# Tension-compression asymmetry of ferritic stainless steel sheet

Sekiya Koki¹,ª, Higashii Hyuga¹,ᵇ, Taoka Akira²,ᶜ and Kuwabara Toshihiko³,ᵈ*

¹Department of Mechanical Systems Engineering, Graduate School of Engineering, Tokyo
University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588,
Japan

²Raw Material Production Engineering Development Section, AISIN Co., 2-1 Asahi-cho, Kariya-
shi, Aichi 448-8650, Japan

³Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo
University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588,
Japan

ªsekiya@st.go.tuat.ac.jp, ᵇs209914v@st.go.tuat.ac.jp, ᶜa taoka@koki.aisin.co.jp,

ᵈkuwabara@cc.tuat.ac.jp

**Keywords:** In-Plane Compression, Strength Differential Effect, Bending Moment

**Abstract.** The strength differential effect (SDE) (i.e., the tension-compression asymmetry of stress-strain curves), for a ferritic stainless steel sheet was measured using an in-plane compression test. It was found that the in-plane compressive flow stress was 4-7% higher than the uniaxial tensile flow stress in both the rolling and transverse directions. In addition, bending moment-curvature curves were obtained from a pure bending test. These curves were found to be in good agreement with those calculated with SDE taken into account. Thus, the validity of the measured SDE was confirmed by the bending experiment.

**Introduction**

Stainless steel sheets are used for automotive exhaust system components because of their superior corrosion and heat resistance. Among the various types of stainless steel, ferritic stainless steel, which is less expensive than austenitic stainless steel and has superior thermal fatigue characteristics, has seen increasing use [1]. Automotive parts are often produced by press forming, in which the materials are subjected to tensile and compressive stresses. The strength differential effect (SDE) has been observed in steel sheets [2]-[9]. Kuwahara et al. [10] measured the SDE in austenitic stainless steel sheets used for electronic components and quantitatively clarified its effect on the prediction of springback in bending. Noma et al. [8] improved the prediction accuracy of springback in forming simulations of curved panels by using a material model that takes into account the SDE of a high-strength steel sheet. Yoon et al. [11] implemented the hydrostatic pressure dependency of the distortional plasticity model HAH₂₀ [12] using a finite element code to account for the SDE observed in advanced high-strength steel sheets. The implementation was validated by comparing the finite element prediction of the material behavior during U-draw bending and B-pillar forming. However, to the authors' knowledge, no studies have measured the SDE of ferritic stainless steel sheets.

The objective of this study is to measure the SDE of a 1.0-mm-thick Fe-18Cr-3Al ferritic stainless steel sheet. First, uniaxial tensile tests and in-plane compression tests (IPCTs) using an in-plane stress reversal test apparatus [7] are conducted to quantitatively evaluate the magnitude of the SDE. Next, bending moment-curvature (M – κ) curves are obtained using a pure bending test apparatus. It is found that the experimental M – κ curves are consistent with those calculated using the tensile and compressive stress-strain (ss) curves in both the rolling direction (RD) and

<figure><img src="image_1051.png" /></figure>
transverse direction (TD). Thus, the validity of the measured SDE is confirmed by the bending experiment.

## Experimental methods

*Test material.* The test material used in this study was a 1.0-mm-thick ferritic stainless steel sheet (Fe-18Cr-3Al). The mechanical properties obtained from uniaxial tensile tests are listed in Table 1.

<table><caption>Table 1 Mechanical properties of test material (Fe-18Cr-3Al).</caption><thead><tr><th>Loading direction</th><th>Young's modulus <i>E</i> [GPa]</th><th>0.2 % proof stress [MPa]</th><th>Tensile strength [MPa]</th><th><i>C</i>* [MPa]</th><th><i>α</i>*</th><th><i>n</i>*</th><th><i>r</i>-value**</th></tr></thead><tbody><tr><td>RD</td><td>195</td><td>428</td><td>567</td><td>960</td><td>0.014</td><td>0.207</td><td>1.12</td></tr><tr><td>TD</td><td>214</td><td>454</td><td>587</td><td>1003</td><td>0.022</td><td>0.216</td><td>1.78</td></tr></tbody></table>

*In-plane compression test.* Fig. 1 shows the IPCT apparatus used in this study. This apparatus was first developed by Kuwabara et al. [6], subsequently modified [10], and used in several studies [7-9, 13]. Fig. 1(a) shows an overview of the dies used for applying in-plane compression to a sheet specimen. Fig. 1(b) shows an overview of the test apparatus. A detailed description of the apparatus is given elsewhere [7]. In this study, the blank holding force applied to the specimen was 1% of the 0.2% proof stress (5.7 MPa). The strain was measured using a strain gauge (YFLA-2-1LJC-F, Tokyo Measuring Instruments Laboratory Co. Ltd.). The strain rate was approximately $5 \times 10^{-4} s^{-1}$ and the testing temperature was room temperature for both tension and compression experiments. Maeda et al. [7] experimentally verified that the effect of frictional force between the upper and lower dies and the specimen on the accuracy of stress measurement can be neglected for the test apparatus shown in Fig. 1. Through in-plane compression tests with a constant gap between the upper and lower dies, Akiyama et al. [14] demonstrated that the blank holding pressure had little effect on the magnitude of compressive flow stress.

<figure><img src="image_1047.png" /><figcaption>Fig. 1 IPCT apparatus [7]: (a) upper and lower dies and (b) overview of test apparatus</figcaption></figure>

Fig. 2 shows the geometry of the specimen used in the in-plane compression test. Noma and Kuwabara [13] performed finite element analyses to evaluate the uniformity of the stress

<footnote>*Approximated using $\sigma = C(\alpha + \varepsilon^P)^n$ at $\varepsilon^P = 0.002 \sim \text{Maximum load point}$</footnote>

<footnote>**Measured at an uniaxial nominal strain of 0.1</footnote>
transverse direction (TD). Thus, the validity of the measured SDE is confirmed by the bending experiment.

## Experimental methods

*Test material.* The test material used in this study was a 1.0-mm-thick ferritic stainless steel sheet (Fe-18Cr-3Al). The mechanical properties obtained from uniaxial tensile tests are listed in Table 1.

<table><caption>Table 1 Mechanical properties of test material (Fe-18Cr-3Al).</caption><thead><tr><th>Loading direction</th><th>Young's modulus E<br>[GPa]</th><th>0.2 % proof stress<br>[MPa]</th><th>Tensile strength<br>[MPa]</th><th>C*<br>[MPa]</th><th>α*</th><th>n*</th><th>r-value**</th></tr></thead><tbody><tr><td>RD</td><td>195</td><td>428</td><td>567</td><td>960</td><td>0.014</td><td>0.207</td><td>1.12</td></tr><tr><td>TD</td><td>214</td><td>454</td><td>587</td><td>1003</td><td>0.022</td><td>0.216</td><td>1.78</td></tr></tbody></table>

*In-plane compression test.* Fig. 1 shows the IPCT apparatus used in this study. This apparatus was first developed by Kuwabara et al. [6], subsequently modified [10], and used in several studies [7-9, 13]. Fig. 1(a) shows an overview of the dies used for applying in-plane compression to a sheet specimen. Fig. 1(b) shows an overview of the test apparatus. A detailed description of the apparatus is given elsewhere [7]. In this study, the blank holding force applied to the specimen was 1% of the 0.2% proof stress (5.7 MPa). The strain was measured using a strain gauge (YFLA-2-1LJC-F, Tokyo Measuring Instruments Laboratory Co. Ltd.). The strain rate was approximately $5 \times 10^{-4} s^{-1}$ and the testing temperature was room temperature for both tension and compression experiments. Maeda et al. [7] experimentally verified that the effect of frictional force between the upper and lower dies and the specimen on the accuracy of stress measurement can be neglected for the test apparatus shown in Fig. 1. Through in-plane compression tests with a constant gap between the upper and lower dies, Akiyama et al. [14] demonstrated that the blank holding pressure had little effect on the magnitude of compressive flow stress.

<figure><img src="image_1052.png" /><figcaption>Fig. 1 IPCT apparatus [7]: (a) upper and lower dies and (b) overview of test apparatus</figcaption></figure>

Fig. 2 shows the geometry of the specimen used in the in-plane compression test. Noma and Kuwabara [13] performed finite element analyses to evaluate the uniformity of the stress

<footnote>*Approximated using $\sigma = C(\alpha + \varepsilon^P)^n$ at $\varepsilon^P = 0.002 \sim \text{Maximum load point}$</footnote>

<footnote>**Measured at an uniaxial nominal strain of 0.1</footnote>
distribution at the gauge section of a specimen subjected to in-plane tension-compression and found that the stress measurement error at the strain gauge location was less than 1%.

<figure><img src="image_1048.png" /><figcaption>Fig. 2 Geometry of specimen for in-plane tension-compression test</figcaption></figure>

Pure bending test. Pure bending tests were performed to verify the accuracy of the SDE test results. Fig. 3(a) shows an overview of the pure bending test apparatus used in this study. An electric cylinder applies a bending force $F$ to steel wires connected to the bending unit. The magnitude of $2F$ is measured using the load cell. Fig. 3(b) shows the details of the bending unit. Both ends of the specimen are fixed to two fan-shaped blocks, as shown in Fig. 3(c). A steel wire is wound around the block with a circular side, the radius of which is $R$. Each block has a rotary shaft and rotates around the axis when $F$ is applied via the steel wire. The rotary shafts are connected to the ball bearings bolted to the upper and lower slide guides; therefore, each block can move freely in the horizontal direction during the bending test, preventing the application of a tensile force to the specimen. Thus, a pure bending moment $M$ can be applied to the specimen. $M$ is calculated as

$$M = FR. \qquad (1)$$

The curvature $\kappa$ of the bent specimen is determined from the values of the tensile and compressive bending strains, $\epsilon_N^T$ and $\epsilon_N^C$ (< 0), at the convex and concave surfaces of the bent specimen, respectively, as shown in Eq. (2):

$$\kappa = \frac{1}{r_n} = \frac{|\epsilon_N^C| + \epsilon_N^T}{t}, \qquad (2)$$

where $r_n$ is the bending radius of the neutral plane at which the bending strain becomes zero (see next section). $\epsilon_N^T$ and $\epsilon_N^C$ are measured using high-elongation strain gauges (Kyowa Electronic Instruments Co., KFG-1N120-C1-11) glued to both sides of the specimen, as shown in Fig. 4.

Fig. 4 shows the geometry of the specimen used in the pure bending test. The specimen was fabricated using wire electric discharge machining. The ratio of the width of the gauge section to the sheet thickness was 1.5. Maeda et al. [7] performed a finite element analysis on this specimen geometry and found that the stress state at the gauge section of the bending specimen could be assumed to be uniaxial in both the tensile and compressive sections. This means that $M$ can be directly calculated using the uniaxial tensile and compressive stress-strain (ss) curves. See the next section for details of the calculation procedure for $M$.
distribution at the gauge section of a specimen subjected to in-plane tension-compression and found that the stress measurement error at the strain gauge location was less than 1%.

<figure><img src="image_1053.png" /><figcaption>Fig. 2 Geometry of specimen for in-plane tension-compression test</figcaption></figure>

Pure bending test. Pure bending tests were performed to verify the accuracy of the SDE test results. Fig. 3(a) shows an overview of the pure bending test apparatus used in this study. An electric cylinder applies a bending force $F$ to steel wires connected to the bending unit. The magnitude of $2F$ is measured using the load cell. Fig. 3(b) shows the details of the bending unit. Both ends of the specimen are fixed to two fan-shaped blocks, as shown in Fig. 3(c). A steel wire is wound around the block with a circular side, the radius of which is $R$. Each block has a rotary shaft and rotates around the axis when $F$ is applied via the steel wire. The rotary shafts are connected to the ball bearings bolted to the upper and lower slide guides; therefore, each block can move freely in the horizontal direction during the bending test, preventing the application of a tensile force to the specimen. Thus, a pure bending moment $M$ can be applied to the specimen. $M$ is calculated as

$$M = FR. \qquad (1)$$

The curvature $\kappa$ of the bent specimen is determined from the values of the tensile and compressive bending strains, $\epsilon_N^T$ and $\epsilon_N^C$ (< 0), at the convex and concave surfaces of the bent specimen, respectively, as shown in Eq. (2):

$$\kappa = \frac{1}{r_n} = \frac{|\epsilon_N^C| + \epsilon_N^T}{t}, \qquad (2)$$

where $r_n$ is the bending radius of the neutral plane at which the bending strain becomes zero (see next section). $\epsilon_N^T$ and $\epsilon_N^C$ are measured using high-elongation strain gauges (Kyowa Electronic Instruments Co., KFG-1N120-C1-11) glued to both sides of the specimen, as shown in Fig. 4.

Fig. 4 shows the geometry of the specimen used in the pure bending test. The specimen was fabricated using wire electric discharge machining. The ratio of the width of the gauge section to the sheet thickness was 1.5. Maeda et al. [7] performed a finite element analysis on this specimen geometry and found that the stress state at the gauge section of the bending specimen could be assumed to be uniaxial in both the tensile and compressive sections. This means that $M$ can be directly calculated using the uniaxial tensile and compressive stress-strain (ss) curves. See the next section for details of the calculation procedure for $M$.
<figure><img src="image_1049.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_1050.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_1051.png" /></figure>

<figure><img src="image_1052.png" /><figcaption>Fig. 3 Experimental apparatus for applying pure bending moment to sheet specimen: (a) overview of test apparatus, (b) overview of bending unit, and (c) pure bending mechanism for bending unit</figcaption></figure>

<figure><img src="image_1053.png" /><figcaption>Fig. 4 Geometry of specimen and mounting position of strain gauges for pure bending test</figcaption></figure>

## Method for calculating bending moment–curvature curve

To validate the SDE, the $M-\kappa$ curves of samples were calculated and compared with the measurements. In the calculation of an $M-\kappa$ curve, we assumed the following.

(i) The specimen sheet thickness was divided into 100 thin layers of equal thickness. The $i$-th layer counted from the innermost concave surface of the bent specimen is denoted as layer $i$. The subscript $i$ is attached to the values related to layer $i$.

(ii) The $x$ and $y$ axes were taken in the longitudinal and normal (through-thickness) directions of the specimen, respectively. $y$ was taken to be zero on the neutral plane.

(iii) The specimen was subjected to pure bending; the resultant force in the longitudinal direction of the specimen was zero.
<figure><img src="image_1054.png" /><figcaption>Fig. 3 Experimental apparatus for applying pure bending moment to sheet specimen: (a) overview of test apparatus, (b) overview of bending unit, and (c) pure bending mechanism for bending unit</figcaption></figure>

<figure><img src="image_1055.png" /><figcaption>Fig. 4 Geometry of specimen and mounting position of strain gauges for pure bending test</figcaption></figure>

## Method for calculating bending moment–curvature curve

To validate the SDE, the $M-\kappa$ curves of samples were calculated and compared with the measurements. In the calculation of an $M-\kappa$ curve, we assumed the following.

(i) The specimen sheet thickness was divided into 100 thin layers of equal thickness. The $i$-th layer counted from the innermost concave surface of the bent specimen is denoted as layer $i$. The subscript $i$ is attached to the values related to layer $i$.

(ii) The $x$ and $y$ axes were taken in the longitudinal and normal (through-thickness) directions of the specimen, respectively. $y$ was taken to be zero on the neutral plane.

(iii) The specimen was subjected to pure bending; the resultant force in the longitudinal direction of the specimen was zero.
(iv) The cross section of the sheet remained planar and normal to the x axis during the bending process.

(v) Each layer in the normal direction of the specimen was subjected to uniaxial tensile or compressive stress, $\sigma_i$; $\sigma_i$ was the only non-zero stress component in the specimen.

(vi) Each layer in the normal direction of the specimen followed the ss curves, as determined by Eq. (3), during the bending process, depending on the magnitude of the bending strain for each layer.

Figs. 5(a) and 5(b) show the measured uniaxial tensile and in-plane compressive ss curves in the RD and TD, respectively. The measured ss curves were approximated using the analytical functions:

$$ \sigma = 1000 E \varepsilon \quad \text{for } |\varepsilon| \le |\varepsilon_E|, \qquad (3a) $$

$$ \sigma = 1000 E \varepsilon_E + \frac{\sigma_{\text{Mid}} - 1000 E \varepsilon_E}{\varepsilon_{\text{Mid}} - \varepsilon_E} (\varepsilon - \varepsilon_E) \quad \text{for } |\varepsilon_E| < |\varepsilon| < |\varepsilon_{\text{Mid}}|, \qquad (3b) $$

$$ \sigma = \frac{\sigma_{\text{Mid}} - \{a - b \exp(-c\varepsilon\}}{\varepsilon_{\text{Mid}} - \varepsilon_E} (\varepsilon - \varepsilon_{\text{Mid}}) + \sigma_{\text{Mid}} \quad \text{for } |\varepsilon_{\text{Mid}}| < |\varepsilon| < |\varepsilon_{\text{P}}|, \qquad (3c) $$

$$ \sigma = \pm \{a - b \exp(-c\varepsilon)\} \quad \text{for } |\varepsilon_{\text{P}}| \le |\varepsilon|, \qquad (3d) $$

where $\varepsilon_E$ is the upper limit of the elastic strain range and $\varepsilon_P$ is the lower limit of the elastic-plastic strain range, for which the ss curve was approximated using Voce's hardening law, $\sigma = a - b \exp(-c\varepsilon)$. The ss curves for $|\varepsilon_E| < |\varepsilon| < |\varepsilon_P|$ sections were approximated by two straight lines, namely one for $\varepsilon_E \le \varepsilon < \varepsilon_{\text{Mid}}$ and one for $\varepsilon_{\text{Mid}} \le \varepsilon < \varepsilon_P$. The plus and minus signs in Eq. (3d) are for tension ($\varepsilon \ge 0$) and compression ($\varepsilon < 0$), respectively. Table 2 summarizes the parameter values used in Eq. (3).

When the curvature of the neutral plane is $\kappa$, the logarithmic longitudinal strain, $\varepsilon$, of a layer at a coordinate $y$ is given by

$$ \varepsilon = \ln(1 + \kappa y). \qquad (4) $$

The force balance equation is

$$ \sum_{j=1}^{100} \sigma_j^T \cdot w_j \cdot \Delta y_j + \sum_{k=1}^{100} \sigma_k^C \cdot w_k \cdot \Delta y_k = 0, \qquad (5) $$

where $w_i$ and $\Delta y_i$ are the width and thickness of the i-th layer, respectively. The position of the neutral plane, at which $y = 0$, was determined to satisfy Eq. (5).

The bending moment $M$ can be calculated as

$$ M = \sum_{j=1}^{100} \sigma_j^T \cdot y_j \cdot w_j \cdot \Delta y_j + \sum_{k=1}^{100} \sigma_k^C \cdot y_k \cdot w_k \cdot \Delta y_k. \qquad (6) $$
(iv) The cross section of the sheet remained planar and normal to the x axis during the bending process.

(v) Each layer in the normal direction of the specimen was subjected to uniaxial tensile or compressive stress, $\sigma_i$; $\sigma_i$ was the only non-zero stress component in the specimen.

(vi) Each layer in the normal direction of the specimen followed the ss curves, as determined by Eq. (3), during the bending process, depending on the magnitude of the bending strain for each layer.

Figs. 5(a) and 5(b) show the measured uniaxial tensile and in-plane compressive ss curves in the RD and TD, respectively. The measured ss curves were approximated using the analytical functions:

$$ \sigma=1000E\varepsilon \quad \text{for } |\varepsilon|\le|\varepsilon_E|, \qquad (3a) $$

$$ \sigma=1000E\varepsilon_E+\frac{\sigma_{\text{Mid}}-1000E\varepsilon_E}{\varepsilon_{\text{Mid}}-\varepsilon_E} (\varepsilon-\varepsilon_E) \quad \text{for } |\varepsilon_E|<|\varepsilon|<|\varepsilon_{\text{Mid}}|, \qquad (3b) $$

$$ \sigma = \frac{\sigma_{\text{Mid}} - \{a - b\exp(-c\varepsilon\}}{\varepsilon_{\text{Mid}} - \varepsilon_E} (\varepsilon - \varepsilon_{\text{Mid}}) + \sigma_{\text{Mid}} \quad \text{for } |\varepsilon_{\text{Mid}}| < |\varepsilon| < |\varepsilon_{\text{P}}|, \qquad (3c) $$

$$ \sigma = \pm\{a-b\exp(-c\varepsilon)\} \quad \text{for } |\varepsilon_P|\le|\varepsilon|, \qquad (3d) $$

where $\varepsilon_E$ is the upper limit of the elastic strain range and $\varepsilon_P$ is the lower limit of the elastic-plastic strain range, for which the ss curve was approximated using Voce's hardening law, $\sigma = a - b\exp(-c\varepsilon)$. The ss curves for $|\varepsilon_E| < |\varepsilon| < |\varepsilon_P|$ sections were approximated by two straight lines, namely one for $\varepsilon_E \le \varepsilon < \varepsilon_{\text{Mid}}$ and one for $\varepsilon_{\text{Mid}} \le \varepsilon < \varepsilon_P$. The plus and minus signs in Eq. (3d) are for tension ($\varepsilon \ge 0$) and compression ($\varepsilon < 0$), respectively. Table 2 summarizes the parameter values used in Eq. (3).

When the curvature of the neutral plane is $\kappa$, the logarithmic longitudinal strain, $\varepsilon$, of a layer at a coordinate $y$ is given by

$$ \varepsilon = \ln(1+\kappa y). \qquad (4) $$

The force balance equation is

$$ \sum_{j=1}^{100} \sigma_j^T \cdot w_j \cdot \Delta y_j + \sum_{k=1}^{100} \sigma_k^C \cdot w_k \cdot \Delta y_k = 0, \qquad (5) $$

where $w_i$ and $\Delta y_i$ are the width and thickness of the i-th layer, respectively. The position of the neutral plane, at which $y = 0$, was determined to satisfy Eq. (5).

The bending moment $M$ can be calculated as

$$ M = \sum_{j=1}^{100} \sigma_j^T \cdot y_j \cdot w_j \cdot \Delta y_j + \sum_{k=1}^{100} \sigma_k^C \cdot y_k \cdot w_k \cdot \Delta y_k. \qquad (6) $$
<table><caption>Table 2 Summary of parameter values used in Eq. (3).</caption><thead><tr><th>Loding direction</th><th>Loading mode</th><th><i>E</i> [GPa]</th><th><i>r</i>-value</th><th><i>a</i>* [MPa]</th><th><i>b</i>* [MPa]</th><th><i>c</i>*</th><th><i>ε</i><sub>E</sub></th><th><i>ε</i><sub>P</sub></th><th><i>ε</i><sub>mid</sub></th><th><i>σ</i><sub>mid</sub> [MPa]</th></tr></thead><tbody><tr><td rowspan="2">RD</td><td>Tension</td><td>195</td><td>1.12</td><td>753.0</td><td>339.3</td><td>8.969</td><td>0.0017</td><td>0.0040</td><td>0.0026</td><td>405</td></tr><tr><td>Compression</td><td>186</td><td>1.15</td><td>808.9</td><td>379.6</td><td>9.677</td><td>0.0016</td><td>0.0035</td><td>0.0025</td><td>400</td></tr><tr><td rowspan="2">TD</td><td>Tension</td><td>214</td><td>1.78</td><td>755.2</td><td>315.9</td><td>9.529</td><td>0.0018</td><td>0.0030</td><td>0.0022</td><td>426</td></tr><tr><td>Compression</td><td>222</td><td>1.93</td><td>854.2</td><td>397.2</td><td>8.745</td><td>0.0017</td><td>0.0025</td><td>0.0022</td><td>446</td></tr></tbody></table>

## Experimental results and discussion

SDE in uniaxial stress states. Figs. 5(a) and 5(b) compare the uniaxial tensile true stress-logarithmic plastic strain curves, $\sigma^T - \varepsilon^P$, with the compressive ones, $|\sigma^C| - |\varepsilon^P|$, measured using the IPCT for the RD and TD, respectively. Each curve is the average of two measurements. The flow stresses are higher in compression than in tension in both directions. Therefore, the SDE was clearly observed. To quantitatively evaluate the magnitude of the SDE in the uniaxial stress state, the rate of increase of the flow stress, $\beta_{SDE}$, was defined as

$$ \beta_{SDE} = 2 \frac{|\sigma^C| - \sigma^T}{|\sigma^C| + \sigma^T}, \qquad (8) $$

where $\sigma^T$ and $|\sigma^C|$ are the measured values of the tensile and compressive flow stresses, respectively, at a given plastic work per unit volume, $w^P$. Fig. 6 shows the variation of $\beta_{SDE}$ with $w^P$ for the RD and TD. It was found that $0.001 < \beta_{SDE} < 0.064$ for the RD and $0.041 < \beta_{SDE} < 0.073$ for the TD; therefore, the test sample clearly exhibited the SDE.

<figure><img src="image_1054.png" /><figcaption>Fig. 5 Comparison of true stress-logarithmic strain curves for tension and compression in (a) RD and (b) TD</figcaption></figure>

<footnote>*Approximated using $\sigma = a - b\exp(-c\varepsilon^p)$</footnote>
<table><caption>Table 2 Summary of parameter values used in Eq. (3).</caption><thead><tr><th>Loding direction</th><th>Loading mode</th><th><i>E</i> [GPa]</th><th><i>r</i>-value</th><th><i>a</i>* [MPa]</th><th><i>b</i>* [MPa]</th><th><i>c</i>*</th><th><i>ε</i><sub>E</sub></th><th><i>ε</i><sub>P</sub></th><th><i>ε</i><sub>mid</sub></th><th><i>σ</i><sub>mid</sub> [MPa]</th></tr></thead><tbody><tr><td rowspan="2">RD</td><td>Tension</td><td>195</td><td>1.12</td><td>753.0</td><td>339.3</td><td>8.969</td><td>0.0017</td><td>0.0040</td><td>0.0026</td><td>405</td></tr><tr><td>Compression</td><td>186</td><td>1.15</td><td>808.9</td><td>379.6</td><td>9.677</td><td>0.0016</td><td>0.0035</td><td>0.0025</td><td>400</td></tr><tr><td rowspan="2">TD</td><td>Tension</td><td>214</td><td>1.78</td><td>755.2</td><td>315.9</td><td>9.529</td><td>0.0018</td><td>0.0030</td><td>0.0022</td><td>426</td></tr><tr><td>Compression</td><td>222</td><td>1.93</td><td>854.2</td><td>397.2</td><td>8.745</td><td>0.0017</td><td>0.0025</td><td>0.0022</td><td>446</td></tr></tbody></table>

## Experimental results and discussion

SDE in uniaxial stress states. Figs. 5(a) and 5(b) compare the uniaxial tensile true stress-logarithmic plastic strain curves, $\sigma^T - \varepsilon^P$, with the compressive ones, $|\sigma^C| - |\varepsilon^P|$, measured using the IPCT for the RD and TD, respectively. Each curve is the average of two measurements. The flow stresses are higher in compression than in tension in both directions. Therefore, the SDE was clearly observed. To quantitatively evaluate the magnitude of the SDE in the uniaxial stress state, the rate of increase of the flow stress, $\beta_{SDE}$, was defined as

$$ \beta_{SDE} = 2 \frac{|\sigma^C| - \sigma^T}{|\sigma^C| + \sigma^T}, \qquad (8) $$

where $\sigma^T$ and $|\sigma^C|$ are the measured values of the tensile and compressive flow stresses, respectively, at a given plastic work per unit volume, $w^P$. Fig. 6 shows the variation of $\beta_{SDE}$ with $w^P$ for the RD and TD. It was found that $0.001 < \beta_{SDE} < 0.064$ for the RD and $0.041 < \beta_{SDE} < 0.073$ for the TD; therefore, the test sample clearly exhibited the SDE.

<figure><img src="image_1056.png" /><figcaption>Fig. 5 Comparison of true stress-logarithmic strain curves for tension and compression in (a) RD and (b) TD</figcaption></figure>

<footnote>*Approximated using $\sigma = a - b\exp(-c\varepsilon^p)$</footnote>
<figure><img src="image_1055.png" /><figcaption>Fig. 6 Change in $\beta_{SDE}$ with plastic work per unit volume</figcaption></figure>

The magnitude of $\beta_{SDE}$ is 0.05-0.06 for a 980 MPa dual phase steel sheet [7] and 0.02-0.10 for a cold rolled steel sheet [9]. Therefore, the magnitude of $\beta_{SDE}$ of the test sample used in this study (Fig. 6) is the same level of the steel sheets.

$M - \kappa$ curve. Fig. 7(a) and 7(b) compare the experimental and calculated $M - \kappa$ curves for the RD and TD, respectively. The three experimental curves, 1, 2 and 3, are consistent with each other in both directions, confirming the reproducibility of the experimental data. The red curves are the $M - \kappa$ curves calculated without considering the SDE; it was assumed that both the tensile and compressive ss curves follow the tensile ss curve approximated by Voce's hardening law shown in Eq. (3). The blue curves are the $M - \kappa$ curves calculated considering the SDE. The blue curves are higher than the red ones for a range of $\kappa \ge 10 \ m^{-1}$ in both directions. This was expected, as the flow stresses were higher in compression than in tension, as shown in Fig. 5. Furthermore, in both directions, the $M - \kappa$ curves calculated considering the SDE reproduced the experimental ones more accurately than did those calculated without considering the SDE. Thus, the SDE shown in Figs. 5 and 6 is validated by the pure bending experiment.
<figure><img src="image_1057.png" /><figcaption>Fig. 6 Change in $\beta_{SDE}$ with plastic work per unit volume</figcaption></figure>

The magnitude of $\beta_{SDE}$ is 0.05-0.06 for a 980 MPa dual phase steel sheet [7] and 0.02-0.10 for a cold rolled steel sheet [9]. Therefore, the magnitude of $\beta_{SDE}$ of the test sample used in this study (Fig. 6) is the same level of the steel sheets.

$M - \kappa$ curve. Fig. 7(a) and 7(b) compare the experimental and calculated $M - \kappa$ curves for the RD and TD, respectively. The three experimental curves, 1, 2 and 3, are consistent with each other in both directions, confirming the reproducibility of the experimental data. The red curves are the $M - \kappa$ curves calculated without considering the SDE; it was assumed that both the tensile and compressive ss curves follow the tensile ss curve approximated by Voce's hardening law shown in Eq. (3). The blue curves are the $M - \kappa$ curves calculated considering the SDE. The blue curves are higher than the red ones for a range of $\kappa \ge 10 \ m^{-1}$ in both directions. This was expected, as the flow stresses were higher in compression than in tension, as shown in Fig. 5. Furthermore, in both directions, the $M - \kappa$ curves calculated considering the SDE reproduced the experimental ones more accurately than did those calculated without considering the SDE. Thus, the SDE shown in Figs. 5 and 6 is validated by the pure bending experiment.
<figure><img src="image_1056.png" /><figcaption>Fig. 7 Comparison of measured M – κ curves with calculated ones in (a1) and (a2) RD and (b1) and (b2) TD. (a2) and (b2) are the enlarged figures of (a1) and (b1), respectively.</figcaption></figure>

**Conclusions**

Uniaxial tensile and in-plane compression tests were performed on a ferritic stainless steel sheet to measure the SDE. A pure bending test was performed to experimentally verify the SDE. The experimental findings obtained in this study can be summarized as follows.

(1) The SDE was observed for the test samples (0.001<$\beta_{SDE}$<0.064 for the RD and 0.041 <$\beta_{SDE}$<0.073 for the TD).

(2) The experimental M-κ curves were consistent with those calculated using the tensile and compressive ss curves obtained from the tension and compression experiments in both the RD and TD. Thus, the validity of the measured SDE was confirmed by the bending experiment.

**References**

[1] J. Hamada, A. Hayashi, N. Kanno, T. Komori, H. Ito, N. Fukuda, Y. Inoue, Development of Heat-resistant Ferritic Stainless Steels “NSSC429NF” and “NSSC448EM” Utilized Dynamic Precipitation Hardening for Automotive Exhaust Systems, Materia Japan 56 (2017) 33-35. (in Japanese) https://doi.org/10.2320/materia.56.33

[2] W.A. Spitzig, R.J. Sober, O. Richmond, Pressure dependence of yielding and associated volume expansion in tempered martensite, Acta Metall. 23 (1975), 885-893.
https://doi.org/10.1016/0001-6160(75)90205-9

[3] W.A. Spitzig, R.J. Sober, O. Richmond, O., The effect of hydrostatic pressure on the deformation behavior of maraging and HY-80 steels and its implication for plasticity, Met. Trans. 7A (1976) 1703-1710. https://doi.org/10.1007/BF02817888
<figure><img src="image_1058.png" /><figcaption>Fig. 7 Comparison of measured M – κ curves with calculated ones in (a1) and (a2) RD and (b1) and (b2) TD. (a2) and (b2) are the enlarged figures of (a1) and (b1), respectively.</figcaption></figure>

**Conclusions**

Uniaxial tensile and in-plane compression tests were performed on a ferritic stainless steel sheet to measure the SDE. A pure bending test was performed to experimentally verify the SDE. The experimental findings obtained in this study can be summarized as follows.

(1) The SDE was observed for the test samples ($0.001 < \beta_{SDE} < 0.064$ for the RD and $0.041 < \beta_{SDE} < 0.073$ for the TD).

(2) The experimental M – κ curves were consistent with those calculated using the tensile and compressive ss curves obtained from the tension and compression experiments in both the RD and TD. Thus, the validity of the measured SDE was confirmed by the bending experiment.

**References**

[1] J. Hamada, A. Hayashi, N. Kanno, T. Komori, H. Ito, N. Fukuda, Y. Inoue, Development of Heat-resistant Ferritic Stainless Steels “NSSC429NF” and “NSSC448EM” Utilized Dynamic Precipitation Hardening for Automotive Exhaust Systems, Materia Japan 56 (2017) 33-35. (in Japanese) https://doi.org/10.2320/materia.56.33

[2] W.A. Spitzig, R.J. Sober, O. Richmond, Pressure dependence of yielding and associated volume expansion in tempered martensite, Acta Metall. 23 (1975), 885-893.
https://doi.org/10.1016/0001-6160(75)90205-9

[3] W.A. Spitzig, R.J. Sober, O. Richmond, O., The effect of hydrostatic pressure on the deformation behavior of maraging and HY-80 steels and its implication for plasticity, Met. Trans. 7A (1976) 1703-1710. https://doi.org/10.1007/BF02817888
[4] O. Richmond, W.A. Spitzig, Pressure dependence and dilatancy of plastic flow, IUTAM Conference, Theoretical and Applied Mechanics, Proc. 15th International Congress of Theoretical and Applied Mechanics. North-Holland Publishers, Amsterdam, 1980, pp. 377-386.

[5] W.A. Spitzig, O. Richmond, The effect of pressure on the flow stress of metals, Acta Metall. 32 (1984) 457-463. https://doi.org/10.1016/0001-6160(84)90119-6

[6] T. Kuwabara, Y. Morita, Y. Miyashita, S. Takahashi, Elastic-Plastic Behavior of Sheet Metal Subjected to In-Plane Reverse Loading, in S. Tanimura, A.S. Khan (Eds.), Proc. Plasticity '95, The fifth international symposium on plasticity and its current applications, Gordon and Breach Publishers, 1995, pp.841-844.

[7] T. Maeda, N. Noma, T. Kuwabara, F. Barlat, Y.P. Korkolis, Measurement of the strength differential effect of DP980 steel sheet and experimental validation using pure bending test, J. Mater. Process. Technol. 256 (2018) 247-253. https://doi.org/10.1016/j.jmatprotec.2018.02.009

[8] N. Noma, K. Hashimoto, T. Maeda, T. Kuwabara, Enhancement of Accuracy of Springback Simulation Using the Material Model Considering the Strength Differential Effect, J. JSTP 61 (2020) 20-25. (in Japanese). https://doi.org/10.9773/sosei.61.20

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

[10] T. Kuwabara, R. Saito, T. Hirano, N. Oohashi, Tension-Compression Asymmetry of SUS304 Stainless Sheet for Electronic Parts and Its Effects on Bending and Springback Behavior, Tetsu-to-Hagané 95 (2009) 732-739. (in Japanese). https://doi.org/10.2355/tetsutohagane.95.732

[11] S.Y. Yoon, F. Barlat, S.Y. Lee, J.H. Kim, M.S. Wi, D.J. Kim, Finite element implementation of hydrostatic pressure-sensitive plasticity and its application to distortional hardening model and sheet metal forming simulations, J. Mater. Processing Technol. 302 (2022) 117494. https://doi.org/10.1016/j.jmatprotec.2022.117494

[12] F. Barlat, S.Y. Yoon, S.Y. Lee, M.S. Wi, J.H. Kim, Distortional plasticity framework with application to advanced high strength steel, Int. J. Solids Struct. 202 (2020) 947-962. https://doi.org/10.1016/j.ijsolstr.2020.05.014

[13] N. Noma, T. Kuwabara, Specimen Geometry Optimization for In-plane Reverse Loading Test of Sheet Metal and Experimental Validation, J. JSTP 53 (2012) 574-579. (in Japanese). https://doi.org/10.9773/sosei.53.574

[14] K. Akiyama, R. Tachibana, T. Sakurai, T. Kuwabara: submitted to J. Materials Processing Technology (2024).
[4] O. Richmond, W.A. Spitzig, Pressure dependence and dilatancy of plastic flow, IUTAM Conference, Theoretical and Applied Mechanics, Proc. 15th International Congress of Theoretical and Applied Mechanics. North-Holland Publishers, Amsterdam, 1980, pp. 377-386.

[5] W.A. Spitzig, O. Richmond, The effect of pressure on the flow stress of metals, Acta Metall. 32 (1984) 457-463. https://doi.org/10.1016/0001-6160(84)90119-6

[6] T. Kuwabara, Y. Morita, Y. Miyashita, S. Takahashi, Elastic-Plastic Behavior of Sheet Metal Subjected to In-Plane Reverse Loading, in S. Tanimura, A.S. Khan (Eds.), Proc. Plasticity '95, The fifth international symposium on plasticity and its current applications, Gordon and Breach Publishers, 1995, pp.841-844.

[7] T. Maeda, N. Noma, T. Kuwabara, F. Barlat, Y.P. Korkolis, Measurement of the strength differential effect of DP980 steel sheet and experimental validation using pure bending test, J. Mater. Process. Technol. 256 (2018) 247-253. https://doi.org/10.1016/j.jmatprotec.2018.02.009

[8] N. Noma, K. Hashimoto, T. Maeda, T. Kuwabara, Enhancement of Accuracy of Springback Simulation Using the Material Model Considering the Strength Differential Effect, J. JSTP 61 (2020) 20-25. (in Japanese). https://doi.org/10.9773/sosei.61.20

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

[10] T. Kuwabara, R. Saito, T. Hirano, N. Oohashi, Tension-Compression Asymmetry of SUS304 Stainless Sheet for Electronic Parts and Its Effects on Bending and Springback Behavior, Tetsu-to-Hagané 95 (2009) 732-739. (in Japanese). https://doi.org/10.2355/tetsutohagane.95.732

[11] S.Y. Yoon, F. Barlat, S.Y. Lee, J.H. Kim, M.S. Wi, D.J. Kim, Finite element implementation of hydrostatic pressure-sensitive plasticity and its application to distortional hardening model and sheet metal forming simulations, J. Mater. Processing Technol. 302 (2022) 117494. https://doi.org/10.1016/j.jmatprotec.2022.117494

[12] F. Barlat, S.Y. Yoon, S.Y. Lee, M.S. Wi, J.H. Kim, Distortional plasticity framework with application to advanced high strength steel, Int. J. Solids Struct. 202 (2020) 947-962. https://doi.org/10.1016/j.ijsolstr.2020.05.014

[13] N. Noma, T. Kuwabara, Specimen Geometry Optimization for In-plane Reverse Loading Test of Sheet Metal and Experimental Validation, J. JSTP 53 (2012) 574-579. (in Japanese). https://doi.org/10.9773/sosei.53.574

[14] K. Akiyama, R. Tachibana, T. Sakurai, T. Kuwabara: submitted to J. Materials Processing Technology (2024).
# Measurement and analysis of the strength differential effect of 6000-series aluminum alloy sheet

AKIYAMA Kaisei¹, a * and KUWABARA Toshihiko², b

¹Department of Mechanical Systems Engineering, Graduate School of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588, Japan

²Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588, Japan

aks-akiyama@st.go.tuat.ac.jp, bkuwabara@cc.tuat.ac.jp

**Keywords:** Compressive Flow Stress, Stacked Compression Test, Equibiaxial Tensile Test, Hydrostatic Stress

**Abstract.** The tension-compression asymmetry (TCA, referred to as the strength differential effect, SDE, for annealed materials) of a 1.1-mm-thick 6000-series aluminum alloy sheet, A6116-T4, which is 3-months age hardened is measured using a uniaxial tensile test and an in-plane compression test. It is found that the in-plane compressive flow stress is 1-7 % higher than the uniaxial tensile flow stress; therefore, the material exhibits the SDE. Moreover, a stacked compression test in the normal (through-thickness) direction (ND) of the test sample is also performed to measure the hydrostatic stress dependence of the yield stress. It is found that the uniaxial compressive flow stress in the ND is 4-9 % higher than the equibiaxial tensile flow stress measured using a cruciform equibiaxial tension test (ISO 16842); therefore, the hydrostatic stress dependence of the yield stress is confirmed. Hence, it is concluded that the SDE observed in A6116-T4 is caused by the hydrostatic stress dependence of the yield stress.

**Introduction**

Aluminum alloy sheets are used for manufacturing automotive parts to reduce the weight of car bodies. Automotive parts are mainly fabricated using press forming, which has superior productivity. During press forming, materials are subjected to compressive stress as well as tensile stress. Some metals have different work-hardening behavior in tension and compression, a phenomenon called the Tension-Compression Asymmetry (TCA). In particular, the TCA observed for the materials having no prestraining is referred to as the Strength Differential Effect (SDE).

The SDE in aluminum alloys has been previously investigated. Spitzig and Richmond [1] performed uniaxial tension and compression tests under hydrostatic pressure on both iron-based materials, including Fe single crystals, and 1100 aluminum to find that these materials exhibit the SDE, and demonstrated that the cause of the SDE is attributed to the fact that the flow stress of these materials is dependent on the hydrostatic pressure. Barlat et al. [2] conducted tension and compression tests on an aluminum alloy sheet (2090-T3) at increments of 15 degrees from the rolling direction (RD) to measure the respective 0.2% proof stresses, $\sigma^T$ and $\sigma^C$. They found that $\sigma^T$ was 12% larger than $\sigma^C$ in the RD and that $\sigma^C$ was several percent larger than $\sigma^T$ in the directions 45°, 60°, 75°, and 90° from the RD, confirming the SDE. Kuwabara et al. [3] performed an in-plane compression test (IPCT) on a 5182-O aluminum alloy sheet and two low-carbon steel sheets using comb-shaped dies. They found that the flow stresses were higher in compression than in tension for the steel sheets, whereas the aluminum alloy sheet did not exhibit the SDE. Kuwabara et al. [4] concluded that the SDE in mild steel sheet is also due to the hydrostatic stress dependence

<figure><img src="image_1057.png" /></figure>
# Measurement and analysis of the strength differential effect of 6000-series aluminum alloy sheet

AKIYAMA Kaisei¹,ᵃ* and KUWABARA Toshihiko²,ᵇ

¹Department of Mechanical Systems Engineering, Graduate School of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588, Japan

²Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi, Tokyo 184-8588, Japan

ᵃks-akiyama@st.go.tuat.ac.jp, ᵇkuwabara@cc.tuat.ac.jp

**Keywords:** Compressive Flow Stress, Stacked Compression Test, Equibiaxial Tensile Test, Hydrostatic Stress

**Abstract.** The tension-compression asymmetry (TCA, referred to as the strength differential effect, SDE, for annealed materials) of a 1.1-mm-thick 6000-series aluminum alloy sheet, A6116-T4, which is 3-months age hardened is measured using a uniaxial tensile test and an in-plane compression test. It is found that the in-plane compressive flow stress is 1-7 % higher than the uniaxial tensile flow stress; therefore, the material exhibits the SDE. Moreover, a stacked compression test in the normal (through-thickness) direction (ND) of the test sample is also performed to measure the hydrostatic stress dependence of the yield stress. It is found that the uniaxial compressive flow stress in the ND is 4-9 % higher than the equibiaxial tensile flow stress measured using a cruciform equibiaxial tension test (ISO 16842); therefore, the hydrostatic stress dependence of the yield stress is confirmed. Hence, it is concluded that the SDE observed in A6116-T4 is caused by the hydrostatic stress dependence of the yield stress.

**Introduction**

Aluminum alloy sheets are used for manufacturing automotive parts to reduce the weight of car bodies. Automotive parts are mainly fabricated using press forming, which has superior productivity. During press forming, materials are subjected to compressive stress as well as tensile stress. Some metals have different work-hardening behavior in tension and compression, a phenomenon called the Tension-Compression Asymmetry (TCA). In particular, the TCA observed for the materials having no pretraining is referred to as the Strength Differential Effect (SDE).

The SDE in aluminum alloys has been previously investigated. Spitzig and Richmond [1] performed uniaxial tension and compression tests under hydrostatic pressure on both iron-based materials, including Fe single crystals, and 1100 aluminum to find that these materials exhibit the SDE, and demonstrated that the cause of the SDE is attributed to the fact that the flow stress of these materials is dependent on the hydrostatic pressure. Barlat et al. [2] conducted tension and compression tests on an aluminum alloy sheet (2090-T3) at increments of 15 degrees from the rolling direction (RD) to measure the respective 0.2% proof stresses, $\sigma^T$ and $\sigma^C$. They found that $\sigma^T$ was 12% larger than $\sigma^C$ in the RD and that $\sigma^C$ was several percent larger than $\sigma^T$ in the directions 45°, 60°, 75°, and 90° from the RD, confirming the SDE. Kuwabara et al. [3] performed an in-plane compression test (IPCT) on a 5182-O aluminum alloy sheet and two low-carbon steel sheets using comb-shaped dies. They found that the flow stresses were higher in compression than in tension for the steel sheets, whereas the aluminum alloy sheet did not exhibit the SDE. Kuwabara et al. [4] concluded that the SDE in mild steel sheet is also due to the hydrostatic stress dependence

<figure><img src="image_1059.png" /></figure>
of flow stress. Holmen et al. [5] investigated the SDE of four 6000-series aluminum alloys (one was rolled and other three were extrude) in several tempers with yield strengths varying from 27 to 373 MPa. They observed that the axial stresses measured in compression tests were significantly higher than corresponding tensile stresses for nearly all material configurations. Ku et al. [6] experimentally verified the TCA in rolled aluminum alloy 7056 in the overaged condition (T7) with two tempers and reported that the TCA was attributed to the microstructural evolution. However, no experimental verification of the hydrostatic stress dependence of aluminum alloys has been reported, except for the work by Spitzig and Richmond [1].

This study investigates the SDE of a 6000-series rolled aluminum alloy sheet, A6116-T4, with a nominal thickness of 1.1 mm, which was age hardened for three months. In-plane tension and compression tests were performed to evaluate the SDE. Moreover, an equibiaxial tension test using a cruciform specimen [7] and a uniaxial compression test in the normal (through-thickness) direction (ND) of the test sample [4] were performed to clarify whether the flow stresses were affected by hydrostatic stress. It is found that the test sample exhibited the SDE and that the observed SDE can be attributed to the hydrostatic stress dependence of the yield stress.

## Effect of hydrostatic stress on flow stress

Hereafter, the RD, transverse (TD), and ND of the material are referred to as the x-, y-, and z-axes, respectively. When the hydrostatic stress, $-\sigma_b$, is superimposed on the equibiaxial tensile stress state, $\sigma_b$, in the xy-plane, the equibiaxial tensile stress $\sigma_b$ is cancelled by the hydrostatic stress $-\sigma_b$ and is equivalent to the uniaxial compressive stress state, $-\sigma_b$, in the z-axis (Fig. 1). Therefore, under the assumption that the flow stress of the material is independent of hydrostatic stress, the $\sigma_b$-$|\epsilon_z^p|$ ($\epsilon_z^p$: plastic strain in the z-axis direction) curve obtained from the equibiaxial tensile test should be identical to the $|\sigma_z^c|$-$|\epsilon_z^p|$ curve obtained from the uniaxial compression test in the ND. On the other hand, if the flow stress is affected by hydrostatic stress, the two curves should be different from each other.

<figure><img src="image_1058.png" /><figcaption>Fig. 1 Equivalence of the equibiaxial tension with the uniaxial compression with an assumption that the flow stress is independent of hydrostatic stress</figcaption></figure>

## Experimental method

Test material. The test sample was a 6000-series aluminum alloy sheet, A6116-T4, with a nominal thickness of 1.1 mm. It was 3-months age hardened. The mechanical properties of the sample are shown in Table 1.
of flow stress. Holmen et al. [5] investigated the SDE of four 6000-series aluminum alloys (one was rolled and other three were extrude) in several tempers with yield strengths varying from 27 to 373 MPa. They observed that the axial stresses measured in compression tests were significantly higher than corresponding tensile stresses for nearly all material configurations. Ku et al. [6] experimentally verified the TCA in rolled aluminum alloy 7056 in the overaged condition (T7) with two tempers and reported that the TCA was attributed to the microstructural evolution. However, no experimental verification of the hydrostatic stress dependence of aluminum alloys has been reported, except for the work by Spitzig and Richmond [1].

This study investigates the SDE of a 6000-series rolled aluminum alloy sheet, A6116-T4, with a nominal thickness of 1.1 mm, which was age hardened for three months. In-plane tension and compression tests were performed to evaluate the SDE. Moreover, an equibiaxial tension test using a cruciform specimen [7] and a uniaxial compression test in the normal (through-thickness) direction (ND) of the test sample [4] were performed to clarify whether the flow stresses were affected by hydrostatic stress. It is found that the test sample exhibited the SDE and that the observed SDE can be attributed to the hydrostatic stress dependence of the yield stress.

## Effect of hydrostatic stress on flow stress

Hereafter, the RD, transverse (TD), and ND of the material are referred to as the x-, y-, and z-axes, respectively. When the hydrostatic stress, $-\sigma_b$, is superimposed on the equibiaxial tensile stress state, $\sigma_b$, in the xy-plane, the equibiaxial tensile stress $\sigma_b$ is cancelled by the hydrostatic stress $-\sigma_b$ and is equivalent to the uniaxial compressive stress state, $-\sigma_b$, in the z-axis (Fig. 1). Therefore, under the assumption that the flow stress of the material is independent of hydrostatic stress, the $\sigma_b$-$|\epsilon_z^p|$ ($\epsilon_z^p$: plastic strain in the z-axis direction) curve obtained from the equibiaxial tensile test should be identical to the $|\sigma_z^c|$-$|\epsilon_z^p|$ curve obtained from the uniaxial compression test in the ND. On the other hand, if the flow stress is affected by hydrostatic stress, the two curves should be different from each other.

<figure><img src="image_1060.png" /><figcaption>Fig. 1 Equivalence of the equibiaxial tension with the uniaxial compression with an assumption that the flow stress is independent of hydrostatic stress</figcaption></figure>

## Experimental method

Test material. The test sample was a 6000-series aluminum alloy sheet, A6116-T4, with a nominal thickness of 1.1 mm. It was 3-months age hardened. The mechanical properties of the sample are shown in Table 1.
<table><caption>Table 1 Mechanical properties of A6116-T4</caption><thead><tr><th>Angle from RD [°]</th><th>Stress direction</th><th>E [GPa]</th><th>σ<sub>0.2</sub> [MPa]</th><th>r</th><th>σ<sub>B</sub> [MPa]</th><th>ε<sub>TS</sub><sup>p</sup> *</th><th>a** [MPa]</th><th>b** [MPa]</th><th>c**</th></tr></thead><tbody><tr><td rowspan="2">0</td><td>Tension</td><td>69.5</td><td>142</td><td>0.707</td><td>251</td><td>0.199</td><td>323</td><td>195</td><td>11.7</td></tr><tr><td>Compression</td><td>67.9</td><td>150</td><td>0.723</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">90</td><td>Tension</td><td>67.7</td><td>137</td><td>0.671</td><td>244</td><td>0.209</td><td>319</td><td>195</td><td>11.0</td></tr><tr><td>Compression</td><td>67.0</td><td>145</td><td>0.729</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

In-plane compression test. Fig. 2a and b show the IPCT apparatus used in this study. This apparatus was first developed in [3], modified in [8], and used in [9]-[11] for steel sheets. Fig. 2a shows an overview of the dies used for applying in-plane compression to a sheet specimen. Fig. 2b shows an overview of the test apparatus. Lower die 1 is bolted to the lower plate of the die set. Lower die 2 is bolted to the slide rail, so that the die can move in the horizontal direction without friction. A specimen is placed on lower dies 1 and 2 with both ends clamped by clamping jigs (not shown in the figure). Upper dies 1 and 2 are placed on the specimen. The positioning pins (not shown in the figure) fixed to upper dies 1 and 2 align with holes in lower dies 1 and 2. Accordingly, upper die 1 is stationary, and the motion of upper die 2 is synchronized with that of lower die 2. Lower die 2 is connected to a load cell to measure the tension-compression force during a test. It is actuated by hydraulic cylinder A to apply an in-plane tension-compression force to the specimen. Hydraulic cylinder B exerts a blank holding force on the specimen to prevent the buckling of the specimen during a test. The blank holding force is kept constant during a test via a hydraulic control valve. Steel cylindrical rollers are inserted between the blank holding platens and upper dies 1 and 2 so that the blank holding force can be transmitted to the specimen during an IPCT without friction. In this study, the blank holding force applied to the specimen was 1.2 % of 0.2 % yield stress (=1.7 MPa). The strain was measured using a strain gauge (YFLA-2-1LJC-F, Tokyo Measuring Instruments Laboratory Co. Ltd.). The strain rate was approximately $5 \times 10^{-4}$ s$^{-1}$ and the testing temperature was room temperature for both the tension and compression experiments.

<figure><img src="image_1059.png" /><figcaption>Fig. 2 IPCT devise [11]: (a) upper and lower dies and (b) overview of test apparatus</figcaption></figure>

<footnote>* $\epsilon_{TS}^p$ is a logarithmic plastic strain giving the maximum tensile load</footnote>

<footnote>** Approximated using $\sigma = a - b\exp(-c\epsilon^p)$ at $\epsilon^p = 0.002 - \epsilon_{TS}^p$</footnote>
<table><caption>Table 1 Mechanical properties of A6116-T4</caption><thead><tr><th>Angle from RD [°]</th><th>Stress direction</th><th>E [GPa]</th><th>σ<sub>0.2</sub> [MPa]</th><th>r</th><th>σ<sub>B</sub> [MPa]</th><th>ε<sub>TS</sub><sup>p</sup> *</th><th>a** [MPa]</th><th>b** [MPa]</th><th>c**</th></tr></thead><tbody><tr><td rowspan="2">0</td><td>Tension</td><td>69.5</td><td>142</td><td>0.707</td><td>251</td><td>0.199</td><td>323</td><td>195</td><td>11.7</td></tr><tr><td>Compression</td><td>67.9</td><td>150</td><td>0.723</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">90</td><td>Tension</td><td>67.7</td><td>137</td><td>0.671</td><td>244</td><td>0.209</td><td>319</td><td>195</td><td>11.0</td></tr><tr><td>Compression</td><td>67.0</td><td>145</td><td>0.729</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

In-plane compression test. Fig. 2a and b show the IPCT apparatus used in this study. This apparatus was first developed in [3], modified in [8], and used in [9]-[11] for steel sheets. Fig. 2a shows an overview of the dies used for applying in-plane compression to a sheet specimen. Fig. 2b shows an overview of the test apparatus. Lower die 1 is bolted to the lower plate of the die set. Lower die 2 is bolted to the slide rail, so that the die can move in the horizontal direction without friction. A specimen is placed on lower dies 1 and 2 with both ends clamped by clamping jigs (not shown in the figure). Upper dies 1 and 2 are placed on the specimen. The positioning pins (not shown in the figure) fixed to upper dies 1 and 2 align with holes in lower dies 1 and 2. Accordingly, upper die 1 is stationary, and the motion of upper die 2 is synchronized with that of lower die 2. Lower die 2 is connected to a load cell to measure the tension-compression force during a test. It is actuated by hydraulic cylinder A to apply an in-plane tension-compression force to the specimen. Hydraulic cylinder B exerts a blank holding force on the specimen to prevent the buckling of the specimen during a test. The blank holding force is kept constant during a test via a hydraulic control valve. Steel cylindrical rollers are inserted between the blank holding platens and upper dies 1 and 2 so that the blank holding force can be transmitted to the specimen during an IPCT without friction. In this study, the blank holding force applied to the specimen was 1.2 % of 0.2 % yield stress (=1.7 MPa). The strain was measured using a strain gauge (YFLA-2-1LJC-F, Tokyo Measuring Instruments Laboratory Co. Ltd.). The strain rate was approximately $5 \times 10^{-4}$ s$^{-1}$ and the testing temperature was room temperature for both the tension and compression experiments.

<figure><img src="image_1061.png" /><figcaption>Fig. 2 IPCT devise [11]: (a) upper and lower dies and (b) overview of test apparatus</figcaption></figure>

<footnote>* $ε_{TS}^p$ is a logarithmic plastic strain giving the maximum tensile load</footnote>

<footnote>** Approximated using $σ = a - b\exp(-cε^p)$ at $ε^p = 0.002 - ε_{TS}^p$</footnote>
Uniaxial compression test of stacked specimen. Fig. 3a shows a schematic diagram of the uniaxial compression testing device for a stacked specimen. This device was incorporated into a regular tensile testing machine (AUTOGRAPH AG-100kNX, Shimadzu Corporation). Fig. 3b shows the details of a compression die. The compression dies have a thickness of 4 mm (a 3-mm-thick super steel base plate coated with 1-mm-thick sintered polycrystal diamond) and a diameter of 19 mm. Polycrystal diamond is effective for reducing the friction between the specimen and the compression dies and thus suppresses the non-uniform deformation of the specimen [12]. The compression dies were positioned on the upper and lower supporting plates using a clamping plate. The supporting plates were bolted to the upper and lower die plates. Since the initial thickness of the specimen was 1.1 mm, the amount of compressive displacement due to compression in the ND was small, making it difficult to measure plastic strain with high accuracy. Therefore, we stacked sheet samples in the ND and increased the initial thickness to 5.5 and 4.4 mm to improve measurement accuracy. The sheets were fixed with an adhesive (CEMEDINE Metal Lock Series Y610) to prevent interlaminar slippage.

Fig. 3c shows a schematic diagram of the uniaxial compression testing device for a stacked specimen. We performed compression tests using two stacked specimens with the same cross-sectional area but different heights, $h_H = 5.5$ mm and $h_L = 4.4$ mm, and measured the nominal stress-displacement for each specimen using a displacement meter (GT2-S5, KEYENCE Co. Ltd.). By denoting the difference in displacement between the two stacked specimens for a given nominal stress as $\Delta u$ and calculating the compressive true strain, $\epsilon_z$, as $\ln(1 - \frac{\Delta u}{h_H - h_L})$, we determined the true stress-true strain curve, $|\sigma_z^c| - |\epsilon_z|$, for a specimen with an initial stack height of $h (= h_H - h_L)$. This approach enabled us to exclude the influences of the elastic deformation of the compression

<figure><img src="image_1060.png" /><figcaption>Fig. 3 SCT device [4]: (a) overview of test apparatus, (b) details of lower part of compression die, and (c) specimen geometry</figcaption></figure>
Uniaxial compression test of stacked specimen. Fig. 3a shows a schematic diagram of the uniaxial compression testing device for a stacked specimen. This device was incorporated into a regular tensile testing machine (AUTOGRAPH AG-100kNX, Shimadzu Corporation). Fig. 3b shows the details of a compression die. The compression dies have a thickness of 4 mm (a 3-mm-thick super steel base plate coated with 1-mm-thick sintered polycrystal diamond) and a diameter of 19 mm. Polycrystal diamond is effective for reducing the friction between the specimen and the compression dies and thus suppresses the non-uniform deformation of the specimen [12]. The compression dies were positioned on the upper and lower supporting plates using a clamping plate. The supporting plates were bolted to the upper and lower die plates. Since the initial thickness of the specimen was 1.1 mm, the amount of compressive displacement due to compression in the ND was small, making it difficult to measure plastic strain with high accuracy. Therefore, we stacked sheet samples in the ND and increased the initial thickness to 5.5 and 4.4 mm to improve measurement accuracy. The sheets were fixed with an adhesive (CEMEDINE Metal Lock Series Y610) to prevent interlaminar slippage.

Fig. 3c shows a schematic diagram of the uniaxial compression testing device for a stacked specimen. We performed compression tests using two stacked specimens with the same cross-sectional area but different heights, $h_H = 5.5$ mm and $h_L = 4.4$ mm, and measured the nominal stress-displacement for each specimen using a displacement meter (GT2-S5, KEYENCE Co. Ltd.). By denoting the difference in displacement between the two stacked specimens for a given nominal stress as $\Delta u$ and calculating the compressive true strain, $\epsilon_z$, as $\ln(1 - \frac{\Delta u}{h_H - h_L})$, we determined the true stress-true strain curve, $|\sigma_z^c| - |\epsilon_z|$, for a specimen with an initial stack height of $h (= h_H - h_L)$. This approach enabled us to exclude the influences of the elastic deformation of the compression

<figure><img src="image_1062.png" /><figcaption>Fig. 3 SCT device [4]: (a) overview of test apparatus, (b) details of lower part of compression die, and (c) specimen geometry</figcaption></figure>
device and potential friction between the specimen and compression plates on the measurement of $|ε_z|$. Thus, Young's modulus is calculated from the slope at unloading. We then determined the compressive true stress vs. compressive plastic strain curve, $|σ_z^C| - |ε_z^P|$.

## Experimental results

Measurement of SDE. Fig. 4a and b compare the uniaxial tensile true stress-logarithmic plastic strain curves, $σ^T - ε^P$, with the compressive ones, $|σ^C| - |ε^P|$, for the RD and TD, respectively. Each curve is the average of two measurements. The compressive stress exceeds the tensile stress in both directions. Therefore, the SDE was clearly observed.

To quantitatively evaluate the magnitude of the SDE in the uniaxial stress state, the increase rate in the flow stress, $β_{SDE-uni}$, was defined as

$$β_{SDE-uni} = 2 \frac{|σ^C| - σ^T}{|σ^C| + σ^T} \qquad (1)$$

where $σ^T$ and $|σ^C|$ are the measured values at a given plastic work per unit volume, $w^P$. In the calculation of $β_{SDE-uni}$, both the uniaxial tension and in-plane compression test curves were averaged over two tests. Fig. 5 shows the variation of $β_{SDE-uni}$ with $w^P$ for the RD and TD. It was found that $0.015 < β_{SDE-uni} < 0.053$ for the RD and $0.036 < β_{SDE-uni} < 0.066$ for the TD; therefore, the test sample clearly exhibited the SDE.

Verification of hydrostatic stress dependence of flow stress. Fig. 6 compares the $σ_b - |ε_z^P|$ curve measured using the equibiaxial tensile test with a cruciform specimen [1122], the average of two measurements, with the $|σ_z^C| - |ε_z^P|$ curve obtained from the uniaxial compression test in the ND of the stacked specimen. It can be clearly seen that the $|σ_z^C|$ exceeds the $σ_b$. Therefore, the flow stress of the test sample is affected by hydrostatic stress.

In order to quantitatively evaluate the effect of hydrostatic stress on flow stress, the plastic flow stress increase rate, $β_{SDE-bs}$, was defined as

$$β_{SDE-bs} = 2 \frac{|σ_z^C| - σ_b}{|σ_z^C| + σ_b} \qquad (2)$$

<figure><img src="image_1061.png" /><figcaption>Fig. 4 Comparison of true stress-true strain curves between tension and compression in (a) RD and (b) TD</figcaption></figure>
device and potential friction between the specimen and compression plates on the measurement of $|ε_z|$. Thus, Young's modulus is calculated from the slope at unloading. We then determined the compressive true stress vs. compressive plastic strain curve, $|σ_z^C| - |ε_z^P|$.

## Experimental results

Measurement of SDE. Fig. 4a and b compare the uniaxial tensile true stress-logarithmic plastic strain curves, $σ^T - ε^P$, with the compressive ones, $|σ^C| - |ε^P|$, for the RD and TD, respectively. Each curve is the average of two measurements. The compressive stress exceeds the tensile stress in both directions. Therefore, the SDE was clearly observed.

To quantitatively evaluate the magnitude of the SDE in the uniaxial stress state, the increase rate in the flow stress, $β_{SDE-uni}$, was defined as

$$β_{SDE-uni} = 2 \frac{|σ^C| - σ^T}{|σ^C| + σ^T} \qquad (1)$$

where $σ^T$ and $|σ^C|$ are the measured values at a given plastic work per unit volume, $w^P$. In the calculation of $β_{SDE-uni}$, both the uniaxial tension and in-plane compression test curves were averaged over two tests. Fig. 5 shows the variation of $β_{SDE-uni}$ with $w^P$ for the RD and TD. It was found that $0.015 < β_{SDE-uni} < 0.053$ for the RD and $0.036 < β_{SDE-uni} < 0.066$ for the TD; therefore, the test sample clearly exhibited the SDE.

Verification of hydrostatic stress dependence of flow stress. Fig. 6 compares the $σ_b - |ε_z^P|$ curve measured using the equibiaxial tensile test with a cruciform specimen [1122], the average of two measurements, with the $|σ_z^C| - |ε_z^P|$ curve obtained from the uniaxial compression test in the ND of the stacked specimen. It can be clearly seen that the $|σ_z^C|$ exceeds the $σ_b$. Therefore, the flow stress of the test sample is affected by hydrostatic stress.

In order to quantitatively evaluate the effect of hydrostatic stress on flow stress, the plastic flow stress increase rate, $β_{SDE-bs}$, was defined as

$$β_{SDE-bs} = 2 \frac{|σ_z^C| - σ_b}{|σ_z^C| + σ_b} \qquad (2)$$

<figure><img src="image_1063.png" /><figcaption>Fig. 4 Comparison of true stress-true strain curves between tension and compression in (a) RD and (b) TD</figcaption></figure>
<figure><img src="image_1062.png" /><figcaption>Fig. 5 Change in $\beta_{SDE-uni}$ and $\beta_{SDE-bs}$ with plastic work per unit volume, $w^p$</figcaption></figure>

<figure><img src="image_1063.png" /><figcaption>Fig. 6 Comparison of $|\sigma_z^C| - |\varepsilon_z^p|$ and $\sigma_b - |\varepsilon_z^p|$ curves</figcaption></figure>

where $\sigma_b$ and $|\sigma_z^C|$ are the measured values at a given plastic work per unit volume, $w^p$. The variation of $\beta_{SDE-bs}$ with $w^p$ is shown in Fig. 5. $\beta_{SDE-bs}$ was in the range from 4 to 9 %. Therefore, we conclude that the flow stress of the test sample could be affected by hydrostatic stress.

## Discussion

The Spitzig-Richmond yield condition [1] is given by

$$ \bar{\sigma}(\boldsymbol{\sigma}) = c - \alpha c \operatorname{tr} \boldsymbol{\sigma} = c(1 - \alpha \operatorname{tr} \boldsymbol{\sigma}) \quad (3) $$

where $\boldsymbol{\sigma}$ is the stress tensor, $\operatorname{tr} \boldsymbol{\sigma}$ is its trace, $\bar{\sigma}$ is a pressure-independent effective stress, $c$ is a strength parameter, and $\alpha$ is the pressure coefficient. Denoting the $\alpha$ measured using tension and
<figure><img src="image_1064.png" /><figcaption>Fig. 5 Change in $\beta_{SDE-uni}$ and $\beta_{SDE-bs}$ with plastic work per unit volume, $w^p$</figcaption></figure>

<figure><img src="image_1065.png" /><figcaption>Fig. 6 Comparison of $|\sigma_z^C| - |\varepsilon_z^p|$ and $\sigma_b - |\varepsilon_z^p|$ curves</figcaption></figure>

where $\sigma_b$ and $|\sigma_z^C|$ are the measured values at a given plastic work per unit volume, $w^p$. The variation of $\beta_{SDE-bs}$ with $w^p$ is shown in Fig. 5. $\beta_{SDE-bs}$ was in the range from 4 to 9 %. Therefore, we conclude that the flow stress of the test sample could be affected by hydrostatic stress.

Discussion

The Spitzig-Richmond yield condition [1] is given by

$$ \bar{\sigma}(\boldsymbol{\sigma}) = c - \alpha c \operatorname{tr} \boldsymbol{\sigma} = c(1 - \alpha \operatorname{tr} \boldsymbol{\sigma}) \quad (3) $$

where $\boldsymbol{\sigma}$ is the stress tensor, $\operatorname{tr} \boldsymbol{\sigma}$ is its trace, $\bar{\sigma}$ is a pressure-independent effective stress, $c$ is a strength parameter, and $\alpha$ is the pressure coefficient. Denoting the $\alpha$ measured using tension and
compression tests as $\alpha_{uni}$ and that measured using an equibiaxial tensile test and an SCT as $\alpha_{bs}$, respectively, $\alpha_{uni}$ and $\alpha_{bs}$ are calculated as (see [4])

$$ \alpha_{uni} = \frac{|\sigma^C| - \sigma^T}{2|\sigma^C|\sigma^T} \qquad (4) $$

$$ \alpha_{bs} = \frac{|\sigma_z^C| - \sigma_b}{3|\sigma_z^C|\sigma_b} \qquad (5) $$

Considering that $\sigma^C \approx \sigma_z^C$ and $\sigma^T \approx \sigma_b$, we obtain the following equation:

$$ \frac{\alpha_{bs}}{\alpha_{uni}} \approx \frac{|\sigma_z^C| - \sigma_b}{|\sigma^C| - \sigma^T} \times \frac{2}{3} \qquad (6) $$

If $\alpha$ is a material constant, $\alpha_{bs}/\alpha_{uni}$ should be unity. It means that $(|\sigma_z^C| - \sigma_b)/(|\sigma^C| - \sigma^T)$ should be 3/2. However, from Fig. 5, it was measured that $(|\sigma_z^C| - \sigma_b)/(|\sigma^C| - \sigma^T) \approx 1$ except for the early stage of plastic deformation, $w^P < 0.2$ MPa. This suggests that there may be other influencing factors besides the hydrostatic pressure effect as a cause of the SDE in this material. Sun et al. [13] investigated the tension-compression (TC) asymmetry of a 6061 aluminum alloy using a strain gradient crystal plasticity model implemented with the finite element method. They concluded that the TC asymmetry for the aluminum alloys is caused by a combination of difference in the GND density and rotation of crystal matrix and particles, which is induced by plasticity even at the yield strain with limited plasticity. However, it is noted that they found that tensile flow stress was higher than compressive one. Therefore, the origin of the SDE in 6000-series aluminum alloys requires further study.

## Conclusions

A uniaxial tensile test and an in-plane compression test were performed on a 6000-series aluminum alloy sheet (A6116-T4) to measure the SDE. In addition, an equibiaxial tensile test using a cruciform specimen and uniaxial compression tests in the ND using stacked specimens were performed to verify the hydrostatic stress dependence of the flow stress. The experimental findings obtained in this study are summarized follows.

(1) The SDE was observed for our test sample ($0.015 < \beta_{SDE-uni} < 0.053$ for the RD and $0.036 < \beta_{SDE-uni} < 0.066$ for the TD).

(2) The compressive stress, $|\sigma_z^C|$, measured in the uniaxial compression test in the ND exceeded the equibiaxial tensile stress, $\sigma_b$, measured using a cruciform specimen. Therefore, the flow stress of the test sample could be affected by hydrostatic stress.

(3) There may be other influencing factors besides the hydrostatic pressure effect as a cause of the SDE in this material, as suggested by Sun et al. [13].

## References

[1] W.A. Spitzig, O. Richmond, The effect of pressure on the flow stress of metals, Acta Metall 32 (1984) 457-463. https://doi.org/10.1016/0001-6160(84)90119-6

[2] F. Barlat, D.J. Lege, J.C. Brem, and C.J. Warren, Constitutive behavior for anisotropic materials and application to a 2090 Al-Li alloy, in: T.C. Lowe, A.D. Rollet, P.S. Follansbee and G.S. Daehn (Eds.), Modeling the Deformation of Crystalline Solids, The Minerals & Materials Society, 1991, pp. 189-203.

[3] T. Kuwabara, Y. Morita, Y. Miyashita, S. Takahashi, Elastic-Plastic Behavior of Sheet Metal Subjected to In-Plane Reverse Loading, in: Proc. Plasticity '95, The fifth international symposium
compression tests as $\alpha_{uni}$ and that measured using an equibiaxial tensile test and an SCT as $\alpha_{bs}$, respectively, $\alpha_{uni}$ and $\alpha_{bs}$ are calculated as (see [4])

$$ \alpha_{uni} = \frac{|\sigma^C| - \sigma^T}{2|\sigma^C|\sigma^T} \qquad (4) $$

$$ \alpha_{bs} = \frac{|\sigma_z^C| - \sigma_b}{3|\sigma_z^C|\sigma_b} \qquad (5) $$

Considering that $\sigma^C \approx \sigma_z^C$ and $\sigma^T \approx \sigma_b$, we obtain the following equation:

$$ \frac{\alpha_{bs}}{\alpha_{uni}} \approx \frac{|\sigma_z^C| - \sigma_b}{|\sigma^C| - \sigma^T} \times \frac{2}{3} \qquad (6) $$

If $\alpha$ is a material constant, $\alpha_{bs}/\alpha_{uni}$ should be unity. It means that $(|\sigma_z^C| - \sigma_b)/(|\sigma^C| - \sigma^T)$ should be 3/2. However, from Fig. 5, it was measured that $(|\sigma_z^C| - \sigma_b)/(|\sigma^C| - \sigma^T) \approx 1$ except for the early stage of plastic deformation, $w^P < 0.2$ MPa. This suggests that there may be other influencing factors besides the hydrostatic pressure effect as a cause of the SDE in this material. Sun et al. [13] investigated the tension-compression (TC) asymmetry of a 6061 aluminum alloy using a strain gradient crystal plasticity model implemented with the finite element method. They concluded that the TC asymmetry for the aluminum alloys is caused by a combination of difference in the GND density and rotation of crystal matrix and particles, which is induced by plasticity even at the yield strain with limited plasticity. However, it is noted that they found that tensile flow stress was higher than compressive one. Therefore, the origin of the SDE in 6000-series aluminum alloys requires further study.

## Conclusions

A uniaxial tensile test and an in-plane compression test were performed on a 6000-series aluminum alloy sheet (A6116-T4) to measure the SDE. In addition, an equibiaxial tensile test using a cruciform specimen and uniaxial compression tests in the ND using stacked specimens were performed to verify the hydrostatic stress dependence of the flow stress. The experimental findings obtained in this study are summarized follows.

(1) The SDE was observed for our test sample ($0.015 < \beta_{SDE-uni} < 0.053$ for the RD and $0.036 < \beta_{SDE-uni} < 0.066$ for the TD).

(2) The compressive stress, $|\sigma_z^C|$, measured in the uniaxial compression test in the ND exceeded the equibiaxial tensile stress, $\sigma_b$, measured using a cruciform specimen. Therefore, the flow stress of the test sample could be affected by hydrostatic stress.

(3) There may be other influencing factors besides the hydrostatic pressure effect as a cause of the SDE in this material, as suggested by Sun et al. [13].

## References

[1] W.A. Spitzig, O. Richmond, The effect of pressure on the flow stress of metals, Acta Metall 32 (1984) 457-463. https://doi.org/10.1016/0001-6160(84)90119-6

[2] F. Barlat, D.J. Lege, J.C. Brem, and C.J. Warren, Constitutive behavior for anisotropic materials and application to a 2090 Al-Li alloy, in: T.C. Lowe, A.D. Rollet, P.S. Follansbee and G.S. Daehn (Eds.), Modeling the Deformation of Crystalline Solids, The Minerals & Materials Society, 1991, pp. 189-203.

[3] T. Kuwabara, Y. Morita, Y. Miyashita, S. Takahashi, Elastic-Plastic Behavior of Sheet Metal Subjected to In-Plane Reverse Loading, in: Proc. Plasticity '95, The fifth international symposium
on plasticity and its current applications, ed. S Tanimura S, Khan AS, (1995) pp.841-844, Gordon and Breach Publishers.

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

[5] J.K. Holmen, B.H. Frodal, O.S. Hopperstad, T. Børvik, Strength differential effect in age hardened aluminum alloys, Int. J. Plasticity 99 (2017) 144-161. https://doi.org/10.1016/j.ijplas.2017.09.004

[6] A.Y. Ku, A.S. Khan, T. Gnaupel-Herold, Quasi-static and dynamic response, and texture evolution of two overaged Al 7056 alloy plates in T761 and T721 tempers: Experiments and modeling, Int. J. Plasticity 130 (2020) 102679. https://doi.org/10.1016/j.ijplas.2020.102679

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

[8] T. Kuwabara, Y. Kumano, J. Ziegelheim, I. Kurosaki, Tension-compression asymmetry of phosphor bronze for electronic parts and its effect on bending behavior, Int. J. Plasticity 25 (2009) 1759-1776. https://doi.org/10.1016/j.ijplas.2009.01.004

[9] N. Noma, T. Kuwabara, Specimen Geometry Optimization for In-plane Reverse Loading Test of Sheet Metal and Experimental Validation, J. Jpn. Soc. Technol. Plast. 53 (617) (2012) 574-579.

[10] N. Noma, K. Hashimoto, T. Maeda, T. Kuwabara, Enhancement of Accuracy of Springback Simulation Using the Material Model Considering the Strength Differential Effect, J. JSTP 61(708) (2020) 20-25 (in Japanese).

[11] T. Maeda, N. Noma, T. Kuwabara, F. Barlat, Y.P. Korkolis, Measurement of the strength differential effect of DP980 steel sheet and experimental validation using pure bending test, J. Mater. Process. Technol. 256 (2018) 247-253. https://doi.org/10.1016/j.jmatprotec.2018.02.009

[12] T. Koizumi, M. Kuroda, Evaluation of tension-compression asymmetry of a low-carbon steel sheet using a modified classical compression test method, IOP Conf. Series, J. Phys. (2018) 1063. https://doi.org/10.1088/1742-6596/1063/1/012167

[13] F. Sun, S. Wang, Q. Xie, Role of particles and lattice rotation in tension-compression asymmetry of aluminium alloys, Int. J. Plasticity 159 (2022) 103464. https://doi.org/10.1016/j.ijplas.2022.103464
on plasticity and its current applications, ed. S Tanimura S, Khan AS, (1995) pp.841-844, Gordon and Breach Publishers.

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

[5] J.K. Holmen, B.H. Frodal, O.S. Hopperstad, T. Børvik, Strength differential effect in age hardened aluminum alloys, Int. J. Plasticity 99 (2017) 144-161. https://doi.org/10.1016/j.ijplas.2017.09.004

[6] A.Y. Ku, A.S. Khan, T. Gnaupel-Herold, Quasi-static and dynamic response, and texture evolution of two overaged Al 7056 alloy plates in T761 and T721 tempers: Experiments and modeling, Int. J. Plasticity 130 (2020) 102679. https://doi.org/10.1016/j.ijplas.2020.102679

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

[8] T. Kuwabara, Y. Kumano, J. Ziegelheim, I. Kurosaki, Tension-compression asymmetry of phosphor bronze for electronic parts and its effect on bending behavior, Int. J. Plasticity 25 (2009) 1759-1776. https://doi.org/10.1016/j.ijplas.2009.01.004

[9] N. Noma, T. Kuwabara, Specimen Geometry Optimization for In-plane Reverse Loading Test of Sheet Metal and Experimental Validation, J. Jpn. Soc. Technol. Plast. 53 (617) (2012) 574-579.

[10] N. Noma, K. Hashimoto, T. Maeda, T. Kuwabara, Enhancement of Accuracy of Springback Simulation Using the Material Model Considering the Strength Differential Effect, J. JSTP 61(708) (2020) 20-25 (in Japanese).

[11] T. Maeda, N. Noma, T. Kuwabara, F. Barlat, Y.P. Korkolis, Measurement of the strength differential effect of DP980 steel sheet and experimental validation using pure bending test, J. Mater. Process. Technol. 256 (2018) 247-253. https://doi.org/10.1016/j.jmatprotec.2018.02.009

[12] T. Koizumi, M. Kuroda, Evaluation of tension-compression asymmetry of a low-carbon steel sheet using a modified classical compression test method, IOP Conf. Series, J. Phys. (2018) 1063. https://doi.org/10.1088/1742-6596/1063/1/012167

[13] F. Sun, S. Wang, Q. Xie, Role of particles and lattice rotation in tension-compression asymmetry of aluminium alloys, Int. J. Plasticity 159 (2022) 103464. https://doi.org/10.1016/j.ijplas.2022.103464
# Residual formability analysis of bent and remanufactured thin steel sheets

FARIOLI Daniele¹,ª*, KAYA Ertuğrul¹,ᵇ, STRANO Matteo¹,ᶜ

¹Politecnico di Milano, Department of Mechanical Engineering, Via La Masa 1, 20156 Milano

ªdaniele.farioli@polimi.it, ᵇertugrul.kaya@polimi.it, ᶜmatteo.strano@polimi.it

**Keywords:** Residual Formability, Automotive Steel, Remanufacturing, V-Bending, Flattening

**Abstract.** This study examines the plastic deformation and remaining formability of DC04 mild steel, a material frequently used in automotive applications, after undergoing V-bending and subsequent cold or warm flattening processes. The research aims to determine if previously bent sheets maintain sufficient ductility for an additional forming cycle, a key factor in advancing circular economy practices. Samples were cold-formed into V-bends of various angles and then reshaped through flattening, with critical process parameters controlled. Tensile tests were performed on both the original sheets and reshaped samples to study the impact of the bending index, dwell force, dwell time, and flattening temperature on mechanical properties. Geometric characterization of the samples before and after flattening showed localized permanent deformation and an “M” profile near the bend line, leading to a spring-forward effect and a “U” shape of the samples. Analysis of tensile stress-strain curves and regression models indicated a decrease in elongation at maximum stress with an increasing bending index, while the ultimate tensile strength remained relatively constant. Yield stress showed a minor increase with a higher bending index and flattening temperature. These results underscore the potential for reusing end-of-life vehicle sheets, providing a more sustainable alternative to conventional recycling methods.

**Introduction.**

Formability is the ability of a given metal workpiece to undergo plastic deformation without failure [1]. According to Small et al., formability can be defined as a probability of failure, proposing the concept of “forming maps”, which showed how failure risk increases with major strain [2]. Automotive industries are interested in the formability of sheets, especially in the context of multistage stamping [3]. Ensuring successful forming, sufficient post-process residual ductility, and enhancing the strength of lightweight car bodies brought the development of special alloys such as aluminum alloys and third-generation advanced high-strength steels [4]. Beyond the total tensile elongation, factors such as presswork hardening, anisotropy ratio, strain rate sensitivity, uniform elongation, edge sensitivity, and internal stress between phases play critical roles in determining the formability of complicated stress and strain distributions [4]. One practical issue is that high-strength sheet materials are often sensible to edge cracks. Because of this, many edge crack testing methods were developed [5]. Despite many new concepts of formability prediction, strain-based Forming Limit Diagrams (FLD) are used most often in engineering practice to assess sheet formability. FLDs can be determined by different methods, including experimental, theoretical, as well as hybrid methods [6]. Theoretical methods are based on criteria of the loss of stability (strain localization) or damage (fracture) of the material. In general, the most appreciated and reliable methods used are Erichsen, Marciniak, and Nakazima tests [6].

Such tests are based on Digital Image Correlation (DIC) performed with automatic strain measurement systems (AutoGrid, ASAME, or ARAMIS) and inverse parameter identification [6]. Other methods are based on stress measurement using X-ray diffraction during the Marciniak punch test [7]. The strain path dependency, a drawback of FLCs, led to the proposal of the

<figure><img src="image_1064.png" /></figure>
# Residual formability analysis of bent and remanufactured thin steel sheets

FARIOLI Daniele¹,ª*, KAYA Ertuğrul¹,ᵇ, STRANO Matteo¹,ᶜ

¹Politecnico di Milano, Department of Mechanical Engineering, Via La Masa 1, 20156 Milano

ªdaniele.farioli@polimi.it, ᵇertugrul.kaya@polimi.it, ᶜmatteo.strano@polimi.it

**Keywords:** Residual Formability, Automotive Steel, Remanufacturing, V-Bending, Flattening

**Abstract.** This study examines the plastic deformation and remaining formability of DC04 mild steel, a material frequently used in automotive applications, after undergoing V-bending and subsequent cold or warm flattening processes. The research aims to determine if previously bent sheets maintain sufficient ductility for an additional forming cycle, a key factor in advancing circular economy practices. Samples were cold-formed into V-bends of various angles and then reshaped through flattening, with critical process parameters controlled. Tensile tests were performed on both the original sheets and reshaped samples to study the impact of the bending index, dwell force, dwell time, and flattening temperature on mechanical properties. Geometric characterization of the samples before and after flattening showed localized permanent deformation and an “M” profile near the bend line, leading to a spring-forward effect and a “U” shape of the samples. Analysis of tensile stress-strain curves and regression models indicated a decrease in elongation at maximum stress with an increasing bending index, while the ultimate tensile strength remained relatively constant. Yield stress showed a minor increase with a higher bending index and flattening temperature. These results underscore the potential for reusing end-of-life vehicle sheets, providing a more sustainable alternative to conventional recycling methods.

**Introduction.**

Formability is the ability of a given metal workpiece to undergo plastic deformation without failure [1]. According to Small et al., formability can be defined as a probability of failure, proposing the concept of “forming maps”, which showed how failure risk increases with major strain [2]. Automotive industries are interested in the formability of sheets, especially in the context of multistage stamping [3]. Ensuring successful forming, sufficient post-process residual ductility, and enhancing the strength of lightweight car bodies brought the development of special alloys such as aluminum alloys and third-generation advanced high-strength steels [4]. Beyond the total tensile elongation, factors such as presswork hardening, anisotropy ratio, strain rate sensitivity, uniform elongation, edge sensitivity, and internal stress between phases play critical roles in determining the formability of complicated stress and strain distributions [4]. One practical issue is that high-strength sheet materials are often sensible to edge cracks. Because of this, many edge crack testing methods were developed [5]. Despite many new concepts of formability prediction, strain-based Forming Limit Diagrams (FLD) are used most often in engineering practice to assess sheet formability. FLDs can be determined by different methods, including experimental, theoretical, as well as hybrid methods [6]. Theoretical methods are based on criteria of the loss of stability (strain localization) or damage (fracture) of the material. In general, the most appreciated and reliable methods used are Erichsen, Marciniak, and Nakazima tests [6].

Such tests are based on Digital Image Correlation (DIC) performed with automatic strain measurement systems (AutoGrid, ASAME, or ARAMIS) and inverse parameter identification [6]. Other methods are based on stress measurement using X-ray diffraction during the Marciniak punch test [7]. The strain path dependency, a drawback of FLCs, led to the proposal of the

<figure><img src="image_1066.png" /></figure>
Generalized Forming Limit concept (GFLC), allowing for easy evaluation of formability in multi-step deformation histories. Stoughton and Yoon proposed a new failure prediction criterion by combining σFLC and the maximum shear stress (MSS) criterion, introducing the "polar diagram of the effective plastic strain" (PEPS) concept, which is path-independent and successfully implemented in commercial Finite Element (FE) codes [7].

Moreover, for most sheet materials under these complex loading processes, a strong Bauschinger effect is observed, where the material hardening behavior tends to be anisotropic [8]. The phenomenon stands in the formation and interaction of dislocation structures on the micro-structural level which can be in turn modeled by evolving structural tensors [9]. Anisotropic hardening demonstrated to predict better the spring-back of parts with respect to isotropic hardening [7]. The experimental validation for anisotropic hardening consists of multiaxial experiments, shear test and plane strain tension test, hydraulic bulge test, biaxial stress test using cruciform or tubular specimen, combined use of simple mechanical tests, in-plane compression and reverse loading tests and hole expansion forming [7].

## Motivation of the research.

Most of the scientific literature presented focuses on the formability of sheets starting from the “as-received” state. At the same time, cold recycling and remanufacturing of sheet metals is an emerging area mainly because of environmental concerns: despite the proven benefits of recycling, sheet metal remelting is still a practice affected by relevant CO₂ emissions. If these “wastes” are recycled, reused, remanufactured, or repurposed by a cold process, i.e., a non-melting process, the amount of CO₂ emission can be reduced significantly [10], [11]. Employing cold recycling also allows for substantial energy savings compared to manufacturing from new materials [12]. Because of this, reusing and reshaping end-of-life sheets or pre-formed sheets by cold metal recycling routines gained significant interest over the past 15 years, with gradual progress being made by several Scientists. Furthermore, it has been demonstrated that regions of minimally deformed panels still retain significant residual life, enabling the extraction of sheets with retained ductility for subsequent remanufacturing processes under a Circular Economy perspective [11]. To deal with the issue of quantifying the formability of remanufactured sheets, Falsafi et al. proposed a reformability index. This index represents possible levels of strains for deformation along different paths, based on Polar Effective Plastic Strain (PEPS) [13]. Takano et al. studied the feasibility of cold recycling of sheet-metal wastes, finding that a forming limit of flattened sheet metals is similar to that of a sheet metal with uniform thickness [14]. Tekkaya et al. analyzed the remanufacturing of a contoured sheet metal part using a hydro-forming technology concluding that this technique applies to a re-use of sheet metal formed parts such as car bonnets [15]. Farioli et al. [11] demonstrated how to reshape successfully curved thin steel sheets into flat parts by cold and warm flattening. The process described in the paper is meant to exploit the residual material formability of automotive sheets under a circular economy perspective.

Following an approach similar to [11], this study investigates the plastic deformation and residual formability of DC04 mild steel, a prevalent material in automotive applications, subjected to V-bending followed by either cold or warm flattening processes. Examining the impact of V-bending on rectangular specimens is meant to replicate at a “small scale” specific geometrical feature typical of automotive sheet metals. Understanding the residual life and formability of these materials is crucial for circular economy applications, where materials are employed for a second life cycle and where sheet metal formability can be affected by the temperature of the dies and geometrical conditions [16]. Therefore, this article aims to test experimentally the residual formability of previously bent sheets, i.e. to assess whether they retain adequate ductility for another forming cycle.
Generalized Forming Limit concept (GFLC), allowing for easy evaluation of formability in multi-step deformation histories. Stoughton and Yoon proposed a new failure prediction criterion by combining σFLC and the maximum shear stress (MSS) criterion, introducing the "polar diagram of the effective plastic strain" (PEPS) concept, which is path-independent and successfully implemented in commercial Finite Element (FE) codes [7].

Moreover, for most sheet materials under these complex loading processes, a strong Bauschinger effect is observed, where the material hardening behavior tends to be anisotropic [8]. The phenomenon stands in the formation and interaction of dislocation structures on the micro-structural level which can be in turn modeled by evolving structural tensors [9]. Anisotropic hardening demonstrated to predict better the spring-back of parts with respect to isotropic hardening [7]. The experimental validation for anisotropic hardening consists of multiaxial experiments, shear test and plane strain tension test, hydraulic bulge test, biaxial stress test using cruciform or tubular specimen, combined use of simple mechanical tests, in-plane compression and reverse loading tests and hole expansion forming [7].

## Motivation of the research.

Most of the scientific literature presented focuses on the formability of sheets starting from the “as-received” state. At the same time, cold recycling and remanufacturing of sheet metals is an emerging area mainly because of environmental concerns: despite the proven benefits of recycling, sheet metal remelting is still a practice affected by relevant CO₂ emissions. If these “wastes” are recycled, reused, remanufactured, or repurposed by a cold process, i.e., a non-melting process, the amount of CO₂ emission can be reduced significantly [10], [11]. Employing cold recycling also allows for substantial energy savings compared to manufacturing from new materials [12]. Because of this, reusing and reshaping end-of-life sheets or pre-formed sheets by cold metal recycling routines gained significant interest over the past 15 years, with gradual progress being made by several Scientists. Furthermore, it has been demonstrated that regions of minimally deformed panels still retain significant residual life, enabling the extraction of sheets with retained ductility for subsequent remanufacturing processes under a Circular Economy perspective [11]. To deal with the issue of quantifying the formability of remanufactured sheets, Falsafi et al. proposed a reformability index. This index represents possible levels of strains for deformation along different paths, based on Polar Effective Plastic Strain (PEPS) [13]. Takano et al. studied the feasibility of cold recycling of sheet-metal wastes, finding that a forming limit of flattened sheet metals is similar to that of a sheet metal with uniform thickness [14]. Tekkaya et al. analyzed the remanufacturing of a contoured sheet metal part using a hydro-forming technology concluding that this technique applies to a re-use of sheet metal formed parts such as car bonnets [15]. Farioli et al. [11] demonstrated how to reshape successfully curved thin steel sheets into flat parts by cold and warm flattening. The process described in the paper is meant to exploit the residual material formability of automotive sheets under a circular economy perspective.

Following an approach similar to [11], this study investigates the plastic deformation and residual formability of DC04 mild steel, a prevalent material in automotive applications, subjected to V-bending followed by either cold or warm flattening processes. Examining the impact of V-bending on rectangular specimens is meant to replicate at a “small scale” specific geometrical feature typical of automotive sheet metals. Understanding the residual life and formability of these materials is crucial for circular economy applications, where materials are employed for a second life cycle and where sheet metal formability can be affected by the temperature of the dies and geometrical conditions [16]. Therefore, this article aims to test experimentally the residual formability of previously bent sheets, i.e. to assess whether they retain adequate ductility for another forming cycle.
Experimental procedure and materials.

Experiments were conducted to understand the potential of reshaping and giving sheet metal components a second life by remanufacturing. Specifically, samples of tensile test shapes (Figure 2) were extracted from 0.8 mm thick cold-rolled DC04 steel using waterjet cutting.

<figure><img src="image_1065.png" /><figcaption>Figure 2. Geometry of the tensile test samples (mm).</figcaption></figure>

<figure><img src="image_1066.png" /><figcaption>Figure 1. V-bending of the tensile test samples.</figcaption></figure>

<figure><img src="image_1067.png" /><figcaption>Figure 3. Flattening bent samples for obtaining a reshaped part.</figcaption></figure>

These samples were cold-formed to produce V-bends of varying angles by varying the punch stroke (Figure 1). The punch performs an air-bending operation without bottoming at a controlled and constant rate equal to 3 mm/min. two stroke values, or “Bending Indexes” BI (mm) were tested in these experiments. The flattening process was then employed as a manufacturing method to reshape these parts for potential reuse. The flattening process is characterized by three main parameters. The first parameter is the dwell force DwF (kN): it is the maximum constant flattening force applied. While the upper flattening tool goes down at a constant and controlled rate (50 mm/min), the V-bent sample initially unbends and when it starts getting flat the force increases steeply. When the force reaches a predefined value, namely the dwell force, such load is kept constant for a set time, called dwell time. The second parameter is the dwell time τ (s): it is the time for which the dwell force is applied. The third parameter is the temperature T (°C): it is the temperature of the tools and of the chamber where flattening occurs. Such value is kept constant and controlled. Both V-bending and flattening have been conducted onto the MTS servo-hydraulic monoaxial testing machine: flat hard tool steel dies have been used for flattening and specific punch and matrix for V-bending operation (Figure 5). Flattening tests have been performed both in cold (20°C) and warm (250°C) conditions, by using a closed chamber with controlled and fixed temperature (Figure 4).

<figure><img src="image_1068.png" /><figcaption>Figure 4. The picture at the beginning of flattening one sample in a controlled-temperature chamber.</figcaption></figure>

<figure><img src="image_1069.png" /><figcaption>Figure 5. 2D drawing of the V-bending tools (mm).</figcaption></figure>
Experimental procedure and materials.

Experiments were conducted to understand the potential of reshaping and giving sheet metal components a second life by remanufacturing. Specifically, samples of tensile test shapes (Figure 2) were extracted from 0.8 mm thick cold-rolled DC04 steel using waterjet cutting.

<figure><img src="image_1067.png" /><figcaption>Figure 2. Geometry of the tensile test samples (mm).</figcaption></figure>

<figure><img src="image_1068.png" /><figcaption>Figure 1. V-bending of the tensile test samples.</figcaption></figure>

<figure><img src="image_1069.png" /><figcaption>Figure 3. Flattening bent samples for obtaining a reshaped part.</figcaption></figure>

These samples were cold-formed to produce V-bends of varying angles by varying the punch stroke (Figure 1). The punch performs an air-bending operation without bottoming at a controlled and constant rate equal to 3 mm/min. two stroke values, or “Bending Indexes” BI (mm) were tested in these experiments. The flattening process was then employed as a manufacturing method to reshape these parts for potential reuse. The flattening process is characterized by three main parameters. The first parameter is the dwell force DwF (kN): it is the maximum constant flattening force applied. While the upper flattening tool goes down at a constant and controlled rate (50 mm/min), the V-bent sample initially unbends and when it starts getting flat the force increases steeply. When the force reaches a predefined value, namely the dwell force, such load is kept constant for a set time, called dwell time. The second parameter is the dwell time τ (s): it is the time for which the dwell force is applied. The third parameter is the temperature T (°C): it is the temperature of the tools and of the chamber where flattening occurs. Such value is kept constant and controlled. Both V-bending and flattening have been conducted onto the MTS servo-hydraulic monoaxial testing machine: flat hard tool steel dies have been used for flattening and specific punch and matrix for V-bending operation (Figure 5). Flattening tests have been performed both in cold (20°C) and warm (250°C) conditions, by using a closed chamber with controlled and fixed temperature (Figure 4).

<figure><img src="image_1070.png" /><figcaption>Figure 4. The picture at the beginning of flattening one sample in a controlled-temperature chamber.</figcaption></figure>

<figure><img src="image_1071.png" /><figcaption>Figure 5. 2D drawing of the V-bending tools (mm).</figcaption></figure>
Tensile tests have been performed onto MTS machine Alliance RT/100 as described in Figure 6 and Figure 7. The elongation rate applied is $\dot{\Delta}l = 5$ mm/min.

<figure><img src="image_1070.png" /><figcaption>Figure 6. Experimental setup for tensile tests.</figcaption></figure>

<figure><img src="image_1071.png" /><figcaption>Figure 7. Zoomed view of the sample with an extensometer.</figcaption></figure>

Such tests have been performed both on the as-received sheets and the V-bent and flattened samples. This procedure allows for characterizing the work-hardening and the residual tensile formability of re-shaped parts when compared to the as-received material.

## Design of experiments.

This paragraph is meant to explain the parameters used during the experimental procedure (V-bending + flattening) and the sequential stages of deformations experienced by the samples (Figure 8). The factors varied during the experiments: the Bending Index (BI), the Dwell Force (DwF), the Dwell time ($\tau$) and the Flattening Temperature (T). For each factor, two levels have been tested. For each combination of levels and factors, one replica was tested for the V-bent and flattened samples. Right after the flattening process, the samples were tested in tension to get the stress-strain curve. The as-received material has been tested in tension without performing any preliminary deformation and re-shaping. In total, 19 samples have been tested according to Figure 9: 16 samples for characterizing the effect of bending + flattening on the work hardening of the material and 3 samples for characterizing the as-received one. The response variables of interest for the tensile tests were the Ultimate Tensile Strength (UTS, [MPa]), the yielding stress (Ys, [MPa]), the Young's modulus (E, [GPa]) and the elongation at maximum stress (A%, [-]). Such response variables, together with the $\sigma - \epsilon$ diagram, can be extracted directly by the Software which controls the tensile test machine, without the need for any post-processing operation on the experimental data.

<figure><img src="image_1072.png" /><figcaption>Figure 8. Deformation stages experienced by the material tested.</figcaption></figure>

<table><caption>Figure 9. Design of Experiments followed.</caption><thead><tr><th>Sample type</th><th>BI [mm]</th><th>τ [s]</th><th>DwF [kN]</th><th>T [°C]</th><th># Replicas</th></tr></thead><tbody><tr><td>Bent and flattened</td><td>3 - 6.5</td><td>2.60</td><td>6.75-13.5</td><td>20-250</td><td>1</td></tr><tr><td>As-received</td><td>-</td><td>-</td><td>-</td><td>-</td><td>3</td></tr></tbody></table>

## Geometrical characterization of the samples.

The V-bending operation induces in the material different levels of plastic deformation according to the stroke applied by the punch. Consequently, in correspondence with the bent zones, different curvatures for the samples are obtained, therefore plastic strain distributions are obtained. To
Tensile tests have been performed onto MTS machine Alliance RT/100 as described in Figure 6 and Figure 7. The elongation rate applied is $\dot{\Delta}l = 5$ mm/min.

<figure><img src="image_1072.png" /><figcaption>Figure 6. Experimental setup for tensile tests.</figcaption></figure>

<figure><img src="image_1073.png" /><figcaption>Figure 7. Zoomed view of the sample with an extensometer.</figcaption></figure>

Such tests have been performed both on the as-received sheets and the V-bent and flattened samples. This procedure allows for characterizing the work-hardening and the residual tensile formability of re-shaped parts when compared to the as-received material.

## Design of experiments.

This paragraph is meant to explain the parameters used during the experimental procedure (V-bending + flattening) and the sequential stages of deformations experienced by the samples (Figure 8). The factors varied during the experiments: the Bending Index (BI), the Dwell Force (DwF), the Dwell time ($\tau$) and the Flattening Temperature (T). For each factor, two levels have been tested. For each combination of levels and factors, one replica was tested for the V-bent and flattened samples. Right after the flattening process, the samples were tested in tension to get the stress-strain curve. The as-received material has been tested in tension without performing any preliminary deformation and re-shaping. In total, 19 samples have been tested according to Figure 9: 16 samples for characterizing the effect of bending + flattening on the work hardening of the material and 3 samples for characterizing the as-received one. The response variables of interest for the tensile tests were the Ultimate Tensile Strength (UTS, [MPa]), the yielding stress (Ys, [MPa]), the Young's modulus (E, [GPa]) and the elongation at maximum stress (A%, [-]). Such response variables, together with the $\sigma - \epsilon$ diagram, can be extracted directly by the Software which controls the tensile test machine, without the need for any post-processing operation on the experimental data.

<figure><img src="image_1074.png" /><figcaption>Figure 8. Deformation stages experienced by the material tested.</figcaption></figure>

<table><caption>Figure 9. Design of Experiments followed.</caption><thead><tr><th>Sample type</th><th>BI [mm]</th><th>τ [s]</th><th>DwF [kN]</th><th>T [°C]</th><th># Replicas</th></tr></thead><tbody><tr><td>Bent and flattened</td><td>3 - 6.5</td><td>2.60</td><td>6.75-13.5</td><td>20-250</td><td>1</td></tr><tr><td>As-received</td><td>-</td><td>-</td><td>-</td><td>-</td><td>3</td></tr></tbody></table>

## Geometrical characterization of the samples.

The V-bending operation induces in the material different levels of plastic deformation according to the stroke applied by the punch. Consequently, in correspondence with the bent zones, different curvatures for the samples are obtained, therefore plastic strain distributions are obtained. To
measure such curvatures and to understand the way the specimen deforms during the forming and re-forming cycle, the samples have been characterized with Alicona Infinite Focus, as described in Figure 10. Alicona is a commercial tool that can be used for surface topography measurement. The output of the measurement is an .STL file, which can be directly analyzed by the Software developed by Alicona Brucker (Figure 11). In this specific case, the measurements were performed in two moments. Firstly, after bending, to understand the curvature and angle of the specimen. This is an indirect measure of the damage experienced by the material because of plastic deformation. The second measurement were performed after flattening, to understand the mechanics of deformation of the specimens.

<figure><img src="image_1073.png" /><figcaption>Figure 10. Positioning the sample for the geometrical analysis.</figcaption></figure>

<figure><img src="image_1074.png" /><figcaption>Figure 11. Interpolation of the external curvature of the sample (in .STL format) directly through the Alicona Brucker Software.</figcaption></figure>

As mentioned previously, after scanning all the bent samples, two geometrical features have been measured: the external curvature radius (Figure 12) and the angle of the bent samples (Figure 13).

<figure><img src="image_1075.png" /><figcaption>Figure 12. The geometrical relationship between external curvature radius and bending stroke.</figcaption></figure>

<figure><img src="image_1076.png" /><figcaption>Figure 13. The geometrical relationship between bending angle and bending stroke.</figcaption></figure>

Clearly, Figure 12 shows how the external curvature radius of the sample decreases (i.e. the bent line gets sharper) by increasing the stroke in the bending phase. This also denotes an increased level of plastic deformation in the material. Similarly, Figure 13 reveals that the angle of the samples after spring-back reduces by increasing the bending stroke. The trends in the two figures are analogous, implying that a correlation between angle and curvature radius exists. Lastly, Alicona has been used to analyze the bent (and unbent) line post-flattening with two aims: firstly, to deduce the deformation history of the sample during flattening and understanding how flattening modifies the shape of the sample by applying a mix of un-bending and localized compression (Figure 15). Secondly, Alicona was used to quantify the spring-back of the samples. The samples
measure such curvatures and to understand the way the specimen deforms during the forming and re-forming cycle, the samples have been characterized with Alicona Infinite Focus, as described in Figure 10. Alicona is a commercial tool that can be used for surface topography measurement. The output of the measurement is an .STL file, which can be directly analyzed by the Software developed by Alicona Brucker (Figure 11). In this specific case, the measurements were performed in two moments. Firstly, after bending, to understand the curvature and angle of the specimen. This is an indirect measure of the damage experienced by the material because of plastic deformation. The second measurement were performed after flattening, to understand the mechanics of deformation of the specimens.

<figure><img src="image_1075.png" /><figcaption>Figure 10. Positioning the sample for the geometrical analysis.</figcaption></figure>

<figure><img src="image_1076.png" /><figcaption>Figure 11. Interpolation of the external curvature of the sample (in .STL format) directly through the Alicona Brucker Software.</figcaption></figure>

As mentioned previously, after scanning all the bent samples, two geometrical features have been measured: the external curvature radius (Figure 12) and the angle of the bent samples (Figure 13).

<figure><img src="image_1077.png" /><figcaption>Figure 12. The geometrical relationship between external curvature radius and bending stroke.</figcaption></figure>

<figure><img src="image_1078.png" /><figcaption>Figure 13. The geometrical relationship between bending angle and bending stroke.</figcaption></figure>

Clearly, Figure 12 shows how the external curvature radius of the sample decreases (i.e. the bent line gets sharper) by increasing the stroke in the bending phase. This also denotes an increased level of plastic deformation in the material. Similarly, Figure 13 reveals that the angle of the samples after spring-back reduces by increasing the bending stroke. The trends in the two figures are analogous, implying that a correlation between angle and curvature radius exists. Lastly, Alicona has been used to analyze the bent (and unbent) line post-flattening with two aims: firstly, to deduce the deformation history of the sample during flattening and understanding how flattening modifies the shape of the sample by applying a mix of un-bending and localized compression (Figure 15). Secondly, Alicona was used to quantify the spring-back of the samples. The samples
have been positioned (through specific supports) in a way that the sample could be observed laterally, i.e. along the plane of the thickness of the blank (Figure 14).

<figure><img src="image_1077.png" /><figcaption>Figure 14. Positioning the sample for the geometrical analysis with Alicona.</figcaption></figure>

<caption>Figure 15. Sample scanned with Alicona with a zoom close to the flattened central area.</caption>

<caption>Figure 16. Laser-based profilometer used for 3D scanning the samples;</caption>

Figure 15 is remarkable since it is possible to detect how the flattened sample, in correspondence with the bent line, assumes an “M” shape. The central part of the “M” is a consequence of the bending process, where the tip of the punch induces a concavity upwards. Instead, the two sides of the “M” have a concavity pointing downwards. This is because the first stage of flattening is unbending of the central portion, where the lateral flanges do not remain linear but get bended and curved because of the friction between the sample and lower die and because of their bending stiffness (which is in turn related to the sample slenderness). This phenomenon can be appreciated in Figure 17.

<figure><img src="image_1078.png" /><figcaption>Figure 17. a) Beginning of flattening; b) the un-bending stage of the sample (the lateral flange bends); c) the dwelling stage of flattening (the material gets compressed through the thickness); d) spring-forward effect.</figcaption></figure>

It is noticeable from Figure 17-d that a small spring-forward effect exists. This is because during flattening the sample un-bends and the flanges get a concavity opposite to the one of the central bent line. When the punch continues pushing down the sample, the central bent opens while the lateral flanges are pushed down. But, when the punch force is released, elastic spring-back of the lateral flanges in the proximity of the central bent line takes place. This spring-back is oriented with a concavity pointing upwards, while the spring-back of the central bent line is oriented downwards. The sample globally assumes a “U” shape because of the elastic spring-back of the two regions adjacent to the bent line, while locally, the central part of the specimen assumes an
have been positioned (through specific supports) in a way that the sample could be observed laterally, i.e. along the plane of the thickness of the blank (Figure 14).

<figure><img src="image_1079.png" /><figcaption>Figure 14. Positioning the sample for the geometrical analysis with Alicona.</figcaption></figure>

<caption>Figure 15. Sample scanned with Alicona with a zoom close to the flattened central area.</caption>

<caption>Figure 16. Laser-based profilometer used for 3D scanning the samples;</caption>

Figure 15 is remarkable since it is possible to detect how the flattened sample, in correspondence with the bent line, assumes an “M” shape. The central part of the “M” is a consequence of the bending process, where the tip of the punch induces a concavity upwards. Instead, the two sides of the “M” have a concavity pointing downwards. This is because the first stage of flattening is unbending of the central portion, where the lateral flanges do not remain linear but get bended and curved because of the friction between the sample and lower die and because of their bending stiffness (which is in turn related to the sample slenderness). This phenomenon can be appreciated in Figure 17.

<figure><img src="image_1080.png" /><figcaption>Figure 17. a) Beginning of flattening; b) the un-bending stage of the sample (the lateral flange bends); c) the dwelling stage of flattening (the material gets compressed through the thickness); d) spring-forward effect.</figcaption></figure>

It is noticeable from Figure 17-d that a small spring-forward effect exists. This is because during flattening the sample un-bends and the flanges get a concavity opposite to the one of the central bent line. When the punch continues pushing down the sample, the central bent opens while the lateral flanges are pushed down. But, when the punch force is released, elastic spring-back of the lateral flanges in the proximity of the central bent line takes place. This spring-back is oriented with a concavity pointing upwards, while the spring-back of the central bent line is oriented downwards. The sample globally assumes a “U” shape because of the elastic spring-back of the two regions adjacent to the bent line, while locally, the central part of the specimen assumes an
“M” shape. The spring-forward effect can be quantified quickly through profilometers based on laser sensor technology (Figure 16). The sensor used is a Wenglor MLWI 131 laser profilometer, with a vertical resolution equal to 4.9 µm and the width of the laser is 52 mm. The sensor is actuated by an electric motor and can translate along the Y axis (see Figure 16) while scanning the samples. Different from Alicona, this system is more suited for quick 3D scanning where high resolution or high magnifications are not required. One example of the scanned profile is reported in Figure 18, where a slight non-planarity or spring-forward effect can be appreciated, while in Figure 19 it is possible to appreciate the “M” profile of a remanufactured sample with a double change of concavity and linear trends far from the bent line.

<figure><img src="image_1079.png" /><figcaption>Figure 18. Colour map of the sample is visualized from the top view or isometric view.</figcaption></figure>

<figure><img src="image_1080.png" /><figcaption>Figure 19. Remanufactured sample scanned with the laser profilometer. This plot is the mean profile of the scan along the X direction.</figcaption></figure>

Figure 17 and Figure 19 might be considered contradictory: in Figure 17-d the flanges are pointing upwards while in Figure 19 downwards. This is because the samples, when measured with the laser profilometer, were placed with the flanges in contact with the support plate, flipping it by 180° after flattening. During the last stage of flattening, when the punch is applying the dwell force, the material gets compressed through the thickness. This can be comprehended by studying the flattening force vs stroke curve of the flattening stage, reported in Figure 20.

<figure><img src="image_1081.png" /><figcaption>Figure 20. Flattening curve: load/sample width (Y axis) vs stroke of the flattening punch (X-axis).</figcaption></figure>

Figure 20 shows the flattening curves performed in the cold (black lines, 20°C) and warm (red lines, 250°C) processes for two different values of bending index (BI) applied (3 and 6,5 mm). The curves are characterized by two regions: at the beginning the force is almost constant for increasing flattening stroke, corresponding to a limited resisting force exerted by the sample during
"M" shape. The spring-forward effect can be quantified quickly through profilometers based on laser sensor technology (Figure 16). The sensor used is a Wenglor MLWI 131 laser profilometer, with a vertical resolution equal to 4.9 µm and the width of the laser is 52 mm. The sensor is actuated by an electric motor and can translate along the Y axis (see Figure 16) while scanning the samples. Different from Alicona, this system is more suited for quick 3D scanning where high resolution or high magnifications are not required. One example of the scanned profile is reported in Figure 18, where a slight non-planarity or spring-forward effect can be appreciated, while in Figure 19 it is possible to appreciate the "M" profile of a remanufactured sample with a double change of concavity and linear trends far from the bent line.

<figure><img src="image_1081.png" /><figcaption>Figure 18. Colour map of the sample is visualized from the top view or isometric view.</figcaption></figure>

<figure><img src="image_1082.png" /><figcaption>Figure 19. Remanufactured sample scanned with the laser profilometer. This plot is the mean profile of the scan along the X direction.</figcaption></figure>

Figure 17 and Figure 19 might be considered contradictory: in Figure 17-d the flanges are pointing upwards while in Figure 19 downwards. This is because the samples, when measured with the laser profilometer, were placed with the flanges in contact with the support plate, flipping it by 180° after flattening. During the last stage of flattening, when the punch is applying the dwell force, the material gets compressed through the thickness. This can be comprehended by studying the flattening force vs stroke curve of the flattening stage, reported in Figure 20.

<figure><img src="image_1083.png" /><figcaption>Figure 20. Flattening curve: load/sample width (Y axis) vs stroke of the flattening punch (X-axis).</figcaption></figure>

Figure 20 shows the flattening curves performed in the cold (black lines, 20°C) and warm (red lines, 250°C) processes for two different values of bending index (BI) applied (3 and 6,5 mm). The curves are characterized by two regions: at the beginning the force is almost constant for increasing flattening stroke, corresponding to a limited resisting force exerted by the sample during
the first stage of flattening, i.e. unbending. Secondly, once the sample is completely unbent and flattened, the material gets compressed through the thickness and the plot force vs stroke becomes steep, reminding a sort of coining operation. The decision to test two values of dwell force (one lower and one higher) was intended to comprehend weather the pressing tonnage affects spring-back (or, in this case, spring-forward), since similarly to bending the bottoming phase is needed to calibrate accurately the bent angle. Despite this original thought, it was noticed that using flat dies, to cancel the spring-forward effect a very high tonnage would be required to induce plastic deformation through the thickness. Too high tonnage is unpractical for large samples and would have a double effect. On one hand, it allows having a uniform thickness distribution canceling out the “M” profile. On the other hand, the sample would “spread” laterally because of the high pressure, thus inducing thinning. In this research, the values of forces tested were not enough to induce plastic deformation through the thickness capable of canceling the “M” profile.

## Tensile test results.

Following this geometrical explanation of the deformation of the sample, it is clear that the damaged zone of the sample is no longer localized only along the bent line, but also close to it because of the flattening stage. To quantify the effect of the bending and flattening process parameters on the formability of the samples, tensile tests of the remanufactured and reshaped parts were performed. In most cases, the samples failed close to the bent line, indicating that the material underwent damage and strain hardening because of bending, but more importantly, because of the flattening process, where the unbending phase induces such peculiar counter-curvature. This is due to a geometrical effect as explained previously. The tensile engineering stress-strain curves performed are reported in Figure 21. As previously mentioned, from each tensile test, four response variables were extracted: the elongation in correspondence of the max stress (A%), Young's modulus (E), the maximum tensile stress (UTS) and the yielding stress (Ys). To appreciate the effect of the factors tested (BI, DwF, $\tau$, T) on the response variables (A%, E, UTS, and Ys), a scatter plot and a regression model were employed in JMP Software to find possible correlations. For the sake of simplicity, the results of the regression are summarized in Table 1, and relevant scatter plots obtained are reported (namely Figure 23, Figure 24, and Figure 25).

<figure><img src="image_1082.png" /><figcaption>Figure 21. Engineering tensile stress-strain curves for as-received samples (violet) and remanufactured ones (black).</figcaption></figure>

<figure><img src="image_1083.png" /><figcaption>Figure 22. Tensile test of cold-formed and cold-flattened tensile test samples, highlighting the effect of BI.</figcaption></figure>

Filtering Figure 21 by excluding warm-flattened samples and as-received ones, it was possible to obtain Figure 22, where the dashed lines correspond to higher BI values (6.5 mm), i.e. higher punch stroke in the bending stage. The continuous black lines correspond to lower BU values (3 mm). Figure 23 shows how the elongation at maximum stress, which can be at first approximation considered as a formability indicator, decreases for the progressive increase of the bending index. More specifically, the variation in formability can be appreciated in Table 2.
the first stage of flattening, i.e. unbending. Secondly, once the sample is completely unbent and flattened, the material gets compressed through the thickness and the plot force vs stroke becomes steep, reminding a sort of coining operation. The decision to test two values of dwell force (one lower and one higher) was intended to comprehend weather the pressing tonnage affects spring-back (or, in this case, spring-forward), since similarly to bending the bottoming phase is needed to calibrate accurately the bent angle. Despite this original thought, it was noticed that using flat dies, to cancel the spring-forward effect a very high tonnage would be required to induce plastic deformation through the thickness. Too high tonnage is unpractical for large samples and would have a double effect. On one hand, it allows having a uniform thickness distribution canceling out the “M” profile. On the other hand, the sample would “spread” laterally because of the high pressure, thus inducing thinning. In this research, the values of forces tested were not enough to induce plastic deformation through the thickness capable of canceling the “M” profile.

## Tensile test results.

Following this geometrical explanation of the deformation of the sample, it is clear that the damaged zone of the sample is no longer localized only along the bent line, but also close to it because of the flattening stage. To quantify the effect of the bending and flattening process parameters on the formability of the samples, tensile tests of the remanufactured and reshaped parts were performed. In most cases, the samples failed close to the bent line, indicating that the material underwent damage and strain hardening because of bending, but more importantly, because of the flattening process, where the unbending phase induces such peculiar counter-curvature. This is due to a geometrical effect as explained previously. The tensile engineering stress-strain curves performed are reported in Figure 21. As previously mentioned, from each tensile test, four response variables were extracted: the elongation in correspondence of the max stress (A%), Young's modulus (E), the maximum tensile stress (UTS) and the yielding stress (Ys). To appreciate the effect of the factors tested (BI, DwF, $\tau$, T) on the response variables (A%, E, UTS, and Ys), a scatter plot and a regression model were employed in JMP Software to find possible correlations. For the sake of simplicity, the results of the regression are summarized in Table 1, and relevant scatter plots obtained are reported (namely Figure 23, Figure 24, and Figure 25).

<figure><img src="image_1084.png" /><figcaption>Figure 21. Engineering tensile stress-strain curves for as-received samples (violet) and remanufactured ones (black).</figcaption></figure>

<figure><img src="image_1085.png" /><figcaption>Figure 22. Tensile test of cold-formed and cold-flattened tensile test samples, highlighting the effect of BI.</figcaption></figure>

Filtering Figure 21 by excluding warm-flattened samples and as-received ones, it was possible to obtain Figure 22, where the dashed lines correspond to higher BI values (6.5 mm), i.e. higher punch stroke in the bending stage. The continuous black lines correspond to lower BU values (3 mm). Figure 23 shows how the elongation at maximum stress, which can be at first approximation considered as a formability indicator, decreases for the progressive increase of the bending index. More specifically, the variation in formability can be appreciated in Table 2.
<table><caption>Table 1. Table indicating the presence of correlations between factors (green cells) and response variables (grey cells) indicating the type of trend: increasing (↑), decreasing (↓), multiple (↑↓), or absent (-).</caption><thead><tr><th></th><th>A%</th><th>E</th><th>Ys</th><th>UTS</th></tr></thead><tbody><tr><th>BI</th><td>↓</td><td>-</td><td>↑</td><td>-</td></tr><tr><th>T</th><td>↓</td><td>↑</td><td>↑</td><td>↑</td></tr><tr><th>τ</th><td>-</td><td>↑↓</td><td>-</td><td>-</td></tr><tr><th>F</th><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

<table><caption>Table 2. Variation of elongation at maximum stress according to different forming and reshaping cycles.</caption><thead><tr><th>Forming and reshaping cycle</th><th>A<sup>*</sup><sub>mean</sub> [-]</th><th>ΔA<sup>**</sup><sub>mean</sub> (%)</th></tr></thead><tbody><tr><td>BI 3 mm - cold flattened</td><td>0,25</td><td>-4,4</td></tr><tr><td>BI 3 mm - warm flattened</td><td>0,23</td><td>-11,8</td></tr><tr><td>BI 6,5 mm - cold flattened</td><td>0,22</td><td>-13,3</td></tr><tr><td>BI 6,5 mm - warm flattened</td><td>0,19</td><td>-25,7</td></tr></tbody></table>

* Mean elongation at maximum stress.

** Variation in percentage of the mean elongation at maximum stress with respect to the “as-received” samples.

The decrease in A% is caused by the progressive plastic deformation and damage accumulated during forming and flattening. Nevertheless, considering the worst case, for DC04, applying a severe bending and flattening is not decreasing the residual formability in a tremendous amount: the decrement percentage in A% with respect to the “as-received” sample corresponds to values lower (in absolute value) than 26% (Table 2). Therefore, it is reasonable to state that such material, even after severe forming cycles, can be reshaped and reused for a second life where potentially another forming stage can be applied.

<figure><img src="image_1084.png" /><figcaption>Figure 23. Elongation in correspondence of the max stress (A%) vs bending index (BI) for cold and warm flattened samples.</figcaption></figure>

<figure><img src="image_1085.png" /><figcaption>Figure 24. Ultimate tensile stress (UTS) vs bending index (BI) for cold and warm flattened samples.</figcaption></figure>

<figure><img src="image_1086.png" /><figcaption>Figure 25. Yielding stress (Ys) vs bending index (BI) for cold and warm flattened samples.</figcaption></figure>

Figure 24 shows how the UTS is approximately constant for increasing bending index, while slightly increasing for warm formed samples. Even though such behavior seems counter-intuitive, it is in line with other studies where temperature can activate thermal aging phenomena, thus modifying the mechanical response of the material [11]. Some mild-steel grades (including DC04) experience also a slight decrease in elongation at fracture when exposed to a range of temperatures between 200 and 400 °C (as observable in Figure 23). Lastly, in Figure 25 it is possible to appreciate the slight increase of yielding stress (Ys) for increasing BI and flattening temperature. The explanation of this behaviour would require some microstructural analysis of the material.
<table><caption>Table 1. Table indicating the presence of correlations between factors (green cells) and response variables (grey cells) indicating the type of trend: increasing (↑), decreasing (↓), multiple (↑↓), or absent (-).</caption><thead><tr><th></th><th>A%</th><th>E</th><th>Ys</th><th>UTS</th></tr></thead><tbody><tr><th>BI</th><td>↓</td><td>-</td><td>↑</td><td>-</td></tr><tr><th>T</th><td>↓</td><td>↑</td><td>↑</td><td>↑</td></tr><tr><th>τ</th><td>-</td><td>↑↓</td><td>-</td><td>-</td></tr><tr><th>F</th><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>

<table><caption>Table 2. Variation of elongation at maximum stress according to different forming and reshaping cycles.</caption><thead><tr><th>Forming and reshaping cycle</th><th>A<sup>*</sup><sub>mean</sub> [-]</th><th>ΔA<sup>**</sup><sub>mean</sub> (%)</th></tr></thead><tbody><tr><td>BI 3 mm - cold flattened</td><td>0,25</td><td>-4,4</td></tr><tr><td>BI 3 mm - warm flattened</td><td>0,23</td><td>-11,8</td></tr><tr><td>BI 6,5 mm - cold flattened</td><td>0,22</td><td>-13,3</td></tr><tr><td>BI 6,5 mm - warm flattened</td><td>0,19</td><td>-25,7</td></tr></tbody></table>

* Mean elongation at maximum stress.

** Variation in percentage of the mean elongation at maximum stress with respect to the “as-received” samples.

The decrease in A% is caused by the progressive plastic deformation and damage accumulated during forming and flattening. Nevertheless, considering the worst case, for DC04, applying a severe bending and flattening is not decreasing the residual formability in a tremendous amount: the decrement percentage in A% with respect to the “as-received” sample corresponds to values lower (in absolute value) than 26% (Table 2). Therefore, it is reasonable to state that such material, even after severe forming cycles, can be reshaped and reused for a second life where potentially another forming stage can be applied.

<figure><img src="image_1086.png" /><figcaption>Figure 23. Elongation in correspondence of the max stress (A%) vs bending index (BI) for cold and warm flattened samples.</figcaption></figure>

<figure><img src="image_1087.png" /><figcaption>Figure 24. Ultimate tensile stress (UTS) vs bending index (BI) for cold and warm flattened samples.</figcaption></figure>

<figure><img src="image_1088.png" /><figcaption>Figure 25. Yielding stress (Ys) vs bending index (BI) for cold and warm flattened samples.</figcaption></figure>

Figure 24 shows how the UTS is approximately constant for increasing bending index, while slightly increasing for warm formed samples. Even though such behavior seems counter-intuitive, it is in line with other studies where temperature can activate thermal aging phenomena, thus modifying the mechanical response of the material [11]. Some mild-steel grades (including DC04) experience also a slight decrease in elongation at fracture when exposed to a range of temperatures between 200 and 400 °C (as observable in Figure 23). Lastly, in Figure 25 it is possible to appreciate the slight increase of yielding stress (Ys) for increasing BI and flattening temperature. The explanation of this behaviour would require some microstructural analysis of the material.
According to Wang et al. in low-carbon steels subjected to warm forming, dynamic strain aging (DSA) phenomena can lead to an increase of micro-hardness and peak stress [17].

## Conclusions

The findings presented in this paper contributed to assessing and understanding whether previously bent sheets retain adequate ductility for another forming cycle, a crucial aspect for promoting circular economy practices.

Specifically, the objective of this study is to investigate the plastic deformation and residual formability of DC04 mild steel, a prevalent material used in car bodies. The main idea is to comprehend if stamped steels extracted from end-of-life vehicles could be reshaped or remanufactured providing environmental benefits concerning conventional recycling. Similarly to a former study [11], the flattening process was employed as a reshaping or remanufacturing strategy for recovering the original planar geometry of a V-bent.

To simulate the potential implementation on real car bodies, the experimental procedure involved cold forming samples of 0.8 mm thick DC04 steel into V-bends of varying angles using a controlled punch stroke (Figure 1).

Subsequently, a flattening process was employed to reshape these parts for potential reuse, with key parameters such as dwell force, dwell time, and temperature carefully controlled. Tensile tests (Figure 6) were conducted on both as-received sheets and reshaped samples to understand how work-hardening and residual tensile formability depends on Bending Index (BI), Dwell Force (DwF), Dwell Time (τ), and Flattening Temperature (T). Two levels were tested for each factor (Figure 9), while the response variables of interest were Ultimate Tensile Strength (UTS), Yielding Stress (Ys), Young's Modulus (E), and Elongation at Maximum Stress (A%).

The samples were measured with different techniques before (Figure 10) and after flattening (Figure 14 and Figure 16) in order to comprehend the deformation mechanisms of the material in proximity of the bent line (i.e. the damaged region) and in order to understand the origin of the spring-forward effect (Figure 17). Before flattening Alicona Infinite Focus was employed to measure the external curvature radius and angles of the bent samples. The measurements provided an understanding into the deformation history of the material, providing conclusions in line with the state of the art of sheet metal bending.

Observations during flattening instead are more interesting since this process is novel and employed for remanufacturing already shaped parts. During flattening the bent line unbends and the flange deforms close to the bent line, eventually resulting in a localized permanent deformation and a “M” profile (Figure 15). This mechanism generates in turn a spring-forward effect and globally the samples assumed a “U” shape when the flattening force is released (i.e. when the punch moves upwards, Figure 17).

Analysis of tensile engineering stress-strain curves, scatter plots, and regression models revealed some correlations between the experimental parameters and response variables (Table 1). Particularly, the elongation at maximum stress (A%) decreased with increasing bending index (BI), indicating reduced formability due to accumulated plastic deformation (Figure 23 and Table 2). Ultimate Tensile Strength (UTS) remained relatively constant (Figure 24), while Yielding Stress (Ys) showed a slight increase with a higher Bending Index and flattening temperature (T), as visible in Figure 25. The results are in line with other studies and are explainable by thermal aging, where microstructural modifications induced by the warm flattening stage take place, thus affecting the mechanical properties of the sample.

In conclusion, the results underscored and confirmed the real and concrete possibility of potentially reusing these high-added-value sheets from end-of-life vehicles, presenting a more environmentally sustainable alternative to the prevailing recycling methodologies since they retain adequate residual formability for a second life. This would ensure avoiding the energy-demanding process of metal fusion typical of conventional recycling.
According to Wang et al. in low-carbon steels subjected to warm forming, dynamic strain aging (DSA) phenomena can lead to an increase of micro-hardness and peak stress [17].

## Conclusions

The findings presented in this paper contributed to assessing and understanding whether previously bent sheets retain adequate ductility for another forming cycle, a crucial aspect for promoting circular economy practices.

Specifically, the objective of this study is to investigate the plastic deformation and residual formability of DC04 mild steel, a prevalent material used in car bodies. The main idea is to comprehend if stamped steels extracted from end-of-life vehicles could be reshaped or remanufactured providing environmental benefits concerning conventional recycling. Similarly to a former study [11], the flattening process was employed as a reshaping or remanufacturing strategy for recovering the original planar geometry of a V-bent.

To simulate the potential implementation on real car bodies, the experimental procedure involved cold forming samples of 0.8 mm thick DC04 steel into V-bends of varying angles using a controlled punch stroke (Figure 1).

Subsequently, a flattening process was employed to reshape these parts for potential reuse, with key parameters such as dwell force, dwell time, and temperature carefully controlled. Tensile tests (Figure 6) were conducted on both as-received sheets and reshaped samples to understand how work-hardening and residual tensile formability depends on Bending Index (BI), Dwell Force (DwF), Dwell Time (τ), and Flattening Temperature (T). Two levels were tested for each factor (Figure 9), while the response variables of interest were Ultimate Tensile Strength (UTS), Yielding Stress (Ys), Young's Modulus (E), and Elongation at Maximum Stress (A%).

The samples were measured with different techniques before (Figure 10) and after flattening (Figure 14 and Figure 16) in order to comprehend the deformation mechanisms of the material in proximity of the bent line (i.e. the damaged region) and in order to understand the origin of the spring-forward effect (Figure 17). Before flattening Alicona Infinite Focus was employed to measure the external curvature radius and angles of the bent samples. The measurements provided an understanding into the deformation history of the material, providing conclusions in line with the state of the art of sheet metal bending.

Observations during flattening instead are more interesting since this process is novel and employed for remanufacturing already shaped parts. During flattening the bent line unbends and the flange deforms close to the bent line, eventually resulting in a localized permanent deformation and a “M” profile (Figure 15). This mechanism generates in turn a spring-forward effect and globally the samples assumed a “U” shape when the flattening force is released (i.e. when the punch moves upwards, Figure 17).

Analysis of tensile engineering stress-strain curves, scatter plots, and regression models revealed some correlations between the experimental parameters and response variables (Table 1). Particularly, the elongation at maximum stress (A%) decreased with increasing bending index (BI), indicating reduced formability due to accumulated plastic deformation (Figure 23 and Table 2). Ultimate Tensile Strength (UTS) remained relatively constant (Figure 24), while Yielding Stress (Ys) showed a slight increase with a higher Bending Index and flattening temperature (T), as visible in Figure 25. The results are in line with other studies and are explainable by thermal aging, where microstructural modifications induced by the warm flattening stage take place, thus affecting the mechanical properties of the sample.

In conclusion, the results underscored and confirmed the real and concrete possibility of potentially reusing these high-added-value sheets from end-of-life vehicles, presenting a more environmentally sustainable alternative to the prevailing recycling methodologies since they retain adequate residual formability for a second life. This would ensure avoiding the energy-demanding process of metal fusion typical of conventional recycling.
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## References

[1] J. Pavan Kumar, R. Uday Kumar, B. Ramakrishna, B. Ramu, and K. Baba Saheb, "Formability of sheet metals - A review," *IOP Conf. Ser. Mater. Sci. Eng.*, vol. 455, no. 1, 2018. https://doi.org/10.1088/1757-899X/455/1/012081

[2] N. R. Small, D. K. Williams, R. Roy, and S. K. Hazra, "Accounting for the effect of heterogeneous plastic deformation on the formability of aluminium and steel sheets," *Int. J. Adv. Manuf. Technol.*, vol. 109, no. 1-2, pp. 397-410, 2020. https://doi.org/10.1007/s00170-020-05372-0

[3] S. Dhara, S. Basak, S. K. Panda, S. Hazra, B. Shollock, and R. Dashwood, "Formability analysis of pre-strained AA5754-O sheet metal using Yld96 plasticity theory: Role of amount and direction of uni-axial pre-strain," *J. Manuf. Process.*, vol. 24, pp. 270-282, 2016. https://doi.org/10.1016/j.jmapro.2016.09.014

[4] H. L. Yi, L. Sun, and X. C. Xiong, "Challenges in the formability of the next generation of automotive steel sheets," *Mater. Sci. Technol. (United Kingdom)*, vol. 34, no. 9, pp. 1112-1117, 2018. https://doi.org/10.1080/02670836.2018.1424383

[5] M. Feistle, A. Kindsmüller, I. Pätzold, R. Golle, and W. Volk, "Influence of Sheet Metal Pre-forming on Edge Crack Sensitivity using an AHSS Steel Grade," *Int. J. Mater. Form.*, vol. 15, no. 4, pp. 1-11, 2022. https://doi.org/10.1007/s12289-022-01669-5

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# Plasticity experiments on heavy gauge S700 steel

COPPIETERS Sam¹,ª*, LAMBRUGHI Alessandro¹,ᵇ, VANCRAEYNEST Niels¹,ᶜ, ZHANG Yi¹,ᵈ, COOREMAN Steven²,ᵉ and STARMAN Bojan¹,³,ᶜ

¹KU Leuven, Department of Materials Engineering, Ghent Campus, 9000 Ghent, Belgium

²ArcelorMittal Global R&D Gent/OCAS NV, Ghent, Belgium

³University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia

ªsam.coppieters@kuleuven.be, ᵇalessandro.lambrughi@kuleuven.be,

ᶜniels.vancraeynest@kuleuven.be, ᵈyi.zhang@kuleuven.be,

ᵉsteven.cooreman@arcelormittal.com, ṷbojan.starman@fs.uni-lj.si

**Keywords:** Heavy Gauge High Strength Steel, Plasticity, DIC, Delamination, IRT

**Abstract.** High Strength Steel grades are indispensable for the development of heavy-duty constructions and components with high specific strength. In these applications, a profound understanding of the plastic material behavior up to fracture is required to assess the structural integrity through numerical simulations. In this paper, we investigate the plastic behavior of S700 with a nominal thickness of 12 mm. The steel production process of a hot rolled, heavy gauge material inherently results in a through-thickness variation of the mechanical properties: i) solidification of the continuously cast slab starts from the outer surface, causing a gradient of the chemical composition across the thickness, ii) subsequent thermomechanical controlled rolling results in a variation of microstructure and texture over the thickness. To enhance the predictive accuracy of numerical simulations, we are examining the manifestation of this inhomogeneity across a range of plasticity experiments.

**Introduction**

The steelmaking process, involving multiple heating and cooling stages, leads to variations in chemical composition and microstructure across the thickness of high-strength steels. This inhomogeneity results in different grain sizes and structures in various parts of the steel, as demonstrated in studies like those by Raabe et al. [1]. Such variations can lead to direction-dependent mechanical properties, especially in the through-thickness direction, a phenomenon documented by numerous researchers. When the inhomogeneity is overlooked when simulating secondary forming processes such as bending, it can lead to inaccuracies in predicting the gradient of material properties throughout the material's thickness. Accurately characterizing the through-thickness variation of material behavior is crucial for creating precise simulations that are vital in assessing the structural integrity of heavy-duty components. Despite extensive research on the planar behavior of sheet metal, methods to evaluate through-thickness behavior are less developed. Through-thickness tensile tests on small samples from thick plates can provide insights, such as through-thickness ductility, but face limitations. These include dependency on plate thickness, the need to manufacture specimens without altering material behavior, and the requirement for suitable testing equipment. Techniques like welding extensions to small tensile specimens, as proposed by Barsom et al. [2], can alter mechanical properties due to the creation of a heat-affected zone. Moreover, such tests typically yield only the average through-thickness material behavior (i.e., across the gauge section of the small tensile sample) and do not provide detailed information on work hardening variation throughout the thickness. Furthermore, Barsom et al. [2] noted that through-thickness tests often reflect the properties of the weakest region across the thickness. In

<figure><img src="image_1087.png" /></figure>
# Plasticity experiments on heavy gauge S700 steel

COPPIETERS Sam¹,ª*, LAMBRUGHI Alessandro¹,ᵇ, VANCRAEYNEST Niels¹,ᶜ, ZHANG Yi¹,ᵈ, COOREMAN Steven²,ᵉ and STARMAN Bojan¹,¹,³,ᶜ

¹KU Leuven, Department of Materials Engineering, Ghent Campus, 9000 Ghent, Belgium

²ArcelorMittal Global R&D Gent/OCAS NV, Ghent, Belgium

³University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia

ªsam.coppieters@kuleuven.be, ᵇalessandro.lambrughi@kuleuven.be,

ᶜniels.vancraeynest@kuleuven.be, ᵈyi.zhang@kuleuven.be,

ᵉsteven.cooreman@arcelormittal.com, ṷbojan.starman@fs.uni-lj.si

**Keywords:** Heavy Gauge High Strength Steel, Plasticity, DIC, Delamination, IRT

**Abstract.** High Strength Steel grades are indispensable for the development of heavy-duty constructions and components with high specific strength. In these applications, a profound understanding of the plastic material behavior up to fracture is required to assess the structural integrity through numerical simulations. In this paper, we investigate the plastic behavior of S700 with a nominal thickness of 12 mm. The steel production process of a hot rolled, heavy gauge material inherently results in a through-thickness variation of the mechanical properties: i) solidification of the continuously cast slab starts from the outer surface, causing a gradient of the chemical composition across the thickness, ii) subsequent thermomechanical controlled rolling results in a variation of microstructure and texture over the thickness. To enhance the predictive accuracy of numerical simulations, we are examining the manifestation of this inhomogeneity across a range of plasticity experiments.

**Introduction**

The steelmaking process, involving multiple heating and cooling stages, leads to variations in chemical composition and microstructure across the thickness of high-strength steels. This inhomogeneity results in different grain sizes and structures in various parts of the steel, as demonstrated in studies like those by Raabe et al. [1]. Such variations can lead to direction-dependent mechanical properties, especially in the through-thickness direction, a phenomenon documented by numerous researchers. When the inhomogeneity is overlooked when simulating secondary forming processes such as bending, it can lead to inaccuracies in predicting the gradient of material properties throughout the material's thickness. Accurately characterizing the through-thickness variation of material behavior is crucial for creating precise simulations that are vital in assessing the structural integrity of heavy-duty components. Despite extensive research on the planar behavior of sheet metal, methods to evaluate through-thickness behavior are less developed. Through-thickness tensile tests on small samples from thick plates can provide insights, such as through-thickness ductility, but face limitations. These include dependency on plate thickness, the need to manufacture specimens without altering material behavior, and the requirement for suitable testing equipment. Techniques like welding extensions to small tensile specimens, as proposed by Barsom et al. [2], can alter mechanical properties due to the creation of a heat-affected zone. Moreover, such tests typically yield only the average through-thickness material behavior (i.e., across the gauge section of the small tensile sample) and do not provide detailed information on work hardening variation throughout the thickness. Furthermore, Barsom et al. [2] noted that through-thickness tests often reflect the properties of the weakest region across the thickness. In

<figure><img src="image_1089.png" /></figure>

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contrast, the indentation hardness test is capable of detecting inhomogeneous material behavior in
steel. However, despite their utility in estimating the initial yield stress, hardness measurements
typically exhibit significant standard deviation. This substantial variability limits their
effectiveness in accurately determining strain hardening behavior. A more direct method to assess
work hardening variation through the thickness of steel is slicing a thick tensile specimen into
thinner specimens for tensile testing, as suggested by Sohn et al. [3]. This method effectively
provides the flow curve and material properties as a function of thickness, as shown in Section 1
of this paper. However, the use of this method is limited to analyzing pre-necking hardening
behavior, only up to the point of maximum uniform tensile strain. Considering that secondary
forming processes often induce strains that exceed this threshold, there arises a need for a
methodology capable of determining the post-necking strain hardening behavior. Zhang et al. [4]
introduced a Finite Element Model Updating (FEMU) technique to characterize the post-necking
hardening behavior of thick, high-strength steel by exploiting the strain fields observed during
diffuse necking in standard tensile tests. This method, however, operates under the assumption that
material damage is negligible in the developing diffuse neck. As demonstrated in Section 2 of this
paper, however, S700 steel is susceptible to delamination splits during diffuse necking, which
compromises the applicability of Zhang et al.'s approach for this material. Consequently, as
discussed in the third section of this paper, we utilize the free-end torsion test to ascertain the large
strain flow curve of S700 steel.

Through-thickness strain hardening variation: the slicing method

An effective method for assessing work hardening behavior across the thickness of a material
involves sectioning a thick tensile specimen into several thinner tensile specimens, as suggested
by Sohn et al. [3]. The choice of sectioning method is critical to minimize its impact on the flow
curve data. In this study, a 12 mm thick standard tensile specimen is sectioned using wire-cut
Electrical Discharge Machining (EDM), a process that, despite its precision, unavoidably removes
some material (wire diameter 0.31 mm) and partially releases residual stresses. Figure 1 illustrates
the specimen being divided into eight distinct layers. It's important to note that the outer layers (1
and 8, see Fig.1), with an average thickness of approximately 0.4 mm (as opposed to the 1.5 mm
of the other layers), are excluded due to a significant thickness variability leading to heterogeneous
strain fields before the maximum uniform tensile strain is reached. Consequently, only layers 2 to
7, which maintain consistent thickness, are used for reliable strain hardening analysis. Quasi-static
tensile tests, with a nominal strain rate of 1.4·10⁻⁴ s⁻¹ in the gauge length of the specimen, are
conducted using a standard tensile machine with a load-capacity of 250 kN. Stereo-DIC (Digital
Image Correlation) is used to capture the strain fields on the surface of the specimens during the
experiments. The DIC setup was configured with 6 MPx Manta G-609 cameras equipped with
Kowa lenses (f=25 mm). The synchronization between the tensile load and the DIC system enabled
to derive the strain hardening behavior. Fig. 2 presents the results, revealing significant differences
in strain hardening behavior between layers 2 and 7 (labelled *sample 1.2* and *1.7*) and the more
central layers 3 to 6. In general, symmetry around the sample's mid-plane is observed, with the
corresponding layers (2 and 7, 3 and 6, 4 and 5) exhibiting similar strain hardening characteristics.
Notably, the initial yield stress strongly increases towards the center of the specimen, with central
layers 4 and 5 showing a distinct yield plateau, leading to Lüders bands visible in the measured
strain fields. Additionally, Fig. 2 compares these results with those from a tensile test on the full
thickness, indicated by the red triangles. Seemingly, the strain hardening behavior from the full
thickness provides an averaged representation of the behaviors observed in the individual slices.
For surface layers 1 and 8, which were not included in the direct measurement, it is assumed that
their strain hardening behavior is analogous to that observed in the subsurface layers 2 and 7. The
latter is verified by simulating the full-thickness tensile test accounting for the eight different
layers. This simulation assumes that the surface layers (1 and 8) mirror the hardening behavior of
contrast, the indentation hardness test is capable of detecting inhomogeneous material behavior in
steel. However, despite their utility in estimating the initial yield stress, hardness measurements
typically exhibit significant standard deviation. This substantial variability limits their
effectiveness in accurately determining strain hardening behavior. A more direct method to assess
work hardening variation through the thickness of steel is slicing a thick tensile specimen into
thinner specimens for tensile testing, as suggested by Sohn et al. [3]. This method effectively
provides the flow curve and material properties as a function of thickness, as shown in Section 1
of this paper. However, the use of this method is limited to analyzing pre-necking hardening
behavior, only up to the point of maximum uniform tensile strain. Considering that secondary
forming processes often induce strains that exceed this threshold, there arises a need for a
methodology capable of determining the post-necking strain hardening behavior. Zhang et al. [4]
introduced a Finite Element Model Updating (FEMU) technique to characterize the post-necking
hardening behavior of thick, high-strength steel by exploiting the strain fields observed during
diffuse necking in standard tensile tests. This method, however, operates under the assumption that
material damage is negligible in the developing diffuse neck. As demonstrated in Section 2 of this
paper, however, S700 steel is susceptible to delamination splits during diffuse necking, which
compromises the applicability of Zhang et al.'s approach for this material. Consequently, as
discussed in the third section of this paper, we utilize the free-end torsion test to ascertain the large
strain flow curve of S700 steel.

Through-thickness strain hardening variation: the slicing method

An effective method for assessing work hardening behavior across the thickness of a material
involves sectioning a thick tensile specimen into several thinner tensile specimens, as suggested
by Sohn et al. [3]. The choice of sectioning method is critical to minimize its impact on the flow
curve data. In this study, a 12 mm thick standard tensile specimen is sectioned using wire-cut
Electrical Discharge Machining (EDM), a process that, despite its precision, unavoidably removes
some material (wire diameter 0.31 mm) and partially releases residual stresses. Figure 1 illustrates
the specimen being divided into eight distinct layers. It's important to note that the outer layers (1
and 8, see Fig.1), with an average thickness of approximately 0.4 mm (as opposed to the 1.5 mm
of the other layers), are excluded due to a significant thickness variability leading to heterogeneous
strain fields before the maximum uniform tensile strain is reached. Consequently, only layers 2 to
7, which maintain consistent thickness, are used for reliable strain hardening analysis. Quasi-static
tensile tests, with a nominal strain rate of 1.4·10⁻⁴ s⁻¹ in the gauge length of the specimen, are
conducted using a standard tensile machine with a load-capacity of 250 kN. Stereo-DIC (Digital
Image Correlation) is used to capture the strain fields on the surface of the specimens during the
experiments. The DIC setup was configured with 6 MPx Manta G-609 cameras equipped with
Kowa lenses (f=25 mm). The synchronization between the tensile load and the DIC system enabled
to derive the strain hardening behavior. Fig. 2 presents the results, revealing significant differences
in strain hardening behavior between layers 2 and 7 (labelled *sample 1.2* and *1.7*) and the more
central layers 3 to 6. In general, symmetry around the sample's mid-plane is observed, with the
corresponding layers (2 and 7, 3 and 6, 4 and 5) exhibiting similar strain hardening characteristics.
Notably, the initial yield stress strongly increases towards the center of the specimen, with central
layers 4 and 5 showing a distinct yield plateau, leading to Lüders bands visible in the measured
strain fields. Additionally, Fig. 2 compares these results with those from a tensile test on the full
thickness, indicated by the red triangles. Seemingly, the strain hardening behavior from the full
thickness provides an averaged representation of the behaviors observed in the individual slices.
For surface layers 1 and 8, which were not included in the direct measurement, it is assumed that
their strain hardening behavior is analogous to that observed in the subsurface layers 2 and 7. The
latter is verified by simulating the full-thickness tensile test accounting for the eight different
layers. This simulation assumes that the surface layers (1 and 8) mirror the hardening behavior of
the subsurface layers (2 and 7), a hypothesis corroborated by the excellent match between the simulated and experimentally obtained strain hardening behaviors in Fig. 2. Finally, Table 1 enumerates the Young's modulus *E*, the initial yield stress $\sigma_0$, and the *r*-value $r_0$ of each layer. It can be observed that *E* and $\sigma_0$ are significantly higher in the central layers (4 and 5), while the $r_0$ exhibits minimal variation across the thickness. The final row of the table displays the results from the full thickness tensile test for comparison purposes.

<table><caption>Table 1. The variation of Young's modulus *E*, the initial yield stress $\sigma_0$ and the *r*-value.</caption><thead><tr><th>Experiment</th><th><i>E</i> [GPa]</th><th>σ<sub>0</sub> [MPa]</th><th><i>r</i><sub>0</sub> [-]</th></tr></thead><tbody><tr><td><em>Sliced specimens</em></td><td></td><td></td><td></td></tr><tr><td>Sample 1.2</td><td>191</td><td>671</td><td>0.738</td></tr><tr><td>Sample 1.3</td><td>194</td><td>707</td><td>0.723</td></tr><tr><td>Sample 1.4</td><td>207</td><td>733</td><td>0.720</td></tr><tr><td>Sample 1.5</td><td>206</td><td>738</td><td>0.715</td></tr><tr><td>Sample 1.6</td><td>186</td><td>678</td><td>0.737</td></tr><tr><td>Sample 1.7</td><td>189</td><td>656</td><td>0.656</td></tr><tr><td>Average Slices</td><td>196</td><td>697</td><td>0.715</td></tr><tr><td>Std Dev Slices</td><td>8.64</td><td>31.08</td><td>0.0276</td></tr><tr><td><em>Full thickness</em></td><td>191</td><td>680</td><td>0.684</td></tr></tbody></table>

<figure><img src="image_1088.png" /><figcaption>Figure 1. Slicing method: a full-thickness standard tensile specimen is sliced into 8 layers.</figcaption></figure>
the subsurface layers (2 and 7), a hypothesis corroborated by the excellent match between the simulated and experimentally obtained strain hardening behaviors in Fig. 2. Finally, Table 1 enumerates the Young's modulus *E*, the initial yield stress $\sigma_0$, and the *r*-value $r_0$ of each layer. It can be observed that *E* and $\sigma_0$ are significantly higher in the central layers (4 and 5), while the $r_0$ exhibits minimal variation across the thickness. The final row of the table displays the results from the full thickness tensile test for comparison purposes.

<table><caption>Table 1. The variation of Young's modulus *E*, the initial yield stress $\sigma_0$ and the *r*-value.</caption><thead><tr><th>Experiment</th><th><i>E</i> [GPa]</th><th>σ<sub>0</sub> [MPa]</th><th><i>r</i><sub>0</sub> [-]</th></tr></thead><tbody><tr><td><em>Sliced specimens</em></td><td></td><td></td><td></td></tr><tr><td>Sample 1.2</td><td>191</td><td>671</td><td>0.738</td></tr><tr><td>Sample 1.3</td><td>194</td><td>707</td><td>0.723</td></tr><tr><td>Sample 1.4</td><td>207</td><td>733</td><td>0.720</td></tr><tr><td>Sample 1.5</td><td>206</td><td>738</td><td>0.715</td></tr><tr><td>Sample 1.6</td><td>186</td><td>678</td><td>0.737</td></tr><tr><td>Sample 1.7</td><td>189</td><td>656</td><td>0.656</td></tr><tr><td>Average Slices</td><td>196</td><td>697</td><td>0.715</td></tr><tr><td>Std Dev Slices</td><td>8.64</td><td>31.08</td><td>0.0276</td></tr><tr><td><em>Full thickness</em></td><td>191</td><td>680</td><td>0.684</td></tr></tbody></table>

<figure><img src="image_1090.png" /><figcaption>Figure 1. Slicing method: a full-thickness standard tensile specimen is sliced into 8 layers.</figcaption></figure>
<figure><img src="image_1089.png" /><figcaption>Figure 2. Variation in strain hardening behaviour (RD) through the thickness.</figcaption></figure>

Occurrence of delamination splits during diffuse necking

A post-mortem inspection of a nonconventional tensile specimen, fractured under a quasi-static strain rate, revealed delamination splits in the 12 mm thick S700 steel [5]. Delamination refers to the material's splitting along its rolling plane when under tension, a phenomenon that notably affects the material's upper shelf energy and transition temperature. Influenced by various factors such as the steel grade, segregation of impurity atoms, and spatial variations in material strength, this issue is particularly evident in S700 steel, as demonstrated in the previous section. To provide a more comprehensive understanding of delamination in thick S700 steel, we performed tensile tests (in the rolling direction) at different nominal strain rates, continuing the experiments until fracture with a constant cross head speed (1.3 mm/min, 58 mm/min and 288 mm/min). The tensile tests were performed on a standard tensile machine with a load capacity of 250kN with hydraulic grips. For temperature assessment, we employed a Long-Waves (LW) infrared thermographic camera (FLIR 615A) equipped with a 24.6 mm lens to measure the local surface temperature of the specimen. The samples, painted black with white speckles, facilitated simultaneous accurate measurement of full-field thermal and kinematic fields. The emissivity of this paint was considered constant and equal to ε = 0.93 [-] as characterized by Cholewa et al. [6]. The latter assumption was verified by comparing the measurements from Infrared Thermography (IRT) and a thermocouple. Local strains were measured using a stereo-DIC setup. The integration of IRT into the stereo-DIC system ensured that the local temperature was available for each DIC data point. Flir BlackFly cameras (5MPx) with Kowa lenses (f=25 mm) were used. Each speckle pattern had an average speckle size of approximately 5px. A performance analysis was conducted to determine the most optimal DIC settings for each experiment. The maximum strain noise floor in the experiments was 323 µStrain. Using the optimal DIC settings, the thermal noise floor was estimated from still images under thermal equilibrium [7] to an average noise level of 58 mK. A value being close to a Noise Equivalent Temperature Difference (NETD) of 50 mK reported by the camera's manufacturer.

Figure 3 illustrates the delaminations observed in the fractured specimens subjected to various strain rates. It is apparent from the figure that delaminations were more pronounced at lower strain rates and reduced with increasing strain rates. Specifically, a high density of delaminations is visible at the lowest strain rate, a central delamination at the medium strain rate, and an absence of delaminations at the highest rate. It is important to note that these occurrences of delaminations
<figure><img src="image_1091.png" /><figcaption>Figure 2. Variation in strain hardening behaviour (RD) through the thickness.</figcaption></figure>

Occurrence of delamination splits during diffuse necking

A post-mortem inspection of a nonconventional tensile specimen, fractured under a quasi-static strain rate, revealed delamination splits in the 12 mm thick S700 steel [5]. Delamination refers to the material's splitting along its rolling plane when under tension, a phenomenon that notably affects the material's upper shelf energy and transition temperature. Influenced by various factors such as the steel grade, segregation of impurity atoms, and spatial variations in material strength, this issue is particularly evident in S700 steel, as demonstrated in the previous section. To provide a more comprehensive understanding of delamination in thick S700 steel, we performed tensile tests (in the rolling direction) at different nominal strain rates, continuing the experiments until fracture with a constant cross head speed (1.3 mm/min, 58 mm/min and 288 mm/min). The tensile tests were performed on a standard tensile machine with a load capacity of 250kN with hydraulic grips. For temperature assessment, we employed a Long-Waves (LW) infrared thermographic camera (FLIR 615A) equipped with a 24.6 mm lens to measure the local surface temperature of the specimen. The samples, painted black with white speckles, facilitated simultaneous accurate measurement of full-field thermal and kinematic fields. The emissivity of this paint was considered constant and equal to ε = 0.93 [-] as characterized by Cholewa et al. [6]. The latter assumption was verified by comparing the measurements from Infrared Thermography (IRT) and a thermocouple. Local strains were measured using a stereo-DIC setup. The integration of IRT into the stereo-DIC system ensured that the local temperature was available for each DIC data point. Flir BlackFly cameras (5MPx) with Kowa lenses (f=25 mm) were used. Each speckle pattern had an average speckle size of approximately 5px. A performance analysis was conducted to determine the most optimal DIC settings for each experiment. The maximum strain noise floor in the experiments was 323 µStrain. Using the optimal DIC settings, the thermal noise floor was estimated from still images under thermal equilibrium [7] to an average noise level of 58 mK. A value being close to a Noise Equivalent Temperature Difference (NETD) of 50 mK reported by the camera's manufacturer.

Figure 3 illustrates the delaminations observed in the fractured specimens subjected to various strain rates. It is apparent from the figure that delaminations were more pronounced at lower strain rates and reduced with increasing strain rates. Specifically, a high density of delaminations is visible at the lowest strain rate, a central delamination at the medium strain rate, and an absence of delaminations at the highest rate. It is important to note that these occurrences of delaminations
were not discernible from the load-displacement curves. Furthermore, the load-displacement curves corresponding to different strain rates indicate that S700 steel exhibits strain rate independence. Figure 4 presents the strain and temperature fields on the surface of the specimen, measured using Digital Image Correlation (DIC) and Infrared Thermography (IRT). All the plots in Figure 4 correspond to a maximum total true axial strain of 0.63. Both strain and temperature fields share the same color bars on the right-hand side of the figure. It can be seen that for the smallest strain rate, the maximal surface temperature only reaches 42.6°C. For the mid and high strain rate, adiabatic conditions in the diffuse neck cause a significant temperature increase. Indeed, the mid and high strain rate yield a maximum surface temperature of 146°C and 160°C, respectively. From the comparison between Fig. 3 and Fig. 4 it can be concluded that an increase in temperature corresponds to a reduction in delamination density. Before reaching the point of fracture, it is essential to highlight the rapid increase in strain rate, leading to adiabatic conditions and a corresponding temperature rise in the necking area. However, accurately measuring this phenomenon poses challenges due to its transient nature. Nevertheless, the current results clearly indicate that the delamination in S700 under tensile load is driven by temperature while the stress-strain response itself is rate independent.

<figure><img src="image_1090.png" /><figcaption>Figure 3. Delamination splits as a function of the nominal strain rate.</figcaption></figure>
were not discernible from the load-displacement curves. Furthermore, the load-displacement curves corresponding to different strain rates indicate that S700 steel exhibits strain rate independence. Figure 4 presents the strain and temperature fields on the surface of the specimen, measured using Digital Image Correlation (DIC) and Infrared Thermography (IRT). All the plots in Figure 4 correspond to a maximum total true axial strain of 0.63. Both strain and temperature fields share the same color bars on the right-hand side of the figure. It can be seen that for the smallest strain rate, the maximal surface temperature only reaches 42.6°C. For the mid and high strain rate, adiabatic conditions in the diffuse neck cause a significant temperature increase. Indeed, the mid and high strain rate yield a maximum surface temperature of 146°C and 160°C, respectively. From the comparison between Fig. 3 and Fig. 4 it can be concluded that an increase in temperature corresponds to a reduction in delamination density. Before reaching the point of fracture, it is essential to highlight the rapid increase in strain rate, leading to adiabatic conditions and a corresponding temperature rise in the necking area. However, accurately measuring this phenomenon poses challenges due to its transient nature. Nevertheless, the current results clearly indicate that the delamination in S700 under tensile load is driven by temperature while the stress-strain response itself is rate independent.

<figure><img src="image_1092.png" /><figcaption>Figure 3. Delamination splits as a function of the nominal strain rate.</figcaption></figure>
<figure><img src="image_1091.png" /><figcaption>Figure 4. Axial true strain and surface temperature in the diffuse neck as a function of the nominal strain rate.</figcaption></figure>

## Large strain flow curve: the free-end torsion test

Figure 2 demonstrates that S700 steel reaches a maximum uniform tensile strain of approximately 0.1. However, beyond this point, S700 steel also shows a significant post-necking region extending up to fracture. To conduct accurate simulations for assessing the formability or structural integrity of S700 steel, it's essential to have data on its large strain flow curve. Zhang et al. [4] suggested an inverse method to extract this large strain flow curve from the diffuse necking observed in a tensile test. This method involves the use of Finite Element Model Updating (FEMU) and the analysis of measured strain fields. However, in the previous section it is shown that S700 exhibits delamination during diffuse necking which prevents the straightforward application of the FEMU approach to acquire the large strain flow curve. Vancraeynest et al. [8] proposed to use a free-end torsion test to inversely extract the post-necking hardening behavior of S700. The advantage is that in the free-end torsion test, large plastic strains can be probed without instability problems. More importantly, the delamination as observed in the tensile test does not occur in the torsion test. More details on the FEMU approach applied to the free-end torsion test can be found in [8].

To assess the robustness of the free-end torsion test in characterizing the large strain flow curve, three torsion samples (with a diameter of 8 mm) cut in the RD from a S700 steel plate were tested. The FEMU approach uses the torque-angle curve to identify the parameters of the selected hardening law. Vancraeynest et al. [8] recommended Voce's hardening law for describing the large strain flow curve of S700. An improved analysis (meshing strategy and explicit integration scheme), however, showed that Swift's hardening law is more appropriate than Voce in the probed strain range. The results of the identified Swift hardening laws are shown in Fig. 5. It can be inferred that the free-end torsion test enables to probe equivalent plastic strains up to 0.9. Despite the careful manufacturing of the specimens to ensure consistency in their through-thickness positions, some variation in the large strain flow curve is evident. The inset of Figure 5 displays a fractured specimen, where the torsion failure does not show a localized reduction of area. Notably, the plane of ductile shear failure is perpendicular to the longitudinal axis. It is evident from the inset of Figure 5 that the fractured parts are not perfectly aligned. This misalignment is attributed to the asymmetrical nature of ductile fracture. Furthermore, a post-mortem analysis of the fracture surface revealed no signs of delaminations. This observation is supported by Direct Current
<figure><img src="image_1093.png" /><figcaption>Figure 4. Axial true strain and surface temperature in the diffuse neck as a function of the nominal strain rate.</figcaption></figure>

## Large strain flow curve: the free-end torsion test

Figure 2 demonstrates that S700 steel reaches a maximum uniform tensile strain of approximately 0.1. However, beyond this point, S700 steel also shows a significant post-necking region extending up to fracture. To conduct accurate simulations for assessing the formability or structural integrity of S700 steel, it's essential to have data on its large strain flow curve. Zhang et al. [4] suggested an inverse method to extract this large strain flow curve from the diffuse necking observed in a tensile test. This method involves the use of Finite Element Model Updating (FEMU) and the analysis of measured strain fields. However, in the previous section it is shown that S700 exhibits delamination during diffuse necking which prevents the straightforward application of the FEMU approach to acquire the large strain flow curve. Vancraeynest et al. [8] proposed to use a free-end torsion test to inversely extract the post-necking hardening behavior of S700. The advantage is that in the free-end torsion test, large plastic strains can be probed without instability problems. More importantly, the delamination as observed in the tensile test does not occur in the torsion test. More details on the FEMU approach applied to the free-end torsion test can be found in [8].

To assess the robustness of the free-end torsion test in characterizing the large strain flow curve, three torsion samples (with a diameter of 8 mm) cut in the RD from a S700 steel plate were tested. The FEMU approach uses the torque-angle curve to identify the parameters of the selected hardening law. Vancraeynest et al. [8] recommended Voce's hardening law for describing the large strain flow curve of S700. An improved analysis (meshing strategy and explicit integration scheme), however, showed that Swift's hardening law is more appropriate than Voce in the probed strain range. The results of the identified Swift hardening laws are shown in Fig. 5. It can be inferred that the free-end torsion test enables to probe equivalent plastic strains up to 0.9. Despite the careful manufacturing of the specimens to ensure consistency in their through-thickness positions, some variation in the large strain flow curve is evident. The inset of Figure 5 displays a fractured specimen, where the torsion failure does not show a localized reduction of area. Notably, the plane of ductile shear failure is perpendicular to the longitudinal axis. It is evident from the inset of Figure 5 that the fractured parts are not perfectly aligned. This misalignment is attributed to the asymmetrical nature of ductile fracture. Furthermore, a post-mortem analysis of the fracture surface revealed no signs of delaminations. This observation is supported by Direct Current
Potential Drop (DCPD) measurements taken during the torsion tests. In the absence of pronounced necking in these tests, any change in the specimen's resistance directly correlates with the occurrence of delamination. Figure 6 indicates that the specimen's resistance during torsion remains approximately constant until fracture, thereby supporting the hypothesis that delamination is not a concern in the torsion test. This finding validates the use of the torsion test for reliably extracting the large strain flow curve. However, it's important to note that this identification assumes plastic isotropic material behavior, and the potential effects of plastic anisotropy warrant further investigation. Finally, it was observed that at a rotation angle of approximately 300°, the torsion sample begins to develop localized superficial spiral deformations.

<figure><img src="image_1092.png" /><figcaption>Figure 5. Strain hardening behavior identified through the free-end torsion test.</figcaption></figure>

The spiral deformations previously mentioned are not clearly visible in the inset of Figure 5. To accurately determine their amplitude and frequency in the longitudinal direction of the torsion sample, which seems to be superficial, additional measurements were undertaken using the Atorn easyContour device equipped with a 120V/33 ceramic probe. This device boasts a maximum measuring range ($L_{z,max}$) of 30 mm and an error margin of ±1.8 µm + 2 ($\frac{L_z}{25}$) in the z-direction. For aligning the torsion sample with the x-axis of the Atorn easyContour device, a V-block was employed. The orientation of the torsion samples in the V-block ensured that the principal material axis in the normal direction aligned with the z-axis of the measuring device. The results are presented in Figure 7. Despite meticulous efforts to align the torsion samples, the deformation patterns observed in tests 2 and 3 exhibit a slight skew. This skewness is attributable to the asymmetrical fracture behavior previously mentioned. At the point of fracture, the maximum amplitude of the spiral deformation is approximately 0.06 mm. The frequency of these deformations along the longitudinal direction of the torsion sample remains consistent across the three experiments. Further investigation is needed to understand the origin of this phenomenon and its potential impact on accurately identifying the large strain flow curve.
Potential Drop (DCPD) measurements taken during the torsion tests. In the absence of pronounced necking in these tests, any change in the specimen's resistance directly correlates with the occurrence of delamination. Figure 6 indicates that the specimen's resistance during torsion remains approximately constant until fracture, thereby supporting the hypothesis that delamination is not a concern in the torsion test. This finding validates the use of the torsion test for reliably extracting the large strain flow curve. However, it's important to note that this identification assumes plastic isotropic material behavior, and the potential effects of plastic anisotropy warrant further investigation. Finally, it was observed that at a rotation angle of approximately 300°, the torsion sample begins to develop localized superficial spiral deformations.

<figure><img src="image_1094.png" /><figcaption>Figure 5. Strain hardening behavior identified through the free-end torsion test.</figcaption></figure>

The spiral deformations previously mentioned are not clearly visible in the inset of Figure 5. To accurately determine their amplitude and frequency in the longitudinal direction of the torsion sample, which seems to be superficial, additional measurements were undertaken using the Atorn easyContour device equipped with a 120V/33 ceramic probe. This device boasts a maximum measuring range ($L_{z,max}$) of 30 mm and an error margin of ±1.8 µm + 2 (Lz/25) in the z-direction. For aligning the torsion sample with the x-axis of the Atorn easyContour device, a V-block was employed. The orientation of the torsion samples in the V-block ensured that the principal material axis in the normal direction aligned with the z-axis of the measuring device. The results are presented in Figure 7. Despite meticulous efforts to align the torsion samples, the deformation patterns observed in tests 2 and 3 exhibit a slight skew. This skewness is attributable to the asymmetrical fracture behavior previously mentioned. At the point of fracture, the maximum amplitude of the spiral deformation is approximately 0.06 mm. The frequency of these deformations along the longitudinal direction of the torsion sample remains consistent across the three experiments. Further investigation is needed to understand the origin of this phenomenon and its potential impact on accurately identifying the large strain flow curve.
<figure><img src="image_1093.png" /><figcaption>Figure 6. Direct Current Potential Drop (DCPD) measurement during the free-end torsion test.</figcaption></figure>

<figure><img src="image_1094.png" /><figcaption>Figure 7. Superficial spiral deformation measured for three fractured torsion samples (S700)</figcaption></figure>

**Summary**

From the experimental campaign on S700 steel, the following conclusions have been drawn:

* The slicing method distinctly reveals a through-thickness variation in the hardening behavior of the S700 steel in the rolling direction. Notably, the yield strength of the material increases significantly towards the mid-plane of the sample. However, there is no significant variation in the r-value observed.

* Tensile tests conducted at different strain rates on the full thickness of S700 steel indicate that its strain hardening behavior is strain rate independent. Nonetheless, temperature has been identified as the primary factor influencing the occurrence of delamination splits during diffuse necking under tension.

* The free-end torsion tests prove effective in probing large strains without inducing delamination splits at room temperature, making this method a viable option for extracting the large strain flow curve of S700 steel. Superficial spiral necking in the torsion sample
<figure><img src="image_1095.png" /><figcaption>Figure 6. Direct Current Potential Drop (DCPD) measurement during the free-end torsion test.</figcaption></figure>

<figure><img src="image_1096.png" /><figcaption>Figure 7. Superficial spiral deformation measured for three fractured torsion samples (S700)</figcaption></figure>

**Summary**

From the experimental campaign on S700 steel, the following conclusions have been drawn:

* The slicing method distinctly reveals a through-thickness variation in the hardening behavior of the S700 steel in the rolling direction. Notably, the yield strength of the material increases significantly towards the mid-plane of the sample. However, there is no significant variation in the r-value observed.

* Tensile tests conducted at different strain rates on the full thickness of S700 steel indicate that its strain hardening behavior is strain rate independent. Nonetheless, temperature has been identified as the primary factor influencing the occurrence of delamination splits during diffuse necking under tension.

* The free-end torsion tests prove effective in probing large strains without inducing delamination splits at room temperature, making this method a viable option for extracting the large strain flow curve of S700 steel. Superficial spiral necking in the torsion sample
was noted at large rotation angles, a phenomenon that warrants further investigation in future studies.

**Acknowledgement**

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

**Disclaimer**

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

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