* (True Thickness Strain at fracture at minimum thickness) and *TFS* (True Fracture Strain from area reduction), to quantify local ductility based on the postmortem microscope analysis of the fracture surface of tensile test samples. In 18 laboratories, more than 1440 tensile tests have been performed on a sample batch of 3 AHSS grades (CR440Y780T-DH, CR780Y980T-CP, and HR660Y760T-CP). Using ASTM, JIS and ISO tensile specimens, each laboratory determined the conventional mechanical properties as well as *TTS* and *TFS*. For comparison HET were also performed. The measurement results were analyzed statistically in terms of gauge repeatability and reproducibility (GRR). Measurement variance for *TTS* is comparable to elongation at breakage and significantly lower than HET. *TTS* results clearly show superior local ductility of the -CP steels over -DH. An analysis of fracture morphology shows differences between the hot-rolled and cold-rolled -CP grades and a tendency for the -DH steel to fail at the edge. For -CP steels, *TTS* and *TFS* results are indistinguishable, for the -DH grade, the *TFS* value is slightly higher than *TTS*, likely a side-effect of higher global ductility.

**Introduction**

The complex nature of advanced high strength steels (AHSS), used to develop high-performance and lightweight components, presents challenges. Microstructures of diverse phases give rise to intricate forming and fracture mechanisms that cannot be encompassed within the broad definition of formability. It is thus vital to differentiate between “local” and “global” formability. Global formability refers to a material's ability to undergo plastic deformation without the emergence of local necking. It can be adequately described by the Forming Limit Curve (FLC) or alternatively by the true uniform strain at necking initiation [5, 6]. Local formability refers to a material's capacity to undergo plastic deformation in a specific region without fracturing [5, 6, 7].

<figure><img src="image_940.png" /></figure>

<footnote n="1">¹Autoliv B.V. & Co KG, Otto Hahn Strasse 4, 25337 Elmshorn, Germany</footnote>

<footnote n="2">²Voestalpine Stahl GmbH, voestalpine-Straße 3, 4020 Linz, Austria</footnote>

<footnote n="3">³Thyssenkrupp Steel Europe AG, 47166 Duisburg, Germany</footnote>

<footnote n="4">⁴ArcelorMittal Maizières, Voie Romaine, BP 30320F-57283 Maizières-lès-Metz Cedex, France</footnote>

<footnote n="5">⁵VOLKSWAGEN AG, Post Box 1437, D-38436 Wolfsburg, Germany</footnote>

<footnote n="6">⁶Salzgitter Mannesmann Forschung GmbH, Eisenhüttenstraße 99, 38239 Salzgitter, Germany</footnote>

<footnote>ªroald.lingbeek@autoliv.com, ᵇpatrick.larour@voestalpine.com, ᶜastrid.ruck@autoliv.com,</footnote>

<footnote>ᵈthorsten.beier@thyssenkrupp.com, ᶜdominique.cornette@arcelormittal.com,</footnote>

<footnote>>fthomas.schadow@volkswagen.de, ᵈs.westhaeuser@sz.szmf.de</footnote>
# Local ductility in steel sheet, towards a practical and robust measurement method

LINGBEEK Roald<fnref n="1" />,ª*, LAROUR Patrick<fnref n="2" />,ᵇ, RUCK Astrid<fnref n="1" />,ᶜ, BEIER Thorsten<fnref n="3" />,ᵈ, CORNETTE Dominique<fnref n="4" />,ᵉ, SCHADOW Thomas<fnref n="5" />,ғ and WESTHÄUSER Sebastian<fnref n="6" />,:g

**Keywords:** Local Ductility, Sheet Metal, AHSS, Measurement Robustness

**Abstract.** Local ductility has become a useful and significant mechanical property of sheet metal, particularly for advanced-high-strength-steels (AHSS) [1, 2]. The widely accepted hole expansion test (HET) [3] has two problems: It mixes up the effects of shear cutting with the intrinsic material ductility, and the variance of test results is large. As a complementary approach, a workgroup consisting of universities, steel producers, automotive suppliers and OEMs have developed VDA-guideline 238-110 [4] for industrial application of the *TTS* (True Thickness Strain at fracture at minimum thickness) and *TFS* (True Fracture Strain from area reduction), to quantify local ductility based on the postmortem microscope analysis of the fracture surface of tensile test samples. In 18 laboratories, more than 1440 tensile tests have been performed on a sample batch of 3 AHSS grades (CR440Y780T-DH, CR780Y980T-CP, and HR660Y760T-CP). Using ASTM, JIS and ISO tensile specimens, each laboratory determined the conventional mechanical properties as well as *TTS* and *TFS*. For comparison HET were also performed. The measurement results were analyzed statistically in terms of gauge repeatability and reproducibility (*GRR*). Measurement variance for *TTS* is comparable to elongation at breakage and significantly lower than HET. *TTS* results clearly show superior local ductility of the -CP steels over -DH. An analysis of fracture morphology shows differences between the hot-rolled and cold-rolled -CP grades and a tendency for the -DH steel to fail at the edge. For -CP steels, *TTS* and *TFS* results are indistinguishable, for the -DH grade, the *TFS* value is slightly higher than *TTS*, likely a side-effect of higher global ductility.

**Introduction**

The complex nature of advanced high strength steels (AHSS), used to develop high-performance and lightweight components, presents challenges. Microstructures of diverse phases give rise to intricate forming and fracture mechanisms that cannot be encompassed within the broad definition of formability. It is thus vital to differentiate between “local” and “global” formability. Global formability refers to a material's ability to undergo plastic deformation without the emergence of local necking. It can be adequately described by the Forming Limit Curve (FLC) or alternatively by the true uniform strain at necking initiation [5, 6]. Local formability refers to a material's capacity to undergo plastic deformation in a specific region without fracturing [5, 6, 7].

<figure><img src="image_947.png" /></figure>

<footnote n="1">¹Autoliv B.V. & Co KG, Otto Hahn Strasse 4, 25337 Elmshorn, Germany</footnote>

<footnote n="2">²Voestalpine Stahl GmbH, voestalpine-Straße 3, 4020 Linz, Austria</footnote>

<footnote n="3">³Thyssenkrupp Steel Europe AG, 47166 Duisburg, Germany</footnote>

<footnote n="4">⁴ArcelorMittal Maizières, Voie Romaine, BP 30320F-57283 Maizières-lès-Metz Cedex, France</footnote>

<footnote n="5">⁵VOLKSWAGEN AG, Post Box 1437, D-38436 Wolfsburg, Germany</footnote>

<footnote n="6">⁶Salzgitter Mannesmann Forschung GmbH, Eisenhüttenstraße 99, 38239 Salzgitter, Germany</footnote>

<footnote>ªroald.lingbeek@autoliv.com, ᵇpatrick.larour@voestalpine.com, ᶜastrid.ruck@autoliv.com,</footnote>

<footnote>ᵈthorsten.beier@thyssenkrupp.com, ᶜdominique.cornette@arcelormittal.com,</footnote>

<footnote>fthomas.schadow@volkswagen.de, gſ.s.westhaeuser@sz.szmf.de</footnote>
Local ductility is an inherent property independent of edge cutting conditions. The difference between both distinct concepts such as local formability and crack resistivity has been highlighted in [8]. A bending test, for example according to [9], is not universally suitable for material differentiation since it often does not fail for materials with tensile strength below 1000MPa [10]. Also, this test only yields a pass/fail outcome, not a quantitative local ductility value. Two other common methods for characterization of local ductility are the hole expansion test (HET) as per ISO 16630 [3], and the hole tensile test (HTT) [11, 12]. The HET has a disadvantage of producing results with high scattering due to the influence of the stamping process and individual interpretation of crack initiation – this results in a poor gauge repeatability and reproducibility (GRR) value [13]. To avoid any potential pre-damage at the edge, it is possible to conduct the HTT using a mechanically machined or electrically discharge machined hole. Strain must be measured locally via digital image correlation (DIC), which is laborious in industrial practice, also, there are measurement resolution issues at the edge. A tensile test with a notched sample is another method employed in material characterization for crash simulations, but this requires a separate sample. TTS and TFS [1] data can be more easily derived from a standard tensile test, which is done anyway when a steel coil is released from production.

## State of the Art for the TTS and TFS measurement method

This paper compares TTS and TFS, in terms of measurement robustness, to HET and classical mechanical properties, in a large-scale round-robin test. The tests and sample types are shown in Table 1. All specimens are prepared via milling or EDM. Additional HET-specimens are prepared with a punched hole, which is done in only one tool shop to keep the cutting-edge conditions equal. The tests were performed in 18 test facilities (at OEMs, tier1-suppliers, universities and steel making companies).

The derivation of TFS has been described in detail by Hance, and the ASTM E8 standard also includes suggestions for evaluation [6, 14]. Hance uses the logarithmic reduction of area at fracture TFS and relates it to the true uniform strain ($A_g$) for a better understanding of relative intrinsic formability characteristics in a so-called formability map. The determination of the fracture surface size, required for TFS, is challenging in terms of microscope techniques and missing automation. Instead of TFS, Heibel at al [5, 7] for instance refer true thickness strain (TTS) at fracture at minimum thickness of the fracture surface. This makes the measurement less challenging, but sufficient to capture a steel's local ductility [15].

<table><caption>Table 1: Scope of round robin test (L: longitudinal, T: transverse direction) [16, 17]</caption><thead><tr><th rowspan="2"></th><th colspan="6">Standard tensile test acc. to</th><th rowspan="2">Notch tensile test</th><th rowspan="2">HET ISO16630</th></tr><tr><th colspan="2">ASTM</th><th colspan="2">DIN ISO</th><th colspan="2">JIS</th></tr><tr><th>Test dir.</th><th>L</th><th>T</th><th>L</th><th>T</th><th>L</th><th>T</th><th>T</th><th>-</th></tr></thead><tbody><tr><th>Edge</th><td colspan="7">Milling/ EDM</td><td>Punch & EDM</td></tr><tr><th>Specimen</th><td>ISO6892 Type 1<br>b<sub>0</sub>= 12.5mm<br>L<sub>0</sub>= 50mm</td><td>ISO6892 Type 2<br>b<sub>0</sub>= 20mm<br>L<sub>0</sub>= 80mm</td><td>ISO6892 Type 3<br>b<sub>0</sub>= 25mm<br>L<sub>0</sub>= 50mm</td><td>R5 notched<br>b<sub>0</sub>= 10mm<br>L<sub>c</sub>=100mm</td><td colspan="2"></td><td>100 x100mm<br>with 10mm<br>hole diameter</td><td></td></tr><tr><th>Material</th><td colspan="7">CR440Y780T-DH 1.5mm / CR780Y980T-CP 1.5mm / HR660Y760T-CP 3mm</td><td></td></tr></tbody></table>

The testing and analysis requirements and recommendations are defined as follows:

**Conditions for the tensile test:**

* 10 samples, 7 valid tests necessary (breakage outside gauge length not accepted)

**Specifications for analysis/measurements:**

* Measurement of the thickness of the fracture surface (right/left edge, right/left
Local ductility is an inherent property independent of edge cutting conditions. The difference between both distinct concepts such as local formability and crack resistivity has been highlighted in [8]. A bending test, for example according to [9], is not universally suitable for material differentiation since it often does not fail for materials with tensile strength below 1000MPa [10]. Also, this test only yields a pass/fail outcome, not a quantitative local ductility value. Two other common methods for characterization of local ductility are the hole expansion test (HET) as per ISO 16630 [3], and the hole tensile test (HTT) [11, 12]. The HET has a disadvantage of producing results with high scattering due to the influence of the stamping process and individual interpretation of crack initiation – this results in a poor gauge repeatability and reproducibility (GRR) value [13]. To avoid any potential pre-damage at the edge, it is possible to conduct the HTT using a mechanically machined or electrically discharge machined hole. Strain must be measured locally via digital image correlation (DIC), which is laborious in industrial practice, also, there are measurement resolution issues at the edge. A tensile test with a notched sample is another method employed in material characterization for crash simulations, but this requires a separate sample. TTS and TFS [1] data can be more easily derived from a standard tensile test, which is done anyway when a steel coil is released from production.

## State of the Art for the TTS and TFS measurement method

This paper compares TTS and TFS, in terms of measurement robustness, to HET and classical mechanical properties, in a large-scale round-robin test. The tests and sample types are shown in Table 1. All specimens are prepared via milling or EDM. Additional HET-specimens are prepared with a punched hole, which is done in only one tool shop to keep the cutting-edge conditions equal. The tests were performed in 18 test facilities (at OEMs, tier1-suppliers, universities and steel making companies).

The derivation of TFS has been described in detail by Hance, and the ASTM E8 standard also includes suggestions for evaluation [6, 14]. Hance uses the logarithmic reduction of area at fracture TFS and relates it to the true uniform strain ($A_g$) for a better understanding of relative intrinsic formability characteristics in a so-called formability map. The determination of the fracture surface size, required for TFS, is challenging in terms of microscope techniques and missing automation. Instead of TFS, Heibel at al [5, 7] for instance refer true thickness strain (TTS) at fracture at minimum thickness of the fracture surface. This makes the measurement less challenging, but sufficient to capture a steel's local ductility [15].

<table><caption>Table 1: Scope of round robin test (L: longitudinal, T: transverse direction) [16, 17]</caption><thead><tr><th rowspan="2"></th><th colspan="6">Standard tensile test acc. to</th><th rowspan="2">Notch tensile test</th><th rowspan="2">HET ISO16630</th></tr><tr><th colspan="2">ASTM</th><th colspan="2">DIN ISO</th><th colspan="2">JIS</th></tr><tr><th>Test dir.</th><th>L</th><th>T</th><th>L</th><th>T</th><th>L</th><th>T</th><th>T</th><th>-</th></tr></thead><tbody><tr><th>Edge</th><td colspan="7">Milling/ EDM</td><td>Punch & EDM</td></tr><tr><th>Specimen</th><td>ISO6892 Type 1<br>b<sub>0</sub>= 12.5mm<br>L<sub>0</sub>= 50mm</td><td>ISO6892 Type 2<br>b<sub>0</sub>= 20mm<br>L<sub>0</sub>= 80mm</td><td>ISO6892 Type 3<br>b<sub>0</sub>= 25mm<br>L<sub>0</sub>= 50mm</td><td>R5 notched<br>b<sub>0</sub>= 10mm<br>L<sub>c</sub>=100mm</td><td>100 x100mm<br>with 10mm<br>hole diameter</td><td></td><td></td><td></td></tr><tr><th>Material</th><td colspan="8">CR440Y780T-DH 1.5mm / CR780Y980T-CP 1.5mm / HR660Y760T-CP 3mm</td></tr></tbody></table>

The testing and analysis requirements and recommendations are defined as follows:

**Conditions for the tensile test:**

* 10 samples, 7 valid tests necessary (breakage outside gauge length not accepted)

**Specifications for analysis/measurements:**

* Measurement of the thickness of the fracture surface (right/left edge, right/left
* Execution according to SEP 1240 [17] at constant strain rate 0.004/s, testing speed of 19,2 mm/min (ISO type 2, L₀=80mm) and 12 mm/min (ASTM, JIS type 1,3, L₀=50mm), R5 notched tensile tests at 5mm/min.

* Recording of force displacement curve

* Inspection of fractured surface

quarter, center, minimum) on one sample half

* Flat tilting angle for imaging, apparent non projected fracture width (at mid thickness plane) from fracture surface ASTM E8 [14] §7.12.3, fracture force acc. ISO 6892-1 appendix A.3.6.1 [16]

The measurement of thickness and width imposes high demands on imaging of the fracture surface. VDA238-110 summarizes this aspect and gives practical examples on specimen holding and the definition of the tilting angle (Fig. 1). Minimum thickness is defined as the lowest measurable thickness within the fracture surface. From Fig.1, the relative position of the thinnest spot is $u_{rel} = \left| \frac{x}{1/2w_a} \right|$, with $w_a$ the width of the sample in the necking zone. From the minimum thickness logarithmic true plastic thickness strain TTS is determined as follows:

$$TTS = \ln\left(\frac{t_0}{t_{min}}\right). \qquad (1)$$

The apparent fracture surface $S_a$ for TFS is determined according to ASTM E8 recommendations from the thickness at the edges and the center, see [6]. Reduction of area respectively TFS are determined from the apparent fracture area $S_a$ follows [4] with horizontal tilted width $w_u$

$$w_u = w_a \cdot \cos \alpha. \qquad (2)$$

$$S_a = 1/6 (t_{left} + 4 \cdot t_{center} + t_{right}) \cdot w_u. \qquad (3)$$

$$TFS = \ln\left(\frac{s_0}{s_a}\right). \qquad (4)$$

<figure><img src="image_941.png" /><figcaption>Figure 1: Measurements of $t_{min}$ on a broken tensile test specimen [4]</figcaption></figure>

## The concept of GRR and analysis a small subset of samples

Gauge Repeatability and Reproducibility (GRR) is a statistical concept to quantify the variance in a measurement procedure. Repeatability is defined as variance in measurements obtained with one measuring instrument when used several times by a single appraiser (here, laboratory) on the same part [18]. It is commonly referred to as “test equipment variation” or “variation within”. Reproducibility is defined as variance in the groups of measurements with respect to appraiser (here, laboratory) when measuring a characteristic on one part. It is commonly referred to as “variation between the appraiser (laboratory)”.

This paragraph analyses GRR on a manually selected subset of 18 broken tensile test samples. It includes a sample for each of the three materials, in ASTM, JIS and ISO specimen shapes, at high (MAX) and low level (MIN) of thickness reduction. Each of these 18 samples has been tested
* Execution according to SEP 1240 [17] at constant strain rate 0.004/s, testing speed of 19,2 mm/min (ISO type 2, L₀=80mm) and 12 mm/min (ASTM, JIS type 1,3, L₀=50mm), R5 notched tensile tests at 5mm/min.

* Recording of force displacement curve

* Inspection of fractured surface

quarter, center, minimum) on one sample half

* Flat tilting angle for imaging, apparent non projected fracture width (at mid thickness plane) from fracture surface ASTM E8 [14] §7.12.3, fracture force acc. ISO 6892-1 appendix A.3.6.1 [16]

The measurement of thickness and width imposes high demands on imaging of the fracture surface. VDA238-110 summarizes this aspect and gives practical examples on specimen holding and the definition of the tilting angle (Fig. 1). Minimum thickness is defined as the lowest measurable thickness within the fracture surface. From Fig.1, the relative position of the thinnest spot is $u_{rel} = \left| \frac{x}{1/2w_a} \right|$, with $w_a$ the width of the sample in the necking zone. From the minimum thickness logarithmic true plastic thickness strain TTS is determined as follows:

$$TTS = \ln\left(\frac{t_0}{t_{min}}\right). \qquad (1)$$

The apparent fracture surface $S_a$ for TFS is determined according to ASTM E8 recommendations from the thickness at the edges and the center, see [6]. Reduction of area respectively TFS are determined from the apparent fracture area $S_a$ follows [4] with horizontal tilted width $w_u$

$$w_u = w_a \cdot \cos \alpha. \qquad (2)$$

$$S_a = 1/6 (t_{left} + 4 \cdot t_{center} + t_{right}) \cdot w_u. \qquad (3)$$

$$TFS = \ln\left(\frac{s_0}{s_a}\right). \qquad (4)$$

<figure><img src="image_948.png" /><figcaption>Figure 1: Measurements of $t_{min}$ on a broken tensile test specimen [4]</figcaption></figure>

## The concept of GRR and analysis a small subset of samples

Gauge Repeatability and Reproducibility (GRR) is a statistical concept to quantify the variance in a measurement procedure. Repeatability is defined as variance in measurements obtained with one measuring instrument when used several times by a single appraiser (here, laboratory) on the same part [18]. It is commonly referred to as “test equipment variation” or “variation within”. Reproducibility is defined as variance in the groups of measurements with respect to appraiser (here, laboratory) when measuring a characteristic on one part. It is commonly referred to as “variation between the appraiser (laboratory)”.

This paragraph analyses GRR on a manually selected subset of 18 broken tensile test samples. It includes a sample for each of the three materials, in ASTM, JIS and ISO specimen shapes, at high (MAX) and low level (MIN) of thickness reduction. Each of these 18 samples has been tested
multiple times, at different laboratories. To obtain maximum information at reduced effort, a “design of experiments” (DOE) scheme was used to distribute the samples across the labs. Fig. 2 defines where samples were tested, and shows the resulting variance on the measurement of $t_{min}$.

<figure><img src="image_942.png" /><figcaption>Figure 2: Measurements of $t_{min}$ for the subset of 18 samples</figcaption></figure>

<figure><img src="image_943.png" /><figcaption>Figure 3: %CV (total) for various thicknesses measured in the subset of 18 samples</figcaption></figure>

Fig. 3 shows measurement variance as %$CV_{total}$, with $s_{total}$, the standard deviation of all measurements, combining repeatability and reproducibility (as well as sample-appraiser interaction).

$$CV_{total} = 100 \frac{s_{total}}{\bar{x}} \qquad (5)$$

Overall, the $CV_{total}$ of $t_{edge}$ is higher than $t_{center}$, $t_{1/4}$ and higher than $t_{min}$ – note, on identical samples. The variance of the latter three quantities is comparable. It is difficult to differentiate between sample type (ASTM, ISO, JIS) and MIN, or MAX samples and material.
multiple times, at different laboratories. To obtain maximum information at reduced effort, a “design of experiments” (DOE) scheme was used to distribute the samples across the labs. Fig. 2 defines where samples were tested, and shows the resulting variance on the measurement of $t_{min}$.

<figure><img src="image_949.png" /><figcaption>Figure 2: Measurements of $t_{min}$ for the subset of 18 samples</figcaption></figure>

<figure><img src="image_950.png" /><figcaption>Figure 3: %CV (total) for various thicknesses measured in the subset of 18 samples</figcaption></figure>

Fig. 3 shows measurement variance as %CV$_{total}$, with $s_{total}$, the standard deviation of all measurements, combining repeatability and reproducibility (as well as sample-appraiser interaction).

$$CV_{total} = 100 \frac{s_{total}}{\bar{x}} \qquad (5)$$

Overall, the $CV_{total}$ of $t_{edge}$ is higher than $t_{center}$, $t_{1/4}$ and higher than $t_{min}$ – note, on identical samples. The variance of the latter three quantities is comparable. It is difficult to differentiate between sample type (ASTM, ISO, JIS) and MIN, or MAX samples and material.
## GRR of measurements, using all 1440 samples

The CV of an $A_{80}$ or HER measurement *cannot* be assessed in the same way as Fig. 3: Each $A_{80}$ or HER determination requires an individual destructive test. Hence, for all the graphs in this section below, the entire measurement procedure, including the execution of the tensile test, is assessed integrally. As no two tensile test samples break in the exact same way, the fundamental stochastic nature of breakage (and variance across the steel coil width) is encapsulated in the presented results: From a statistical standpoint, specimen variation and repeatability are mixed up. In ASTM A691 [19] terminology, the *reproducibility* standard deviation is $s_R$ (which in fact describes the *total variance*) is:

$$s_R = \sqrt{s_L^2 + s_R^2} \qquad (6)$$

Where $s_L^2$ is the *between-laboratory variance*, and $s_R$ the repeatability standard deviation. In following graphs, the coefficient of variation $CV_r$ will be used for repeatability, $CV_R$ for reproducibility:

$$CV_r = 100 \frac{s_R}{\bar{x}}, \quad CV_R = 100 \frac{s_R}{\bar{x}} \qquad (7)$$

Table 2 shows the outcome of tensile tests for all three tested steel grades. In Fig. 4, clearly the tensile strength $R_m$ can be determined with the lowest CV (the lowest scatter). All breakage strain values, $A_{80}$, TTS, TFS and HER show higher coefficients of variation, which can be explained in terms of the inherently stochastic nature of breakage.

<table><caption>Table 2: Mean values of measured quantities over all labs results.</caption><thead><tr><th rowspan="3">Material</th><th rowspan="3">Test Dir.</th><th colspan="6">ISO 6892 Tensile Type 2 L0=80mm</th><th colspan="2">HER</th></tr><tr><th>R<sub>m</sub><br>[MPa]</th><th>R<sub>p02</sub><br>[MPa]</th><th>A<sub>80</sub></th><th>TTS<br>(log)</th><th>TFS<br>(log)</th><th>TTS<br>(log)</th><th>EDM</th><th>Punched</th></tr><tr><th></th><th></th><th></th><th></th><th></th><th></th><th>Notch</th><th></th><th></th></tr></thead><tbody><tr><td rowspan="2">CR440Y780T-DH</td><td>L</td><td>785</td><td>482</td><td>22%</td><td>0.65</td><td>0.76</td><td>-</td><td rowspan="2">87%</td><td rowspan="2">29%</td></tr><tr><td>T</td><td>793</td><td>493</td><td>19%</td><td>0.60</td><td>0.72</td><td>0.58</td></tr><tr><td rowspan="2">CR780Y980T-CP</td><td>L</td><td>1008</td><td>917</td><td>8%</td><td>1.09</td><td>1.06</td><td>-</td><td rowspan="2">150%</td><td rowspan="2">75%</td></tr><tr><td>T</td><td>1006</td><td>918</td><td>8%</td><td>1.05</td><td>1.01</td><td>0.75</td></tr><tr><td rowspan="2">HR660Y760T-CP</td><td>L</td><td>868</td><td>694</td><td>12%</td><td>1.06</td><td>0.97</td><td>-</td><td rowspan="2">121%</td><td rowspan="2">27%</td></tr><tr><td>T</td><td>896</td><td>784</td><td>11%</td><td>0.97</td><td>0.91</td><td>0.76</td></tr></tbody></table>

Fig 5 shows the values of the TTS calculation for every of the 1440 tensile tests, as well as the relative location $u_{rel} = \left| \frac{x}{1/2w_a} \right|$ of the thinnest point on the x-axis (value 0 means, in sample center line, 1.0 means at the edge, see Fig. 1). Both CP grades show a significantly higher local ductility than the DH grade. The CR780Y980T-CP grade breaks consistently in a shear mode and the location of the thinnest point of the broken sample appears random. In contrast, the HR660Y780T-CP consistently breaks in the middle of the sample, but for this steel grade, the morphology of the breakage zone shows no tendency. CR440Y780T-DH shows the location of the thinnest spot either in the middle, or on the edge of the sample.
## GRR of measurements, using all 1440 samples

The CV of an $A_{80}$ or HER measurement *cannot* be assessed in the same way as Fig. 3: Each $A_{80}$ or HER determination requires an individual destructive test. Hence, for all the graphs in this section below, the entire measurement procedure, including the execution of the tensile test, is assessed integrally. As no two tensile test samples break in the exact same way, the fundamental stochastic nature of breakage (and variance across the steel coil width) is encapsulated in the presented results: From a statistical standpoint, specimen variation and repeatability are mixed up. In ASTM A691 [19] terminology, the *reproducibility* standard deviation is $s_R$ (which in fact describes the *total variance*) is:

$$s_R = \sqrt{s_L^2 + s_R^2} \qquad (6)$$

Where $s_L^2$ is the *between-laboratory variance*, and $s_R$ the repeatability standard deviation. In following graphs, the coefficient of variation $CV_r$ will be used for repeatability, $CV_R$ for reproducibility:

$$CV_r = 100 \frac{s_R}{\bar{x}}, \quad CV_R = 100 \frac{s_R}{\bar{x}} \qquad (7)$$

Table 2 shows the outcome of tensile tests for all three tested steel grades. In Fig. 4, clearly the tensile strength $R_m$ can be determined with the lowest CV (the lowest scatter). All breakage strain values, $A_{80}$, TTS, TFS and HER show higher coefficients of variation, which can be explained in terms of the inherently stochastic nature of breakage.

<table><caption>Table 2: Mean values of measured quantities over all labs results.</caption><thead><tr><th rowspan="2">Material</th><th rowspan="2">Test Dir.</th><th rowspan="2">R<sub>m</sub><br>[MPa]</th><th rowspan="2">R<sub>p02</sub><br>[MPa]</th><th rowspan="2">A<sub>80</sub></th><th rowspan="2">TTS<br>(log)</th><th rowspan="2">TFS<br>(log)</th><th rowspan="2">TTS<br>(log)</th><th colspan="2">HER</th></tr><tr><th>ISO 6892 Tensile Type 2 L0=80mm</th><th>Notch</th><th>EDM</th><th>Punched</th></tr></thead><tbody><tr><td rowspan="2">CR440Y780T-DH</td><td>L</td><td>785</td><td>482</td><td>22%</td><td>0.65</td><td>0.76</td><td>-</td><td rowspan="2">87%</td><td rowspan="2">29%</td></tr><tr><td>T</td><td>793</td><td>493</td><td>19%</td><td>0.60</td><td>0.72</td><td>0.58</td></tr><tr><td rowspan="2">CR780Y980T-CP</td><td>L</td><td>1008</td><td>917</td><td>8%</td><td>1.09</td><td>1.06</td><td>-</td><td rowspan="2">150%</td><td rowspan="2">75%</td></tr><tr><td>T</td><td>1006</td><td>918</td><td>8%</td><td>1.05</td><td>1.01</td><td>0.75</td></tr><tr><td rowspan="2">HR660Y760T-CP</td><td>L</td><td>868</td><td>694</td><td>12%</td><td>1.06</td><td>0.97</td><td>-</td><td rowspan="2">121%</td><td rowspan="2">27%</td></tr><tr><td>T</td><td>896</td><td>784</td><td>11%</td><td>0.97</td><td>0.91</td><td>0.76</td></tr></tbody></table>

Fig 5 shows the values of the TTS calculation for every of the 1440 tensile tests, as well as the relative location $u_{rel} = \left| \frac{x}{1/2w_a} \right|$ of the thinnest point on the x-axis (value 0 means, in sample center line, 1.0 means at the edge, see Fig. 1). Both CP grades show a significantly higher local ductility than the DH grade. The CR780Y980T-CP grade breaks consistently in a shear mode and the location of the thinnest point of the broken sample appears random. In contrast, the HR660Y780T-CP consistently breaks in the middle of the sample, but for this steel grade, the morphology of the breakage zone shows no tendency. CR440Y780T-DH shows the location of the thinnest spot either in the middle, or on the edge of the sample.
<figure><img src="image_944.png" /><figcaption>Figure 4: CV~r~ and CV~R~ scattering for various tests</figcaption></figure>

<figure><img src="image_945.png" /><figcaption>Figure 5: TTS vs. location of thinnest point, failure mode, split by material and test orientation</figcaption></figure>

<figure><img src="image_946.png" /><figcaption>Figure 6: Morphology of the breakage zone for the three steel grades</figcaption></figure>
<figure><img src="image_951.png" /><figcaption>Figure 4: CV~r~ and CV~R~ scattering for various tests</figcaption></figure>

<figure><img src="image_952.png" /><figcaption>Figure 5: TTS vs. location of thinnest point, failure mode, split by material and test orientation</figcaption></figure>

<figure><img src="image_953.png" /><figcaption>Figure 6: Morphology of the breakage zone for the three steel grades</figcaption></figure>
Fig. 6 and 7 show the influence of the chosen sample type on morphology of the breakage zone, and the location of the thinnest spot as absolute relative distance in more detail. For the DH and HR660Y760T-CP, the ASTM sample type shows a larger tendency to a “normal” failure.

<figure><img src="image_947.png" /><figcaption>Figure 7: TTS location of thinnest spot (Rel. distance = 1.0 is at sample edge, 0.0 at center line)</figcaption></figure>

Fig. 8 shows the *TTS* for each tensile test, sorted by laboratory. This gives an impression of reproducibility: overall spread of the measurement appears similar for all laboratories. The cold rolled materials do not show a clear rolling-direction (material orientation) effect, but the hot rolled steel does. Overall, the sample type (ASTM, JIS, ISO) does not make a clear difference to the measured TTS value.

Fig. 9 below shows *t*min value measured on one of the broken sample sections (“side1”) statistically yields the same result when measured at the corresponding other broken section (“side2”). Note, this also visualizes the repeatability variance.

Fig. 10 shows the correlation between *TTS* and *TFS*, calculated with the same broken sample as basis. This statistically shows, *TTS* and *TFS* are quasi-identical, although the -DH steel grade shows a slight offset: *TFS* is marginally higher than *TTS*.

<figure><img src="image_948.png" /><figcaption>Figure 8: TTS by lab, sample type and orientation</figcaption></figure>
Fig. 6 and 7 show the influence of the chosen sample type on morphology of the breakage zone, and the location of the thinnest spot as absolute relative distance in more detail. For the DH and HR660Y760T-CP, the ASTM sample type shows a larger tendency to a “normal” failure.

<figure><img src="image_954.png" /><figcaption>Figure 7: TTS location of thinnest spot (Rel. distance = 1.0 is at sample edge, 0.0 at center line)</figcaption></figure>

Fig. 8 shows the *TTS* for each tensile test, sorted by laboratory. This gives an impression of reproducibility: overall spread of the measurement appears similar for all laboratories. The cold rolled materials do not show a clear rolling-direction (material orientation) effect, but the hot rolled steel does. Overall, the sample type (ASTM, JIS, ISO) does not make a clear difference to the measured TTS value.

Fig. 9 below shows *t*min value measured on one of the broken sample sections (“side1”) statistically yields the same result when measured at the corresponding other broken section (“side2”). Note, this also visualizes the repeatability variance.

Fig. 10 shows the correlation between *TTS* and *TFS*, calculated with the same broken sample as basis. This statistically shows, *TTS* and *TFS* are quasi-identical, although the -DH steel grade shows a slight offset: *TFS* is marginally higher than *TTS*.

<figure><img src="image_955.png" /><figcaption>Figure 8: TTS by lab, sample type and orientation</figcaption></figure>
<figure><img src="image_949.png" /><figcaption>Figure 9: $t_{min}$ at two sides of broken specimen</figcaption></figure>

<figure><img src="image_950.png" /><figcaption>Figure 10: TTS vs TFS correlation</figcaption></figure>

<figure><img src="image_951.png" /><figcaption>Figure 11: HER results, sorted by laboratory and colored by sample-manufacturing method</figcaption></figure>

Finally, Fig. 11 visualizes the well-reported issues with the *HET*: the difference between punched and wire eroded (EDM) samples is large. The variance for the results in a single lab is considerable already, but the difference from laboratory to laboratory is very large, also for EDM manufactured samples – particularly when compared to the *TTS* results in Fig. 10.

## Discussion

To assess the robustness of *TTS* and *TFS* measurements, firstly, the total variance of repeated thickness measurements on single specimens has been analyzed. Fig. 3 shows, for *t*<sub>min</sub>, *CV*<sub>total</sub> is in order of magnitude of 5%, which represents the “bottom line variance”, of a thickness measurement on a broken tensile test specimen. *CV*<sub>total</sub> is higher for *t*<sub>edge</sub>.

Using the entire data set of 1440 tensile tests, Fig. 4 shows that *CV*<sub>R</sub> values around 8% for *TTS* and *TFS* are comparable to those for breakage elongation (*A*<sub>80</sub>). This demonstrates *TTS* and *TFS* as industrially usable measurement quantities. Only for CR440Y780T-DH, overall variance *CV*<sub>R</sub> (and particularly, the repeatability *CV*<sub>r</sub>) is significantly worse for *TTS*. This is due to the high mean value of *A*<sub>80</sub> and low mean *TTS* value for the (global ductility focused) dual-phase steel.

While *CV*<sub>r</sub> (single lab variance) of *HET* with EDM samples is similar to *TTS* and *TFS*, the *CV*<sub>R</sub> (overall variance) for punched-sample *HET* is by far the highest of all material quantities. Fig. 11 shows: Specimens with EDM edges have similar spread in absolute values, but the mean value of *HER* is lower for punched samples. This leads to such a high coefficient of variation, that the
<figure><img src="image_956.png" /><figcaption>Figure 9: $t_{min}$ at two sides of broken specimen</figcaption></figure>

<figure><img src="image_957.png" /><figcaption>Figure 10: TTS vs TFS correlation</figcaption></figure>

<figure><img src="image_958.png" /><figcaption>Figure 11: HER results, sorted by laboratory and colored by sample-manufacturing method</figcaption></figure>

Finally, Fig. 11 visualizes the well-reported issues with the *HET*: the difference between punched and wire eroded (EDM) samples is large. The variance for the results in a single lab is considerable already, but the difference from laboratory to laboratory is very large, also for EDM manufactured samples – particularly when compared to the *TTS* results in Fig. 10.

Discussion

To assess the robustness of *TTS* and *TFS* measurements, firstly, the total variance of repeated thickness measurements on single specimens has been analyzed. Fig. 3 shows, for *t*<sub>min</sub>, *CV*<sub>total</sub> is in order of magnitude of 5%, which represents the “bottom line variance”, of a thickness measurement on a broken tensile test specimen. *CV*<sub>total</sub> is higher for *t*<sub>edge</sub>.

Using the entire data set of 1440 tensile tests, Fig. 4 shows that $CV_R$ values around 8% for *TTS* and *TFS* are comparable to those for breakage elongation ($A_{80}$). This demonstrates *TTS* and *TFS* as industrially usable measurement quantities. Only for CR440Y780T-DH, overall variance $CV_R$ (and particularly, the repeatability $CV_r$) is significantly worse for *TTS*. This is due to the high mean value of $A_{80}$ and low mean *TTS* value for the (global ductility focused) dual-phase steel.

While $CV_r$ (single lab variance) of *HET* with EDM samples is similar to *TTS* and *TFS*, the $CV_R$ (overall variance) for punched-sample *HET* is by far the highest of all material quantities. Fig. 11 shows: Specimens with EDM edges have similar spread in absolute values, but the mean value of *HER* is lower for punched samples. This leads to such a high coefficient of variation, that the
usefulness of the hole expansion test must be questioned. The lab-by-lab differences for HR660Y760T-CP stand out regarding this aspect.

Fig. 10 shows that *TTS* and *TFS* are basically equivalent: hence, the authors favor the simpler *TTS* measurement. Interestingly, in case of the -DH steel, *TFS* is slightly higher than *TTS*: In high global formability grades (such as -DH) the width reduction contribution to the overall fracture area increases (more diffuse and local necking with higher work hardening capability). *TFS* includes a contribution of both thickness and width reduction, leading to a comparatively high value, while *TTS* is only representing the thickness reduction. *TTS* may be a more conservative, but more representative value of local formability material capability [5].

Assessing all *TTS* results in a single graph (Fig. 5), the -CP steel grades clearly have a higher local ductility than the -DH grade. The -DH steel shows thinnest spots on both the center line of the sample and at the edge. This hints to the edge-crack sensitivity of dual phase steel. The CR780Y980T-CP consistently fails in shear mode. In such a shear mode, the thinnest point is spread nearly randomly.

Zooming in on fracture morphology: ASTM specimens lead to the most consistent results (Fig. 6 and 7). Even for CR780Y980T-CP, which always breaks in shear, ASTM samples cluster the thinnest point in the center of the sample. The ASTM sample shape helps to avoid failure starting from the edges for edge-crack sensitive material, such as the -DH grade. Those results are in line with [20] investigations on sub size ASTM sample geometry, which emphasize the strong influence of sample width on fracture morphology and local ductility results. The authors note that ASTM specimens do not represent the standard in Europe, which would imply additional effort. Due to the lower anisotropy of cold-rolled strips, no clear influence of the rolling direction (material orientation) on *TTS* can be determined for either CR440Y780T-DH or CR780Y980T-CP (Fig. 5 and 10). This directional influence is more pronounced for HR660Y760T-CP. Tensile specimens in the transverse direction show lower *TTS* values than longitudinal.

## Conclusion and outlook

This paper demonstrates that the newly published recommendation for local ductility measurement [4] leads to repeatable and reproducible results across various laboratories. This was shown in large-scale test campaigns using samples of three AHSS grades: CR440Y780T-DH, CR780T980T-CP, and HR660Y760T-CP. The results show advantages of *TTS* over *HET*: There is less measurement scatter, the measurement is simpler, and the influence of shear cutting is avoided. Nevertheless, the behavior of AHSS in shear cutting is a very interesting subject itself, which should be characterized in another way.

During this cooperation several questions turned up, that will be focus for a new ESTEP (European Steel Technology Platform) project:

1. How do sample dimensions influence breakage mode, *TTS* and *TFS* values? Can the measurement method be applied to steel sheets with larger thickness?

2. What is the resolution of the described method of local ductility measurement in the differentiation of AHSS types (global, local, or balanced type)?

3. What is the influence of the testing speed?

4. Can local ductility information be derived from stress/strain curves in tensile testing?

## References

[1] Frómeta, D., Lara, A., Grifé, L. et al. Fracture Resistance of Advanced High-Strength Steel Sheets for Automotive Applications. Metal Mater Trans A 52, 840–856 (2021).
https://doi.org/10.1007/s11661-020-06119-y

[2] WorldAutoSteel AHSS Guidelines Online Version 7.0 (2021):
https://ahssinsights.org/forming/formability/true-fracture-strain/
usefulness of the hole expansion test must be questioned. The lab-by-lab differences for HR660Y760T-CP stand out regarding this aspect.

Fig. 10 shows that *TTS* and *TFS* are basically equivalent: hence, the authors favor the simpler *TTS* measurement. Interestingly, in case of the -DH steel, *TFS* is slightly higher than *TTS*: In high global formability grades (such as -DH) the width reduction contribution to the overall fracture area increases (more diffuse and local necking with higher work hardening capability). *TFS* includes a contribution of both thickness and width reduction, leading to a comparatively high value, while *TTS* is only representing the thickness reduction. *TTS* may be a more conservative, but more representative value of local formability material capability [5].

Assessing all *TTS* results in a single graph (Fig. 5), the -CP steel grades clearly have a higher local ductility than the -DH grade. The -DH steel shows thinnest spots on both the center line of the sample and at the edge. This hints to the edge-crack sensitivity of dual phase steel. The CR780Y980T-CP consistently fails in shear mode. In such a shear mode, the thinnest point is spread nearly randomly.

Zooming in on fracture morphology: ASTM specimens lead to the most consistent results (Fig. 6 and 7). Even for CR780Y980T-CP, which always breaks in shear, ASTM samples cluster the thinnest point in the center of the sample. The ASTM sample shape helps to avoid failure starting from the edges for edge-crack sensitive material, such as the -DH grade. Those results are in line with [20] investigations on sub size ASTM sample geometry, which emphasize the strong influence of sample width on fracture morphology and local ductility results. The authors note that ASTM specimens do not represent the standard in Europe, which would imply additional effort. Due to the lower anisotropy of cold-rolled strips, no clear influence of the rolling direction (material orientation) on *TTS* can be determined for either CR440Y780T-DH or CR780Y980T-CP (Fig. 5 and 10). This directional influence is more pronounced for HR660Y760T-CP. Tensile specimens in the transverse direction show lower *TTS* values than longitudinal.

## Conclusion and outlook

This paper demonstrates that the newly published recommendation for local ductility measurement [4] leads to repeatable and reproducible results across various laboratories. This was shown in large-scale test campaigns using samples of three AHSS grades: CR440Y780T-DH, CR780T980T-CP, and HR660Y760T-CP. The results show advantages of *TTS* over *HET*: There is less measurement scatter, the measurement is simpler, and the influence of shear cutting is avoided. Nevertheless, the behavior of AHSS in shear cutting is a very interesting subject itself, which should be characterized in another way.

During this cooperation several questions turned up, that will be focus for a new ESTEP (European Steel Technology Platform) project:

1. How do sample dimensions influence breakage mode, *TTS* and *TFS* values? Can the measurement method be applied to steel sheets with larger thickness?

2. What is the resolution of the described method of local ductility measurement in the differentiation of AHSS types (global, local, or balanced type)?

3. What is the influence of the testing speed?

4. Can local ductility information be derived from stress/strain curves in tensile testing?

## References

[1] Frómeta, D., Lara, A., Grifé, L. et al. Fracture Resistance of Advanced High-Strength Steel Sheets for Automotive Applications. Metal Mater Trans A 52, 840–856 (2021).
https://doi.org/10.1007/s11661-020-06119-y

[2] WorldAutoSteel AHSS Guidelines Online Version 7.0 (2021):
https://ahssinsights.org/forming/formability/true-fracture-strain/
[3] ISO 16630:2017 (E): Metallic materials-sheet and strip-hole expanding test.

[4] VDA 238-110 Test specification: Ermittlung der Lokalen Duktilität aus der Bruchfläche von Zugproben / Determination of Local Ductility by Fracture Surface Analyses of Tensile Test Specimens 08/2023.

[5] Heibel S et al Damage Mechanisms and Mechanical Properties of High-Strength Multi-phase Steels. Materials. 2018; 11(5):761. https://doi.org/10.3390/ma11050761

[6] Hance B 2016: Advanced High Strength Steel: Deciphering Local and Global Formability. Int. Automotive Body Congress, Sept. 28-29, Dearborn, MI, USA.

[7] Gruenbaum M, Aydin G, Dettinger T, and Heibel S 2019 Local formability of AHSS: Measurement technique, specimen types and robustness IOP Conf. Ser. Mater. Sci. Eng., vol. 651, no. 1. https://doi.org/10.1088/1757-899X/651/1/012056

[8] Denks, I.A. et al. (2019), On the Correlation between Suitable Material Parameters for the Prediction of Local Formability of Advanced High Strength Steels. steel research int., 90: 1800460. https://doi.org/10.1002/srin.201800460

[9] VDA238-100 Test specification: Plättchen-Biegeversuch für metallische Werkstoffe / Plate bending test for metallic materials. 07/2020.

[10] Wagner L. et al 2020 Experimental issues in the instrumented 3 point bending VDA238-100 test. IOP Conf. Ser.: Mater. Sci. Eng. 967 012079. https://doi.org/10.1088/1757-899X/967/1/012079

[11] P. Dietsch et al. Methodology to assess fracture during crash simulation: fracture strain criteria and their calibration. Metall. Res. Technol. 114, 607 (2017). https://doi.org/10.1051/metal/2016065

[12] Watanabe K Simple prediction method for the edge fracture of steel sheet during vehicle collision (1st report) 5. LS-DYNA Anwenderforum, Ulm 2006.

[13] Atzema E, Borsutzki M, Braun M, Brockmann S, Bülter M, Carlsson B, Larour P and Richter A 2012 A European round robin test for the Hole Expansion test according to ISO 16630 Int. Conf. Proc. New Development in Sheet Metal Forming Germany Volume: pp. 171-84

[14] ASTM E8/E8M-22: Standard Test Methods for Tension Testing of Metallic Materials.

[15] T. Beier et al 2023 Local and global forming potential of present dual-phase steel developments IOP Conf. Ser.: Mater. Sci. Eng. 1284 012017. https://doi.org/10.1088/1757-899X/1284/1/012017

[16] ISO 6892-1:2019(E): Metallic materials - Tensile testing - Part 1: Method of test at room temperature.

[17] SEP 1240:2006-07: Testing and Documentation Guideline for the Experimental Determination of Mechanical Properties of Steel Sheets for CAE-Calculations.

[18] A.I.A.G. - Chrysler Corp., Ford Motor Co., General Motors Corp: Measurement Systems Analysis, Reference Manual, 4th edition, Michigan (USA) 2010

[19] ASTM E691-23: Standard Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method.

[20] Hance B.M and Link T.M. 2019 Effects of fracture area measurement method and tension test specimen type on fracture strain values of 980 class AHSS IOP Conf. Ser.: Mater. Sci. Eng. 651 012061. https://doi.org/10.1088/1757-899X/651/1/012061
[3] ISO 16630:2017 (E): Metallic materials-sheet and strip-hole expanding test.

[4] VDA 238-110 Test specification: Ermittlung der Lokalen Duktilität aus der Bruchfläche von Zugproben / Determination of Local Ductility by Fracture Surface Analyses of Tensile Test Specimens 08/2023.

[5] Heibel S et al Damage Mechanisms and Mechanical Properties of High-Strength Multi-phase Steels. Materials. 2018; 11(5):761. https://doi.org/10.3390/ma11050761

[6] Hance B 2016: Advanced High Strength Steel: Deciphering Local and Global Formability. Int. Automotive Body Congress, Sept. 28-29, Dearborn, MI, USA.

[7] Gruenbaum M, Aydin G, Dettinger T, and Heibel S 2019 Local formability of AHSS: Measurement technique, specimen types and robustness IOP Conf. Ser. Mater. Sci. Eng., vol. 651, no. 1. https://doi.org/10.1088/1757-899X/651/1/012056

[8] Denks, I.A. et al. (2019), On the Correlation between Suitable Material Parameters for the Prediction of Local Formability of Advanced High Strength Steels. steel research int., 90: 1800460. https://doi.org/10.1002/srin.201800460

[9] VDA238-100 Test specification: Plättchen-Biegeversuch für metallische Werkstoffe / Plate bending test for metallic materials. 07/2020.

[10] Wagner L. et al 2020 Experimental issues in the instrumented 3 point bending VDA238-100 test. IOP Conf. Ser.: Mater. Sci. Eng. 967 012079. https://doi.org/10.1088/1757-899X/967/1/012079

[11] P. Dietsch et al. Methodology to assess fracture during crash simulation: fracture strain criteria and their calibration. Metall. Res. Technol. 114, 607 (2017). https://doi.org/10.1051/metal/2016065

[12] Watanabe K Simple prediction method for the edge fracture of steel sheet during vehicle collision (1st report) 5. LS-DYNA Anwenderforum, Ulm 2006.

[13] Atzema E, Borsutzki M, Braun M, Brockmann S, Bülter M, Carlsson B, Larour P and Richter A 2012 A European round robin test for the Hole Expansion test according to ISO 16630 Int. Conf. Proc. New Development in Sheet Metal Forming Germany Volume: pp. 171-84

[14] ASTM E8/E8M-22: Standard Test Methods for Tension Testing of Metallic Materials.

[15] T. Beier et al 2023 Local and global forming potential of present dual-phase steel developments IOP Conf. Ser.: Mater. Sci. Eng. 1284 012017. https://doi.org/10.1088/1757-899X/1284/1/012017

[16] ISO 6892-1:2019(E): Metallic materials - Tensile testing - Part 1: Method of test at room temperature.

[17] SEP 1240:2006-07: Testing and Documentation Guideline for the Experimental Determination of Mechanical Properties of Steel Sheets for CAE-Calculations.

[18] A.I.A.G. - Chrysler Corp., Ford Motor Co., General Motors Corp: Measurement Systems Analysis, Reference Manual, 4th edition, Michigan (USA) 2010

[19] ASTM E691-23: Standard Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method.

[20] Hance B.M and Link T.M. 2019 Effects of fracture area measurement method and tension test specimen type on fracture strain values of 980 class AHSS IOP Conf. Ser.: Mater. Sci. Eng. 651 012061. https://doi.org/10.1088/1757-899X/651/1/012061
# Crystal plasticity-based forming limit analysis for two types of 5052 aluminum alloy sheets with different heat treatment conditions

SATO Sho<fnref n="1" />,ᵃ*, TSUKAMOTO Maya<fnref n="1" />, MAEDA Yasuhiro<fnref n="2" />, MAEDA Yasushi<fnref n="2" /> and HAMA Takayuki<fnref n="1" />,ᵇ

**Keywords:** Crystal-Plasticity Analysis, Forming Limit Diagram, Aluminum Alloy, Nakajima Test, Marciniak-Kuczyński Analysis

**Abstract.** 5000 series aluminum (Al) alloy sheets are increasingly applied to press forming products, such as components used in railroad vehicles and some automobiles, due to their advantages of high specific strength and good corrosion resistance. However, in press forming processes, forming defects, such as failure and wrinkles, often appear. Therefore, it is essential to understand the formability of sheet materials under various stress states and to design appropriate processes to prevent forming-defects occurrence. To evaluate occurrence of failure under various strain states, a Forming Limit Diagram (FLD) is widely used. Although the effects of texture and work-hardening behavior of materials on FLD have been widely studied, their experimental validations are not sufficient. In this study, A5052-O and -H32 Al alloy sheets, which have similar texture and show different work-hardening behavior, are used to investigate the effect of work hardening on the FLD. In the FLDs obtained experimentally, the limit strain under plane-strain tension was larger in the A5052-O sheet with larger work hardening than that of the A5052-H32 sheet with smaller work hardening, whereas that under equibiaxial tension was similar in the two sheets. These trends were reproduced qualitatively well by crystal-plasticity forming limit analyses. The mechanism that yielded these trends were discussed using the simulation results.

**Introduction**

To facilitate weight reduction of transportation equipment, 5000 series aluminum (Al) alloy sheets are increasingly applied to press forming products, such as components used in railroad vehicles and some automobiles, due to their advantages of high specific strength and good corrosion resistance. However, the ductility of Al alloy sheets is usually not large; thus, the sheets are often suffered from failure during press forming processes. Therefore, it is essential to understand the formability of sheet materials under various stress conditions and to design appropriate forming processes to prevent occurrence of failure.

To evaluate occurrence of failure under various strain conditions, a Forming Limit Diagram (FLD) is widely used [1], in which the safe zone and failure zone is divided by a forming-limit curve (FLC). FLC is obtained by measuring major and minor strains when localized necking occurs under different strain states. FLC is known to be influenced by different factors, including strain rate [2], strain path [3], stress state [4], and temperature [5]. The effects of texture [6] and work hardening [7] of materials have also been widely discussed. However, these studies were usually concerned with virtual materials; thus, experimental validations of numerical results are not sufficient.

In this study, the effects of work hardening on FLCs are investigated experimentally using two types of 5000 series Al alloy sheets with different heat treatment conditions: A5052-O and A5052-H32 sheets. The two sheets have similar texture, whereas they exhibit different work-hardening

<figure><img src="image_952.png" /></figure>

<footnote n="1">¹Graduate School of Energy Science, Kyoto University, Kyoto 606-8501, Japan</footnote>

<footnote n="2">²Technical Development Group, KOBE STEEL, LTD., Kobe 651-2271, Japan</footnote>

<footnote>ᵃsato.syo.62e@st.kyoto-u.ac.jp, ᵇhama@energy.kyoto-u.ac.jp</footnote>
# Crystal plasticity-based forming limit analysis for two types of 5052 aluminum alloy sheets with different heat treatment conditions

SATO Sho¹,ᵃ*, TSUKAMOTO Maya¹, MAEDA Yasuhiro², MAEDA Yasushi² and HAMA Takayuki¹,ᵇ

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

²Technical Development Group, KOBE STEEL, LTD., Kobe 651-2271, Japan

ᵃ sato.syo.62e@st.kyoto-u.ac.jp, ᵇhama@energy.kyoto-u.ac.jp

**Keywords:** Crystal-Plasticity Analysis, Forming Limit Diagram, Aluminum Alloy, Nakajima Test, Marciniak-Kuczyński Analysis

**Abstract.** 5000 series aluminum (Al) alloy sheets are increasingly applied to press forming products, such as components used in railroad vehicles and some automobiles, due to their advantages of high specific strength and good corrosion resistance. However, in press forming processes, forming defects, such as failure and wrinkles, often appear. Therefore, it is essential to understand the formability of sheet materials under various stress states and to design appropriate processes to prevent forming-defects occurrence. To evaluate occurrence of failure under various strain states, a Forming Limit Diagram (FLD) is widely used. Although the effects of texture and work-hardening behavior of materials on FLD have been widely studied, their experimental validations are not sufficient. In this study, A5052-O and -H32 Al alloy sheets, which have similar texture and show different work-hardening behavior, are used to investigate the effect of work hardening on the FLD. In the FLDs obtained experimentally, the limit strain under plane-strain tension was larger in the A5052-O sheet with larger work hardening than that of the A5052-H32 sheet with smaller work hardening, whereas that under equibiaxial tension was similar in the two sheets. These trends were reproduced qualitatively well by crystal-plasticity forming limit analyses. The mechanism that yielded these trends were discussed using the simulation results.

**Introduction**

To facilitate weight reduction of transportation equipment, 5000 series aluminum (Al) alloy sheets are increasingly applied to press forming products, such as components used in railroad vehicles and some automobiles, due to their advantages of high specific strength and good corrosion resistance. However, the ductility of Al alloy sheets is usually not large; thus, the sheets are often suffered from failure during press forming processes. Therefore, it is essential to understand the formability of sheet materials under various stress conditions and to design appropriate forming processes to prevent occurrence of failure.

To evaluate occurrence of failure under various strain conditions, a Forming Limit Diagram (FLD) is widely used [1], in which the safe zone and failure zone is divided by a forming-limit curve (FLC). FLC is obtained by measuring major and minor strains when localized necking occurs under different strain states. FLC is known to be influenced by different factors, including strain rate [2], strain path [3], stress state [4], and temperature [5]. The effects of texture [6] and work hardening [7] of materials have also been widely discussed. However, these studies were usually concerned with virtual materials; thus, experimental validations of numerical results are not sufficient.

In this study, the effects of work hardening on FLCs are investigated experimentally using two types of 5000 series Al alloy sheets with different heat treatment conditions: A5052-O and A5052-H32 sheets. The two sheets have similar texture, whereas they exhibit different work-hardening

<figure><img src="image_959.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
behavior. Moreover, crystal-plasticity Marciniak-Kuczyński (MK) analyses are also conducted considering the two Al sheets to investigate the deformation mechanism in detail.

## Methods of experiment and simulation

Experimental methods. In this study, the A5052-O and A5052-H32 Al alloy sheets with a thickness of 1.0 mm were used. Hereinafter, these sheets are respectively termed as the O and H32 sheets. The stress-strain curves under uniaxial tension in the rolling direction and pole figures of the two sheets are respectively shown in Figs. 1 and 2. Both sheets show similar cube texture. By contrast, they exhibit distinctly different work-hardening behaviors: the yield stress was larger in the H32 sheet than in the O sheet, whereas the work hardening was larger in the O sheet than in the H32 sheet with the work-hardening exponents of 0.239 and 0.157, respectively.

The Nakajima tests [8] were conducted to evaluate FLCs of these sheets. The schematic diagrams of experimental setup and specimen geometry are shown in Fig.3. The test conditions were as follows: after a sheet specimen was clamped between the upper and lower dies, the sheet was stretched using a hemispherical punch with a punch speed of approximately 1 mm/s until failure. To vary the strain ratio, several dumbbell-shape specimens with different width W shown in Fig.3(b) were used. To reduce friction between the punch and specimen, a lubrication layer consisting of Vaseline grease, Teflon sheets and PVC sheets was used. A white-black random pattern applied to the specimen surface was captured by CCD cameras, and strains were measured using the Digital Image Correlation (DIC) method (ARAMIS, GOM Corporation). The initiation point of local necking was determined using the "time-dependent" method [9], which identifies the moment when the thickness-strain rate significantly changes at the location where local

<figure><img src="image_953.png" /><figcaption>Fig. 1 True stress-true strain curves under uniaxial tension of O and H32 sheets.</figcaption></figure>
behavior. Moreover, crystal-plasticity Marciniak-Kuczyński (MK) analyses are also conducted considering the two Al sheets to investigate the deformation mechanism in detail.

## Methods of experiment and simulation

Experimental methods. In this study, the A5052-O and A5052-H32 Al alloy sheets with a thickness of 1.0 mm were used. Hereinafter, these sheets are respectively termed as the O and H32 sheets. The stress-strain curves under uniaxial tension in the rolling direction and pole figures of the two sheets are respectively shown in Figs. 1 and 2. Both sheets show similar cube texture. By contrast, they exhibit distinctly different work-hardening behaviors: the yield stress was larger in the H32 sheet than in the O sheet, whereas the work hardening was larger in the O sheet than in the H32 sheet with the work-hardening exponents of 0.239 and 0.157, respectively.

The Nakajima tests [8] were conducted to evaluate FLCs of these sheets. The schematic diagrams of experimental setup and specimen geometry are shown in Fig.3. The test conditions were as follows: after a sheet specimen was clamped between the upper and lower dies, the sheet was stretched using a hemispherical punch with a punch speed of approximately 1 mm/s until failure. To vary the strain ratio, several dumbbell-shape specimens with different width W shown in Fig.3(b) were used. To reduce friction between the punch and specimen, a lubrication layer consisting of Vaseline grease, Teflon sheets and PVC sheets was used. A white-black random pattern applied to the specimen surface was captured by CCD cameras, and strains were measured using the Digital Image Correlation (DIC) method (ARAMIS, GOM Corporation). The initiation point of local necking was determined using the "time-dependent" method [9], which identifies the moment when the thickness-strain rate significantly changes at the location where local

<figure><img src="image_960.png" /><figcaption>Fig. 1 True stress-true strain curves under uniaxial tension of O and H32 sheets.</figcaption></figure>
<figure><img src="image_954.png" /><figcaption>Fig.2 Pole figures of (a)O and (b)H32 sheets.</figcaption></figure>

<figure><img src="image_955.png" /><figcaption>Fig.3 Schematic diagrams of (a)experimental setup and (b)specimen geometry in mm.</figcaption></figure>
<figure><img src="image_961.png" /><figcaption>Fig.2 Pole figures of (a)O and (b)H32 sheets.</figcaption></figure>

<figure><img src="image_962.png" /><figcaption>Fig.3 Schematic diagrams of (a)experimental setup and (b)specimen geometry in mm.</figcaption></figure>
necking occurs. The tests were performed at least three times for each condition to assure the reproducibility.

Method of simulation. The MK model [10] coupled with an elastic-viscoplastic crystal-plasticity constitutive model was used to evaluate the onset of necking. In the crystal-plasticity model [11,12], the slip rate $\dot{\gamma}^{(\alpha)}$ of slip system $\alpha$ is defined by the following rate-dependent equation:

$$ \dot{\gamma}^{(\alpha)} = \dot{\gamma}_0 \left| \frac{\tau^{(\alpha)}}{\tau_Y^{(\alpha)}} \right|^{\frac{1}{m}} \operatorname{sgn}(\tau^{(\alpha)}) \quad (1) $$

where $\tau_Y^{(\alpha)}$ is the current resistance of the $\alpha$th slip system. $\dot{\gamma}_0$ is the reference slip rate, and $m$ is the strain rate sensitivity exponent. The rate of slip resistance is given by

$$ \dot{t}_Y^{(\alpha)} = \sum_{\beta} h_{\alpha\beta} |\dot{\gamma}^{(\beta)}| \quad (2) $$

where $h_{\alpha\beta}$ is the hardening moduli matrix expressing the interaction of slip systems.

The following dislocation-density-based hardening model was utilized:

$$ h_{\alpha\beta} = \frac{\mu}{2} g_{\alpha\beta} \left( \sum_{\kappa} g_{\alpha\kappa} \rho^{(\kappa)} \right)^{-\frac{1}{2}} \left[ \frac{1}{K} \left( \sum_{\kappa} g_{\alpha\kappa} \rho^{(\kappa)} \right)^{\frac{1}{2}} - 2y_c \rho^{(\beta)} \right] \quad (3) $$

where $\mu$ is the shear modulus, $y_c$ is characteristic length that represents the annihilation process of dislocation dipoles, $K$ is a material parameter, and $g_{\alpha\beta}$ is the interaction matrix. $\rho^{(\alpha)}$ is the total dislocation density of the $\alpha$th slip system and its evolution is given by

$$ \dot{\rho}^{(\alpha)} = \frac{1}{b} \left( \frac{1}{L^{(\alpha)}} - 2y_c \rho^{(\alpha)} \right) |\dot{\gamma}^{(\alpha)}| \quad (4) $$

with $b$ being the magnitude of the Burgers vector. $L^{(\alpha)}$ represents the mean free path and is expressed as $L^{(\alpha)} = K/\sqrt{\sum_{\kappa} g_{\alpha\kappa} \rho^{(\kappa)}}$.

In the analysis, 5000 crystal orientations were considered, and a polycrystalline constitutive equation was derived based on the Taylor assumption [13]. The material parameters were determined to fit the stress-strain curves under uniaxial tension in the rolling direction within uniform elongation ranges, as shown in Fig. 1. An initial imperfection $f$, which is a parameter of the MK model, was determined to match a limit strain under plane-strain tension obtained in experiments. Specifically, $f$ was set to 0.9978 and 0.9989 respectively for the O and H32 sheets. The MK analyses were conducted assuming linear strain paths. Although detailed results are presented, the simulation results remained almost unchanged when simulations were conducted under nonlinear strain paths that approximated the experiments.

## Results

The experimental and simulation results of FLD are shown in Fig. 4. The horizontal and vertical axes show respectively the minor and major limit strains. In the experimental results, the two sheets exhibited similar V-shaped transitions: the limit strains tended to increase with the increase in the absolute minor strain with a minimum presenting in the first quadrant. By contrast, the transition slopes differed depending on the sheets. The two sheets exhibited similar slopes in the second quadrant, whereas the H32 sheet clearly exhibited a larger slope in the first quadrant than the O sheet. To evaluate the differences in the variation trends between the two materials in more detail, the major limit strain ratio $\varepsilon^*/\varepsilon^*_{\rho=0}$ were calculated to normalize the FLDs, where $\varepsilon^*$ is the major limit strain and $\varepsilon^*_{\rho=0}$ is the major limit strain under plane-strain tension. Fig.5 shows the normalized FLDs. It should be noted that the strain ratio $\rho$, which is the ratio of the major strain to the minor strain, is used for the horizontal axis to represent clearly the relationship between the biaxial tension states and the limit strain ratios. The two sheets did not show clear differences in
necking occurs. The tests were performed at least three times for each condition to assure the reproducibility.

Method of simulation. The MK model [10] coupled with an elastic-viscoplastic crystal-plasticity constitutive model was used to evaluate the onset of necking. In the crystal-plasticity model [11,12], the slip rate $\dot{\gamma}^{(\alpha)}$ of slip system $\alpha$ is defined by the following rate-dependent equation:

$$ \dot{\gamma}^{(\alpha)} = \dot{\gamma}_0 \left| \frac{\tau^{(\alpha)}}{\tau_Y^{(\alpha)}} \right|^{\frac{1}{m}} \operatorname{sgn}(\tau^{(\alpha)}) \quad (1) $$

where $\tau_Y^{(\alpha)}$ is the current resistance of the $\alpha$th slip system. $\dot{\gamma}_0$ is the reference slip rate, and $m$ is the strain rate sensitivity exponent. The rate of slip resistance is given by

$$ \dot{t}_Y^{(\alpha)} = \sum_{\beta} h_{\alpha\beta} |\dot{\gamma}^{(\beta)}| \quad (2) $$

where $h_{\alpha\beta}$ is the hardening moduli matrix expressing the interaction of slip systems.

The following dislocation-density-based hardening model was utilized:

$$ h_{\alpha\beta} = \frac{\mu}{2} g_{\alpha\beta} \left( \sum_{\kappa} g_{\alpha\kappa} \rho^{(\kappa)} \right)^{-\frac{1}{2}} \left[ \frac{1}{K} \left( \sum_{\kappa} g_{\alpha\kappa} \rho^{(\kappa)} \right)^{\frac{1}{2}} - 2y_c \rho^{(\beta)} \right] \quad (3) $$

where $\mu$ is the shear modulus, $y_c$ is characteristic length that represents the annihilation process of dislocation dipoles, $K$ is a material parameter, and $g_{\alpha\beta}$ is the interaction matrix. $\rho^{(\alpha)}$ is the total dislocation density of the $\alpha$th slip system and its evolution is given by

$$ \dot{\rho}^{(\alpha)} = \frac{1}{b} \left( \frac{1}{L^{(\alpha)}} - 2y_c \rho^{(\alpha)} \right) |\dot{\gamma}^{(\alpha)}| \quad (4) $$

with $b$ being the magnitude of the Burgers vector. $L^{(\alpha)}$ represents the mean free path and is expressed as $L^{(\alpha)} = K/\sqrt{\sum_{\kappa} g_{\alpha\kappa} \rho^{(\kappa)}}$.

In the analysis, 5000 crystal orientations were considered, and a polycrystalline constitutive equation was derived based on the Taylor assumption [13]. The material parameters were determined to fit the stress-strain curves under uniaxial tension in the rolling direction within uniform elongation ranges, as shown in Fig. 1. An initial imperfection $f$, which is a parameter of the MK model, was determined to match a limit strain under plane-strain tension obtained in experiments. Specifically, $f$ was set to 0.9978 and 0.9989 respectively for the O and H32 sheets. The MK analyses were conducted assuming linear strain paths. Although detailed results are presented, the simulation results remained almost unchanged when simulations were conducted under nonlinear strain paths that approximated the experiments.

## Results

The experimental and simulation results of FLD are shown in Fig. 4. The horizontal and vertical axes show respectively the minor and major limit strains. In the experimental results, the two sheets exhibited similar V-shaped transitions: the limit strains tended to increase with the increase in the absolute minor strain with a minimum presenting in the first quadrant. By contrast, the transition slopes differed depending on the sheets. The two sheets exhibited similar slopes in the second quadrant, whereas the H32 sheet clearly exhibited a larger slope in the first quadrant than the O sheet. To evaluate the differences in the variation trends between the two materials in more detail, the major limit strain ratio $\varepsilon^*/\varepsilon^*_{\rho=0}$ were calculated to normalize the FLDs, where $\varepsilon^*$ is the major limit strain and $\varepsilon^*_{\rho=0}$ is the major limit strain under plane-strain tension. Fig.5 shows the normalized FLDs. It should be noted that the strain ratio $\rho$, which is the ratio of the major strain to the minor strain, is used for the horizontal axis to represent clearly the relationship between the biaxial tension states and the limit strain ratios. The two sheets did not show clear differences in
the second quadrant, whereas, in the first quadrant, the limit strain ratios are larger in the H32 sheet than in the O sheet.

The simulation results reproduced qualitatively well the V-shaped transitions and the trend that the limit strain ratios in the first quadrant were larger in the H32 sheet that had smaller work hardening than in the O sheet that had larger work hardening. This result is consistent with a previous study [7]. However, the differences between the two materials observed in the first quadrant were much less pronounced than those of the experimental results.

Discussion

It was reported that the major limit strain ratios in the first quadrant decrease as the yield surface becomes sharper [14-16]. Fig. 6 shows the normalized contours and stress plots of equal plastic work measured at the plastic work of 6.0 MJ/m³. The stress plots under equibiaxial tension were measured experimentally using an equibiaxial tension test using a cruciform specimen [17]. The contours were determined using the Yld2000-2d function with the exponent M = 8.0 [18]. Although there are only three stress plots on each contour, the contour near equibiaxial tension tended to be sharper in the H32 sheet than in the O sheet in the experiments, whereas the results were almost independent of the material in the simulations, showing that no clear correlation is observed between the FLDs and the contours obtained in this study.

Because of the large difference in the work hardening between the O and H32 sheets, it is hypothesized that the difference in work-hardening behavior between the two sheets is a possible factor that yielded the difference in the FLDs in the first quadrant. To support this hypothesis, the following numerical experiments were conducted. We prepared a virtual O sheet that had the texture of the O sheet and the work-hardening properties of the H32 sheet. It is expected that the virtual O sheet gives higher limit strain ratios in the first quadrant than the original O sheet. f for the virtual O sheet was set to the same value as that for the H32 sheet.

The results are shown in Fig. 7. The original and virtual O sheets exhibited similar limit strain ratios in the second quadrant, whereas the virtual O sheet showed larger limit strain ratios in the first quadrant than the original O sheet, confirming the aforementioned hypothesis. These results show that, even with the same texture, the work-hardening behavior of the H32 sheet leads to larger limit strain ratios in the first quadrant compared to that of the O sheet, indicating that the difference in the work-hardening behavior played an important role in the difference in the first quadrant of the FLC between the O and H32 sheets.

<figure><img src="image_956.png" /><figcaption>Fig. 4 Experimental and simulation results of FLDs.</figcaption></figure>
the second quadrant, whereas, in the first quadrant, the limit strain ratios are larger in the H32 sheet than in the O sheet.

The simulation results reproduced qualitatively well the V-shaped transitions and the trend that the limit strain ratios in the first quadrant were larger in the H32 sheet that had smaller work hardening than in the O sheet that had larger work hardening. This result is consistent with a previous study [7]. However, the differences between the two materials observed in the first quadrant were much less pronounced than those of the experimental results.

## Discussion

It was reported that the major limit strain ratios in the first quadrant decrease as the yield surface becomes sharper [14-16]. Fig. 6 shows the normalized contours and stress plots of equal plastic work measured at the plastic work of 6.0 MJ/m³. The stress plots under equibiaxial tension were measured experimentally using an equibiaxial tension test using a cruciform specimen [17]. The contours were determined using the Yld2000-2d function with the exponent M = 8.0 [18]. Although there are only three stress plots on each contour, the contour near equibiaxial tension tended to be sharper in the H32 sheet than in the O sheet in the experiments, whereas the results were almost independent of the material in the simulations, showing that no clear correlation is observed between the FLDs and the contours obtained in this study.

Because of the large difference in the work hardening between the O and H32 sheets, it is hypothesized that the difference in work-hardening behavior between the two sheets is a possible factor that yielded the difference in the FLDs in the first quadrant. To support this hypothesis, the following numerical experiments were conducted. We prepared a virtual O sheet that had the texture of the O sheet and the work-hardening properties of the H32 sheet. It is expected that the virtual O sheet gives higher limit strain ratios in the first quadrant than the original O sheet. f for the virtual O sheet was set to the same value as that for the H32 sheet.

The results are shown in Fig. 7. The original and virtual O sheets exhibited similar limit strain ratios in the second quadrant, whereas the virtual O sheet showed larger limit strain ratios in the first quadrant than the original O sheet, confirming the aforementioned hypothesis. These results show that, even with the same texture, the work-hardening behavior of the H32 sheet leads to larger limit strain ratios in the first quadrant compared to that of the O sheet, indicating that the difference in the work-hardening behavior played an important role in the difference in the first quadrant of the FLC between the O and H32 sheets.

<figure><img src="image_963.png" /><figcaption>Fig. 4 Experimental and simulation results of FLDs.</figcaption></figure>
<figure><img src="image_957.png" /><figcaption>Fig. 5 Normalized FLDs</figcaption></figure>

<figure><img src="image_958.png" /><figcaption>Fig. 6 Normalized contours and stress plots of equal plastic work.</figcaption></figure>
<figure><img src="image_964.png" /><figcaption>Fig. 5 Normalized FLDs</figcaption></figure>

<figure><img src="image_965.png" /><figcaption>Fig. 6 Normalized contours and stress plots of equal plastic work.</figcaption></figure>
<figure><img src="image_959.png" /><figcaption>Fig. 7 Simulation results of nomalized FLDs for original and virtual O sheets.</figcaption></figure>

Conclusion

This study evaluated FLCs for A5052-O and A5052-H32 Al alloy sheets which had similar texture and exhibit distinctly different work-hardening behaviors through experiments and numerical analyses. The obtained results are summarized as follows:

1) Experimentally-obtained FLCs showed that the two sheets exhibited similar limit strain ratios in the second quadrant, whereas the H32 sheet clearly exhibited larger limit strain ratios than the O sheet in the first quadrant.

2) Crystal-plasticity MK analyses reproduced qualitatively well the differences in FLCs between the two sheets observed in the experimental results.

3) Numerical experiments showed that the differences in work hardening resulted in the difference in the limit strain ratios in the first quadrant between the two sheets, exhibiting that larger work hardening induces smaller limit strain ratios.

References

[1] S.P. Keeler and W.A. Backofen, Plastic instability and fracture in sheet stretched over rigid punches, ASM Tran. Q. 56 (1963) 25-48.

[2] S.M. Mirfalah-Nasiri, A. Basti and R. Hashemi, Forming limit curves analysis of aluminum alloy considering the through-thickness normal stress, anisotropic yield functions and strain rate, Int. J. Mech. Sci., 117 (2016) 93-101. https://doi.org/10.1016/j.ijmecsci.2016.08.011

[3] A. Werber, W. Nester, M. Grünbaum, K. Wiegand, J. Simon, J. Timm and W. Hotz, Prod. Eng., Assessment of forming limit stress curves as failure criterion for non-proportional forming processes, 7 (2013) 213-221. https://doi.org/10.1007/s11740-013-0446-6

[4] B. Ma, K. Diao, X. Wu, X. Li, M. Wan and Z. Cai, The effect of the through-thickness normal stress on sheet formability, J. Manuf. Process., 21 (2016) 134-140. https://doi.org/10.1016/j.jmapro.2015.12.006

[5] B.L. Ma, M. Wan, X.J. Li, X.D. Wu and S.K. Diao, Evaluation of limit strain and temperature history in hot stamping of advanced high strength steels (AHSS), Int. J. Mech. Sci., 128-129 (2017) 607-613. https://doi.org/10.1016/j.ijmecsci.2017.05.026
<figure><img src="image_966.png" /><figcaption>Fig. 7 Simulation results of nomalized FLDs for original and virtual O sheets.</figcaption></figure>

**Conclusion**

This study evaluated FLCs for A5052-O and A5052-H32 Al alloy sheets which had similar texture and exhibit distinctly different work-hardening behaviors through experiments and numerical analyses. The obtained results are summarized as follows:

1) Experimentally-obtained FLCs showed that the two sheets exhibited similar limit strain ratios in the second quadrant, whereas the H32 sheet clearly exhibited larger limit strain ratios than the O sheet in the first quadrant.

2) Crystal-plasticity MK analyses reproduced qualitatively well the differences in FLCs between the two sheets observed in the experimental results.

3) Numerical experiments showed that the differences in work hardening resulted in the difference in the limit strain ratios in the first quadrant between the two sheets, exhibiting that larger work hardening induces smaller limit strain ratios.

**References**

[1] S.P. Keeler and W.A. Backofen, Plastic instability and fracture in sheet stretched over rigid punches, ASM Tran. Q. 56 (1963) 25-48.

[2] S.M. Mirfalah-Nasiri, A. Basti and R. Hashemi, Forming limit curves analysis of aluminum alloy considering the through-thickness normal stress, anisotropic yield functions and strain rate, Int. J. Mech. Sci., 117 (2016) 93-101. https://doi.org/10.1016/j.ijmecsci.2016.08.011

[3] A. Werber, W. Nester, M. Grünbaum, K. Wiegand, J. Simon, J. Timm and W. Hotz, Prod. Eng., Assessment of forming limit stress curves as failure criterion for non-proportional forming processes, 7 (2013) 213-221. https://doi.org/10.1007/s11740-013-0446-6

[4] B. Ma, K. Diao, X. Wu, X. Li, M. Wan and Z. Cai, The effect of the through-thickness normal stress on sheet formability, J. Manuf. Process., 21 (2016) 134-140. https://doi.org/10.1016/j.jmapro.2015.12.006

[5] B.L. Ma, M. Wan, X.J. Li, X.D. Wu and S.K. Diao, Evaluation of limit strain and temperature history in hot stamping of advanced high strength steels (AHSS), Int. J. Mech. Sci., 128-129 (2017) 607-613. https://doi.org/10.1016/j.ijmecsci.2017.05.026
[6] K. Yoshida, T. Ishizaki, M. Kuroda and S. Ikawa, The effects of texture on formability of aluminum alloy sheet, Acta. Mater. 55 (2007) 4499-4506.
https://doi.org/10.1016/j.actamat.2007.04.014

[7] M. Mohammadi, A. P. Brahme, R. K. Mishra and K. Inal, Effect of post-necking behavior and equivalent stress-strain curves on the accuracy of M-K based forming limit diagrams, Compt. Mater. Sci., 85 (2014) 316-323. https://doi.org/10.1016/j.commatsci.2014.01.017

[8] ISO12004-2:2021 Metallic materials -determination of forming-limit curves for sheet and strip- Part 2: Determination of forming-limit curves in the laboratory.

[9] W. Hotz, M. Merklein, A. Kuppert, H. Friebe and M. Klein, Time dependent FLC determination – Comparison of different algorithms to detect the onset of unstable necking before fracture, Key Eng. Mater., 549 (2013) 397-404.
https://doi.org/10.4028/www.scientific.net/KEM.549.397

[10] Z. Marciniak and K. Kuczyński, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci., 9 (1967) 609-620. https://doi.org/10.1016/0020-7403(67)90066-5

[11] T. Hama, S. Yagi, K. Tatsukawa, Y. Maeda, Y. Maeda, and H. Takuda, Evolution of plastic deformation behavior upon strain-path changes in an A6022-T4 Al alloy sheet, Int. J. Plast., 137 (2021) 102913. https://doi.org/10.1016/j.ijplas.2020.102913

[12] T. Hama, R. Namakawa, Y. Maeda, and Y. Maeda, Prediction of work-hardening behavior under various loading paths in 5083-O aluminum alloy sheet using crystal plasticity models, Mater. Trans., 62-8 (2021) 1124-1132. https://doi.org/10.2320/matertrans.MT-M2021020

[13] G. I. Taylor, Plastic strain in metals, J. Inst. Metals 62 (1938) 307-324.

[14] F. Barlat, Crystallographic texture, anisotropic yield surfaces and forming limits of sheet metals, Mater. Sci. Eng. 91 (1987) 55-72. https://doi.org/10.1016/0025-5416(87)90283-7

[15] K.W. Neale and E. Chater, Limit strain predictions for strain-rate sensitive anisotropic sheets, Int. J. Mech. Sci. 22 (1980) 563-574. https://doi.org/10.1016/0020-7403(80)90018-1

[16] P.D. Wu, S.R. MacEwen, D.J. Lloyd and K.W. Neale, Effect of cube texture on sheet metal formability, Mater. Sci. Eng. A 364 (2004), 182-187. https://doi.org/10.1016/j.msea.2003.08.020

[17] ISO 16842:2021 Metallic materials – Sheet and strip – Biaxial tensile testing method using cruciform test piece

[18] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.-H. Choi, and E. Chu, Plane stress yield function for aluminum alloy sheets – part 1: theory, Int. J. Plast. 19 (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0
[6] K. Yoshida, T. Ishizaki, M. Kuroda and S. Ikawa, The effects of texture on formability of aluminum alloy sheet, Acta. Mater. 55 (2007) 4499-4506.
https://doi.org/10.1016/j.actamat.2007.04.014

[7] M. Mohammadi, A. P. Brahme, R. K. Mishra and K. Inal, Effect of post-necking behavior and equivalent stress-strain curves on the accuracy of M-K based forming limit diagrams, Compt. Mater. Sci., 85 (2014) 316-323. https://doi.org/10.1016/j.commatsci.2014.01.017

[8] ISO12004-2:2021 Metallic materials -determination of forming-limit curves for sheet and strip- Part 2: Determination of forming-limit curves in the laboratory.

[9] W. Hotz, M. Merklein, A. Kuppert, H. Friebe and M. Klein, Time dependent FLC determination – Comparison of different algorithms to detect the onset of unstable necking before fracture, Key Eng. Mater., 549 (2013) 397-404.
https://doi.org/10.4028/www.scientific.net/KEM.549.397

[10] Z. Marciniak and K. Kuczyński, Limit strains in the processes of stretch-forming sheet metal, Int. J. Mech. Sci., 9 (1967) 609-620. https://doi.org/10.1016/0020-7403(67)90066-5

[11] T. Hama, S. Yagi, K. Tatsukawa, Y. Maeda, Y. Maeda, and H. Takuda, Evolution of plastic deformation behavior upon strain-path changes in an A6022-T4 Al alloy sheet, Int. J. Plast., 137 (2021) 102913. https://doi.org/10.1016/j.ijplas.2020.102913

[12] T. Hama, R. Namakawa, Y. Maeda, and Y. Maeda, Prediction of work-hardening behavior under various loading paths in 5083-O aluminum alloy sheet using crystal plasticity models, Mater. Trans., 62-8 (2021) 1124-1132. https://doi.org/10.2320/matertrans.MT-M2021020

[13] G. I. Taylor, Plastic strain in metals, J. Inst. Metals 62 (1938) 307-324.

[14] F. Barlat, Crystallographic texture, anisotropic yield surfaces and forming limits of sheet metals, Mater. Sci. Eng. 91 (1987) 55-72. https://doi.org/10.1016/0025-5416(87)90283-7

[15] K.W. Neale and E. Chater, Limit strain predictions for strain-rate sensitive anisotropic sheets, Int. J. Mech. Sci. 22 (1980) 563-574. https://doi.org/10.1016/0020-7403(80)90018-1

[16] P.D. Wu, S.R. MacEwen, D.J. Lloyd and K.W. Neale, Effect of cube texture on sheet metal formability, Mater. Sci. Eng. A 364 (2004), 182-187. https://doi.org/10.1016/j.msea.2003.08.020

[17] ISO 16842:2021 Metallic materials – Sheet and strip – Biaxial tensile testing method using cruciform test piece

[18] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.-H. Choi, and E. Chu, Plane stress yield function for aluminum alloy sheets – part 1: theory, Int. J. Plast. 19 (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0
# Evaluation of correlations between principal axes in uniaxial tensile tests of aluminum based on Mohr's strain circle

IIZUKA Takashi¹,ᵃ*

¹Faculty of Mechanical Engineering, Kyoto Institute of Technology, Matsugasaki Goshokaido-cho, Sakyo-ku, Kyoto 606-8585, Japan

ᵃtiizuka@kit.ac.jp

**Keywords:** Anisotropic Metals, Principal Axes, The Associated Flow Rule, Mohr'S Strain Circle

**Abstract.** Anisotropic elasticity is expected to contribute significantly to material deformation and should be considered in nature. Therefore, in this study, the correlations between elastic and plastic strains are investigated, with emphasis on the principal axes. In an experiment, strains are measured in eight directions during tensile tests, and Mohr's circles are derived for elastic and plastic strains. The planar anisotropy of elasticity and plasticity for some mechanical properties is confirmed, and the angle deviations between the principal axes of stress and strain are compared for elasticity and plasticity. In the case of aluminum, for both elasticity and plasticity, the planar anisotropy of its mechanical properties indicates four-fold symmetry, and the principal axes of elastic and plastic strains are identical, with some angle deviations from the principal axes of stress.

**Introduction**

The fundamental theory of plasticity is applicable to isotropic materials and is summarized based on the Mises' yield criterion and the associated flow rule. The von Mises yield criterion presents a yield surface with a circular cylinder shape in the principal stress field; however, Bishop and Hill indicated that yield criteria that present pillar-type surfaces with a cross-section of six-fold symmetry may be suitable for isotropic materials if the Bauschinger effect is disregarded [1]. This implies that all yield functions for isotropic materials can be expressed by those proposed by Hosfold [2].

Hill first proposed a theory of plasticity for anisotropic materials in 1948 [3]. The criterion is extremely simple and easy to adopt for numerical simulations; however, it presents several drawbacks, such as the inability to express the yield stress and Lanford value simultaneously. After the introduction of the theory above, the anisotropy of yield surfaces in the principal stress field was discussed extensively; however, substantial new proposals regarding the yield criterion for anisotropic materials did not appear until the work of Barlat et al. in 1991 [4]. Barlat et al. developed a criterion in a step-by-step manner and presented a substantial final version in 2003 [5-7]. This yield criterion is extremely flexible and is considered the best for expressing the yielding of anisotropic materials.

Additionally, the plasticity of anisotropic materials has been investigated based on crystal plasticity. Bishop and Hill derived anisotropic yield surfaces by considering the plastic deformation of crystal grains in polycrystalline aggregates [1, 8]. Such attempts to obtain a yield surface based on crystal plasticity were similarly performed by Hutchinson, and Logan and Hosford [9, 10]; however, the method to obtain the yield surface is extremely complicated and thus not applicable to general-purpose numerical simulations, such as finite-element analysis. Meanwhile, Asaro established a crystal-plasticity method that incorporated inclusions and self-consistent using the finite-element method (FEM) [11]. This method is extremely useful and widely used; however, the isotropic elasticity was initially assumed even in CP-FEMs. In recent years, the anisotropic elasticity has been considered in most studies using CP-FEM. However, the

<figure><img src="image_960.png" /></figure>
# Evaluation of correlations between principal axes in uniaxial tensile tests of aluminum based on Mohr's strain circle

IIZUKA Takashi¹,ᵃ*

¹Faculty of Mechanical Engineering, Kyoto Institute of Technology, Matsugasaki Goshokaido-cho, Sakyo-ku, Kyoto 606-8585, Japan

ᵃtiizuka@kit.ac.jp

**Keywords:** Anisotropic Metals, Principal Axes, The Associated Flow Rule, Mohr'S Strain Circle

**Abstract.** Anisotropic elasticity is expected to contribute significantly to material deformation and should be considered in nature. Therefore, in this study, the correlations between elastic and plastic strains are investigated, with emphasis on the principal axes. In an experiment, strains are measured in eight directions during tensile tests, and Mohr's circles are derived for elastic and plastic strains. The planar anisotropy of elasticity and plasticity for some mechanical properties is confirmed, and the angle deviations between the principal axes of stress and strain are compared for elasticity and plasticity. In the case of aluminum, for both elasticity and plasticity, the planar anisotropy of its mechanical properties indicates four-fold symmetry, and the principal axes of elastic and plastic strains are identical, with some angle deviations from the principal axes of stress.

**Introduction**

The fundamental theory of plasticity is applicable to isotropic materials and is summarized based on the Mises' yield criterion and the associated flow rule. The von Mises yield criterion presents a yield surface with a circular cylinder shape in the principal stress field; however, Bishop and Hill indicated that yield criteria that present pillar-type surfaces with a cross-section of six-fold symmetry may be suitable for isotropic materials if the Bauschinger effect is disregarded [1]. This implies that all yield functions for isotropic materials can be expressed by those proposed by Hosfold [2].

Hill first proposed a theory of plasticity for anisotropic materials in 1948 [3]. The criterion is extremely simple and easy to adopt for numerical simulations; however, it presents several drawbacks, such as the inability to express the yield stress and Lanford value simultaneously. After the introduction of the theory above, the anisotropy of yield surfaces in the principal stress field was discussed extensively; however, substantial new proposals regarding the yield criterion for anisotropic materials did not appear until the work of Barlat et al. in 1991 [4]. Barlat et al. developed a criterion in a step-by-step manner and presented a substantial final version in 2003 [5-7]. This yield criterion is extremely flexible and is considered the best for expressing the yielding of anisotropic materials.

Additionally, the plasticity of anisotropic materials has been investigated based on crystal plasticity. Bishop and Hill derived anisotropic yield surfaces by considering the plastic deformation of crystal grains in polycrystalline aggregates [1, 8]. Such attempts to obtain a yield surface based on crystal plasticity were similarly performed by Hutchinson, and Logan and Hosford [9, 10]; however, the method to obtain the yield surface is extremely complicated and thus not applicable to general-purpose numerical simulations, such as finite-element analysis. Meanwhile, Asaro established a crystal-plasticity method that incorporated inclusions and self-consistent using the finite-element method (FEM) [11]. This method is extremely useful and widely used; however, the isotropic elasticity was initially assumed even in CP-FEMs. In recent years, the anisotropic elasticity has been considered in most studies using CP-FEM. However, the

<figure><img src="image_967.png" /></figure>
effect of anisotropic elasticity obtained from such simulations have not been reflected in macroscopic yield functions.

Therefore, in this study, macroscopic correlations between elastic and plastic strains are investigated, with emphasis on the principal axes. Tensile tests are conducted to measure the strains in eight directions, and the development of Mohr's circles for elastic and plastic strains is investigated. Some elastic and plastic properties are confirmed, and the angle deviations between the principal axes of stress and strain are compared for elastic and plastic cases.

Measurement of strain in multiple directions

In this study, eight strain gauges were used to measure Mohr's strain circle via a uniaxial tensile test. The strain gauges were fixed every 22.5°, and the angle of direction for each strain gauge from the tensile direction was termed the measuring direction θ. This indicates that the angular distribution of the normal strain is measured from 0° to 157.5° during the tensile test. The JIS No.5 standard tensile specimen configuration was adopted, and the positions for fixing the strain gauges were determined based on a base point, which was the center point of the tensile specimen, as shown in Fig. 1.

<figure><img src="image_961.png" /><figcaption>Figure 1 Fixing configuration of eight strain gauges.</figcaption></figure>

Pure aluminum (JIS A1050P-O) with a thickness of 1 mm was used. Tensile tests were conducted by varying the tensile direction every 15° from the rolling direction, and this direction was defined as the tensile direction *φ*, as shown in Fig. 2(a). A Cu-Ni-type strain gauge with a gauge length of 5 mm (Tokyo Measuring Instruments Laboratory, YEFLAB-5) was used to measure the strain in the plastic-deformation region. A strain of approximately 10% can be measured using this strain gauge. In such strain measurements, material rotation is assumed to

<figure><img src="image_962.png" /><figcaption>Figure 2 Geometrical parameters and material rotation in uniaxial tensile test.</figcaption></figure>
effect of anisotropic elasticity obtained from such simulations have not been reflected in macroscopic yield functions.

Therefore, in this study, macroscopic correlations between elastic and plastic strains are investigated, with emphasis on the principal axes. Tensile tests are conducted to measure the strains in eight directions, and the development of Mohr's circles for elastic and plastic strains is investigated. Some elastic and plastic properties are confirmed, and the angle deviations between the principal axes of stress and strain are compared for elastic and plastic cases.

Measurement of strain in multiple directions

In this study, eight strain gauges were used to measure Mohr's strain circle via a uniaxial tensile test. The strain gauges were fixed every 22.5°, and the angle of direction for each strain gauge from the tensile direction was termed the measuring direction θ. This indicates that the angular distribution of the normal strain is measured from 0° to 157.5° during the tensile test. The JIS No.5 standard tensile specimen configuration was adopted, and the positions for fixing the strain gauges were determined based on a base point, which was the center point of the tensile specimen, as shown in Fig. 1.

<figure><img src="image_968.png" /><figcaption>Figure 1 Fixing configuration of eight strain gauges.</figcaption></figure>

Pure aluminum (JIS A1050P-O) with a thickness of 1 mm was used. Tensile tests were conducted by varying the tensile direction every 15° from the rolling direction, and this direction was defined as the tensile direction *φ*, as shown in Fig. 2(a). A Cu-Ni-type strain gauge with a gauge length of 5 mm (Tokyo Measuring Instruments Laboratory, YEFLAB-5) was used to measure the strain in the plastic-deformation region. A strain of approximately 10% can be measured using this strain gauge. In such strain measurements, material rotation is assumed to

<figure><img src="image_969.png" /><figcaption>Figure 2 Geometrical parameters and material rotation in uniaxial tensile test.</figcaption></figure>
<figure><img src="image_963.png" /><figcaption>Figure 3 Appearance of tensile test based on eight strain gauges.</figcaption></figure>

occur continuously with increasing plastic elongation. Because the material rotation is assumed to be caused by plastic deformation, the ideal variation in each measurement direction was calculated from the measured plastic strain in both the tensile and transverse directions, as shown in Fig. 2(b). The appearance of a tensile specimen before and during the tensile test is shown in Fig. 3. The occurrence and extent of the material rotation can be confirmed from this figure. In the Results section, the first effect of this material rotation is shown.

## Incremental uniaxial tensile test

To measure the elastic and plastic strains concurrently, a uniaxial tensile test was conducted by incrementally adding plastic strain. An example of a tensile load-stroke diagram is shown in Fig. 4(a). The initial distance between the chucks was set to 100 mm, and loading and unloading were repeated after adding the displacement prescribed in advance as a stroke between the chucks. As shown in Fig. 4(a), the displacement was prescribed to be smaller in the early stage, and it was increased with respect to a plastic elongation. The additional stroke was 0.12 mm when the stroke between the chucks, $u$, was 0–1.2 mm, and it was increased to 0.24 and 0.60 mm until the stroke reached 2.4 and 6.0 mm, respectively. Although the results for further loading are also shown in the figure, the discussion herein is based on only the results obtained for strokes from 0 to 6.0 mm.

<figure><img src="image_964.png" /><figcaption>Figure 4 Uniaxial tensile test and evaluation of stress and strain.</figcaption></figure>

In the measurement, a contact-type displacement meter was used together, and the displacement between gauge lengths of 50 mm was measured more precisely. As shown in Fig. 4(b), because some plastic elongation, $\delta_p$, appeared after each unloading, the current cross-sectional area, $A$, was estimated from the current gauge length, $l$, based on the volume constancy law. The true stress was calculated using the current cross-sectional area, and the elastic strains measured by the strain gauges were re-evaluated by considering the plastic strain given by that time. The relationship between the true stress, $\sigma$, and elastic strain in each measuring direction, $\varepsilon_e(\theta)$, can be approximated as a linear function, as follows:
<figure><img src="image_970.png" /><figcaption>Figure 3 Appearance of tensile test based on eight strain gauges.</figcaption></figure>

occur continuously with increasing plastic elongation. Because the material rotation is assumed to be caused by plastic deformation, the ideal variation in each measurement direction was calculated from the measured plastic strain in both the tensile and transverse directions, as shown in Fig. 2(b). The appearance of a tensile specimen before and during the tensile test is shown in Fig. 3. The occurrence and extent of the material rotation can be confirmed from this figure. In the Results section, the first effect of this material rotation is shown.

## Incremental uniaxial tensile test

To measure the elastic and plastic strains concurrently, a uniaxial tensile test was conducted by incrementally adding plastic strain. An example of a tensile load-stroke diagram is shown in Fig. 4(a). The initial distance between the chucks was set to 100 mm, and loading and unloading were repeated after adding the displacement prescribed in advance as a stroke between the chucks. As shown in Fig. 4(a), the displacement was prescribed to be smaller in the early stage, and it was increased with respect to a plastic elongation. The additional stroke was 0.12 mm when the stroke between the chucks, $u$, was 0–1.2 mm, and it was increased to 0.24 and 0.60 mm until the stroke reached 2.4 and 6.0 mm, respectively. Although the results for further loading are also shown in the figure, the discussion herein is based on only the results obtained for strokes from 0 to 6.0 mm.

<figure><img src="image_971.png" /><figcaption>Figure 4 Uniaxial tensile test and evaluation of stress and strain.</figcaption></figure>

In the measurement, a contact-type displacement meter was used together, and the displacement between gauge lengths of 50 mm was measured more precisely. As shown in Fig. 4(b), because some plastic elongation, $\delta_p$, appeared after each unloading, the current cross-sectional area, $A$, was estimated from the current gauge length, $l$, based on the volume constancy law. The true stress was calculated using the current cross-sectional area, and the elastic strains measured by the strain gauges were re-evaluated by considering the plastic strain given by that time. The relationship between the true stress, $\sigma$, and elastic strain in each measuring direction, $\varepsilon_e(\theta)$, can be approximated as a linear function, as follows:
$$ \sigma = E(\theta)\varepsilon_c(\theta), \qquad (1) $$

where E is the linear coefficient in each direction; notably, this coefficient is no longer a component of the elastic modulus tensor or matrix. Meanwhile, the inverse of coefficient C is related to the component of the elastic compliance tensor or matrix. Because the value of the true stress is the one in tensile direction and common in every θ,

$$ \frac{\varepsilon_c(\theta)}{\sigma} = C(\theta) \qquad (2) $$

is obtained by dividing the measured elastic strain by the true stress, which can be used as an elastic variable instead of the elastic strain. When linear elastic theory is assumed, this elastic variable is not affected directly by the amount of elastic strain; thus, it can be used conveniently for examining elastic properties. In addition, it can be correlated with the elastic compliances, such as

$$ C(0^{\circ})=C_{11}(\phi), \quad C(90^{\circ})=C_{12}(\phi), \qquad (3) $$

where C11 and C12 are the components of the elastic compliance matrix, denoted by the Voigt notation. The plastic strain in each direction was calculated as the logarithmic strain from the strain measured using strain gauge after unloading.

## Estimation of Mohr's strain circle

Based on the elastic and plastic strains measured in the measuring directions, each Mohr's strain circle was estimated, and the elastic/plastic properties and angle deviation of the principal strain direction from tensile direction, ψ, was calculated. The Mohr's circle was estimated using two different methods: the least-squares approximation (LSA) and the discrete Fourier transformation (DFT). Using the LSA, the coefficient of determination,

$$ R^2 = \frac{\sum_{i}^{n} (y_i - f(x_i))^2}{\sum_{i}^{n} (y_i - \bar{y}_i)^2}, \qquad (4) $$

was used to determine the diameter and center coordinates of the circle and the angle deviation. In the equation above, xᵢ and yᵢ are the experimental values, and f(x) and yᵢ are the approximated linear function and average value of yᵢ, respectively. If the angle deviation is provided, then the normal strain is expressed as a first-order function of cos2(θ - ψ); hence, the intercept and coefficient can be decided based on LSA. The intercept is the center coordinate of the normal strain, and the coefficient is the radius of the circle. The R² was calculated by varying the angle deviation from -45° to 45°, and the angle deviation with the largest R² was adopted.

Meanwhile, using the DFT, terms other than the constant and a period of 180° were disregarded after frequency resolution. The value of the constant term is the center coordinate of the normal strain, and the coefficient of the term with a period of 180° is the radius of the circle. The angle deviation was calculated by synthesizing sine and cosine functions. However, the DFT method is intrinsically valid only for data with a constant angle interval.
$$ \sigma = E(\theta)\varepsilon_c(\theta), \qquad (1) $$

where E is the linear coefficient in each direction; notably, this coefficient is no longer a component of the elastic modulus tensor or matrix. Meanwhile, the inverse of coefficient C is related to the component of the elastic compliance tensor or matrix. Because the value of the true stress is the one in tensile direction and common in every θ,

$$ \frac{\varepsilon_c(\theta)}{\sigma} = C(\theta) \qquad (2) $$

is obtained by dividing the measured elastic strain by the true stress, which can be used as an elastic variable instead of the elastic strain. When linear elastic theory is assumed, this elastic variable is not affected directly by the amount of elastic strain; thus, it can be used conveniently for examining elastic properties. In addition, it can be correlated with the elastic compliances, such as

$$ C(0^{\circ})=C_{11}(\phi), \quad C(90^{\circ})=C_{12}(\phi), \qquad (3) $$

where C11 and C12 are the components of the elastic compliance matrix, denoted by the Voigt notation. The plastic strain in each direction was calculated as the logarithmic strain from the strain measured using strain gauge after unloading.

## Estimation of Mohr's strain circle

Based on the elastic and plastic strains measured in the measuring directions, each Mohr's strain circle was estimated, and the elastic/plastic properties and angle deviation of the principal strain direction from tensile direction, ψ, was calculated. The Mohr's circle was estimated using two different methods: the least-squares approximation (LSA) and the discrete Fourier transformation (DFT). Using the LSA, the coefficient of determination,

$$ R^2 = \frac{\sum_{i}^{n} (y_i - f(x_i))^2}{\sum_{i}^{n} (y_i - \bar{y}_i)^2}, \qquad (4) $$

was used to determine the diameter and center coordinates of the circle and the angle deviation. In the equation above, xᵢ and yᵢ are the experimental values, and f(x) and yᵢ are the approximated linear function and average value of yᵢ, respectively. If the angle deviation is provided, then the normal strain is expressed as a first-order function of cos2(θ - ψ); hence, the intercept and coefficient can be decided based on LSA. The intercept is the center coordinate of the normal strain, and the coefficient is the radius of the circle. The R² was calculated by varying the angle deviation from -45° to 45°, and the angle deviation with the largest R² was adopted.

Meanwhile, using the DFT, terms other than the constant and a period of 180° were disregarded after frequency resolution. The value of the constant term is the center coordinate of the normal strain, and the coefficient of the term with a period of 180° is the radius of the circle. The angle deviation was calculated by synthesizing sine and cosine functions. However, the DFT method is intrinsically valid only for data with a constant angle interval.
## Material rotation in uniaxial tensile test

As mentioned previously, the measurement directions of strain were rotated by plastic deformation. Therefore, the effect of material rotation on strain measurement was investigated. The direction of each strain gauge was measured from photographs captured after each unloading process using graphics software. As an example, variations with stroke in the direction of eight strain gauges for $\phi = 0°$ is shown in Fig. 5. In this figure, the curves drawn for each direction were approximated from those estimated using the volume constancy law. However, the angles were measured by assuming the vertical direction on the photograph as the tensile direction. As shown in the figure, each measurement direction varied slightly, whereas the rotation minimally affected the strain measurement. The maximum rotation was 3°, and the variation was reflected in the LSA. In the DFT approximation, this variation was disregarded.

<figure><img src="image_965.png" /><figcaption>Figure 5 Variations in measuring direction with uniaxial tension ($\phi=0°$).</figcaption></figure>

## Elastic properties and deviation between principal axes of stress and strain

Fig. 6 shows the variations in the elastic variables defined in Eq. 2 for measuring the direction from 0° to 157.5° under $\phi = 30°$ with increasing stroke between chucks. As shown in this figure, the variables increase significantly in the early stage; however, the variation became almost linear and moderate beginning from $u = 1$ mm.

<figure><img src="image_966.png" /><figcaption>Figure 6 Variations in elastic variable C for incremental uniaxial tension ($\phi = 30°$).</figcaption></figure>

Variations in the linear coefficient *E* and elastic variable *C* with the measuring direction are shown for the case of $\phi = 30°$ in Fig. 7. In this figure, the variations at $u = 0.6$, 1.2, and 2.4 mm are compared. As shown in Fig. 6, almost no differences were observed between the strokes in both cases. For the linear coefficient *E*, no clear relationships were indicated, whereas for the elastic variable, clear cosine-like curves depicting the relationship between *C* and $\theta$ were expected. This implies that Mohr's strain circle can be estimated from these results for the elastic variable.
## Material rotation in uniaxial tensile test

As mentioned previously, the measurement directions of strain were rotated by plastic deformation. Therefore, the effect of material rotation on strain measurement was investigated. The direction of each strain gauge was measured from photographs captured after each unloading process using graphics software. As an example, variations with stroke in the direction of eight strain gauges for $\phi = 0°$ is shown in Fig. 5. In this figure, the curves drawn for each direction were approximated from those estimated using the volume constancy law. However, the angles were measured by assuming the vertical direction on the photograph as the tensile direction. As shown in the figure, each measurement direction varied slightly, whereas the rotation minimally affected the strain measurement. The maximum rotation was 3°, and the variation was reflected in the LSA. In the DFT approximation, this variation was disregarded.

<figure><img src="image_972.png" /><figcaption>Figure 5 Variations in measuring direction with uniaxial tension ($\phi=0°$).</figcaption></figure>

## Elastic properties and deviation between principal axes of stress and strain

Fig. 6 shows the variations in the elastic variables defined in Eq. 2 for measuring the direction from 0° to 157.5° under $\phi = 30°$ with increasing stroke between chucks. As shown in this figure, the variables increase significantly in the early stage; however, the variation became almost linear and moderate beginning from $u = 1$ mm.

<figure><img src="image_973.png" /><figcaption>Figure 6 Variations in elastic variable C for incremental uniaxial tension ($\phi = 30°$).</figcaption></figure>

Variations in the linear coefficient *E* and elastic variable *C* with the measuring direction are shown for the case of $\phi = 30°$ in Fig. 7. In this figure, the variations at $u = 0.6$, 1.2, and 2.4 mm are compared. As shown in Fig. 6, almost no differences were observed between the strokes in both cases. For the linear coefficient *E*, no clear relationships were indicated, whereas for the elastic variable, clear cosine-like curves depicting the relationship between *C* and *θ* were expected. This implies that Mohr's strain circle can be estimated from these results for the elastic variable.
<figure><img src="image_967.png" /><figcaption>Figure 7 Variations in linear coefficient *E* and elastic compliance *C* with *θ* (φ =30°).</figcaption></figure>

As mentioned previously, Mohr's circle was estimated using the LSA and DFT. The angular deviation between the principal axes of stress and strain can be obtained from these approximations as well. For the elastic strain, the anisotropy of the elastic properties can be discussed using *C* instead of using the elastic strain, as shown in Eq. 2. Furthermore, if *θ* = 0° and *θ* = 90° are adopted for the obtained Mohr's strain circle, then the elastic compliances *C*₁₁ and *C*₁₂ can be expressed as shown in Eq. 3, respectively.

As for the tensile direction φ = 75°, variations in C₁₁ and C₁₂ and the angle deviation ψ with the stroke *u* are shown in Fig. 8. In this figure, only the results with a high R² exceeding 0.9 for the LSA are plotted, and the results obtained using the two approximations are compared. Fig. 8(a) shows that the value of C₁₁ is approximately 15Pa⁻¹. The initial value of C₁₁ was approximately 13 Pa⁻¹, which then increased nonlinearly and significantly until approximately *u* = 1 mm. When the stroke was greater than 1 mm, the increase diminished, and the value remained almost constant. The two approximation methods did not indicate any difference.

<figure><img src="image_968.png" /><figcaption>Figure 8 Variations in elastic compliances C₁₁ and C₁₂, and deviation angle ψ for φ = 75°.</figcaption></figure>

As shown in Fig. 8(b), variations in $C_{12}$ with stroke were not observed in either approximation. The value of $C_{12}$ was approximately -8 Pa⁻¹ for both approximation methods. Meanwhile, the angle deviated by 6° to 7°. This deviation, although small in magnitude, had a significant meaning in mechanics. Thus, one can conclude that the principal axis of elastic strain no longer agreed with that of stress when the tensile direction was not the symmetric axis of the sheet metal. Variations in the angle deviation were not observed, even when the stroke increased. The DFT estimated a slightly larger deviation than the LSA; however, the results can be regarded are similar.

Next, the anisotropies of the elastic properties and angle deviation are shown for *u* = 1.2 mm in Fig. 9. In this figure, the horizontal axis represents the tensile direction, and the planar anisotropies estimated using the two approximations are compared. First, for elastic compliance $C_{11}$ shown in Fig. 9(a), the planar anisotropy showed four-fold symmetry, as expected. $C_{11}$ indicated a high value of approximately 16.5 Pa⁻¹ at φ = 0° and φ = 90°, i.e., in the rolling and transverse directions,
<figure><img src="image_974.png" /><figcaption>Figure 7 Variations in linear coefficient *E* and elastic compliance *C* with *θ* (φ = 30°).</figcaption></figure>

As mentioned previously, Mohr's circle was estimated using the LSA and DFT. The angular deviation between the principal axes of stress and strain can be obtained from these approximations as well. For the elastic strain, the anisotropy of the elastic properties can be discussed using *C* instead of using the elastic strain, as shown in Eq. 2. Furthermore, if *θ* = 0° and *θ* = 90° are adopted for the obtained Mohr's strain circle, then the elastic compliances *C*₁₁ and *C*₁₂ can be expressed as shown in Eq. 3, respectively.

As for the tensile direction φ = 75°, variations in C₁₁ and C₁₂ and the angle deviation ψ with the stroke *u* are shown in Fig. 8. In this figure, only the results with a high R² exceeding 0.9 for the LSA are plotted, and the results obtained using the two approximations are compared. Fig. 8(a) shows that the value of C₁₁ is approximately 15Pa⁻¹. The initial value of C₁₁ was approximately 13 Pa⁻¹, which then increased nonlinearly and significantly until approximately *u* = 1 mm. When the stroke was greater than 1 mm, the increase diminished, and the value remained almost constant. The two approximation methods did not indicate any difference.

<figure><img src="image_975.png" /><figcaption>Figure 8 Variations in elastic compliances C₁₁ and C₁₂, and deviation angle ψ for φ = 75°.</figcaption></figure>

As shown in Fig. 8(b), variations in $C_{12}$ with stroke were not observed in either approximation. The value of $C_{12}$ was approximately -8 Pa⁻¹ for both approximation methods. Meanwhile, the angle deviated by 6° to 7°. This deviation, although small in magnitude, had a significant meaning in mechanics. Thus, one can conclude that the principal axis of elastic strain no longer agreed with that of stress when the tensile direction was not the symmetric axis of the sheet metal. Variations in the angle deviation were not observed, even when the stroke increased. The DFT estimated a slightly larger deviation than the LSA; however, the results can be regarded as similar.

Next, the anisotropies of the elastic properties and angle deviation are shown for $u = 1.2$ mm in Fig. 9. In this figure, the horizontal axis represents the tensile direction, and the planar anisotropies estimated using the two approximations are compared. First, for elastic compliance $C_{11}$ shown in Fig. 9(a), the planar anisotropy showed four-fold symmetry, as expected. $C_{11}$ indicated a high value of approximately 16.5 Pa⁻¹ at φ = 0° and φ = 90°, i.e., in the rolling and transverse directions,
respectively, and the lowest value of approximately 15 Pa⁻¹ near φ = 45°. This tendency was similar in the two approximations; however, the LSA estimated a slightly better ordered symmetry.

<figure><img src="image_969.png" /><figcaption>Figure 9 Variations in elastic compliances $C_{11}$ and $C_{12}$, and deviation angle $\psi$ with $\phi$.</figcaption></figure>

$C_{12}$ did not vary significantly, and its value ranged from -6 to -9 Pa⁻¹. Meanwhile, its planar anisotropy was similar to that of $C_{11}$, and the highest value was recorded at $\phi = 0°$ and $\phi = 90°$. Although a slight deviation was indicated between the two approximations, the effect was insignificant. The angle deviation shown in Fig. 9(c) was almost 0° at $\phi = 0°$ and $\phi = 90°$, where one of the structural symmetric axes was assumed to be in the tensile direction. The planar anisotropy exhibited four-fold symmetry, and the largest deviation of approximately 8° was indicated at $\phi = 90°$. As mentioned previously, this deviation, although small in magnitude, should be regarded as a significant deviation. The value estimated via the DFT was slightly larger than that estimated using LSA; however, the difference was not significant.

## Plastic properties and deviation between principal axes of stress and strain

Fig. 10 shows the variations in plastic strains in the length and width directions and the angle deviation between the principal axes of stress and strain with increasing stroke under a tensile direction of 75°. Based on this figure, the results obtained from the two approximations are compared. In Fig. 10(a), the variations in plastic strain in the length direction, i.e., the tensile direction, are compared. As shown, the plastic strain increased linearly with the stroke between the chucks. Meanwhile, the plastic strain in the width direction decreased linearly as the stroke increased, as shown in Fig. 10(b). In both cases, almost no difference was observed between the results obtained using the two approximation methods. However, as shown in Fig. 10(c), the angle deviation increased significantly in the early stage by $u = 1$ mm. This suggests rotation in the

<figure><img src="image_970.png" /><figcaption>Figure 10 Variations in length and width plastic strains and deviation angle $\psi$ for $\phi = 75°$. principal axis of the plastic strain from the tensile direction to a certain stationary direction.</figcaption></figure>

After the direction of the principal axis of the plastic strain became stationary, the angle deviation increased only slightly from 4° to 6°. Such a difference in the principal direction between
respectively, and the lowest value of approximately 15 Pa⁻¹ near φ = 45°. This tendency was similar in the two approximations; however, the LSA estimated a slightly better ordered symmetry.

<figure><img src="image_976.png" /><figcaption>Figure 9 Variations in elastic compliances $C_{11}$ and $C_{12}$, and deviation angle $\psi$ with $\phi$.</figcaption></figure>

$C_{12}$ did not vary significantly, and its value ranged from -6 to -9 Pa⁻¹. Meanwhile, its planar anisotropy was similar to that of $C_{11}$, and the highest value was recorded at $\phi = 0°$ and $\phi = 90°$. Although a slight deviation was indicated between the two approximations, the effect was insignificant. The angle deviation shown in Fig. 9(c) was almost 0° at $\phi = 0°$ and $\phi = 90°$, where one of the structural symmetric axes was assumed to be in the tensile direction. The planar anisotropy exhibited four-fold symmetry, and the largest deviation of approximately 8° was indicated at $\phi = 90°$. As mentioned previously, this deviation, although small in magnitude, should be regarded as a significant deviation. The value estimated via the DFT was slightly larger than that estimated using LSA; however, the difference was not significant.

## Plastic properties and deviation between principal axes of stress and strain

Fig. 10 shows the variations in plastic strains in the length and width directions and the angle deviation between the principal axes of stress and strain with increasing stroke under a tensile direction of 75°. Based on this figure, the results obtained from the two approximations are compared. In Fig. 10(a), the variations in plastic strain in the length direction, i.e., the tensile direction, are compared. As shown, the plastic strain increased linearly with the stroke between the chucks. Meanwhile, the plastic strain in the width direction decreased linearly as the stroke increased, as shown in Fig. 10(b). In both cases, almost no difference was observed between the results obtained using the two approximation methods. However, as shown in Fig. 10(c), the angle deviation increased significantly in the early stage by $u = 1$ mm. This suggests rotation in the

<figure><img src="image_977.png" /><figcaption>Figure 10 Variations in length and width plastic strains and deviation angle $\psi$ for $\phi = 75°$. principal axis of the plastic strain from the tensile direction to a certain stationary direction.</figcaption></figure>

After the direction of the principal axis of the plastic strain became stationary, the angle deviation increased only slightly from 4° to 6°. Such a difference in the principal direction between
the stress and strain naturally generates shear strain in the tensile direction, and the strains in the tensile and vertical directions are no longer the principal values. In this case, the Lankford value evaluated via the ordinary method is not the ratio between the principal strains. In addition, an increase in the angular deviation in the early stage indicates that the deformation pattern varied continuously within a stroke of 1 mm. One can assume that such transitions in the deformation state occur consistently in the early stages after the stress state changes. The difference between the results yielded by the LSA and DFT ranged from 1° to 2°, which was considered insignificant.

The representative plastic properties include the planar strain ratio $\beta$ and Lankford value $r$. They are defined as

$$ \beta = \frac{\varepsilon_p(90^\circ)}{\varepsilon_p(0^\circ)}, \quad r = -\frac{\varepsilon_p(90^\circ)}{\varepsilon_p(0^\circ) + \varepsilon_p(90^\circ)}. \qquad (5) $$

The planar anisotropies of the planar strain ratio, the Lankford value, and the angular deviation between the principal axes of stress and strain at $u = 1.2$ mm are shown in Fig. 11. Although the shape of the symmetry was not particularly neat, the planar anisotropy of the planar strain ratio exhibited four-fold symmetry, as shown in Fig. 11(a). Similar to the case of $C_{11}$, the highest value was indicated at $\phi = 0^\circ$ and $\phi = 90^\circ$, and the lowest value near $\phi = 45^\circ$. The value of $\beta$ was approximately 0.5 in a wide range, which implies that an almost isotropic deformation occurred within the range. The results obtained from the two approximations differed only slightly.

As shown in Fig. 11(b), the Lankford value was inversely proportional to the planar strain ratio. This tendency was as expected, and the lowest value was approximately 0.7 at $\phi = 0^\circ$ and $\phi = 90^\circ$, and the maximum value was 1.2 near $\phi = 45^\circ$. The angle deviation was approximately 0 at $\phi = 0^\circ$ and $\phi = 90^\circ$, as expected. The planar anisotropy was similar to the four-fold symmetry, i.e. similar to Mohr's circle of elastic strain. The maximum deviation was approximately 6° and was indicated near $\phi = 45^\circ$, as similarly observed for the elastic strain in Fig. 9.

<figure><img src="image_971.png" /><figcaption>Figure 11 Variations in plastic strain ratios $\beta$ and $r$, and deviation angle $\psi$ with $\phi$.</figcaption></figure>

## Correlations between elastic properties and plastic properties

Here, the correlations between elastic and plastic properties are discussed. First, the angle deviations obtained from Figs. 9(c) and 11(c) are compared in Fig. 12. In this figure, the angular deviation between the principal axes of the stress and plastic strain increments is shown as well. The plastic strain increment was calculated using the plastic strain results. As shown in this figure, the planar anisotropies of the angle deviation were similar those with four-fold symmetry. The value differed only slightly by 1° to 2° and thus exerted minimal effects. This indicates that the directions of the principal axes were identical for the three strains.
the stress and strain naturally generates shear strain in the tensile direction, and the strains in the tensile and vertical directions are no longer the principal values. In this case, the Lankford value evaluated via the ordinary method is not the ratio between the principal strains. In addition, an increase in the angular deviation in the early stage indicates that the deformation pattern varied continuously within a stroke of 1 mm. One can assume that such transitions in the deformation state occur consistently in the early stages after the stress state changes. The difference between the results yielded by the LSA and DFT ranged from 1° to 2°, which was considered insignificant.

The representative plastic properties include the planar strain ratio $\beta$ and Lankford value $r$. They are defined as

$$ \beta = \frac{\varepsilon_p(90^\circ)}{\varepsilon_p(0^\circ)}, \quad r = -\frac{\varepsilon_p(90^\circ)}{\varepsilon_p(0^\circ) + \varepsilon_p(90^\circ)}. \qquad (5) $$

The planar anisotropies of the planar strain ratio, the Lankford value, and the angular deviation between the principal axes of stress and strain at $u = 1.2$ mm are shown in Fig. 11. Although the shape of the symmetry was not particularly neat, the planar anisotropy of the planar strain ratio exhibited four-fold symmetry, as shown in Fig. 11(a). Similar to the case of $C_{11}$, the highest value was indicated at $\phi = 0^\circ$ and $\phi = 90^\circ$, and the lowest value near $\phi = 45^\circ$. The value of $\beta$ was approximately 0.5 in a wide range, which implies that an almost isotropic deformation occurred within the range. The results obtained from the two approximations differed only slightly.

As shown in Fig. 11(b), the Lankford value was inversely proportional to the planar strain ratio. This tendency was as expected, and the lowest value was approximately 0.7 at $\phi = 0^\circ$ and $\phi = 90^\circ$, and the maximum value was 1.2 near $\phi = 45^\circ$. The angle deviation was approximately 0 at $\phi = 0^\circ$ and $\phi = 90^\circ$, as expected. The planar anisotropy was similar to the four-fold symmetry, i.e. similar to Mohr's circle of elastic strain. The maximum deviation was approximately 6° and was indicated near $\phi = 45^\circ$, as similarly observed for the elastic strain in Fig. 9.

<figure><img src="image_978.png" /><figcaption>Figure 11 Variations in plastic strain ratios $\beta$ and $r$, and deviation angle $\psi$ with $\phi$.</figcaption></figure>

## Correlations between elastic properties and plastic properties

Here, the correlations between elastic and plastic properties are discussed. First, the angle deviations obtained from Figs. 9(c) and 11(c) are compared in Fig. 12. In this figure, the angular deviation between the principal axes of the stress and plastic strain increments is shown as well. The plastic strain increment was calculated using the plastic strain results. As shown in this figure, the planar anisotropies of the angle deviation were similar those with four-fold symmetry. The value differed only slightly by 1° to 2° and thus exerted minimal effects. This indicates that the directions of the principal axes were identical for the three strains.
<figure><img src="image_972.png" /><figcaption>Fig. 12 Comparison of deviation angle $\psi$ during elastic and plastic strains (LSA).</figcaption></figure>

Finally, the undecided elastic compliance, $C_{13}$, was estimated from the previously obtained elastic and plastic properties. If the associated flow rule and coaxiality between the elastic-strain deviation and plastic strain can be assumed simultaneously, then $C_{13}$ can be calculated. Here, if the elastic-strain deviation is expressed as the ordinary relation, then the relationship

$$C_{13} = 3 \frac{C_{12} - \beta C_{11}}{1 - \beta} - C_{11} - C_{12} \quad (6)$$

can be established. Based on the relationship above, the planar anisotropy of $C_{13}$ for $u = 1.2$ mm was calculated using the results of the two approximations, and the results are shown in Fig. 13. As shown in this figure, the planar anisotropy of $C_{13}$ showed four-fold symmetry as expected, and the maximum value was indicated near $\phi = 45^{\circ}$. The value was similar to that of $C_{12}$, and the variation in the direction was insignificant. No significant differences were observed between the two approximations.

<figure><img src="image_973.png" /><figcaption>Fig. 13 Estimation of elastic compliance $C_{13}$ from plastic strain ratio.</figcaption></figure>

## Conclusions

In this study, an incremental tensile test was conducted using eight strain gauges, and the variation in Mohr's strain circle with elongation was investigated to determine the elastic and plastic properties. For the elastic properties, both $C_{11}$ and $C_{12}$ as well as the angle deviation showed four-fold symmetry; meanwhile, the elastic compliances and angle deviation showed the minimum and the maximum values near the $45^{\circ}$ direction from the rolling direction, respectively. The largest angle deviation was approximately $8^{\circ}$, which was not large but sufficiently significant. Similarly, the planar strain ratio and Lankford value showed four-fold symmetry as well as contrary tendencies. The planar anisotropy of the Lankford value was that typically observed. The angular deviation between the principal axes of stress and plastic strain showed four-fold symmetry, with a maximum value of approximately $6^{\circ}$. Almost no differences were observed between the two approximations. The direction of the principal strain can be regarded as identical for all elastic,
<figure><img src="image_979.png" /><figcaption>Fig. 12 Comparison of deviation angle $\psi$ during elastic and plastic strains (LSA).</figcaption></figure>

Finally, the undecided elastic compliance, $C_{13}$, was estimated from the previously obtained elastic and plastic properties. If the associated flow rule and coaxiality between the elastic-strain deviation and plastic strain can be assumed simultaneously, then $C_{13}$ can be calculated. Here, if the elastic-strain deviation is expressed as the ordinary relation, then the relationship

$$C_{13} = 3 \frac{C_{12} - \beta C_{11}}{1 - \beta} - C_{11} - C_{12} \quad (6)$$

can be established. Based on the relationship above, the planar anisotropy of $C_{13}$ for $u = 1.2$ mm was calculated using the results of the two approximations, and the results are shown in Fig. 13. As shown in this figure, the planar anisotropy of $C_{13}$ showed four-fold symmetry as expected, and the maximum value was indicated near $\phi = 45^{\circ}$. The value was similar to that of $C_{12}$, and the variation in the direction was insignificant. No significant differences were observed between the two approximations.

<figure><img src="image_980.png" /><figcaption>Fig. 13 Estimation of elastic compliance $C_{13}$ from plastic strain ratio.</figcaption></figure>

## Conclusions

In this study, an incremental tensile test was conducted using eight strain gauges, and the variation in Mohr's strain circle with elongation was investigated to determine the elastic and plastic properties. For the elastic properties, both $C_{11}$ and $C_{12}$ as well as the angle deviation showed four-fold symmetry; meanwhile, the elastic compliances and angle deviation showed the minimum and the maximum values near the $45^{\circ}$ direction from the rolling direction, respectively. The largest angle deviation was approximately $8^{\circ}$, which was not large but sufficiently significant. Similarly, the planar strain ratio and Lankford value showed four-fold symmetry as well as contrary tendencies. The planar anisotropy of the Lankford value was that typically observed. The angular deviation between the principal axes of stress and plastic strain showed four-fold symmetry, with a maximum value of approximately $6^{\circ}$. Almost no differences were observed between the two approximations. The direction of the principal strain can be regarded as identical for all elastic,
plastic, and plastic strain increments. Finally, the undecided elastic compliance $C_{13}$ was estimated using the results for both the elastic and plastic properties.

## Acknowledgement

This study was funded by the Amada Foundation (AF-2023055-X1). We are grateful for the support provided.

## References

[1] J.F.W. Bishop, R. Hill, A theory of the Plastic Distortion of a Polycrystalline Aggregate under Combined Stresses, Phil. Mag. 42 (1951) 414-427. https://doi.org/10.1080/14786445108561065

[2] W.F. Hosfold, A Generalized Isotropic Yield Criterion, J. Appl. Mech. 39 (1972) 607-609. https://doi.org/10.1115/1.3422732

Reference to a chapter in an edited book:

[3] R. Hill, A theory of the yielding and plastic flow of Anisotropic Metals, Proc. Royal Soc. London A193 (1948) 281-297. https://doi.org/10.1098/rspa.1948.0045

[4] F. Barlat, D.J. Lege, J.C. Brem: A Six-component Yield Function for Anisotropic Materials, Int. J. Plast. 7 (1991) 693-712. https://doi.org/10.1016/0749-6419(91)90052-Z

[5] F. Barlat, R.C. Becker, Y. Hayashida, Y. Maeda, M. Yanagawa, K. Chung, J.C. Brem, D.J. Lege, K. Marsui, S.J. Murtha, S. Hattori, Yielding Description for Solution Strengthened Aluminum Alloys, Int. J. Plast. 13 (1997) 385-401. https://doi.org/10.1016/S0749-6419(97)80005-8

[6] F. Barlat, Y. Maeda, K. Chung, M. Yanagawa, J.C. Brem, Y. Hayashida, D.J. Lege, K. Matsui, S.J. Murtha, S. Hattori, R.C. Becker, S. Makosey, Yield Function Development for Aluminum Alloy Sheets, J. Mech. Phys. Solids 45 (1997) 1727-1763. https://doi.org/10.1016/S0022-5096(97)00034-3

[7] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.-H. Choi, E. Chu, Plane Stress Yield Function for Aluminum Alloy Sheets -Part I:Theory, Int. J. Plast. 19 (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0

[8] J.F.W. Bishop, R. Hill, A Theoretical Derivation of the Plastic Properties of a Polycrystalline Face-centred Metal, Phil. Mag. 42 (1951) 1298-1397. https://doi.org/10.1080/14786444108561385

[9] J.W. Hutchinson, Plastic Stress-strain Relations of F.C.C Polycrystalline Metals Hardening According to Taylor's Rule, J. Mech. Phy. Solids 12 (1963) 11-24. https://doi.org/10.1016/0022-5096(64)90003-1

[10] R.W. Logan, W.F. Hosford, Upper-bound Anisotropic Yield Locus Calculations Assuming <111>-pencil Glide, Int. J. Mech. Sci. 22 (1980) 419-430. https://doi.org/10.1016/0020-7403(80)90011-9

[11] R.J. Asaro, Micromechanics of Crystals and Polycrystals, Adv. Appl. Mech. 23 (1983) 1-115. https://doi.org/10.1016/S0065-2156(08)70242-4
plastic, and plastic strain increments. Finally, the undecided elastic compliance $C_{13}$ was estimated using the results for both the elastic and plastic properties.

## Acknowledgement

This study was funded by the Amada Foundation (AF-2023055-X1). We are grateful for the support provided.

## References

[1] J.F.W. Bishop, R. Hill, A theory of the Plastic Distortion of a Polycrystalline Aggregate under Combined Stresses, Phil. Mag. 42 (1951) 414-427. https://doi.org/10.1080/14786445108561065

[2] W.F. Hosfold, A Generalized Isotropic Yield Criterion, J. Appl. Mech. 39 (1972) 607-609. https://doi.org/10.1115/1.3422732

Reference to a chapter in an edited book:

[3] R. Hill, A theory of the yielding and plastic flow of Anisotropic Metals, Proc. Royal Soc. London A193 (1948) 281-297. https://doi.org/10.1098/rspa.1948.0045

[4] F. Barlat, D.J. Lege, J.C. Brem: A Six-component Yield Function for Anisotropic Materials, Int. J. Plast. 7 (1991) 693-712. https://doi.org/10.1016/0749-6419(91)90052-Z

[5] F. Barlat, R.C. Becker, Y. Hayashida, Y. Maeda, M. Yanagawa, K. Chung, J.C. Brem, D.J. Lege, K. Marsui, S.J. Murtha, S. Hattori, Yielding Description for Solution Strengthened Aluminum Alloys, Int. J. Plast. 13 (1997) 385-401. https://doi.org/10.1016/S0749-6419(97)80005-8

[6] F. Barlat, Y. Maeda, K. Chung, M. Yanagawa, J.C. Brem, Y. Hayashida, D.J. Lege, K. Matsui, S.J. Murtha, S. Hattori, R.C. Becker, S. Makosey, Yield Function Development for Aluminum Alloy Sheets, J. Mech. Phys. Solids 45 (1997) 1727-1763. https://doi.org/10.1016/S0022-5096(97)00034-3

[7] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.-H. Choi, E. Chu, Plane Stress Yield Function for Aluminum Alloy Sheets -Part I:Theory, Int. J. Plast. 19 (2003) 1297-1319. https://doi.org/10.1016/S0749-6419(02)00019-0

[8] J.F.W. Bishop, R. Hill, A Theoretical Derivation of the Plastic Properties of a Polycrystalline Face-centred Metal, Phil. Mag. 42 (1951) 1298-1397. https://doi.org/10.1080/14786444108561385

[9] J.W. Hutchinson, Plastic Stress-strain Relations of F.C.C Polycrystalline Metals Hardening According to Taylor's Rule, J. Mech. Phy. Solids 12 (1963) 11-24. https://doi.org/10.1016/0022-5096(64)90003-1

[10] R.W. Logan, W.F. Hosford, Upper-bound Anisotropic Yield Locus Calculations Assuming <111>-pencil Glide, Int. J. Mech. Sci. 22 (1980) 419-430. https://doi.org/10.1016/0020-7403(80)90011-9

[11] R.J. Asaro, Micromechanics of Crystals and Polycrystals, Adv. Appl. Mech. 23 (1983) 1-115. https://doi.org/10.1016/S0065-2156(08)70242-4
# Development of the cross-section moment in air-bending

POKKA Aki-Petteri¹,ª*, KESTI Vili²,ᵇ, TROIVE Lars²,ᶜ and KAIJALAINEN Antti¹,ᵈ

¹University of Oulu, Materials and Mechanical Engineering, P.O. Box 4200, 90014 Oulu, Finland

²SSAB Europe, Rautaruukintie 115, 92101 Raahe, Finland

ªaki-petteri.pokka@oulu.fi, ᵇvili.kesti@ssab.com, ᶜlars.troive@ssab.com,

ᵈantti.kaijalainen@oulu.fi

**Keywords:** Air Bending, Steel, DIC, Strain Distribution, Bending Moment

**Abstract.** The engineering potential of air-bending as a sheet-metal forming process has been limited by certain challenges related to high-strength steels, strain localization, surface defects, “multi-breakage” and bend shape. The phenomena related to these challenges are not yet fully understood, as the conventional test methods have not provided sufficient data for in-depth analysis of the material behavior in air-bending. In this study, nine thermomechanically rolled steel grades are bent in an air-bending test setup using three different punch radii, and Digital Image Correlation for strain measurement on the outer surface. The development of the cross-section moment is measured from the force-displacement curve. A connection is found between the developments of the cross-section moment, strain distribution and multi-breakage, as well as the strain-hardening properties of the material. The presented results illustrate the potential for predicting bending behavior based on the force-displacement curve, that could be achieved with better understanding of the physics related to the air-bending process.

## Introduction

Air-bending is a widely used forming method for sheet metals because it is fast, cost-effective, and flexible. A wide range of bending angles can be obtained without tool modifications just by controlling punch displacement. The radius of the bend is usually controlled by altering the punch radius. However, the final shape of the bend also depends on the material properties. The distribution of strain and curvature on the outer bend surface can vary greatly depending on the strain-hardening properties of the material. Materials with a low strain-hardening tend to produce more localized strain distributions, leading to increased peak strain [1] and punch-sheet separation at the centerline (“multi-breakage”) due to decreased inner radius [2].

The multi-breakage and punch-sheet separation have been known as a phenomenon for a long time and studied by several authors [3-10]. However, research on the root of the phenomenon, i.e., the development of the strain and moment distributions, has been relatively scarce. Strain distributions in air-bending have been studied by several authors [1,2,10-17], as is the case for the development of the bending moment in air-bending [4,18,19]. However, to the knowledge of the authors, development of the strain and bending moment have not been studied simultaneously before. Therefore, research in this area is necessary.

In this study, 6 mm thick steel sheets (with yield strength between 355 – 1700 MPa) are bent in a 3-point bending test setup in a Zwick universal tensile test machine. Three different punch radii are used. Digital Image Correlation (DIC) is used for measuring the strain on the outer bend surface. The aim of the paper is to investigate the connection between the cross-section moment, strain distribution development and multi-breakage. The effects of tool geometry and material properties on the development of the cross-section moment are also investigated.

<figure><img src="image_974.png" /></figure>
# Development of the cross-section moment in air-bending

POKKA Aki-Petteri¹,ᵃ*, KESTI Vili²,ᵇ, TROIVE Lars²,ᶜ and KAIJALAINEN Antti¹,ᵈ

¹University of Oulu, Materials and Mechanical Engineering, P.O. Box 4200, 90014 Oulu, Finland

²SSAB Europe, Rautaruukintie 115, 92101 Raahe, Finland

ᵃaki-petteri.pokka@oulu.fi, ᵇvili.kesti@ssab.com, ᶜlars.troive@ssab.com,

ᵈantti.kaijalainen@oulu.fi

**Keywords:** Air Bending, Steel, DIC, Strain Distribution, Bending Moment

**Abstract.** The engineering potential of air-bending as a sheet-metal forming process has been limited by certain challenges related to high-strength steels, strain localization, surface defects, “multi-breakage” and bend shape. The phenomena related to these challenges are not yet fully understood, as the conventional test methods have not provided sufficient data for in-depth analysis of the material behavior in air-bending. In this study, nine thermomechanically rolled steel grades are bent in an air-bending test setup using three different punch radii, and Digital Image Correlation for strain measurement on the outer surface. The development of the cross-section moment is measured from the force-displacement curve. A connection is found between the developments of the cross-section moment, strain distribution and multi-breakage, as well as the strain-hardening properties of the material. The presented results illustrate the potential for predicting bending behavior based on the force-displacement curve, that could be achieved with better understanding of the physics related to the air-bending process.

## Introduction

Air-bending is a widely used forming method for sheet metals because it is fast, cost-effective, and flexible. A wide range of bending angles can be obtained without tool modifications just by controlling punch displacement. The radius of the bend is usually controlled by altering the punch radius. However, the final shape of the bend also depends on the material properties. The distribution of strain and curvature on the outer bend surface can vary greatly depending on the strain-hardening properties of the material. Materials with a low strain-hardening tend to produce more localized strain distributions, leading to increased peak strain [1] and punch-sheet separation at the centerline (“multi-breakage”) due to decreased inner radius [2].

The multi-breakage and punch-sheet separation have been known as a phenomenon for a long time and studied by several authors [3-10]. However, research on the root of the phenomenon, i.e., the development of the strain and moment distributions, has been relatively scarce. Strain distributions in air-bending have been studied by several authors [1,2,10-17], as is the case for the development of the bending moment in air-bending [4,18,19]. However, to the knowledge of the authors, development of the strain and bending moment have not been studied simultaneously before. Therefore, research in this area is necessary.

In this study, 6 mm thick steel sheets (with yield strength between 355 – 1700 MPa) are bent in a 3-point bending test setup in a Zwick universal tensile test machine. Three different punch radii are used. Digital Image Correlation (DIC) is used for measuring the strain on the outer bend surface. The aim of the paper is to investigate the connection between the cross-section moment, strain distribution development and multi-breakage. The effects of tool geometry and material properties on the development of the cross-section moment are also investigated.

<figure><img src="image_981.png" /></figure>
Experimental procedure

Nine thermomechanically hot-rolled steel grades are tested in this study. Tensile and mechanical properties are provided in Table 1. Both longitudinal and transverse orientations are included in this study, referring to the orientation of the material’s rolling direction within the test setup. In this study, the longitudinal (0°) tensile test is defined by the principal strain being parallel to the rolling direction, while the longitudinal (RD) bend test refers to the bend axis being parallel with the rolling direction. Consequently, the direction of the major strain on the outer bend surface is aligned with the rolling direction in the transverse bend orientation, meaning that transverse (TD) bends are compared to longitudinal (0°) tensile properties, and longitudinal (RD) bends to the transverse (90°) tensile properties. A schematic representation of the orientations is presented in Fig. 1.

<table><caption>Table 1. Sheet thicknesses (t) and mechanical properties of the tested materials. The yield strength (R<sub>e</sub>), ultimate tensile strength (R<sub>m</sub>), uniform elongation (A<sub>gt</sub>), total elongation with 40 mm gage length (A<sub>40</sub>), and strain hardening exponent (n<sub>1-Ag</sub>) were measured from tensile tests in a previous publication by the authors [1]. The minimum bend radii to 90° bend angle (R<sub>min</sub>) were provided by a steel manufacturer for corresponding grades.</caption><thead><tr><th>Material</th><th>t [mm]</th><th>R<sub>e</sub> [MPa]</th><th>R<sub>m</sub> [MPa]</th><th>A<sub>gt</sub> [%]</th><th>A<sub>40</sub> [%]</th><th>n<sub>1-Ag</sub> [-]</th><th>R<sub>min</sub></th></tr></thead><tbody><tr><td>St355 (0°/90°)</td><td>5.94</td><td>438/480</td><td>496/502</td><td>16.7/15.7</td><td>32.7/31.2</td><td>0.162/0.150</td><td>0.3 t</td></tr><tr><td>St500 (0°/90°)</td><td>5.92</td><td>585/628</td><td>653/669</td><td>12.1/10.8</td><td>26.9/25.4</td><td>0.120/0.106</td><td>0.8 t</td></tr><tr><td>St700 (0°/90°)</td><td>6.04</td><td>785/818</td><td>864/907</td><td>6.40/4.95</td><td>18.2/15.2</td><td>0.064/0.057</td><td>1 t</td></tr><tr><td>St900_1 (0°/90°)</td><td>5.99</td><td>974/1024</td><td>1042/1142</td><td>3.67/2.73</td><td>13.4/8.2</td><td>0.038/0.041</td><td>3 t</td></tr><tr><td>St900_2 (0°/90°)</td><td>5.93</td><td>1034/1067</td><td>1130/1151</td><td>4.38/3.52</td><td>13.9/11.4</td><td>0.036/0.032</td><td>3.5 t</td></tr><tr><td>St1100_1 (0°/90°)</td><td>6.01</td><td>1132/1127</td><td>1161/1172</td><td>5.20/5.20</td><td>15.4/13.8</td><td>0.044/0.045</td><td>3.5 t</td></tr><tr><td>St1100_2 (0°/90°)</td><td>5.93</td><td>1098/1111</td><td>1253/1271</td><td>3.62/3.11</td><td>12.1/10.8</td><td>0.046/0.042</td><td>3 t / 4 t</td></tr><tr><td>St1500 (0°/90°)</td><td>5.68</td><td>1537/1500</td><td>1761/1780</td><td>4.37/4.00</td><td>10.6/9.4</td><td>0.051/0.054</td><td>6 t</td></tr><tr><td>St1700 (0°/90°)</td><td>6.09</td><td>1740/1680</td><td>1958/1975</td><td>3.77/3.65</td><td>9.6/9.0</td><td>0.041/0.052</td><td>6 t</td></tr></tbody></table>

The bend specimens were cut from the 6 mm thick sheets into 80 mm wide rectangular strips using a water jet cutter. The specimen width of 80 mm is enough to ensure plane strain condition at the center of the bend. The specimens were bent at room temperature using a Zwick 100 kN universal tensile test machine equipped with a purpose-built bending tool setup, shown in Fig. 2a. The setup allows measurement of the bending force and punch stroke, as well as allowing a continuous unobstructed line-of-sight between the DIC cameras and the measured specimen surface, allowing measurement of strain and displacement throughout the test, as illustrated in Fig. 2b. A schematic representation of the bending geometry is presented in Fig. 3, and the parameters of the three tool setups are presented in Table 2. The shoulders of the die are lubricated using a PTFE spray, and they rotate freely in their sockets. In each setup, the specimens were bent until they either fractured or reached the bend angle of α<sub>end</sub>. Two repeat tests were conducted for each combination of tool setup, material, and bend orientation.

The bend angle was calculated according to ISO 7438 [20], taking into account the vertical elasticity of the setup (rigidity of both the tensile test machine and the bending tools), measured at 51.9 kN/mm. The cross-section moment M is determined from the bending force (vertical load) F, bend angle α, die-width W<sub>d</sub>, punch radius R<sub>p</sub> and die shoulder radius R<sub>d</sub> according to Eq. 1 [18].

$$M = \frac{F \cdot [W_d/2 - (R_p + R_d) \sin(\alpha/2)]}{2 \cos^2(\alpha/2)} \quad (1)$$
## Experimental procedure

Nine thermomechanically hot-rolled steel grades are tested in this study. Tensile and mechanical properties are provided in Table 1. Both longitudinal and transverse orientations are included in this study, referring to the orientation of the material's rolling direction within the test setup. In this study, the longitudinal (0°) tensile test is defined by the principal strain being parallel to the rolling direction, while the longitudinal (RD) bend test refers to the bend axis being parallel with the rolling direction. Consequently, the direction of the major strain on the outer bend surface is aligned with the rolling direction in the transverse bend orientation, meaning that transverse (TD) bends are compared to longitudinal (0°) tensile properties, and longitudinal (RD) bends to the transverse (90°) tensile properties. A schematic representation of the orientations is presented in Fig. 1.

<table><caption>Table 1. Sheet thicknesses (t) and mechanical properties of the tested materials. The yield strength (R<sub>e</sub>), ultimate tensile strength (R<sub>m</sub>), uniform elongation (A<sub>gt</sub>), total elongation with 40 mm gage length (A<sub>40</sub>), and strain hardening exponent (n<sub>1-Ag</sub>) were measured from tensile tests in a previous publication by the authors [1]. The minimum bend radii to 90° bend angle (R<sub>min</sub>) were provided by a steel manufacturer for corresponding grades.</caption><thead><tr><th>Material</th><th>t [mm]</th><th>R<sub>e</sub> [MPa]</th><th>R<sub>m</sub> [MPa]</th><th>A<sub>gt</sub> [%]</th><th>A<sub>40</sub> [%]</th><th>n<sub>1-Ag</sub> [-]</th><th>R<sub>min</sub></th></tr></thead><tbody><tr><td>St355 (0°/90°)</td><td>5.94</td><td>438/480</td><td>496/502</td><td>16.7/15.7</td><td>32.7/31.2</td><td>0.162/0.150</td><td>0.3 t</td></tr><tr><td>St500 (0°/90°)</td><td>5.92</td><td>585/628</td><td>653/669</td><td>12.1/10.8</td><td>26.9/25.4</td><td>0.120/0.106</td><td>0.8 t</td></tr><tr><td>St700 (0°/90°)</td><td>6.04</td><td>785/818</td><td>864/907</td><td>6.40/4.95</td><td>18.2/15.2</td><td>0.064/0.057</td><td>1 t</td></tr><tr><td>St900_1 (0°/90°)</td><td>5.99</td><td>974/1024</td><td>1042/1142</td><td>3.67/2.73</td><td>13.4/8.2</td><td>0.038/0.041</td><td>3 t</td></tr><tr><td>St900_2 (0°/90°)</td><td>5.93</td><td>1034/1067</td><td>1130/1151</td><td>4.38/3.52</td><td>13.9/11.4</td><td>0.036/0.032</td><td>3.5 t</td></tr><tr><td>St1100_1 (0°/90°)</td><td>6.01</td><td>1132/1127</td><td>1161/1172</td><td>5.20/5.20</td><td>15.4/13.8</td><td>0.044/0.045</td><td>3.5 t</td></tr><tr><td>St1100_2 (0°/90°)</td><td>5.93</td><td>1098/1111</td><td>1253/1271</td><td>3.62/3.11</td><td>12.1/10.8</td><td>0.046/0.042</td><td>3 t / 4 t</td></tr><tr><td>St1500 (0°/90°)</td><td>5.68</td><td>1537/1500</td><td>1761/1780</td><td>4.37/4.00</td><td>10.6/9.4</td><td>0.051/0.054</td><td>6 t</td></tr><tr><td>St1700 (0°/90°)</td><td>6.09</td><td>1740/1680</td><td>1958/1975</td><td>3.77/3.65</td><td>9.6/9.0</td><td>0.041/0.052</td><td>6 t</td></tr></tbody></table>

The bend specimens were cut from the 6 mm thick sheets into 80 mm wide rectangular strips using a water jet cutter. The specimen width of 80 mm is enough to ensure plane strain condition at the center of the bend. The specimens were bent at room temperature using a Zwick 100 kN universal tensile test machine equipped with a purpose-built bending tool setup, shown in Fig. 2a. The setup allows measurement of the bending force and punch stroke, as well as allowing a continuous unobstructed line-of-sight between the DIC cameras and the measured specimen surface, allowing measurement of strain and displacement throughout the test, as illustrated in Fig. 2b. A schematic representation of the bending geometry is presented in Fig. 3, and the parameters of the three tool setups are presented in Table 2. The shoulders of the die are lubricated using a PTFE spray, and they rotate freely in their sockets. In each setup, the specimens were bent until they either fractured or reached the bend angle of α<sub>end</sub>. Two repeat tests were conducted for each combination of tool setup, material, and bend orientation.

The bend angle was calculated according to ISO 7438 [20], taking into account the vertical elasticity of the setup (rigidity of both the tensile test machine and the bending tools), measured at 51.9 kN/mm. The cross-section moment M is determined from the bending force (vertical load) F, bend angle α, die-width W<sub>d</sub>, punch radius R<sub>p</sub> and die shoulder radius R<sub>d</sub> according to Eq. 1 [18].

$$M = \frac{F \cdot [W_d/2 - (R_p + R_d) \sin(\alpha/2)]}{2 \cos^2(\alpha/2)} \quad (1)$$
<figure><img src="image_975.png" /><figcaption>Fig. 1. The longitudinal and transverse orientations in tensile and bending tests.</figcaption></figure>

The specimens were painted with a black-and-white speckle pattern to improve the DIC measurement accuracy and reliability. The DIC system used in this study was LaVision Strainmaster, with DaVis 8.4 software, and two monochrome CCD cameras of 2456 × 2058 pixel resolution. The DIC recording and processing parameters are presented in Table 3. Fig. 2c provides an example of a strain map of the bent surface measured by DIC. For each captured DIC image, the principal strain values were extracted from three sections across the bend, as illustrated in Fig. 2c. The peak bending strain εb_max at each point in time was calculated as the average of the peak values of the three sections. More information of the test setup and the determination of the peak strain can be found in earlier work by the authors [1,2,17].

In order to track the punch-sheet separation associated with the multi-breakage, the displacement of the sheet in the out-of-plane direction (Z-displacement) is measured by DIC. The values are extracted from a 1 × 1 mm² area on the outer bend surface at the centerline, as illustrated in Fig. 2c. As the punch is stationary in the setup (after adjusting for the vertical elasticity), the measured Z-displacement is a result of a combination of thinning and punch-sheet separation. The derivatives of the peak strain, cross-section moment and Z-displacement were calculated using the central difference method, smoothed by taking the moving average of 9 datapoints.
<figure><img src="image_982.png" /><figcaption>Fig. 1. The longitudinal and transverse orientations in tensile and bending tests.</figcaption></figure>

The specimens were painted with a black-and-white speckle pattern to improve the DIC measurement accuracy and reliability. The DIC system used in this study was LaVision Strainmaster, with DaVis 8.4 software, and two monochrome CCD cameras of 2456 × 2058 pixel resolution. The DIC recording and processing parameters are presented in Table 3. Fig. 2c provides an example of a strain map of the bent surface measured by DIC. For each captured DIC image, the principal strain values were extracted from three sections across the bend, as illustrated in Fig. 2c. The peak bending strain ε<sub>b_max</sub> at each point in time was calculated as the average of the peak values of the three sections. More information of the test setup and the determination of the peak strain can be found in earlier work by the authors [1,2,17].

In order to track the punch-sheet separation associated with the multi-breakage, the displacement of the sheet in the out-of-plane direction (Z-displacement) is measured by DIC. The values are extracted from a 1 × 1 mm² area on the outer bend surface at the centerline, as illustrated in Fig. 2c. As the punch is stationary in the setup (after adjusting for the vertical elasticity), the measured Z-displacement is a result of a combination of thinning and punch-sheet separation. The derivatives of the peak strain, cross-section moment and Z-displacement were calculated using the central difference method, smoothed by taking the moving average of 9 datapoints.
<figure><img src="image_976.png" /><figcaption>Fig. 2. a) The bending test setup, b) positioning of the DIC cameras, and c) areas for extracting the strain (sections A – C) and Z-displacement (Vz) on a DIC-measured strain map of the St1100_1 (RD) at 90° bend angle, using a punch radius R<sub>p</sub> = 2 t.</figcaption></figure>

<table><caption>Table 2. Bending tool setup parameters: Punch radius (R<sub>p</sub>), die width (W<sub>d</sub>), radius of the die shoulders (R<sub>d</sub>), vertical speed of the punch (V<sub>p</sub>) and the final bend angle before unloading (α<sub>end</sub>).</caption><thead><tr><td>Tool setup</td><td>R<sub>p</sub></td><td>W<sub>d</sub> [mm]</td><td>R<sub>d</sub> [mm]</td><td>V<sub>p</sub> [mm/s]</td><td>α<sub>end</sub> [°]</td></tr></thead><tbody><tr><td>1</td><td>4 mm ≈ 0.66 t</td><td>90</td><td>6</td><td>1</td><td>120</td></tr><tr><td>2</td><td>12 mm ≈ 2 t</td><td>90</td><td>6</td><td>1</td><td>100</td></tr><tr><td>3</td><td>24 mm ≈ 4 t</td><td>110</td><td>6</td><td>1</td><td>110</td></tr></tbody></table>

<table><caption>Table 3. Recording and processing parameters for the DIC system.</caption><tr><td>DIC system and software</td><td>LaVision Strainmaster (Stereo DIC), DaVis 8.4</td></tr><tr><td>Sensor, digitization, imaging rate</td><td>2456 × 2058, 12-bit, 2 Hz</td></tr><tr><td>Lens, imaging distance, image scale</td><td>35 mm C-mount, 0.37 – 0.57 m, 21 – 30 pixels/mm</td></tr><tr><td>Subset size, step size, VSG</td><td>15x15 pixels, 5 pixels, 35 pixels</td></tr><tr><td>Strain window, smoothing method</td><td>5x5 data points, 2nd order polynomial fit</td></tr><tr><td>St.dev of principle strain</td><td>450 - 1700 microstrain</td></tr><tr><td>Interpolation, shape function, algorithms</td><td>6th order spline interpolation; affine shape function; LSM (iterative least squares matching) algorithm based on optical flow estimation</td></tr></table>
<figure><img src="image_983.png" /><figcaption>Fig. 2. a) The bending test setup, b) positioning of the DIC cameras, and c) areas for extracting the strain (sections A – C) and Z-displacement (Vz) on a DIC-measured strain map of the St1100_1 (RD) at 90° bend angle, using a punch radius R<sub>p</sub> = 2 t.</figcaption></figure>

<table><caption>Table 2. Bending tool setup parameters: Punch radius (R<sub>p</sub>), die width (W<sub>d</sub>), radius of the die shoulders (R<sub>d</sub>), vertical speed of the punch (V<sub>p</sub>) and the final bend angle before unloading (α<sub>end</sub>).</caption><thead><tr><td>Tool setup</td><td>R<sub>p</sub></td><td>W<sub>d</sub> [mm]</td><td>R<sub>d</sub> [mm]</td><td>V<sub>p</sub> [mm/s]</td><td>α<sub>end</sub> [°]</td></tr></thead><tbody><tr><td>1</td><td>4 mm ≈ 0.66 t</td><td>90</td><td>6</td><td>1</td><td>120</td></tr><tr><td>2</td><td>12 mm ≈ 2 t</td><td>90</td><td>6</td><td>1</td><td>100</td></tr><tr><td>3</td><td>24 mm ≈ 4 t</td><td>110</td><td>6</td><td>1</td><td>110</td></tr></tbody></table>

<table><caption>Table 3. Recording and processing parameters for the DIC system.</caption><tr><td>DIC system and software</td><td>LaVision Strainmaster (Stereo DIC), DaVis 8.4</td></tr><tr><td>Sensor, digitization, imaging rate</td><td>2456 × 2058, 12-bit, 2 Hz</td></tr><tr><td>Lens, imaging distance, image scale</td><td>35 mm C-mount, 0.37 – 0.57 m, 21 – 30 pixels/mm</td></tr><tr><td>Subset size, step size, VSG</td><td>15x15 pixels, 5 pixels, 35 pixels</td></tr><tr><td>Strain window, smoothing method</td><td>5x5 data points, 2nd order polynomial fit</td></tr><tr><td>St.dev of principle strain</td><td>450 - 1700 microstrain</td></tr><tr><td>Interpolation, shape function, algorithms</td><td>6th order spline interpolation; affine shape function; LSM (iterative least squares matching) algorithm based on optical flow estimation</td></tr></table>
<figure><img src="image_977.png" /><figcaption>Fig. 3. The geometry and parameters of the bending tools and the workpiece.</figcaption></figure>

## Results and discussion

As seen in Fig. 4, the strain distributions for St700 are similar at 30° bend angle, regardless of the punch radius. However, as the bend angle increases, the impact of the punch radius on the strain distribution becomes apparent. For punch radius $R_p = 0.66$ t, the deformation is mainly concentrated at the center, creating a narrow strain distribution with a large peak. In contrast, using a larger punch produces a wider strain distribution with a lower peak. In fact, when using the largest punch ($R_p = 4$ t), the peak strain does not increase at all between the bend angles of 60° and 90°, since the deformation is only increasing from the sides.

<figure><img src="image_978.png" /><figcaption>Fig. 4. Strain distributions from section B of St700 (RD) at 30°, 60° and 90° bend angle, bent using a) $R_p = 0.66$ t, b) $R_p = 2$ t and c) $R_p = 4$ t.</figcaption></figure>

Fig. 5 presents the development of the peak strain $\epsilon_{b\_max}$, cross-section moment M, Z-displacement $V_z$ and their first derivatives for the St355, St700 and St900_1 grades bent using punch radius $R_p = 2$ t. From Fig. 5a and Fig. 5d it can be seen that the strain rate ($d\epsilon_{b\_max}/d\alpha$) for the St355 grade starts out higher compared to the St700 and St900_1 grades, but starts to gradually decrease after 10° bend angle, before reaching a steady state ($d\epsilon_{b\_max}/d\alpha = 0$) at 90° bend angle.

The development of the peak strain and strain rate for the St700 and St900_1 grades starts out roughly identical but diverge after 10° bend angle. For both St700 and St900_1, the peak of the strain rate is reached between 30 – 40° bend angle, after which the strain rate starts to decrease. For St700, the strain rate decreases gradually until the steady state is reached at around 80° bend angle. For the St900_1 grade, the strain rate first decreases after around 40° bend angle, then stays roughly constant between 50° – 80° before starting to decrease again.
<figure><img src="image_984.png" /><figcaption>Fig. 3. The geometry and parameters of the bending tools and the workpiece.</figcaption></figure>

## Results and discussion

As seen in Fig. 4, the strain distributions for St700 are similar at 30° bend angle, regardless of the punch radius. However, as the bend angle increases, the impact of the punch radius on the strain distribution becomes apparent. For punch radius $R_p = 0.66$ t, the deformation is mainly concentrated at the center, creating a narrow strain distribution with a large peak. In contrast, using a larger punch produces a wider strain distribution with a lower peak. In fact, when using the largest punch ($R_p = 4$ t), the peak strain does not increase at all between the bend angles of 60° and 90°, since the deformation is only increasing from the sides.

<figure><img src="image_985.png" /><figcaption>Fig. 4. Strain distributions from section B of St700 (RD) at 30°, 60° and 90° bend angle, bent using a) $R_p = 0.66$ t, b) $R_p = 2$ t and c) $R_p = 4$ t.</figcaption></figure>

Fig. 5 presents the development of the peak strain $\epsilon_{b\_max}$, cross-section moment M, Z-displacement $V_z$ and their first derivatives for the St355, St700 and St900_1 grades bent using punch radius $R_p = 2$ t. From Fig. 5a and Fig. 5d it can be seen that the strain rate ($d\epsilon_{b\_max}/d\alpha$) for the St355 grade starts out higher compared to the St700 and St900_1 grades, but starts to gradually decrease after 10° bend angle, before reaching a steady state ($d\epsilon_{b\_max}/d\alpha = 0$) at 90° bend angle.

The development of the peak strain and strain rate for the St700 and St900_1 grades starts out roughly identical but diverge after 10° bend angle. For both St700 and St900_1, the peak of the strain rate is reached between 30 – 40° bend angle, after which the strain rate starts to decrease. For St700, the strain rate decreases gradually until the steady state is reached at around 80° bend angle. For the St900_1 grade, the strain rate first decreases after around 40° bend angle, then stays roughly constant between 50° – 80° before starting to decrease again.
<figure><img src="image_979.png" /><figcaption>Fig. 5. Development of a) the peak strain, b) cross-section moment, c) Z-displacement, d) peak strain derivative, e) moment derivative and f) Z-displacement derivative, measured from St355, St700 and St900_1 bent in the transverse (TD) orientation, using punch radius $R_p = 2$ t.</figcaption></figure>

Looking at Fig. 5b and Fig. 5e, it is seen that both St700 and St900_1 reach a minimum in the moment derivative $dM/d\alpha$ at around 30 – 40° bend angle, after which the moment derivative starts to increase. For St900_1, the rapid increase of the moment derivative is followed by a steady decrease between 50° – 80°, before starting to increase again towards the end of the test. For the St700 grade, the moment derivative increases gradually between 40° – 90°. For the St355 grade, the changes in the moment are more gradual, and the minimum of the moment derivative is reached around 60°.

Fig. 5c and Fig. 5f show the development of the Z-displacement and its derivative. For the St900_1 grade, the displacement increases at a roughly constant rate until around 40° bend angle, meaning that the outside surface moves towards the punch due to thinning. After 40°, the Z-displacement starts to decrease, meaning that the outside surface of the sheet moves away from the punch, i.e., there is a loss of contact between the punch and the sheet at the centerline ($x = 0$ mm). After around 80° bend angle, the derivative of the Z-displacement starts to increase again. Similar observations can be made for the St700 and St355 grades, albeit on a smaller scale.

By comparing the derivative curves in Fig. 5, an observation can be made that the moment derivative $dM/d\alpha$ minimums (at around 30°, 40° and 60° for St900_1, St700 and St355, respectively) roughly coincide with the points at which the derivatives of the peak strain and the Z-displacement start to decrease. These can all be attributed to the wrap-around, and the consequent change in the loading scheme. At the start of the bending process, the deformation is mostly localized at the centerline. As the peak strain increases, the inside radius of the sheet decreases. After a certain level of peak strain is reached, the inside radius of the sheet becomes equal to the punch radius and the sheet “wraps around” the punch, meaning that the loading scheme changes from 3-point bending to something closer to 4-point bending. As the contact points shift further apart, the moment distribution changes, and the bending moment is distributed to a larger area. The contact point shift is also related to the so-called “multi-breakage” effect, where the inside radius of the sheet becomes smaller than the punch radius, leading to a loss of the punch-sheet contact at the centerline, as well as a quick shift in the loading scheme and moment distribution [4]. Due to the implications of the wrap-around on the moment distribution, it is assumed that the wrap-around angle coincides with the derivative $dM/d\alpha$ minimums.
<figure><img src="image_986.png" /><figcaption>Fig. 5. Development of a) the peak strain, b) cross-section moment, c) Z-displacement, d) peak strain derivative, e) moment derivative and f) Z-displacement derivative, measured from St355, St700 and St900_1 bent in the transverse (TD) orientation, using punch radius $R_p = 2$ t.</figcaption></figure>

Looking at Fig. 5b and Fig. 5e, it is seen that both St700 and St900_1 reach a minimum in the moment derivative dM/dα at around 30 – 40° bend angle, after which the moment derivative starts to increase. For St900_1, the rapid increase of the moment derivative is followed by a steady decrease between 50° – 80°, before starting to increase again towards the end of the test. For the St700 grade, the moment derivative increases gradually between 40° – 90°. For the St355 grade, the changes in the moment are more gradual, and the minimum of the moment derivative is reached around 60°.

Fig. 5c and Fig. 5f show the development of the Z-displacement and its derivative. For the St900_1 grade, the displacement increases at a roughly constant rate until around 40° bend angle, meaning that the outside surface moves towards the punch due to thinning. After 40°, the Z-displacement starts to decrease, meaning that the outside surface of the sheet moves away from the punch, i.e., there is a loss of contact between the punch and the sheet at the centerline (x = 0 mm). After around 80° bend angle, the derivative of the Z-displacement starts to increase again. Similar observations can be made for the St700 and St355 grades, albeit on a smaller scale.

By comparing the derivative curves in Fig. 5, an observation can be made that the moment derivative dM/dα minimums (at around 30°, 40° and 60° for St900_1, St700 and St355, respectively) roughly coincide with the points at which the derivatives of the peak strain and the Z-displacement start to decrease. These can all be attributed to the wrap-around, and the consequent change in the loading scheme. At the start of the bending process, the deformation is mostly localized at the centerline. As the peak strain increases, the inside radius of the sheet decreases. After a certain level of peak strain is reached, the inside radius of the sheet becomes equal to the punch radius and the sheet “wraps around” the punch, meaning that the loading scheme changes from 3-point bending to something closer to 4-point bending. As the contact points shift further apart, the moment distribution changes, and the bending moment is distributed to a larger area. The contact point shift is also related to the so-called “multi-breakage” effect, where the inside radius of the sheet becomes smaller than the punch radius, leading to a loss of the punch-sheet contact at the centerline, as well as a quick shift in the loading scheme and moment distribution [4]. Due to the implications of the wrap-around on the moment distribution, it is assumed that the wrap-around angle coincides with the derivative dM/dα minimums.
The increase in the dM/dα after the wrap-around can be explained from two perspectives – the internal and external bending moments. The internal bending moment of the material must be the same as the external moment applied by the tools. Due to strain hardening and the shifting of the neutral axis towards the inside surface, the internal bending moment must increase throughout the bending process to continue deforming the material. Consequently, the external moment applied to the material must also increase. However, due to the contact point shift in the wrap-around, which “forces” a wider moment distribution with a “dull” peak (as in 4-point bending), achieving the increased moment requires more force compared to the 3-point bending prior to wrap-around. This is seen as the increase in the calculated moment derivative dM/dα.

The change of the moment distribution in the wrap-around also promotes distribution of strain into a larger area, which inhibits strain localization at the centerline, indicated by the decrease of the strain rate dε<sub>b_max</sub>/dα after the wrap-around angle (dM/dα minimum) for all the tested grades. The decrease in the Z-displacement after wrap-around, as seen most clearly for the St900_1 grade, is an indication of punch-sheet separation at the centerline, which is characteristic to a pronounced multi-breakage effect. The increase in the derivative of the Z-displacement of the St900_1 grade after 80° can be explained by looking at the derivatives of the strain and moment. The decrease in the strain rate and increase of the moment derivative indicate that there is again a change in the moment distribution due to the contact points moving further away from the centerline. This “second wrap-around” means that the sheet at the centerline starts to approach a steady state as the derivatives of the peak strain and Z-displacement start to approach zero. The unique development of the contact point positions may also be the reason behind the polygonal “nut-like” shape of the bent sheet that has been observed for the St900_1 and other ultra-high strength steel grades in previous studies [2, 5, 7, 8].

The differences between the materials in the development of the strain, moment and Z-
displacement can be attributed to differences in the strain-hardening properties. Materials with
greater strain-hardening have greater resistance to strain localization. This leads to wider strain
distributions with lower peak strain. As the peak strain is lower, the inside radius also decreases at
a slower rate. Consequently, the wrap-around happens later. This is observed in the derivative
curves in Fig. 5. For St900_1, the grade with the lowest strain-hardening, the effects related to the
wrap-around start around 30 – 40°, for the St700 at around 40 – 50°, and for the St355 grade
(greatest strain-hardening) around 50 – 60°.

The dM/dα minimum values seem to be linked to the strain rate (dε<sub>b_max</sub>/dα) maximum values at the wrap-around angle (30 – 40°), as well as the strain-hardening properties of the material. St900_1 (lowest strain-hardening) produces the most localized strain distribution with the highest peak strain and strain rate maximum, while also showing the lowest minimum dM/dα value. On the other hand, St355, the grade with the highest strain-hardening, produces the lower peak strain and strain rate, while also showing the highest minimum dM/dα value at the wrap-around angle. The negative dM/dα before wrap-around (as seen for the St900_1 grade) may be compared to strain softening, in that it is clearly connected to accelerated strain localization at the centerline, which is not a desirable property in terms of bendability. This “softening” effect could be due to development of shear-bands below the surface. However, even though defects (shear bands and notches) could be seen on the surface of the St900_1 (TD) specimen after the test, the investigations performed for this paper cannot confirm that the shear bands initiated before the wrap-around angle. To confirm this conjecture, the tests should be interrupted before the wrap-around angle to inspect the cross-sections of the specimens.

A localized strain distribution has been connected with the more pronounced multi-breakage effect [2]. The strain distribution development has also been previously found to correlate with the global formability properties of the material (e.g., strain-hardening exponent) [1]. As can be seen from Fig. 5b and Fig. 5e, the increase of cross-section moment between 30 – 50° is fastest for the
The increase in the dM/dα after the wrap-around can be explained from two perspectives – the internal and external bending moments. The internal bending moment of the material must be the same as the external moment applied by the tools. Due to strain hardening and the shifting of the neutral axis towards the inside surface, the internal bending moment must increase throughout the bending process to continue deforming the material. Consequently, the external moment applied to the material must also increase. However, due to the contact point shift in the wrap-around, which “forces” a wider moment distribution with a “dull” peak (as in 4-point bending), achieving the increased moment requires more force compared to the 3-point bending prior to wrap-around. This is seen as the increase in the calculated moment derivative dM/dα.

The change of the moment distribution in the wrap-around also promotes distribution of strain into a larger area, which inhibits strain localization at the centerline, indicated by the decrease of the strain rate dε<sub>b_max</sub>/dα after the wrap-around angle (dM/dα minimum) for all the tested grades. The decrease in the Z-displacement after wrap-around, as seen most clearly for the St900_1 grade, is an indication of punch-sheet separation at the centerline, which is characteristic to a pronounced multi-breakage effect. The increase in the derivative of the Z-displacement of the St900_1 grade after 80° can be explained by looking at the derivatives of the strain and moment. The decrease in the strain rate and increase of the moment derivative indicate that there is again a change in the moment distribution due to the contact points moving further away from the centerline. This “second wrap-around” means that the sheet at the centerline starts to approach a steady state as the derivatives of the peak strain and Z-displacement start to approach zero. The unique development of the contact point positions may also be the reason behind the polygonal “nut-like” shape of the bent sheet that has been observed for the St900_1 and other ultra-high strength steel grades in previous studies [2, 5, 7, 8].

The differences between the materials in the development of the strain, moment and Z-
displacement can be attributed to differences in the strain-hardening properties. Materials with
greater strain-hardening have greater resistance to strain localization. This leads to wider strain
distributions with lower peak strain. As the peak strain is lower, the inside radius also decreases at
a slower rate. Consequently, the wrap-around happens later. This is observed in the derivative
curves in Fig. 5. For St900_1, the grade with the lowest strain-hardening, the effects related to the
wrap-around start around 30 – 40°, for the St700 at around 40 – 50°, and for the St355 grade
(greatest strain-hardening) around 50 – 60°.

The dM/dα minimum values seem to be linked to the strain rate (dε<sub>b_max</sub>/dα) maximum values at the wrap-around angle (30 – 40°), as well as the strain-hardening properties of the material. St900_1 (lowest strain-hardening) produces the most localized strain distribution with the highest peak strain and strain rate maximum, while also showing the lowest minimum dM/dα value. On the other hand, St355, the grade with the highest strain-hardening, produces the lower peak strain and strain rate, while also showing the highest minimum dM/dα value at the wrap-around angle. The negative dM/dα before wrap-around (as seen for the St900_1 grade) may be compared to strain softening, in that it is clearly connected to accelerated strain localization at the centerline, which is not a desirable property in terms of bendability. This “softening” effect could be due to development of shear-bands below the surface. However, even though defects (shear bands and notches) could be seen on the surface of the St900_1 (TD) specimen after the test, the investigations performed for this paper cannot confirm that the shear bands initiated before the wrap-around angle. To confirm this conjecture, the tests should be interrupted before the wrap-around angle to inspect the cross-sections of the specimens.

A localized strain distribution has been connected with the more pronounced multi-breakage effect [2]. The strain distribution development has also been previously found to correlate with the global formability properties of the material (e.g., strain-hardening exponent) [1]. As can be seen from Fig. 5b and Fig. 5e, the increase of cross-section moment between 30 – 50° is fastest for the
St900_1 grade, which has the lowest strain-hardening. This indicates that the contact point shift and multi-breakage happen faster for materials with lower strain-hardening. This conclusion is also supported by the fact that the decrease of the Z-displacement after 40° is also fastest for the St900_1 grade, meaning that the punch-sheet separation is largest, due to the pronounced multi-breakage effect.

Fig. 6 presents the peak strain, cross-section moment, Z-displacement and their first derivatives measured from 500 MPa and 700 MPa steels bent using punch radii $R_p = 0.66 \text{ t}$, $R_p = 2 \text{ t}$, and $R_p = 4 \text{ t}$. As can be seen from Fig. 6a and Fig. 6d, the development of the peak strain and strain rate for the presented tests start out roughly identical, as most of the deformation is still elastic, but diverge after around 10°–20° bend angle when the work-hardening properties come into effect. The peak strain increases at a higher rate for the St700 grade in all tool setups. For the St700 grade bent with the largest punch ($R_p = 4 \text{ t}$) the strain rate starts to gradually decrease after 30° bend angle, and a steady state is reached at around 50°. For the St500 grade bent with the largest punch, the decrease of the strain rate starts already at around 15°, but the steady state is still reached at around 50°. For both materials bent with the medium punch ($R_p = 2 \text{ t}$), the strain rate starts to gradually decrease after 40°, reaching a steady state around after 80°. With the smallest punch ($R_p = 0.66 \text{ t}$), the difference between the materials is noticeable. For the St700 grade, the strain rate keeps increasing until around 85°. Whereas for the St500 grade, the strain rate decreases between 15–20°, then remains roughly constant before starting to decrease again after around 75°.

<figure><img src="image_980.png" /><figcaption>Fig. 6. Development of a) the peak strain, b) cross-section moment, c) Z-displacement, d) peak strain derivative, e) moment derivative and f) Z-displacement derivative, from St500 and St700 grades bent in transverse (TD) orientation with punch radii $R_p = 0.66 \text{ t}$, $R_p = 2 \text{ t}$ and $R_p = 4 \text{ t}$.</figcaption></figure>

From Fig. 6e it is seen that for both materials, the moment derivative $dM/dα$ reaches a minimum and starts to increase again at around 30°, 40° and 80° for the largest punch ($R_p = 4 \text{ t}$), medium punch ($R_p = 2 \text{ t}$) and smallest punch ($R_p = 0.66 \text{ t}$), respectively. For the St700 grade, these angles correspond to the angles at which the strain rate first starts to decrease. For the St500 grade, the connection between the strain rate curve and the moment derivative curve is not as clear, due to the changes in the strain rate being more gradual and due to the initial decrease in strain rate after 15°. However, it is still assumed that the wrap-around angle coincides with the $dM/dα$ minimum for both materials. It can also be seen here that for all three tool setups, the St700 grade has a lower $dM/dα$ value at the wrap-around angle compared to the St500 grade, which again demonstrates that materials with lower strain-hardening, that produce more localized strain distribution, will also produce a lower $dM/dα$ minimum.