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# Using miniature cuboidal specimens to determine compression Stress–strain curve of sheet metal under static and dynamic loading

Arthur Girard, Vincent Grolleau, Dirk Mohr

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Arthur Girard, Vincent Grolleau, Dirk Mohr. Using miniature cuboidal specimens to determine compression Stress–strain curve of sheet metal under static and dynamic loading. International Journal of Solids and Structures, 2025, 320, pp.113527. 10.1016/j.ijsolstr.2025.113527. hal-05333569

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# Using miniature cuboidal specimens to determine compression
Stress–strain curve of sheet metal under static and dynamic loading

Arthur Girard<sup>a</sup>, Vincent Grolleau<sup>a,b</sup>, Dirk Mohr<sup>a,*</sup>

## ARTICLE INFO

Keywords:

Upsetting

Compression

Dual-phase steel

Strength-differential effect

Hopkinson bar

Strain rate effect

## ABSTRACT

Characterizing the stress–strain response of sheet metal under both tension and compression is important to provide accurate predictions of spring-back in sheet metal forming. Data on compression is scarce as anti-buckling devices typically need to be used to attain large strains in compression experiments on sheet metal. As a technically simpler alternative, we propose upsetting experiments with miniature cuboidal specimens as these can be conveniently extracted from sheet metal. In addition to static experiments, we design a Dropkinson bar system for the dynamic compression testing of miniature specimens. We demonstrate through finite element simulations that upsetting specimens with square cross-section are equally-suitable for measuring the stress–strain curve as traditional cylindrical specimens with circular cross-sections. Even though the local surface strains can be conveniently measured with planar DIC, detailed analysis revealed that using the average strain along the full specimen height yields the most accurate stress–strain curve estimation. Static and dynamic demonstration experiments are performed on miniature cuboidal specimens extracted from DP780, DP980 and DP1180 steel sheets. The comparison with the results from uniaxial tension experiments reveals a systematic strength differential effect with a yield stress for compression that is up to 4% higher than that for tension. The dynamic experiments at strain rates of about 750/s suggest a yield strength increase of up to 25% (as compared to static experiments).

## 1. Introduction

Upsetting experiments provide an effective means to determine the stress–strain curve of metals under compressive loading conditions. They have been extensively used to determine the so-called flow curve (e.g. Loizou and Sims, 1953) to understand the thermo-mechanical material behavior in forging operations at ambient and high temperatures (e.g. (Pillinger et al. (1985), Lovato and Stout (1992), Khoddam et al., (2021)), to quantify strength differential effects (e.g. Cazacu et al., 2006) or to gain insight into the fracture response of metals at negative stress-triaxialities (e.g Bao and Wierzbicki, (2005), Paredes and Wierzbicki (2020)). Furthermore, the results from upsetting test have been exploited to characterize friction (e.g. Yao et al. (2013), Fan et al., (2017)).

Many studies analyzed the mechanical fields in large strain compression experiments on cylindrical specimens in great detail. It is well established that friction at the specimen-loading platen interface causes the development of a barrel type of shape as the specimen

deforms (e.g. Fan et al. (2017)). With the formation of the barrel shape, the mechanical fields in an upsetting specimen become highly heterogeneous which results in potentially significant errors in the determined stress–strain curve. Irrespective of the tested material, the friction at the specimen boundaries is causing heterogeneous mechanical fields (Schey et al., 1982) which results in potentially significant errors in the determined stress–strain curve. Severe barreling is often accompanied by fold-over, where the lateral surfaces begin to contact the platens (e.g. Hou et al., 1992).

Already more than 70 years ago, Rastegaev (1940) machined the top and bottom surfaces of upsetting specimens to create a reservoir for lubricants. The concept has been refined further by Loizou and Sims (1953) who introduced concentric grooves to the top and bottom surfaces of 1 in. diameter and 1.5 in. high cylindrical specimens to reduce the friction by entrapping lubricant during compression testing. Other lubrication techniques such as using graphite grease (e.g. Scudino et al. (2010)) or Teflon sheets have only partially mitigated the influence of friction on the flow stress. A comprehensive review of the work on

<footnote><sup>a</sup> Department of Mechanical and Process Engineering, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland</footnote>

<footnote><sup>b</sup> Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: dmohr@ethz.ch (D. Mohr).</footnote>
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Sketch of upsetting specimens: (a) cuboidal specimens with square cross-sections; (b) Standard cylindrical geometry with circular cross-section.</figcaption></figure>

friction and its effect on barreling is provided by Khoddam et al. (2021).

As friction effects appear to be inevitable in practice, correction techniques have been developed. For example, Fardi et al. (2017) made use of exponential profile modeling (EPM) which allows for estimating an instantaneous friction coefficient and the strain rate fields in a compression specimen. In the EPM, the barreled shape is described through an exponential relationship between the specimen radius and the height coordinate. The method of Cook and Larke (1945) exploits the results from experiments on specimens of different height-to-diameter ratios to estimate the flow curve from a hypothetical experiment on a specimen with an infinite ratio. Further enhancements of this method have been described in the open literature (e.g. Alexander et al. (1963), Woodward (1977), Bao (2003)). Another family of methods for correcting of barreling makes use of inverse FE-based approaches. For this, the objective function may be based on the global force-displacement response and/or the evolving shape of the specimen (e.g. Osakada et al., (1981), Xinbo et al. (2002), Hartlen and Doman (2019)). Modern processing methods even account for changes in friction throughout the upsetting experiment (e.g. Christiansen et al., 2016).

Cylindrical compression specimens are also widely used for characterizing the high strain rate response of metals. Compression experiments at high loading velocities can be conveniently performed with Split Hopkinson pressure bar (SHPB) systems. Different from static experiments, the specimen height must be kept small to ensure that inertia effects at the specimen level are negligible. With elastic wave propagation speeds of about 5000 m/s in most metals, specimen heights of less than 10 mm are typically chosen when targeting strain rates of about 1000/s. The specimen transit time for an elastic wave is then much shorter than the time scale associated with significant stress changes. For example, Wu et al. (2023) used cylindrical specimens of 3 mm height and 5 mm diameter to characterize the high strain rate response of Ti-6Al-4 V alloy. A similar height-to-diameter ratio had been chosen by Prakash et al. (2023) when performing SHPB experiments on titanium. Kopec et al. (2024) used the 5 mm diameter and 5 mm long specimens when testing aluminum 6061-T6; Soares and Hokka (2021) used specimens of a height of about 8 mm and height-to-diameter ratios ranging from 0.43 to 0.8. Kariem et al. (2019) used specimens of 4 mm height and 6 mm diameter in their SHPB round Robin on stainless steel, copper, aluminum and magnesium. As for static testing, analytical formulas have been proposed to correct for friction effects in dynamic experiments (e.g. Lu et al., 2015).

Traditionally, the application of upsetting experiments is limited to bulk materials. To characterize the flow behavior of sheet metal under biaxial (in-plane) stretching conditions, Barlat et al. (2003) proposed through-thickness compression tests on 12.7 mm diameter discs extracted from sheet metal. Stackings of multiple square specimens (e.g. 12 mm high stacking of ten square samples extracted from 1.2 mm thick

sheets) is also often used for that purpose (e.g. Maeda et al. (1998), An and Vegter (2005), Kuwabara et al. (2022)). Coppieters et al. (2010) developed a FE-based inverse method to identify the out-of-plane hardening curve from stack compression experiments. To characterize the stress-strain response under in-plane compression for sheet metal, anti-buckling devices are typically employed. Boger et al. (2005) designed a hydraulically actuated lateral support system to enable the large strain compression testing of dogbone shaped specimens in a universal testing machine, while a compact floating anti-buckling device has been proposed by Mohr and Marcadet (2015). The challenge of these experiments is to prevent the buckling of the specimen gage section in a way that the friction at the specimen surfaces does not affect the experimental results. Kopec (2023) proposed a device enabling the continuous monitoring of the friction load. Chang et al. (2020) have also explored alternatives to correct the effects of friction-disturbed axial fields by a special device designed to prevent buckling with a friction-countering mechanism. Overall, the effectiveness of anti-buckling devices is limited; they often delay buckling rather than preventing it. Consequently, reliable stress-strain data from these experiments is generally attainable only for axial strains of less than 10 %.

As an alternative to using anti-buckling devices, miniature upsetting specimens may be used to perform in-plane compression experiments. This idea had already been pursued by Pawelski (1967) and recently been refined by Hetz et al. (2022). They used EDM machining to extract 2.25 mm high cylindrical specimens of 1.5 mm diameter from DC04 steel and aluminum 7075-T6. In the context of high strain rate testing, Li and Guo (2006) proposed a micro-Hokkinson bar system for the testing of 1.5 mm high copper and aluminum alloy specimens of 1.5 mm diameter. While the use of miniature specimens appears to be rare for sheet metal testing, micro- and nano-compression specimens are widely used for in-situ SEM testing in material science (e.g. Hemker and Sharpe, 2007, Kiener et al. (2009)). The associated length scale is yet much smaller, with specimen diameters of about 5 to 100 µm (e.g. Hemker and Sharpe, 2007). The specimen shape is typically applied through focused ion beam (FIB) milling. The bottom specimen boundary remains attached to the substrate thereby forming a monolithic structure with the base plate. The loading is applied to the top specimen boundary using a nano-indenter equipped with a flat tip (e.g. Uchic and Dimiduk, 2005). As the lateral expansion is hindered at the bottom boundary, high height-to-diameter ratios are needed to achieve approximately uniaxial loading conditions, which limits the range of validity of the experiments to small strains.

In the present work, we are concerned with the development of a miniature specimen testing technique to characterize the large strain hardening response of sheet metal for in-plane compression loading. While most upsetting testing techniques appear to focus on specimens of cylindrical shape only, we demonstrate that cuboidal specimens are equally suitable for static and dynamic compression testing. After formulating the detailed problem statement in Section 2, we present the experimental procedures in Section 3 which includes the development of a new Dropkinson bar procedure for dynamic testing. Section 4 details the numerical procedures for simulating uniaxial compression experiments. Candidate techniques for deriving stress-strain curves from experimental measurements are outlined in Section 5. The numerical and experimental results are presented and discussed in Section 6, before presenting the main conclusions in Section 7.

## 2. Problem statement

The goals of the present study are to demonstrate that:

(1) Cuboidal specimens with square cross-sections (Fig. 1a) are suitable for measuring the stress-strain curve for compression, thereby providing a viable alternative to standard cylindrical specimens with round cross-sections (Fig. 1b); in particular, the
<table><caption>**Table 1**

Direction-dependent Lankford ratios of selected sheet materials (0°-direction corresponds to the rolling-direction).</caption><thead><tr><th>Material</th><th>r<sub>0</sub></th><th>r<sub>45</sub></th><th>r<sub>90</sub></th></tr></thead><tbody><tr><td>DP590</td><td>0.98</td><td>0.81</td><td>1.13</td></tr><tr><td>DP780</td><td>0.82</td><td>0.86</td><td>0.88</td></tr><tr><td>DP980</td><td>0.78</td><td>1.11</td><td>0.91</td></tr><tr><td>DP1180</td><td>0.76</td><td>1.00</td><td>0.91</td></tr><tr><td>AA5754</td><td>0.62</td><td>0.84</td><td>0.70</td></tr><tr><td>AA6014</td><td>0.77</td><td>0.51</td><td>0.82</td></tr><tr><td>AA2024</td><td>0.66</td><td>0.96</td><td>0.55</td></tr><tr><td>AA7075</td><td>0.57</td><td>1.08</td><td>0.80</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** (a) Photograph of specimen with micro-engraved surface as extracted from a DP980 sheet using wire electro-discharge machining (EDM); (b) micrograph showing the roughness of the EDM machined surface and the etched microstructure below; (c) detail of the micro-engraved surface.</figcaption></figure>

switching from cylindrical to cuboidal specimens will facilitate the extraction of compression specimens from sheet metal.

(2) Good estimates of the stress-strain curve may be obtained up to strains of 0.25 despite the barreling due to friction; different data processing techniques will be assessed for that purpose; in particular, a new correction technique (Lambda method) will be presented to account for the effect of friction-induced barreling;

(3) It is technically feasible to extract and test cuboidal specimens with a cross-section as small as 1mm × 1mm;

(4) DP steels exhibit a tension-compression strength differential effect; characterizing the latter is expected to lead to constitutive models that provide more accurate predictions in standard engineering problems such as estimating the springback in sheet metal forming;

(5) High strain rate compression experiments may be performed on

(6) Quasi-static loading conditions are rapidly attained in dynamic experiments on miniature specimens. To demonstrate this advantage over conventional compression specimens for split Hopkinson pressure bar testing, we present a new “Dropkinson” device comprised of a gravity-accelerated striker bar and a custom-made sensing block for force measurement.

## 3. Experimental procedures

### 3.1. Materials

Experiments are performed on three different grades of dual-phase (DP) steels: DP1180, DP980 and DP780. The materials are provided in the form of sheets of thickness 1.05 mm, 1.56 mm, and 1.52 mm, respectively. These are mildly anisotropic as shown by the Lankford ratios (Table 1). For validation purposes, we also perform experiments on specimens extracted from a C45 steel bar of 15 mm diameter. Hardness measurements confirmed the uniformity of the mechanical properties across the bar diameter. Micrographs taken of all materials (after polishing and chemical etching) showed that the grain size is significantly smaller than the specimen dimensions (i.e. we have at least 10 grains along the smallest specimen dimension).

### 3.2. Static experiments

*Specimens.*

For all DP steels, the width *W* of the square cross-sections of the

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Experimental setup for static compression: (a) Photograph showing 10kN load-cell ⊙, spherical bearing ⊕ for initial alignment machine, base connector ⊥ to hydraulic piston, LVDTs ⊘, optical camera ⊙, telecentric lens ⊙, vertical rail ⊥ for camera positioning, illumination system ⊙. (b) Detailed view of a cuboidal DP980 specimen as sandwiched between two tungsten plates.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** Experimental setup for dynamic compression (Dropkinson bar system): (a) overview drawing with magnified detail showing vertical support structure ©, tube bearing Ø, striker bar ⊙, base block Ø; (b) Photograph showing speckle-painted striker bar tip ⊙, cuboidal specimen Ø, dynamic load cell Ø, base block Ø, telecentric lens Ø, and illumination system ⊙.</figcaption></figure>

cuboidal specimens is equal to the respective sheet thickness. Following the recommendation from earlier studies for preventing the premature buckling of cylindrical specimens (e.g. Chait and Curll (1976), Kuhn (2000)), the specimen height *H* is chosen such that *H/W* = 1.5, i.e. *H* = 2.27mm, *H* = 2.32mm and *H* = 1.57mm, respectively, for the DP780, DP980 (Fig. 2a) and DP1180 steels. From the C45 bar stock, we extracted cuboidal specimens with an aspect ratio of *H/W* = 1.5 and heights ranging from 3 to 8 mm. All specimens are extracted using wire electro-discharge machining (EDM) resulting in a surface roughness of *R*<sub>a</sub> ≈ 10µm (Fig. 2b). The longitudinal axis of the specimens is always aligned with the rolling direction (or bar axis). Using a laser micro-engraving system (Keyence, MD-X2500), a random speckle pattern with an average speckle size of about 50 µm in diameter is applied onto the specimen surface (Fig. 2c).

In addition to compression specimens, we also extracted standard tensile specimens with a 10 mm wide and 40 mm long gage section from the DP steel sheets. As for the compression specimens, the rolling direction is aligned with the axis of loading in the tension experiments. A random speckle pattern featuring black speckles with an average diameter of about 100 µm is applied on the specimen gage section using the same laser micro-engraving system (as for the compression specimens).

*Miniature compression testing procedure.*

Static compression experiments are performed using a universal testing machine (Instron. Model 5980) equipped with two parallel stainless steel loading platens (Fig. 3a). The lower plate is rigidly connected ⊙ to the base of the testing machine. The upper plate is connected to the movable cross-head through a modified axial spherical plain bearing Ø. This bearing allows for the self- alignment of the upper and lower plates; its position is locked after a platen-to-platen compression test. Small TiN tungsten carbide plates (3.18 mm thick, 10 mm edge length) are placed between the specimen and the stainless steel platens (Fig. 3b); the tungsten inserts prevent the local indentation of the steel platens, reduce friction and facilitate the observation of the specimen. In addition, we apply a grease-type of lubricant (Kluber Iso-flex NBU 15) between the specimen and the loading platens.

A telecentric lens © with a magnification of 3x and a focal aperture of F20.1 is used to monitor the specimen front surface during the experiment. This lens features a distortion below 0.1 % and a working distance of 110 mm. The 28 × 20mm camera sensor Ø has a resolution of 12 megapixels, providing a spatial resolution of approximately 1 µm/pixel. Two LED lighting systems with parallel rays between the two plates ensure illumination during the entire test. The cross-head speed is chosen in accordance with the specimen height such that an engineering strain rate of about 10⁻³/s prevails in all experiments.

*Tension experiments.*

Standard tension experiments are conducted at room temperature on a robot-assisted universal tensile testing system equipped with a 25 kN load cell and pneumatic wedge grips. A 5MP camera with Nikon 100 mm lenses recorded images at a frequency of 2 Hz with a spatial resolution of approximately 33.5 µm/pixel. The true stress–strain curve is extracted based on the measurements up to force maximum (point of onset of necking).

### 3.3. *Dynamic experiments*

A first Dropkinson bar configuration has been presented by Song et al. (2018) for the dynamic tensile testing at intermediate strain rates. Here, a new Dropkinson bar set-up (Fig. 4) is conceived to perform direct impact experiments on cuboidal miniature specimens, achieving strain rates of the order of 10²–10³/s. A vertical configuration is adopted to accelerate a striker bar through gravity only. For this, the striker bar ⊙ is dropped at a vertical distance *h* from the specimen's top surface. It is guided by a linear velocity *V*₀, generating a compressive loading pulse upon impact. The specimen is positioned on top of a sensing block which consists of a custom-made dynamic load cell Ø that forms a monolithic part with a large stationary mass (base block Ø).

*Mechanical analysis of Dropkinson bar system.*

The main features considered in the design of the Dropkinson bar system are:
<figure><img src="image_8.png" /><figcaption>Fig. 5. Virtual compression experiments: (a) finite element mesh of 1/8th of the cuboidal specimen comprised of 187,500 first-order solid elements; the open dot in the details shows the output node for computing the local strain according to Eq. (11); (b) Swift hardening curves and (c) friction coefficients for 100 experiments comprised in training and validation dataset; Swift-Voce hardening curves for (d) dual phase steels, and (e) aluminum alloys comprised in the testing datasets.</figcaption></figure>

<table><caption>**Table 2**

Mixed Swift-Voce model hardening parameters for four dual phase steels and for four aluminum alloys.</caption><thead><tr><th>Material</th><th>α [-]</th><th>A [MPa]</th><th>ε<sub>0</sub> [-]</th><th>n [-]</th><th>k<sub>0</sub> [MPa]</th><th>Q [MPa]</th><th>β [-]</th></tr></thead><tbody><tr><td>DP590</td><td>0.68</td><td>1031</td><td>0.0013</td><td>0.199</td><td>324.2</td><td>25.9</td><td>350</td></tr><tr><td>DP780</td><td>0.80</td><td>1272.6</td><td>0.0007</td><td>0.167</td><td>403.9</td><td>29.0</td><td>476.6</td></tr><tr><td>DP980</td><td>0.58</td><td>1448</td><td>0</td><td>0.097</td><td>275</td><td>16.2</td><td>715</td></tr><tr><td>DP1180</td><td>0.475</td><td>1987.7</td><td>0</td><td>0.131</td><td>457.8</td><td>17.4</td><td>733.5</td></tr><tr><td>AA5754</td><td>0.34</td><td>419.6</td><td>0.01</td><td>0.268</td><td>138.1</td><td>14.57</td><td>126.5</td></tr><tr><td>AA2024</td><td>0.368</td><td>798.5</td><td>0.015</td><td>0.202</td><td>240.0</td><td>10.53</td><td>363.8</td></tr><tr><td>AA7075</td><td>0.5</td><td>855.3</td><td>0.026</td><td>0.151</td><td>194.1</td><td>10.08</td><td>502.4</td></tr></tbody></table>

* Duration *T* of the compressive loading pulse generated by the striker bar. It is set by the length *L*<sub>str</sub> of the striker and the longitudinal wave propagation speed *c* = √(*E*/*ρ* in the bar material (Young's modulus *E*, density *ρ*). It equates to the round-trip travel time,

$$T = \frac{2L_{str}}{c} \qquad (1)$$

* Impact velocity *V*<sub>0</sub>. Due to the free fall of the striker bar over a distance *h* before impact, we have

$$V_0 = \sqrt{2gh} \qquad (2)$$

with the gravitational acceleration *g*.

* Impact energy *E*<sub>str</sub>: the kinetic energy of the striker bar at the instant of impact is
<figure><img src="image_9.png" /><figcaption>Fig. 6. (a) Computed axial force as a function of the normalized applied displacement δ/H for cuboidal and cylindrical specimens (friction coefficient μ = 0.15). (b) hardening curve used for the simulations; (c) Deformed configurations at normalized displacements δ/H = 0.10, δ/H = 0.20 and δ/H = 0.30, highlighting the characteristic barreling effect. Contours indicate the von Mises stress distribution, with principal stress directions represented for one elements at the center and two elements along the middle-line at the specimen's surface.</figcaption></figure>

$$E_{str} = mgh \qquad (3)$$

with m denoting the mass of the striker bar.

* Plastic work $W_{pl}$: the plastic work performed when deforming an ideal specimen (height H, cross-sectional area A_s) of an incompressible, perfectly-plastic material of yield stress $\sigma_y$ to a strain $\epsilon_{min}$ reads

$$W_{pl} = \sigma_y \epsilon_{min} H A_s \qquad (4)$$

* Engineering strain rate $|\dot{\epsilon}|$. Neglecting the deformation of the sensing block and assuming that $E_{str} \gg W_{pl}$, the expected engineering strain rate at the specimen level is

$$|\dot{\epsilon}| = \frac{V_0}{H} \qquad (5)$$

Due to the small size of the miniature compression specimens, the condition of quasi-static equilibrium is expected to be readily satisfied when testing metallic materials. The measurement of the force history $F[t]$ of the load cell just below the specimen may thus also be used to reconstruct the elastic wave that is generated in the striker bar upon

impact. Near the striker bar-specimen interface, the axial strain history $\epsilon_{str}[t]$ in the striker bar (cross-sectional area A, Young's modulus E, density ρ, wave speed c = $\sqrt{E/\rho}$) reads

$$\epsilon_{str}[t] = \frac{F[t]}{EA} \qquad (6)$$

The compressive strain wave ($\epsilon_{str} < 0$) reduces the velocity of the downward velocity $v[t]$ of the striker-specimen interface,

$$v[t] = V_0 + c\epsilon_{str}[t] \qquad (7)$$

Assuming the displacement of the bottom specimen boundary (i.e. the specimen-load cell boundary) to be negligible, we may thus compute the relative displacement of the specimen boundaries ($\delta > 0$ for specimen shortening),

$$\delta[t] = V_0 t + \frac{1}{c\rho A} \int_{0}^{t} F[\tau] d\tau \qquad (8)$$

In other words, based on the measurement of force history and the drop height only (to calculate $V_0$), the full force-displacement curve may be extracted from a Dropkinson bar experiment (provided that the striker bar properties are known).

Specific system configuration
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Mechanical fields in compression specimens at three different stages of the experiment ($\delta/H = 0.1, 0.2, 0.3$): Variation of (a) von Mises stress, (b) equivalent plastic strain, (c) radial stress, (d) axial stress, (e) circumferential stress, and (f) stress triaxiality along a line path from the specimen center to the closest point on the specimen surface. For the cuboidal specimens, we also show the variation of (g) the axial strain, and (h) the axial stress along the equatorial path on the specimen surface.</figcaption></figure>

Our target is to perform dynamic compression experiments up to engineering strains of -0.3 at average engineering strain rates $|\dot{\epsilon}|$ of 500 - 1000/s. For this, we will make use of the same cuboidal miniature specimens as for static compression testing with a maximum cross-sectional area of $A_s = 1.5 \times 1.5 \text{mm}^2$, maximum length $H = 2.5 \text{mm}$ and a strength of up to 2000MPa.

1. Choice of drop height: According to Eq. (5), deforming a 2.5 mm high specimen at strain rates of 500-1000/s requires impact velocities between about 1.25-2.5m/s. In terms of drop height, this range corresponds to 78-312mm.

2. Choice of the striker bar length: To attain an engineering strain about -0.4 at the above strain rates, the required duration of the loading pulse is 400-800µs which translates into a required steel striker bar length of 1000-2000mm (Eq. (1)). We thus chose a 2 m long striker bar which still fits (vertically) into a standard lab room if it is positioned at distance of about 300 mm from the specimen. The striker

bar features a 20° chamfer on its impact face (Fig. 4b) leaving a perfectly flat area of 10 mm diameter.

3. Choice of the striker bar diameter: Deforming the specimen up to a strain of -0.3, requires a plastic work of about $W_{pl} \approx 3.4J$ (Eq. (4)). We chose a steel striker bar of 30 mm diameter; it weighs about 11 kg and provides an impact energy of $E_{str} = 8.6J$ when dropping from a height of 78 mm. We thus satisfy the condition $E_{str} \gg W_{pl}$ for the targeted range of applications.

4. Load cell diameter: For the above maximum specimen dimensions, the load cell must be able to measure forces of up to 4.5 kN. We chose a load cell diameter of about 11 mm to be able to machine two parallel planes for the gluing of strain gages (Kyowa KFGS-5-350-C1-11) of the size 5 mm by 2 mm. Neglecting the cross-sectional area reduction due to the machining, the expected axial deformation at a force of 4.5 kN is about $2.4 \times 10^{-4}$ (which corresponds to an average axial stress in the load cell of about 47MPa. With an anticipated noise level of the order of $10^{-6}$ for the combination of strain gages, wiring
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Lambda method: (a) effect of $\lambda$ on the stress–strain curve estimate; (b) Mean relative error per experiment comprised in the training dataset; scatter plots of the optimal as a function of (c) friction coefficient, and (d) stress ratio (i.e. strain hardening). (e) Neural network predicting $\lambda$ as a function of both the friction coefficient and strain hardening. (f) Prediction vs. ground truth plot for the neural network model.</figcaption></figure>

and amplifier system (model Kyowa CDA-900A-DC), the load cell is expected to provide sufficient accuracy for our purposes. Furthermore, the load cell made from precipitation hardened W720 maraging steel is expected to remain in the elastic domain in case of accidentally dropping the striker bar without any specimen in place. We note that the maximum force generated by an impact from a height of 312mm is about 63kN (theoretical value for an impact onto a rigid target).

5. Load cell height: The height of the cylindrical load cell should be as low as possible to ensure its quasi-static equilibrium. We chose a height of 15 mm to facilitate the gluing of standard strain gages on its lateral surfaces and to ensure a short round trip duration of the wave within the dynamic load cell (about 6 µs).

6. Base block size: We chose a diameter of 120 mm and a height of 100 mm for the base block; the resulting mass is 8.8kg. Preliminary numerical simulations suggested that increasing the diameter or height did not improve the overall performance of the sensing block. The main purpose of the base block is to provide a stationary support with minimal displacement of the lower end of the load cell. As

discussed in the open literature (e.g. Tanimura (1984), Chuman et al. (1997), Galpin et al. (2008)), the large impedance mismatch between the load cell and the base block promotes the reflection of elastic waves, thereby allowing the load cell to remain in quasi-equilibrium.

The load cell is calibrated under static loading conditions by placing the sensing block into a universal testing machine (that is equipped with a certified load cell). To eliminate potential errors in the impact velocity estimate based on Eq. (2) (e.g. due to friction in bearings guiding the striker bar), the striker tip displacement history $\delta[t]$ is also measured using digital image correlation (DIC). For our system, preliminary measurements revealed that Eq. (2) overestimated the velocity by more than 20 % as compared to the DIC results. For this, a speckle pattern is applied onto the bar's lateral surface and monitored with a high-speed camera (Photron, model SA-Z) equipped with the same lenses as the quasi-static experimental setup (magnification of 3x and a focal aperture of F20.1). The high-speed camera is triggered by the load cell signal to guarantee the synchronized recording; it acquires images of 48 × 512
<figure><img src="image_12.png" /><figcaption>Fig. 9. Randomly selected examples from training dataset illustrating the quality of the fit obtained when using an optimal value of λ.</figcaption></figure>

pixels at a frequency of 504 kHz with a shutter speed of 1/600,000 s. The resolution is 6.65µm per pixel.

## 4. Finite element models

Numerical simulations of static uniaxial compression experiments are performed using isotropic and anisotropic plasticity models with hardening curves that are representative for DP steels. The main purpose of the simulations is to assess the accuracy and range of validity of the stress-strain curve estimates obtained from uniaxial compression experiments.

### 4.1. Material models

For all models, the yield condition,

$$f[\sigma, \bar{\epsilon}_p] = \bar{\sigma}[\sigma] - k[\bar{\epsilon}_p] = 0, \quad (9)$$

is expressed in the form of the difference of an equivalent stress measure $\bar{\sigma}$ of the stress tensor $\sigma$ and the yield stress $k = k(\bar{\epsilon}_p)$. The latter

is defined as a function of the equivalent plastic strain $\bar{\epsilon}_p$ (which is the integral of the equivalent plastic strain rate that is power-conjugate to the equivalent stress). Specifically, we make use of the Swift-Voce hardening law to define self-similar hardening,

$$k = \alpha A(\bar{\epsilon}_p + \epsilon_0)^n + (1 - \alpha) \{k_0 + Q(1 - \exp(-\beta\bar{\epsilon}_p))\}, \quad (10)$$

with the Swift parameters ($A, \epsilon_0, n$), the Voce parameters ($k_0, Q, \beta$) and the weighting factor $\alpha$.

For the isotropic material model, the von Mises stress is chosen as equivalent stress measure. For the orthotropic models, we adopt the equivalent stress definition of (Hill, 1948) along with a non-associated flow rule (Mohr et al., 2010).

### 4.2. Finite element model

The stress and strain results from the numerical simulations of static experiments are theoretically length scale independent. For the simulations of compression experiments on cuboidal specimens, we always chose $W = 1mm$ and $H = 1.5mm$. In case of simulations with cylindrical
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Method evaluation on testing data for $\mu = 0.10$. The ground truth (red curve) is given by the input hardening curve. Each subplot shows the results for a different material.</figcaption></figure>

specimens, we chose the specimen diameter $D = 1.13\,\text{mm}$ such that it features the same initial cross-sectional area $A_0$ as the cuboidal specimen. We exploit the orthogonal symmetry of the mechanical problem by discretizing only one-eighth of the specimens. The meshes (Fig. 5a) consist of first-order brick elements (element type C3D8R from the Abaqus library) with an edge length of approximately 0.01 mm. The origin of the Cartesian coordinate frame is placed at the very center of the full specimen. To enforce the symmetry in the solutions, we impose the conditions $u(0, y, z) = 0$, $v(x, 0, z) = 0$ and $w(x, y, 0) = 0$ to the boundary nodes of our one-eighth model, with $u$, $v$, $w$, denoting the displacements along the x-, y- and z-directions, respectively. The loading $\delta/2$ is applied at the top boundary ($z = H/2$) of the finite element model through a flat moving rigid surface. The displacement $\delta/2$ of the rigid wall along the z-axis ($\delta > 0$ for a downward motion) is defined with respect to its initial position at $z = H/2$. A kinematic contact model with Coulomb friction (friction coefficient $\mu$) is defined for the interaction of the specimen surface nodes with the rigid surface. Quasi-static loading conditions are ensured by scaling the material density such that the

simulations required 100,000 explicit stable time steps. Furthermore, the displacement loading is progressively applied as a “smooth” ramp. The software Abaqus/explicit is used to perform all simulations.

### 4.3. Dataset for training and validation

A first dataset comprised of results from finite element simulations (“virtual compression experiments”) is created using a von Mises model with associated plastic flow and Swift hardening. We generate 100 Mises-Swift materials by randomly drawing the Swift model parameters from the intervals $A \in [300, 1500]\,\text{MPa}$, $\epsilon_0 \in [10^{-5}, 10^{-2}]$ and $n \in [0.05, 0.3]$. In the FE model, the hardening curves are represented as piece-wise linear with a strain increment of 0.01. Fig. 5b shows the ensemble of all 100 hardening curves. Furthermore, we randomly draw 100 friction coefficients from the interval $\mu \in [0.02, 0.15]$ (Fig. 5c). Simulations are performed up to a displacement of $\delta/2 = 0.1944\,\text{mm}$ which equates to an average logarithmic axial strain of $-0.3$.

We save the input hardening curves as “ground truth”. After
<figure><img src="image_14.png" /><figcaption>Fig. 11. Method evaluation on testing data for $\mu = 0.05$. The ground truth (red curve) is given by the input hardening curve. Each subplot shows the results for a different material.</figcaption></figure>

completing a simulation, the applied displacement history $\delta[t]$, the axial force history $F[t]$ and the local axial strain at the center of the specimen surface,

$$ \epsilon_{\text{loc}}[t] = \ln[1 + w_c[t]/l_e], \quad (11) $$

are recorded at about 200 equally-spaced time increments. In Eq. (11), $w_c$ denotes the z-displacement of the upper node of the first element on the specimen surface (as highlighted in the insert in Fig. 5a), while $l_e = 0.01 \text{mm}$ denotes the corresponding element edge length.

## 4.4. Testing datasets

While the above dataset will be used for method development, an additional testing dataset is created to evaluate the model capabilities. We make use of the finite element model as before, but different material models. Specifically, we take the anisotropic material models that have been identified through experiments in earlier studies: four steels (DP590, DP780, DP980, DP1190) and four aluminum alloys (AA2024-

T351, AA5754, AA6014-T4, AA7075-T6). Table 1 summarizes the Lankford ratios along with all model parameters. Fig. 5d and 5e shows the corresponding equivalent stress versus equivalent strain curves. For each material, virtual compression experiments are performed for the friction coefficients $\mu = 0.05, 0.10$ and $0.15$. From each simulation, we extract the same data $\{\delta[t], F[t], w_c[t]\}$ as from the training and validation simulations. Note that the loading is applied along the rolling direction, while the displacement $w_c$ is extracted from a specimen surface that is perpendicular to the thickness direction of the sheet materials. Table 2..

## 5. Procedures for stress-strain curve extraction

In both the static and the dynamic upsetting experiments, the main measurements are: (i) the relative displacement of the loading platens, (ii) the force acting on the specimen, and (iii) surface strain fields. The challenge is to obtain a meaningful stress-strain curve from these measurements. This involves estimating an equivalent plastic strain history $\bar{\epsilon}_p[t]$ and the corresponding equivalent stress history $\bar{\sigma}[t]$ before
<figure><img src="image_15.png" /><figcaption>Fig. 12. Method evaluation on testing data for $\mu = 0.15$. The ground truth (red curve) is given by the input hardening curve. Each subplot shows the results for a different material.</figcaption></figure>

extracting a stress–strain curve by combining the two. Formally, we write

$$ \bar{\sigma}[\bar{\varepsilon}_p] = \bar{\sigma}[t]^{\circ} \bar{\varepsilon}_p[t], \qquad (12) $$

with the composition operator symbol $^{\circ}$. Provided that quasi-static conditions prevail at the specimen level, the same specimen and processing procedure may be used to extract the stress–strain curve from static and dynamic experiments.

## 5.1. Global method: Estimate based on applied displacement

A first set of strain and stress estimates is obtained based on global force and displacement measurements only. Denoting the relative displacement of the loading platens as $\Delta w = -\delta$ (with $\delta > 0$ for specimen shortening), the average axial strain reads,

$$ \varepsilon_{glob} = \ln\left(\frac{\Delta w}{H} + 1\right) \qquad (13) $$

Assuming incompressibility, we have the associated true stress estimate,

$$ \sigma_{glob} = \frac{F}{A_0} \exp[\varepsilon_{glob}] \qquad (14) $$

In (12), we then use the approximations,

$$ \bar{\sigma} \approx |\sigma_{glob}| \qquad (15) $$

and

$$ \bar{\varepsilon}_p \approx |\varepsilon_{glob} - \sigma_{glob}/E_s| \qquad (16) $$

with the Young’s modulus $E_s$. Equations (13) to (16) define the “global method” for extracting the hardening curve from a compression experiment.

## 5.2. Local method: Estimate based on surface strain measurement

As an alternative to deriving the stress–strain curve from global
<table><caption>**Table 3**
Comparison of the mean relative error (MRE) and maximum relative error (MaxRE) on the testing dataset.</caption><thead><tr><th>Material</th><th>Friction</th><th>Global MRE</th><th>Global MaxRE</th><th>Local MRE</th><th>Local MaxRE</th><th>Lambda MRE</th><th>Lambda MaxRE</th></tr></thead><tbody><tr><td>AA2024</td><td>0.05</td><td>1.2 %</td><td>1.4 %</td><td>3.2 %</td><td>5.1 %</td><td>0.7 %</td><td>0.8 %</td></tr><tr><td>AA5754</td><td>0.05</td><td>1.2 %</td><td>1.3 %</td><td>4.8 %</td><td>7.6 %</td><td>0.8 %</td><td>0.9 %</td></tr><tr><td>AA6014</td><td>0.05</td><td>1.0 %</td><td>1.1 %</td><td>1.3 %</td><td>2.7 %</td><td>1.0 %</td><td>1.2 %</td></tr><tr><td>AA7075</td><td>0.05</td><td>1.0 %</td><td>1.0 %</td><td>7.0 %</td><td>10.8 %</td><td>0.9 %</td><td>1.0 %</td></tr><tr><td>DP590</td><td>0.05</td><td>1.3 %</td><td>1.5 %</td><td>0.2 %</td><td>0.3 %</td><td>0.7 %</td><td>0.8 %</td></tr><tr><td>DP780</td><td>0.05</td><td>1.2 %</td><td>1.4 %</td><td>1.5 %</td><td>3.4 %</td><td>0.7 %</td><td>0.9 %</td></tr><tr><td>DP980</td><td>0.05</td><td>1.0 %</td><td>1.0 %</td><td>2.2 %</td><td>4.4 %</td><td>0.8 %</td><td>0.8 %</td></tr><tr><td>DP1180</td><td>0.05</td><td>1.0 %</td><td>1.1 %</td><td>1.6 %</td><td>3.6 %</td><td>0.8 %</td><td>0.9 %</td></tr><tr><td>averages</td><td>0.05</td><td>1.1 %</td><td>1.2 %</td><td>2.7 %</td><td>4.7 %</td><td>0.8 %</td><td>0.9 %</td></tr><tr><td>AA2024</td><td>0.1</td><td>2.5 %</td><td>3.4 %</td><td>6.4 %</td><td>8.5 %</td><td>0.6 %</td><td>1.4 %</td></tr><tr><td>AA5754</td><td>0.1</td><td>2.6 %</td><td>3.3 %</td><td>6.8 %</td><td>9.3 %</td><td>0.6 %</td><td>1.2 %</td></tr><tr><td>AA6014</td><td>0.1</td><td>2.6 %</td><td>3.2 %</td><td>1.7 %</td><td>2.4 %</td><td>0.5 %</td><td>1.0 %</td></tr><tr><td>AA7075</td><td>0.1</td><td>1.8 %</td><td>2.5 %</td><td>9.1 %</td><td>11.9 %</td><td>0.4 %</td><td>0.8 %</td></tr><tr><td>DP590</td><td>0.1</td><td>3.0 %</td><td>3.0 %</td><td>11.3 %</td><td>15.3 %</td><td>1.0 %</td><td>1.1 %</td></tr><tr><td>DP780</td><td>0.1</td><td>2.8 %</td><td>2.9 %</td><td>3.2 %</td><td>6.3 %</td><td>0.8 %</td><td>0.9 %</td></tr><tr><td>DP980</td><td>0.1</td><td>1.7 %</td><td>2.0 %</td><td>3.6 %</td><td>5.9 %</td><td>0.2 %</td><td>0.6 %</td></tr><tr><td>DP1180</td><td>0.1</td><td>2.0 %</td><td>2.2 %</td><td>3.5 %</td><td>5.9 %</td><td>0.2 %</td><td>0.4 %</td></tr><tr><td>averages</td><td>0.1</td><td>2.4 %</td><td>2.8 %</td><td>5.7 %</td><td>8.2 %</td><td>0.6 %</td><td>0.9 %</td></tr><tr><td>AA2024</td><td>0.15</td><td>2.5 %</td><td>3.4 %</td><td>6.4 %</td><td>8.5 %</td><td>0.6 %</td><td>1.4 %</td></tr><tr><td>AA5754</td><td>0.15</td><td>2.6 %</td><td>3.3 %</td><td>6.8 %</td><td>9.3 %</td><td>0.6 %</td><td>1.2 %</td></tr><tr><td>AA6014</td><td>0.15</td><td>2.6 %</td><td>3.2 %</td><td>1.7 %</td><td>2.4 %</td><td>0.5 %</td><td>1.0 %</td></tr><tr><td>AA7075</td><td>0.15</td><td>1.8 %</td><td>2.5 %</td><td>9.1 %</td><td>11.9 %</td><td>0.4 %</td><td>0.8 %</td></tr><tr><td>DP590</td><td>0.15</td><td>3.0 %</td><td>3.0 %</td><td>0.5 %</td><td>0.6 %</td><td>1.0 %</td><td>1.1 %</td></tr><tr><td>DP780</td><td>0.15</td><td>2.8 %</td><td>2.9 %</td><td>3.2 %</td><td>6.3 %</td><td>0.8 %</td><td>0.9 %</td></tr><tr><td>DP980</td><td>0.15</td><td>1.7 %</td><td>2.0 %</td><td>3.6 %</td><td>5.9 %</td><td>0.6 %</td><td>0.6 %</td></tr><tr><td>DP1180</td><td>0.15</td><td>2.0 %</td><td>2.2 %</td><td>3.5 %</td><td>5.9 %</td><td>0.2 %</td><td>0.4 %</td></tr><tr><td>averages</td><td>0.15</td><td>2.4 %</td><td>2.8 %</td><td>4.4 %</td><td>6.3 %</td><td>0.2 %</td><td>0.9 %</td></tr><tr><td>total average</td><td>0.1</td><td>2.0 %</td><td>2.3 %</td><td>4.3 %</td><td>6.4 %</td><td>0.6 %</td><td>0.9 %</td></tr></tbody></table>

displacement measurements, it may also be computed based on local strain measurements. Note that with digital image correlation (DIC), the strain field is measurable on the specimen front surface (y-z-plane). Assuming that possible perturbations of the mechanical fields are the lowest at the specimen center (i.e. far away from laterally-constraining frictional boundaries), we derive stress-strain curve estimates from the measured axial strain $\epsilon_{loc}$ at the center of the specimen surface. In close analogy with Eqs. (14)–(16), the stress-strain curve according to the “local method” is defined by the equations,

$$ \sigma_{loc} = \frac{F}{A_0} \exp[\epsilon_{loc}], \qquad (17) $$

$$ \bar{\sigma} \approx |\sigma_{loc}|, \qquad (18) $$

and

$$ \bar{\epsilon}_p \approx |\epsilon_{loc} - \sigma_{loc}/E_s| \qquad (19) $$

### 5.3. Lambda method: Global estimate with barreling correction

An enhanced estimate is obtained by introducing a multiplier $\lambda \ge 1$ when calculating the equivalent plastic strain according to the global method,

$$ \bar{\epsilon}_p \approx \lambda |\epsilon_{glob} - \sigma_{glob}/E_s| \qquad (20) $$

while using its inverse when estimating the equivalent stress,

$$ \bar{\sigma} \approx \frac{1}{\lambda} |\sigma_{glob}| \qquad (21) $$

This empirical modification is proposed after observing (from simulations) that stress-strain curve estimates with the global method were always slightly higher than the ground truth. Using the above mathematical form systematically improved the estimates.

The multiplier $\lambda$ indirectly accounts for the effect of barreling. While the degree of barreling is expected to scale with the friction at the

specimen boundary, strain hardening is also expected to affect the barreling. As a first rough measure of strain hardening, we introduce the stress ratio $\chi_{sig}$ of the global stress $\sigma_{glob}$ at a global strain of $\epsilon_{glob} = -0.2$ and that at global strain of $\epsilon_{glob} = -0.1$,

$$ \chi_{sig} = \frac{\sigma_{glob} |\epsilon_{glob} = -0.2|}{\sigma_{glob} |\epsilon_{glob} = -0.1|} \qquad (22) $$

Note that $\chi_{sig} - 1$ is equal to the secant modulus over the strain interval $[0.1, 0.2]$ as normalized by the yield stress at a strain of 0.1.

## 6. Results

### 6.1. Numerical results (static loading)

#### Strain and stress fields in cylindrical and cuboidal specimens.

We discuss the results from a numerical analysis of the uniaxial compression of cylindrical and cuboidal specimens. In our simulations, both specimens have the same height $H = 1mm$ and cross-sectional area of $A_0 = 1mm^2$. The upper loading platen (rigid surface) moves downwards, thereby compressing the specimen in a quasi-static manner. Fig. 6a shows the computed axial force as a function of the applied displacement $\delta$. Up to a displacement of about $\delta/H = 0.25$, the curves for the two specimens lie approximately on top of each other, with a difference in force of less than 1 %. Both curves are monotonically increasing, with a transition from a concave (negative curvature) to a convex (positive curvature) shape. This transition is due to the growing contribution of the cross-section area increase to the increase in total force.

Fig. 6b depicts deformed configurations at a displacement of $\delta/H = 0.10$, $\delta/H = 0.20$ and $\delta/H = 0.30$. In both the cylindrical and cuboidal specimens, we observe the characteristic barreling, with a more significant increase in cross-sectional area at the specimen center ($z = 0$) than at the specimen-to-platen contact surface ($z = H/2$). At a displacement of $\delta/H = 0.30$, the cross-sectional area ratio (center-to-top) is 1.23 and 1.31 for the cylindrical and cuboidal specimens, respectively. Along
<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Barreled cuboidal specimens (FEA) after applying a displacement of $\delta/H = 0.259$. The contour shows the magnitude of the transverse displacement for specimens with $H = 1.5mm$.</figcaption></figure>

with the barreling, friction causes significant deviations of the stress field from the ideal uniaxial state as multi-axial stress states develop. In our eighth models, the stress state perturbations are initially confined to regions near the top boundary. However, as the displacement increases, a state of triaxial compression builds up at the very center of the specimen, while biaxial stress states (compression-tension) develop in the equatorial zones of the specimens.

To gain more insight into the stress and strain distributions in compression specimens, we report the variation of individual components along the radial path from the specimen center towards the front surface (i.e. along the x- and r-axes of the cuboidal and cylindrical specimens, respectively). We refer to the point with $r = 0$ ($x = 0$) as the “center” and the point at $r = R$ ($x = w/2$) as the “surface”. The corresponding plots in Fig. 7 show similar trends for both specimens. The axial stress (Fig. 7d) is the highest at the center, reaching –970 MPa, –1150 MPa and –1350 MPa, at a displacement of $\delta/H = 0.1$, $\delta/H = 0.2$ and $\delta/H = 0.3$, respectively. At the surface, the corresponding axial stresses are significantly lower and in the reverse order: –800 MPa, –750 MPa and –640 MPa. Though these decreasing values suggest unloading, the von Mises stress keeps on increasing at the surface (Fig. 7a), i.e. the stress state is evolving and the stress point moving on the yield surface. While the radial stress (Fig. 7c) must remain zero on the surface ($\sigma_r = 0, \sigma_x = 0$), a tensile hoop stress ($\sigma_\theta > 0, \sigma_y > 0$) builds up at the surface (Fig. 7e). In the cuboidal specimen, we have 230 MPa, 350 MPa and 580 MPa, respectively.

The stress state (Fig. 7f) at the surface, we consider here the stress triaxiality, transitions from initially uniaxial compression ($\eta = -0.33$) to a shear ($\eta = 0$), i.e. a biaxial stress state with principal stresses of

opposite sign $\sigma_z \approx -\sigma_y$. At the center, we observe the build-up of compressive stresses along the radial and hoop direction, with $\sigma_r \approx \sigma_\theta$ and $\sigma_x \approx \sigma_y$ for the cylindrical and cuboidal specimens, respectively. At a displacement of $\delta/H = 0.3$, the stress triaxiality reaches about $\eta \approx -0.5$ at the center in both specimens. When looking at the equivalent plastic strain (Fig. 7b), the highest values are attained at the specimen center; the magnitude of the gradient along the radial direction increases as the applied displacement increases. However, the center-to-surface ratio of the equivalent strains is close to 1.5 for the three cases depicted in Fig. 6b. Due to the concave nature of the hardening curve, the differences in the corresponding von Mises stresses (Fig. 7a) are smaller than those in equivalent plastic strain. For $\delta/H = 0.1$, the von Mises stress in the cuboidal specimen decreases from 925 MPa to 895 MPa (ratio 1.03) when moving along the radial path. At $\delta/H = 0.3$, we observe a decrease from 1120 MPa to 1050 MPa (ratio 1.06). Also, the Mises gradient for the cylindrical specimen is larger than that for the cuboidal specimen.

For the cuboidal specimens, we also plotted the variations in the mechanical fields along the equatorial path on the specimen surface of the central cross-section. Note that the cuboidal specimen comprises four edges along which the stress state remains uniaxial compression throughout the entire experiments. We observe that the magnitude of the axial stress (Fig. 7h) increases when moving from the center towards the edge. Similarly, we observe a decrease in the magnitude of the axial strain along that path (Fig. 7g), with the strain magnitude dropping from 0.36 to 0.24 along the equatorial distance of 0.5 mm.

Overall, we conclude that the stress and strain fields in both specimens are similar. Even though cylindrical specimens are the standard
<figure><img src="image_17.png" /><figcaption>Fig. 14. Results from experiments on specimens of different shape and size: (a) Photographs of the deforming cylindrical specimens at δ/H = 0.2 showing only mild barreling; (b) Recorded force-displacement curves, (c) extracted stress-strain curves (Lambda method).</figcaption></figure>

choice when determining the stress-strain response of metals under compression, there is no obvious mechanical argument for excluding cuboidal specimens for that purpose. In the sequel, we will focus on extracting stress-stain curves from compression experiments on cuboidal specimens as these can be conveniently be extracted from sheet metal. The above analysis of the mechanical fields in cuboidal specimens shows a high level of heterogeneity: the axial stress varies by as much as ± 15 %, significant in-plane stresses build up (both tension and compression) attaining magnitudes of 40 % of the axial stress, and relative differences in the equivalent plastic strain of more than 100 % at a given state of loading. For standard testing conditions, even with proper lubrication, barreling appears to be impossible to prevent. Different from uniaxial tension experiments that feature uniform stress fields in an early phase of the experiment (up to the onset of necking), the mechanical fields in a uniaxial compression specimen are heterogeneous right from the beginning of the experiment. With the exception of fully inverse FE approaches (e.g. FEMU), any analytical technique for extracting the stress-strain curve from compression experiments may thus only yield an approximation.

### Training of the Lambda-method.

The Lambda method is proposed as an attempt of systematically improving the stress-strain curve estimates obtained with the global method. Given a stress-strain curve estimate of the global method, introducing the multipliers λ and 1/λ for the strain and stress, respectively, leads to a potentially improved estimate. For λ > 1, the curve estimate by the Lambda method always lies below that of the global method (Fig. 8a). Here, we use the training data set comprised of the results from 100 virtual experiments to demonstrate the benefits of the Lambda method before making a specific recommendation of λ for the processing of real experiments.

Fig. 9 shows the stress-strain curves for eight randomly picked experiments from our dataset. The ground truth, i.e. the input stress-strain curve, is always depicted as red solid line. The corresponding estimates obtained by the global method, i.e. by computing the stress-strain curve based on the assumption of a uniform stress and strain distribution throughout the specimen, are represented by green curves. The average difference in stress between the ground truth and global estimate over the strain interval 0.05, 0.25 is typically, a few percent (see values indicated in the plot legends in Fig. 9 and the blue bars in Fig. 8b). The average error for all 100 experiments is 2.0 %, while the worst experiments in our dataset feature an average error of 3.5 % for the global method (see plot with i = 50 in Fig. 9 for an example of a large mean relative error).

For each virtual experiment, we run an optimizer to select λ such as to minimize the least square distance between the two curves for each experiment. The corresponding optimized stress-strain curve estimates (black dashed curves in Fig. 9) are in perfect agreement with the ground truth (red solid curves). For each of the 100 experiments, we could find a multiplier that reduces the average error in stress to less than 0.1 % (see red bars in Fig. 8b). A scatter plot of the optimal values of λ as a function of the friction coefficients μ (Fig. 8c) reveals a strong correlation of these variables. In a first approximation, it may be described by the function

$$ \lambda = 1 + p_{1}\mu + p_{2}\mu^{2} \qquad (23) $$

with $p_1 = 0.269$ and $p_2 = -0.705$ (red curve in Fig. 8c). The same data is also plotted as a function of the stress ratio $\chi_{sig}$ (dimensionless measure of the strain hardening), but the correlation is weak (Fig. 8d). However, the successful training of a hyperbolic tangent activated neural network (comprised of 3 layers with 40 neurons per layer) suggests that some of the λ variations at high friction coefficients may be
<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Dimensions of wire-EDMed specimen (DP980 steel). Photographs taken with scale calibrated digital microscope. The bottom row shows the original sheet surface (right and left view), while the specimen surfaces shown in the two top rows have been machined (top, bottom, front and rear view). The color dots had been applied with a felt pen to distinguish the surfaces during manual specimen positioning.</figcaption></figure>

related to differences in strain hardening (Fig. 8e). Note that the $\lambda$ predictions of the network are close to the optimal $\lambda$ values (Fig. 8f).

In practice, the friction coefficient is usually unknown. Considering the friction coefficient range of [0.02, 0.15] as reasonable bounds for real experiments with proper lubrication, we ran an optimization over our full training dataset that minimized the mean square error between the $\lambda$-corrected estimate and the ground truth stress-strain curve. The minimization revealed that for $\lambda = 1.017$ the average error in stress over all 100 virtual experiments is as low as 0.7% (versus 2.0% for the global method). The yellow bars in Fig. 8b depict the average error per experiment for the Lambda method, which reaches a maximum of 1.58 %. Based on our training, we thus recommend using $\lambda = 1.017$ as default value for the processing of experimental results with the Lambda method.

### Comparison of stress-strain curve extraction methods.

After using the training dataset for calibrating the Lambda method based on hypothetical materials, we make use of the testing data set to compare the accuracy of all three stress-strain curve extraction techniques: (1) global method, (2) local method, and (3) Lambda method. The testing dataset comprises the results from numerical simulations for four dual-phase steels DP590, DP780, DP980, and DP1180 and four aluminum alloys (AA2024, AA5754, AA6014-T4, and AA7075-T6 for three platen-to-specimen friction coefficients ($\mu = 0.05, 0.10, 0.15$). Fig. 10 shows the corresponding stress-strain curves (equivalent stress

versus equivalent plastic strain) for the eight materials for a friction coefficient of $\mu = 0.10$. The results for the two other friction coefficients are shown in Fig. 11 ($\mu = 0.05$) and Fig. 12 ($\mu = 0.15$). Each subplot shows the input stress-strain curve (ground truth) as red solid line along with the estimates of the global method (green solid line), local method (blue solid line) and the Lambda method (black dashed line). In the plot legend, we also report the Mean Relative Error (MRE) in stress over the strain interval [0.05, 0.25]. Table 3 presents a more detailed summary for all 24 virtual experiments that includes the maximum relative error (MaxRE) in stress for each stress-strain curve estimate.

The comparison of the MaxRE for all materials and friction coefficients for the global method reveals that corresponding stress estimate is always within 3.4 % of the ground truth over the strain interval from 0.05 to 0.25. This error decreases to 1.5 % for the lowest friction coefficient ($\mu = 0.05$). It is worth noting that the global method is overestimating the stress for all experiments comprised in our dataset, it seems to predict a tight upper bound for the stress-strain curve. Among the DP steels and aluminum alloys, the highest differences are observed for the DP590 steel and the aluminum alloy AA2024 respectively. The local method provides less accurate and less consistent estimates than the global method. Even for low friction coefficients, we observe very substantial errors. For example, for the AA7075, we have an MRE of 7.0 % and a MaxRE of 10.8 % for $\mu = 0.05$. The worst prediction by the local method with a MaxRE of 15.3 % is obtained for the DP590 steel at a
<figure><img src="image_19.png" /><figcaption>Fig. 16. Axial (vertical) and transverse (horizontal) displacement fields observed during static compression experiment on DP980 miniature specimens with $H = 2.3mm$.</figcaption></figure>

friction coefficient of $\mu = 0.10$. Clearly, the global method is the better choice among the two, as using the local method seems to come at the risk of potentially amplifying the effect of the heterogeneous mechanical fields in compression specimens.

The estimates of the Lambda method (black dashed curves) feature the same robustness as those provided by the global method, but their accuracy is significantly higher. On our testing dataset, the Lambda method provides predictions with a mean error in stress of 0.6% which is a substantial improvement from 2.0% and 4.3% for the global method and local method, respectively. Also, the maximum relative error per experiment drops to 1.4% when using the Lambda method. Recall from Table 3 that the corresponding MaxRE for the global and local methods are 3.4% and 15.3%, respectively. Given that Lambda method yielded similar low errors for the training and testing datasets, it is recommended for extracting stress–strain curves from compression experiments on cuboidal specimens.

As friction coefficients below 0.15 appear to be achievable with standard experimental practices, no additional effort is made to come up with a refined friction correction technique. For reference, we plotted the barreled specimen shapes for all eight materials in Fig. 13. It may be useful for comparison with experimentally observed profiles in case of doubt on the lubrication conditions. We also note that small shape differences are observed between the front view and side view due to anisotropy. We refrain from exploring the quality of the predictions beyond true strains of 0.25; fold-over is expected to initiate at strains of 0.30 (friction and material dependent) which adds substantial complexity to the processing of experimental results.

We note that the effect of strain rate has been omitted on our numerical study. Due to barreling, the strain rate varies throughout the specimen, but the rate differences are negligible given that the stress–strain response of metals requires a change of the order of magnitude of

the strain rate to become noticeable (e.g. Meyers, 1994).

## 6.2. *Experimental results*

### Confirmation of shape and scale independence.

Cylindrical specimens with a diameter of 9.94 mm are extracted from the C45 bar. Furthermore, we extract cuboidal specimens of different sizes from the C45 steel bar. The smallest specimen extracted has the characteristic dimensions of the small specimens from sheet metal: 3 mm height and 2 mm wide cross-section. The intermediate and large specimens share the same height-to-width ratio of 1.5 in combination with a height of 6 mm and 8 mm, respectively (Fig. 14a). All experiments have been carried out at a strain rate of $10^{-3}/s$. The force–displacement curves up to a displacement of $\delta/H = 0.25$ are shown for the large cylindrical and cuboidal specimens in Fig. 14b. The results show that all curves lie approximately on top of each other, reconfirming the size and shape indifference of the specimen response through experiments. For reference, we also computed the corresponding stress–strain curve from the experiments on cuboidal specimens using the Lambda-method (Fig. 14c).

### Application I: measuring compression-tension strength differential in DP steels.

We apply the Lambda method to compression experiments on DP steel. Cuboidal specimens are extracted from DP780, DP980, and DP1180 sheet metals. With the sheet thickness defining the specimen width (and hence also the specimen height due to the fixed aspect ratio of 3:2), the targeted specimen dimensions for wire EDM machining are 1.51 × 1.51 × 2.27mm, 1.55 × 1.55 × 2.32mm and 1.05 × 1.05 × 1.57mm, respectively. Examples of digital microscopy images of the front (y-z-plane), side (x-z-plane) and top (x-y-plane) surfaces of a DP980 specimens are shown in Fig. 15. For the large specimens
<figure><img src="image_20.png" /><figcaption>**Fig. 17.** Results from uniaxial compression experiments on cuboidal specimens: (a) processing of force-displacement data; Hardening curves for (b) DP780, (c) DP980 and (d) DP1180. The percentage values shown next to the zoom detail is the average difference among the three curves over the interval [0.10, 0.25].</figcaption></figure>

$(H \approx 2.3mm)$, height measurements on the different specimen faces indicate maximum variations of $\pm 0.4\%$. For the small DP1180 specimens $(H \approx 1.6mm)$, the height variations are up to $\pm 0.7\%$. Note that for all specimens, the z-direction corresponds to the rolling direction and the y-direction to the transverse direction of the sheet metal. All experiments are performed at a quasi-static strain rate of $10^{-3}/s$, with three repeat experiments per material. The relative displacement of the loading platens is measured through three LVDTs that are positioned in a diametral set-up.

Fig. 16 provides an example of the surface displacement fields (as normalized by the initial specimen height) that are typically observed in static compression experiments on cuboidal specimens. An important qualitative feature is the symmetry of the surface displacement field, as it demonstrates the proper alignment of the test set-up during loading. Also, the comparison of Fig. 16c with the computed specimen contours in Fig. 13 suggests that the friction is reasonably low.

Fig. 17a illustrates the processing of the experimental recordings. Note that we are using the global LVDT to record the applied displacement.<fnref n="1" /> Starting with the recorded force-displacement curve (blue dots), we cut off all points for forces below 500 N and add a linear extrapolation with the slope measured between 500 and 1000 N. Furthermore, we shift the curve along the displacement axis to ensure that it starts at

the origin (red curve). Subsequently, we identify the machine stiffness such that the machine stiffness-corrected force-displacement curve (black curve) yields a stress-strain curve with an initial slope of 200 GPa (typical modulus for steel). Both axes of the obtained curve are multiplied by -1 to define the $F-\Delta w$-input for the formulas for the Lambda method (Eqs. (13)–(16), (19), (20)). The rationale for the stiffness correction is as follows: after establishing specimen-machine contact (e. g. for forces above 500 N), the experimental set-up may be considered as an elastic system in the small deformation regime. The force-displacement response for the local indentation problem as a well as the elastic part of the specimen response are linear. We hence apply a simple springs-in-series model to remove the displacement associated with the elastic deformation of our experimental system.

Fig. 17b to 17d summarize the obtained stress-strain curves for all experiments. We observe remarkably good agreement with an average difference among the three curves per material of less than 0.7 % over the equivalent plastic strain interval from 0.1 to 0.25. Fig. 18a shows the direct comparison of the stress-strain curves extracted from miniature compression experiments next to the corresponding curves extracted from uniaxial tension experiments (on regular large-scale specimens). For all three materials, we observe a small, but positive compression-tension strength differential effect, that is, the stress-strain curve for compression always lies above that for tension. When comparing the average compression and tension curves for the DP780 steel (Fig. 18b), the difference is only about 1 %, i.e. within the range of uncertainty of the Lambda method. It is far more significant for the DP980 steel (Fig. 18c) where we observe differences of about 4 %. Differences of about 2 % are observed for the DP1180 steel. Aside from the differences

<footnote n="1">¹ Specimen boundary displacement measurements would be difficult to achieve with DIC. Due to the roll-over effect at the boundary, the points to be observed disappear as the specimen is deformed; consequently, the displacement would only be measurable at a small distance from the boundary, introducing a non-negligible approximation).</footnote>
<figure><img src="image_21.png" /><figcaption>**Fig. 18.** Compression-tension strength differential effect in DP steels. (a) summary of all experimental curves (18 curves in total); for each material, the plot (a) shows three curves for compression (blue curves) and three curves for tension (orange curves) that are mostly lying on top of each other; the tensile curves have been truncated at force maximum (onset of necking); the average curves for compression and tension are depicted in (b) to (d) along with the computed differences between the average curves at selected strain points: (b) DP 780 steel, (c) DP980, (d) DP1180.</figcaption></figure>

in stress level, we note that the compression experiments provide access to the stress–strain curve for plastic strains that are at least twice as high as the necking strain in the tensile experiments (see orange curve end points in Fig. 18a). It is also worth noting that the obtained results are aligned with the findings of Maeda et al. (2018) who reported a strength differential of 5.2 % after testing a DP980 steel with an in-plane tension–compression technique.

*Application II: measuring strain rate effect on hardening of DP steels.*

Two specimens per material are tested in the custom-made Dropkinson device under dynamic loading conditions. The striker bar is always dropped from a height of about 160 mm which equates to a theoretical impact velocity of 1.77m/s. Fig. 19a shows the DIC-measured displacement history right after impact. All curves are very close which indicates a good repeatability of the launching mechanism (pin pull-out). Furthermore, we observe an approximately linear displacement time curve (with slope 1.75m/s) which implies that the striker bar velocity is barely reduced by the interaction with the specimen. At the instant of impact, the striker bar has a kinetic energy of about 17 J which is significantly larger than the plastic work required to shorten our specimens by 30 % (e.g. 1.9 J for the DP980 specimen under static loading conditions).

The force-history as measured by the sensing block is depicted in Fig. 19b for the DP980 and DP1180 specimens. It is remarkable to see a signal that is free from pronounced oscillations. We only see a high frequency noise with an amplitude of less than 50 N on the curves. It can be successfully removed by applying a low pass filter (butterworth filter with a cut-off frequency of 20 kHz). Note that the filtered signals (black dashed curves) lie on the “as measured” unfiltered force histories (solid

lines). In the sequel, we hence use the filtered force signals to compute the stress–strain curves. As for the static experiments, we follow the processing procedures shown in Fig. 17a to correct for the system stiffness (local elastic indentation of the striker bar and elastic deformation of the sensing block). For a 2 mm long steel specimen, the specimen transit time for an elastic wave is only 0.4µs. With an initial force build up at a rate of about 70N/µs (slope of the blue curve in Fig. 18b), the forces at the two sides of the specimen are always expected to be very close. Quasi-static conditions are likely to prevail even during the elastic phase of loading.

Fig. 19c-e show the obtained stress–strain curves after applying the Lambda method. For the 2.3 mm high specimens (experiments on the DP780 and DP980 steels), we observe an excellent repeatability, with the curves from repeat experiment lying approximately on top of each other. For the 1.5 mm high DP1180 specimens (Fig. 18e), the repeatability is still satisfactory, but less good than for the 2.3 mm specimens. The comparison with the static experiments reveals that all materials exhibit a pronounced positive strain rate effect on the yield stress at small strains. However, the strain hardening is significantly lower (close to zero) in the dynamic experiments than in the static experiments, leading to an intersection of the static and dynamic curves for the DP980 and DP1180 steels at an equivalent plastic strain of about 0.28. While the difference between the curves at low strains may be clearly attributed to a strain rate effect, the dynamic curves at high strains are also likely to be affected by the adiabatic heating. It is speculated that the reduced hardening is due to thermal softening. This could also explain why hardening ceases earlier as the material strength increases (and hence, as more heat is generated up to a given strain level). Observe that
<figure><img src="image_22.png" /><figcaption>**Fig. 19.** Results from dynamic compression experiments: (a) striker tip displacement history; (b) recorded force histories (raw and filtered signals); extracted stress-strain curves for (c) DP780, (d) DP980 and (e) DP1180 steels; (f) equivalent plastic strain rate histories from experiments with same striker velocity but different specimen heights.</figcaption></figure>

the apparent tangent modulus is already close to zero at a strain of 0.10 in the DP1180 steel, while a horizontal tangent is only observed at strains greater than 0.20 in the DP780 steel.

Fig. 19f shows the evolution of the equivalent plastic strain rate throughout the experiments. It is computed from taking the time derivative (central difference scheme) of the equivalent plastic strain history. In case of the DP780 and DP980 steels, it increases monotonically from 700/s to 900/s over the strain interval [0, 0.25]. Over the same interval, the strain rate of the DP1180 experiments increases from about 1200 to 1400/s. Note that the strain rate increases are due to the logarithmic definition of the strain. When taking the time derivative of Eq.

(13), we obtain that the logarithmic strain rate $|\dot{e}_{glob}|$ increases as the magnitude of the compressive strain, $|e_{glob}|$, increases (even if the relative velocity $\Delta\dot{w}$ of the loading platens is constant).

## 7. Conclusions

Based on the results from numerical simulations and compression experiments at different length scales and on different materials, the following conclusions are drawn:
1. In the case of barreling (due to the inevitable effect of friction), the mechanical fields in compression specimens become highly heterogeneous. For example, already at moderate strains, the stress field features triaxial compression states at the very specimen center, biaxial compression-tension states at the specimen surface, and uniaxial compression along the specimen edges (in case of cuboidal specimens). Unless inverse FEMU-type of techniques are used, the extraction of stress-strain curves from compression experiments become highly approximative even when using advanced DIC based surface strain measurement techniques.

2. There is no mechanical argument favoring the use of either cylindrical or cuboidal specimen geometries to determine the stress-strain curve for compression. However, miniature cuboidal specimens bring the advantage that they can be conveniently extracted from sheet metal (thickness of the order of 1 mm) with a dimensional uncertainty of less than ±10µm using wire EDM machining.

3. Deriving the stress-strain curve from force-displacement measurements only (global method) yields more accurate and stable estimates than the local method which is based on the strain measured at the center of the specimen's front face. Based on empirical evidence, an enhanced global method (Lambda method) is established which provides the stress-strain curve estimates with an average error in the stress of less than 1 % for equivalent plastic strains of up to 0.25.

4. Static in-plane compression experiments on specimens extracted from DP780, DP980 and DP1120 steel sheets revealed a good repeatability of the obtained stress-strain curves. The comparison with tensile experiments on the same materials revealed a yield strength differential of up to 4 %, with the stress-strain curve for compression systematically lying above that for tension.

5. Using a Dropkinson bar system (comprised of a gravity-accelerated striker bar and a stationary sensing block) in conjunction with miniature cuboidal specimens yields the stress-strain curve for strain rates of the order of 10²–10³/s. The obtained curves are exceptionally smooth (as compared to typical SHPB measurements) which is attributed to the rapid reaching of quasi-static equilibrium of miniature specimens. In the case of the DP steels, the yield stress at small strains increases by more than 20 % when changing from low to high stain rate loading. At the same time the hardening is significantly reduced.

Using cuboidal specimens along with the Lambda method is recommended as a robust technique to determine the stress-strain curve for compression – irrespective of the specimen size and loading velocity. Miniature cuboidal specimens are particularly suitable for characterizing the in-plane compression response of sheet metal. The use of miniature specimens is also recommended to characterize the high strain rate response of bulk materials as they yield remarkably noise-free stress-strain curves despite the dynamic nature of the loading conditions.

### CRediT authorship contribution statement

**Arthur Girard:** Writing – original draft, Visualization, Validation, Software, Investigation, Formal analysis. **Vincent Grolleau:** Writing – original draft, Methodology, Investigation, Conceptualization. **Dirk Mohr:** Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgements

The help with the experiments as well as the discussions with Dr. Christian Roth and Dr. Thomas Tancogne-Dejean are gratefully acknowledged. Special thanks to Professor Bertrand Galpin (UBS) for his advice on the design of the sensing block.

### Data availability

No data was used for the research described in the article.

### References

Alexander, J.M., Brewer, R.C., et al., 1963. Manufacturing properties of materials. (no Title).

An, Y.G., Vegter, H., 2005. Analytical and experimental study of frictional behavior in through-thickness compression test. J. Mater. Process. Technol. 160, 148–155.

Bao, Y., 2003. Prediction of ductile crack formation in uncracked bodies. Massachusetts Institute of Technology.

Bao, Y., Wierzbicki, T., 2005. On the Cut-off Value of negative Triaxiality for Fracture. Eng. Fract. Mech. 72, 1049–1069. https://doi.org/10.1016/j.engfracmech.2004.07.011.

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

Boger, R.K., Wagoner, R.H., Barlat, F., Lee, M.G., Chung, K., 2005. Continuous, Large Strain, Tension/Compression Testing of Sheet Material. Int. J. Plast. 21, 2319–2343. https://doi.org/10.1016/j.ijplas.2004.12.002.

Cazacu, O., Plunkett, B., Barlat, F., 2006. Orthotropic Yield Criterion for Hexagonal Closed Packed Metals. Int. J. Plast. 22, 1171–1194. https://doi.org/10.1016/j.ijplas.2005.06.001.

Chait, R., Curll, C.H., 1976. Evaluating Engineering Alloys in Compression, in: Recent Developments in Mechanical Testing. ASTM International. https://doi.org/10.1520/STP27852S.

Chang, Y., Wang, B.T., Li, X.D., Wang, C.Y., Zhao, K.M., Dong, H., 2020. A New Continuous Tensile-Compressive Testing Device with Friction-Counteracting and Anti-Buckling supporting Mechanism for Large Strain. J. Mater. Process. Technol. 278, 116540. https://doi.org/10.1016/j.jmatprotec.2019.116540.

Christiansen, P., Martins, P.A.F., Bay, N., 2016. Friction compensation in the upsetting of cylindrical test specimens. Exp. Mech. 56, 1271–1279.

Chuman, Y., Mimura, K., Kaizu, K., Tanimura, S., 1997. A sensing block method for measuring impact force generated at a contact part. Int. J. Impact Eng 19, 165–174.

Cook, M., Larke, E., 1945. Resistance of copper and copper alloys to homogeneous deformation in compression. J. Inst. Metals 71, 371–390.

Coppietters, S., Lava, P., Sol, H., Van Bael, A., Van Houtte, P., Debruyne, D., 2010. Determination of the Flow stress and Contact Friction of Sheet Metal in a Multi-Layered Upsetting Test. J. Mater. Process. Technol. 210, 1290–1296. https://doi.org/10.1016/j.jmatprotec.2010.03.017.

Fan, X.G., Dong, Y.D., Yang, H., Gao, P.F., Zhan, M., 2017. Friction Assessment in Uniaxial Compression Test: (a) New Evaluation Method based on Local Bulge Profile. J. Mater. Process. Technol. 243, 282–290. https://doi.org/10.1016/j.jmatprotec.2016.12.023.

Fardi, M., Abraham, R., Hodgson, P.D., Khoddam, S., 2017. A (New Horizon) for {{Barreling Compression Test}}: {{Exponential Profile Modeling}}. Adv. Eng. Mater. 19, 1700328. https://doi.org/10.1002/adem.201700328.

Galpin, B., Grolleau, V., Umiastowski, S., Rio, G., Mahéo, L., 2008. Design and application of an instrumented projectile for load measurements during impact. Int. J. Crashworthiness 13, 139–148.

Hartlen, D.C., Doman, D.A., 2019. A constitutive model fitting methodology for ductile metals using cold upsetting tests and numeric optimization techniques. J. Eng. Mater. Technol. 141, 11008.

Hemker, K.J., Sharpe Jr, W.N., 2007. Microscale characterization of mechanical properties. Annu. Rev. Mater. Res. 37, 93–126.

Hetz, P., Kraus, M., Merklein, M., 2022. Characterization of Sheet Metal Components by using an Upsetting Test with Miniaturized Cylindrical Specimen. CIRP Ann. 71, 233–236. https://doi.org/10.1016/j.cirp.2022.03.010.

Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. London. Ser. A. Math. Phys. Sci. 193, 281–297.

Hou, J., Ståhlberg, U., Keife, H., 1992. Bulging and folding-over in plane-strain upset forging. J. Mater. Process. Technol. 35, 199–212.

Kariem, M.A., Santiago, R.C., Govender, R., Shu, D.W., Ruan, D., Nurick, G., Alves, M., Lu, G., Langdon, G.S., 2019. Round-Robin test of split Hopkinson pressure bar. Int. J. Impact Eng 126, 62–75.

Khoddam, S., Solhjoo, S., Hodgson, P.D., 2021. State of the Art Methods to Post-Process Mechanical Test Data to Characterize the Hot Deformation Behavior of Metals. Adv. Mech. Eng. 13, 168781402110610. https://doi.org/10.1177/16878140211061026.

Kiener, D., Motz, C., Dehm, G., 2009. Micro-Compression Testing: (a) critical Discussion of Experimental Constraints. Mater. Sci. Eng. A 505, 79–87. https://doi.org/10.1016/j.msea.2009.01.005.

Kopec, M., 2023. Compression and fatigue testing of high-strength thin metal sheets by using an anti-buckling device. Exp. Mech. 63, 1003–1013.
Kopec, M., Liu, X., Gorniewicz, D., Modrzejewski, P., Zasada, D., Jóźwiak, S., Janiszewski, J., Kowalewski, Z.L., 2024. Mechanical response of 6061-T6 aluminium alloy subjected to dynamic testing at low temperature: experiment and modelling. Int. J. Impact Eng 185, 104843.

Kuhn, H.A., 2000. Uniaxial compression testing, Mechanical Testing and Evaluation. ASM. International 143–151.

Kuwabara, T., Tachibana, R., Takada, Y., Koizumi, T., Coppietters, S., Barlat, F., 2022. Effect of hydrostatic stress on the strength differential effect in low-carbon steel sheet. Int. J. Mater. Form. 15, 13.

Loizou, N., Sims, R.B., 1953. The yield stress of pure lead in compression. J. Mech. Phys. Solids 1, 234–243.

Lovato, M.L., Stout, M.G., 1992. Compression testing techniques to determine the stress/ strain behavior of metals subject to finite deformation. Metall. Trans. A 23, 935–951.

Maeda, T., Noma, N., Kuwabara, T., Barlat, F., Korkolis, Y.P., 2018. Measurement of the strength differential effect of DP980 steel sheet and experimental validation using pure bending test. J. Mater. Process. Technol. 256, 247–253.

Maeda, Y., Yanagawa, M., Barlat, F., Chung, K., Hayashida, Y., Hattori, S., Matsui, K., Brem, J.C., Lege, D.J., Murtha, S.J., et al., 1998. Experimental analysis of aluminum yield surface for binary Al: Mg alloy sheet samples. Int. J. Plast. 14, 301–318.

Meyers, M.A., 1994. Dynamic Behavior of Materials, Wiley.

Mohr, D., Dunand, M., Kim, K., 2010. Evaluation of associated and non-associated quadratic plasticity models for advanced high strength steel sheets under multi-axial loading. Int. J. Plast. 26, 939–956. https://doi.org/10.1016/j.ijplas.2009.11.006.

Mohr, D., Marcadet, S.J., 2015. International Journal of Solids and Structures Micromechanically-motivated phenomenological Hosford – Coulomb model for predicting ductile fracture initiation at low stress triaxialities. Int. J. Solids Struct. 67–68, 40–55. https://doi.org/10.1016/j.ijsolstr.2015.02.024.

Osakada, K., Kawasaki, T., Mori, K., Taniguchi, N., 1981. A method of determining flow stress under forming conditions. CIRP Ann. 30, 135–138.

Paredes, M., Wierzbicki, T., 2020. On mechanical response of Zircaloy-4 under a wider range of stress states: from uniaxial tension to uniaxial compression. Int. J. Solids Struct. 206, 198–223.

Pawelski, O., 1967. Über das Stauchen von Hohlzylindern und seine Eignung zur Bestimmung der Formänderungsfestigkeit dünner Bleche. Arch. Für Das Eisenhüttenwes. 38, 437–442.

Pillinger, I., Hartley, P., Sturgess, C.E.N., Rowe, G.W., 1985. An Elastic-Plastic Three-Dimensional Finite-Element Analysis of the Upsetting of Rectangular Blocks and Experimental Comparison. Int. J. Mach. Tool Des. Res. 25, 229–243. https://doi.org/10.1016/0020-7357(85)90003-4.

Prakash, G., Singh, N.K., Gupta, N.K., 2023. Flow behaviour of Ti-6Al-4V alloy in a wide range of strain rates and temperatures under tensile, compressive and flexural loads. Int. J. Impact Eng 176, 104549.

Rastegaev, M.V., 1940. A new method of homogeneous compression of specimens for determining flow stress and the coefficient of internal friction. Zadovsk Lab 6, 345.

Schey, J.A., Venner, T.R., Takomana, S.L., 1982. The effect of Friction on pressure in Upsetting at Low Diameter-to-Height Ratios. J. Mech. Work. Technol. 6, 23–33. https://doi.org/10.1016/0378-3804(82)90017-1.

Scudino, S., Surreddi, K.B., Wang, G., Eckert, J., 2010. Enhanced plastic deformation of Zr41. 2Ti13. 8Cu12. 5Ni10Be22. 5 bulk metallic glass by the optimization of frictional boundary restraints. Scr. Mater. 62, 750–753.

Soares, G.C., Hokka, M., 2021. The Taylor-Quinney coefficients and strain hardening of commercially pure titanium, iron, copper, and tin in high rate compression. Int. J. Impact Eng 156, 103940.

Song, B., Sanborn, B., Heister, J., Everett, R., Martinez, T., Groves, G., Johnson, E., Kenney, D., Knight, M., Spletzer, M., 2018. Development of “Dropkinson” bar for intermediate strain-rate testing, in: EPJ Web of Conferences. p. 2004.

Tanimura, S., 1984. A new method for measuring impulsive force at contact parts. Exp. Mech. 24, 271–276.

Uchic, M.D., Dimiduk, D.M., 2005. A Methodology to Investigate size Scale Effects in Crystalline Plasticity using Uniaxial Compression Testing. Mater. Sci. Eng. A 400–401, 268–278. https://doi.org/10.1016/j.msea.2005.03.082.

Woodward, R.L., 1977. Communication: a note on the determination of accurate flow properties from simple compression tests. Metall. Trans. A 8, 1833–1834.

Wu, S., Zhou, C., Shi, Y., Hu, A., Xiao, X., 2023. Plasticity, ductile fracture and ballistic impact behavior of Ti-6Al-4V Alloy. Int. J. Impact Eng 174, 104493.

Xinbo, L., Fubao, Z., Zhiliang, Z., 2002. Determination of metal material flow stress by the method of C-FEM. J. Mater. Process. Technol. 120, 144–150.

Yao, Z., Mei, D., Shen, H., Chen, Z., 2013. A {Friction Evaluation Method based} on {{Barrel Compression Test}}. Tribol. Lett. 51, 525–535. https://doi.org/10.1007/s11249-013-0164-4.

Li, Y.L., Guo, W.-G., 2006. Miniature-Hopkinson bar technique. 爆炸与冲击 26, 303–308.