# Assessing the Potential of Heterogeneous Mechanical Tests for Sheet Metals Through Experimentally Measured Full-Fields

M. Gonçalves, Sandrine Thuillier, A. Andrade-Campos

► To cite this version:

M. Gonçalves, Sandrine Thuillier, A. Andrade-Campos. Assessing the Potential of Heterogeneous Mechanical Tests for Sheet Metals Through Experimentally Measured Full-Fields. Experimental Mechanics, 2025, 10.1007/s11340-025-01168-z. http://dx.doi.org/10.1007/s11340-025-01168-z

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# Assessing the Potential of Heterogeneous Mechanical Tests for Sheet Metals Through Experimentally Measured Full-Fields

M. Gonçalves<fnref n="1" /> ID · S. Thuillier<fnref n="2" /> · A. Andrade-Campos<fnref n="1" />

Received: 6 August 2024 / Accepted: 25 February 2025

© The Author(s) 2025

## Abstract

**Background** Numerical simulation is becoming essential in the mechanical design of sheet metal components, requiring advanced material models, composed of many unknown parameters, to accurately describe complex material behavior. Traditionally, these parameters are identified through multiple quasi-homogeneous tests, each providing specific mechanical data on a particular strain state. The emergence of heterogeneous mechanical tests has revolutionized this process by enabling the capture of a wide range of strain states in a single experiment.

**Objective** This study focuses on the experimental analysis of three heterogeneous mechanical tests, previously studied numerically. The main objective is to confirm the quality and relevance of the mechanical deformation observed when using real data and evaluate the sensitivity of these tests to different high-strength steels.

**Methods** Uniaxial loading tests were conducted on three different specimen designs, using Stereo Digital Image Correlation to capture the mechanical fields on the surface. Multi-DIC systems were used to measure the out-of-plane behavior observed for a specimen design to increase the strain richness provided by the test. The repeatability of these tests is checked due to their complex designs.

**Results** The results show that the potential of heterogeneous mechanical tests remains unchanged when tested in real-world experimental settings.

**Conclusions** When combined with full-field measurement techniques, these can provide a wide range of mechanical behavior data from a single test, reducing the number of tests needed for advanced material characterization.

**Keywords** Heterogeneous tests · Experimental analysis · Digital Image Correlation · Material behavior · Sheet metal

## Introduction

Numerical simulation has become an essential tool for the reliable and efficient mechanical design of new metallic components. Finite element simulations have made it possible to digitalize sheet metal forming operations, reducing the

number of failed processes, and shortening the development lead time of new parts. However, for the numerical model to accurately reproduce the most common complex material behaviors present in sheet metal processes, an advanced material model is required. Such a model usually involves many parameters to be identified and therefore requests mechanical data from many quasi-homogeneous tests. Each one of these tests requires a homogeneous mechanical state for the direct derivation of strain and stress [1]. Mechanical tests with heterogeneous fields have been investigated to speed up this task and reduce the experimental effort. Due to their complex configurations or loading conditions, multiple strain and stress states may occur in the specimens, increasing the mechanical behavior information provided by a single test. However, when dealing with heterogeneous tests, full-field measurement techniques and inverse identification methods are mandatory. With the development of full-field measurement techniques, such as Digital Image

<figure><img src="image_2.png" /></figure>

M. Gonçalves
mafalda.goncalves@ua.pt

S. Thuillier
sandrine.thuillier@univ-ubs.fr

A. Andrade-Campos
gilac@ua.pt

<footnote n="1">¹ Department of Mechanical Engineering, TEMA - Centre for Mechanical Technology and Automation, LASI - Intelligent Systems Associate Laboratory, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal</footnote>

<footnote n="2">² Univ. Bretagne Sud, UMR CNRS, IRDL, F-56100 Lorient, France</footnote>
Correlation [2], it is theoretically possible to retrieve a het-
erogeneous strain or displacement field on the surface of the
specimen [3]. A large number of material model parameters
can therefore be identified using the full-field data from a sin-
gle heterogeneous test, reducing the number of tests required
for the procedure. Since it is not possible to compute the stress
and strain components directly from the experimental data,
the use of inverse methods [4] is required to solve the identi-
fication problem iteratively. This integrated approach, often
referred to as Material Testing 2.0 [5], is showcased in recent
literature [1, 6, 7].

Numerous studies, mostly numerical ones, have explored
the use of heterogeneous mechanical tests along with full-
field measurements for inverse identification procedures, as
evidenced by the following works [3, 8–16]. These works
address the use of different test configurations for improv-
ing material models’ accuracy. Several attempts have already
been made in recent years to propose a test configuration
capable of providing sufficient mechanical information for
the model calibration procedure on its own [17]. A few
works have numerically analyzed in detail the potential of
heterogeneous mechanical tests for model calibration based
on mechanical indicators [18–20]. From the strain field
richness to the model parameters’ identifiability, the perfor-
mance of several mechanical tests was compared, showing
the importance of the test designs for the accuracy of the cali-
bration. From these works, it is possible to establish two main
approaches for test design that can be summarized under two
objectives: the maximization of the strain/stress states present
in a test [8, 9, 16, 21–30] and the accuracy of the identified
material model parameters [12, 31–36]. On the one hand, the
presence of a wider strain/stress state range leads to a richer
mechanical test, from which a larger quantity of material
behavior information can be extracted. The maximization of
the strain state diversity has been made through empirical
[12, 16, 21, 22] and optimization approaches [8, 9, 27–30].
On the other hand, the use of the identification error as a cri-
terion for the test design process has been addressed more
recently, with the possibility of introducing the metrologi-
cal aspects of DIC in the optimization process [12, 34, 35].
Most of these works propose numerical methodologies for
the design of new test configurations. However, only some of
these obtained geometries have been applied to experiments,
especially for thin sheets.

Kim et al. [11] designed a specimen shape to show het-
erogeneous stress fields under uniaxial loading. Specimen
geometries were retrieved from a rolled sheet of 1.2 mm
thickness of DP780 steel and analyzed experimentally. The
full-field data extracted by stereo digital image correlation
was used to identify the hardening and anisotropic material
model parameters using the Virtual Fields Method (VFM). A
mechanical test designed via a shape optimization approach
was also designed and tested experimentally by Aquino et

al. [3] to calibrate a yield function via FEMU. The mate-
rial investigated was a DC04 mild steel with a thickness of
0.70 mm. Notched specimens are often used due to their easy
machinability and rather heterogeneous strain field. Rossi et
al. [37] tested notched specimens of stainless steel of 0.75
mm thickness at different material orientations to inversely
identify a complex yield criterion. Also, Guner et al. [38]
studied several variations of radius and widths for notched
specimens, and the selected geometry was machined in the
aluminum alloy AA6016-T4 with 1.0 mm of thickness and
tested in several orientations to calibrate the Yld2000-2D
yield criterion. Several works focused on using a cruci-
form specimen under biaxial loading to identify anisotropic
yield function parameters. Cooreman et al. [39] performed a
biaxial tensile test on a perforated cruciform specimen of a
DC06 steel of 0.8 mm thickness to determine the hardening
behavior and the yield locus. Teaca et al. [15] used standard
mechanical tests to identify most of the material parameters
and then non-conventional biaxial specimens to extract the
information related to the yield surface shape in the biaxial
tension range for an ES steel and a 1050 A aluminum, both
of 1 mm thickness. Moreover, Zhang et al. [40] combined
two non-conventional experiments: a geometry obtained via
a shape optimization procedure and a perforated cruciform
specimen, the same used in [21], which can generate the
required equibiaxial tension. The two experiments were con-
ducted on a universal tensile machine, one under uniaxial
tensile loading and the other using a biaxial apparatus. A dif-
ferent approach to obtaining the biaxial stress state without
the need for additional testing equipment was addressed by
Macek et al. [10], who designed a heterogeneous mechan-
ical test with a pronounced biaxial stress state close to the
center of the specimen. Using a 304 stainless steel of 0.68
mm thickness, the data provided by the heterogeneous test
was complemented with data provided by standard tensile
tests performed in different directions to calibrate plastic
anisotropy. With the same goal, Thoby et al. [41], tested
experimentally four specimen designs at 45° to the rolling
direction to characterize the plastic anisotropic behavior of
the AA2198-T351 alloy with 4 mm thickness.

Some of the aforementioned works have shown promis-
ing results in terms of test richness. However, there is
still a need for further exploration and application of these
advanced techniques in real experimental settings. Issues
regarding specimen alignment and hence uniformity of the
load application may arise particularly when dealing with
non-symmetric samples. Also, from DIC, the sources of mea-
surement error are numerous and related to the measurement
environment: lighting, the CCD sensor noise, and lens qual-
ity, and the configuration of the workplace [25]. Also, errors
from the speckle pattern and spatial resolution limitations
can add complexity to the measurement of the displacement
and strain fields. There is likely an impact on the repeata-
bility of the strain measurement due to the varying quality of the applied speckle pattern, which may propagate into variations in the inversely identified parameters. Strain information may be lost during the optical measurement mainly near the free edges of the geometries. Due to the lack of material points in those areas, data extraction can be challenging. Several issues may appear and impact the results, resulting in non-comparable numerical and experimental data. The introduction of noise in numerical data through the use of synthetic images is already working towards that goal [42]. However, analyzing experimentally the mechanical tests is crucial to verify the potential given to them by numerical models and to analyze their stability with regard to the materials as previous studies were only conducted for one material and Martins et al. [26] has shown the dependency of the strain field on the material used.

In a previous study, the potential of heterogeneous mechanical tests was numerically investigated and compared [20]. Five advanced heterogeneous mechanical tests were analyzed numerically and the mechanical information was used to compute key performance indicators. At the end of the work, a final ranking of the mechanical tests was proposed based on the richness and quality of the information they can provide. To validate the results, three of the five mechanical tests are chosen to be experimentally investigated. The first one, entitled Notched, was studied numerically and experimentally by Rossi et al. [12] and, consequently, used to identify the constitutive parameters of anisotropic plasticity models. With good machinability, this design presents a single geometric singularity that produces a heterogeneous strain field and a large plastic strain area. In turn, the D specimen, designed by Jones et al. [22], was obtained via an iterative process. The initial shape was designed by engineering intuition and then iteratively updated to present the widest range of strain states with a single test. Lastly, the TopOpt specimen, whose design methodology was first proposed in elasticity [28] and then improved for elastoplasticity [43]. Buckling is evidenced experimentally and predicted numerically for the TopOpt geometry during a uniaxial loading test [20]. Characterized by an out-of-plane displacement, this behavior is usually avoided in sheet metal forming works and is associated with compressive states. However, it was proposed to take advantage of the unusual behavior by using multi-DIC systems to increase the strain richness provided by the test. The Notched and Dshape were also analyzed along with a specimen geometry designed by topology optimization in elasticity [19]. Despite the similarity of the design technique between this design and the TopOpt, the TopOpt was designed assuming elastoplastic behavior of the material. Results showed that the specimen designed with topology optimization presented the most heterogeneous strain field, however, it was classified as the least robust when it comes to experimental bias due to its complex design and consequent

mechanical fields. Therefore, the performance of a specimen with such a complex mechanical behavior needs to be investigated.

This work focuses on the experimental analysis of three heterogeneous mechanical tests, previously investigated numerically in [20], which are now subjected to an experimental procedure. Uniaxial loading tests are performed with the three specimen designs and Stereo Digital Image Correlation is employed to extract the mechanical fields from the tests' surface. Due to their complex design, their repeatability is checked. As a follow-up of the previous work [20], the main goal is to confirm the richness of the mechanical deformation when using real data as well as its sensitivity to different high-strength steels.

This paper is organized as follows: the next section, titled Experimental Setup, provides a detailed description of the selected specimen geometries, materials, and configuration of the test and DIC setups. The results are then presented and discussed in Section Results and Discussion, where an analysis of the repeatability of the tests' behavior is made in the first place. Then, the performance of each test is investigated, taking into account the strain field richness through a mechanical indicator. The sensitivity of the TopOpt test to different high-strength steels is addressed. Finally, the paper culminates with the conclusions, where the main findings and limitations of the work are analyzed in the context of sheet metal characterization.

# Experimental Setup and Testing Procedure

## Specimen geometries

In this work, three advanced mechanical tests are investigated: Notched, D, and TopOpt specimens, whose geometries and overall dimensions are shown in Fig. 1. More details can be found in the original papers. In order to consider the repeatability of the tests, the dimensions of each geometry were measured and are presented in Table 1. In the case of the TopOpt specimen, since it resulted from a topology optimization design process, a smoothing procedure needed to be applied to adapt the geometry into a form suitable for machining. For practical testing purposes, the width of the specimen was scaled to match the grip dimensions used in the experimental setup, maintaining proportionality and experimental consistency.

For each one of the specimen geometries, a rectangular section is added on each side for the grips of the testing machine: 40, 50, and 55 mm of length for the Notched, Dshape and TopOpt specimens, respectively. These complex shapes are cut by water jet. The uniaxial external load is applied in the Y direction of the specimen frame, which is at 45° to the rolling direction.
<figure><img src="image_3.png" /><figcaption>**Fig. 1** Geometries and dimensions of the heterogeneous mechanical tests: (a) Notched [12], (b) D [22], and (c) TopOpt[28]. The test frame is defined for each test geometry. The $\bar{Y}$ is the tensile direction and it is oriented at 45° to the rolling direction</figcaption></figure>

## Materials

Three different high-strength steels are investigated in this study: a dual-phase steels DP600 and DP780, and HF1050, provided in sheets of thickness 0.8, 1.5, and 1.0 mm, respectively. The mechanical properties of the three materials were measured with tensile tests at several orientations to the rolling direction and are given in Table 2. The three steels exhibit values of the standard deviation of the plastic anisotropy coefficients, $\bar{r}$, closer to the isotropic value. Also, the $\Delta r$ property, computed as an average of the plastic anisotropy coefficients, shows a weak in-plane anisotropy

associated with the three materials. The HF1050 steel is the one with the highest ultimate yield stress, $R_m$. Although highly anisotropic materials have not been used, any resulting changes in strain states were expected to be moderate, similar to observations in uniaxial tension tests.

## Test Conditions

The experiments are conducted on an Instron 5969 testing machine, equipped with a load cell of 50 kN maximum capacity. The specimens are clamped on both sides and to ensure quasi-static conditions, a test speed of 1 mm/min for the

<table><caption>**Table 1** Thickness (t) and width (W) measurements in mm for the experimental runs of the three specimens (Notched, Dshape and TopOpt) for the DP600, DP780 and HF1050 steels</caption><thead><tr><th>Material</th><th></th><th>Notched<br>t [mm]</th><th>W [mm]</th><th>Dshape<br>t [mm]</th><th>W [mm]</th><th>TopOpt<br>t [mm]</th><th>W [mm]</th></tr></thead><tbody><tr><td rowspan="3">DP600</td><td>Ref.</td><td>0.800</td><td>40.00</td><td>0.800</td><td>50.00</td><td>0.800</td><td>93.00</td></tr><tr><td>Run 1</td><td>0.788</td><td>37.85</td><td>0.792</td><td>47.32</td><td>0.772</td><td>87.94</td></tr><tr><td>Run 2</td><td>0.785</td><td>37.85</td><td>0.799</td><td>47.31</td><td>0.812</td><td>88.07</td></tr><tr><td rowspan="3">DP780</td><td>Ref.</td><td>—</td><td>—</td><td>—</td><td>—</td><td>1.500</td><td>93.00</td></tr><tr><td>Run 1</td><td>—</td><td>—</td><td>—</td><td>—</td><td>1.487</td><td>93.02</td></tr><tr><td>Run 2</td><td>—</td><td>—</td><td>—</td><td>—</td><td>1.500</td><td>93.06</td></tr><tr><td rowspan="3">HF1050</td><td>Ref.</td><td>—</td><td>—</td><td>—</td><td>—</td><td>1.000</td><td>93.00</td></tr><tr><td>Run 1</td><td>—</td><td>—</td><td>—</td><td>—</td><td>1.014</td><td>93.03</td></tr><tr><td>Run 2</td><td>—</td><td>—</td><td>—</td><td>—</td><td>0.978</td><td>93.04</td></tr></tbody></table>
<table><caption>**Table 2** Mechanical properties of the DP600, DP780 and HF1050 high-strength steels</caption><thead><tr><th></th><th>R<sub>m</sub> [MPa]</th><th>r&#773;</th><th>&Delta;r</th><th>t [mm]</th></tr></thead><tbody><tr><td>DP600</td><td>661.9</td><td>0.999</td><td>0.085</td><td>0.8</td></tr><tr><td>DP780</td><td>786.4</td><td>0.906</td><td>0.048</td><td>1.5</td></tr><tr><td>HF1050</td><td>1093.0</td><td>0.912</td><td>0.069</td><td>1.0</td></tr></tbody></table>

Notched and 3 mm/min for the D and TopOpt specimens are used. Moreover, these values are set to have a similar average strain state, though it may be less meaningful for heterogeneous tests. For the sake of example, for the DP600 steel, the tensile force recorded for the Notched specimen to achieve rupture was 10.6 kN while for the D specimen was 9.56 kN. The TopOpt test, due to its dimension compared to the others, requires a tensile force of 18.95 kN.

A stereo Digital Image Correlation (DIC) system is used to measure the deformation on the specimen surface during the test. Stereo DIC allows the recording of out-of-plane movements, which can be caused by misalignments of the camera to the specimen surface or by instabilities such as buckling [37]. This is done by pointing two cameras at the surface of the specimens. A reference image is taken when the specimen is at rest. Then, sets of images are recorded by both cameras simultaneously at a constant image capture rate of 1 Hz. By comparing two recorded pictures of the specimen captured by both cameras, it is possible to obtain the displacement and strain fields. The optical measurement requires the application of a random speckle pattern on the specimen surface that follows the deformation of the test. In this work, a spray-painting technique is used consisting of a uniformly distributed mate white coat and a layer of random

black dots. An example of the speckle pattern before the test is illustrated in Fig. 2 for each one of the specimen designs.

Experimental data was collected using the commercial software, VIC-3D [44], along with two 5-megapixel Basler acA2440-35um cameras, each having a maximum resolution of 2048 x 2448 pixels. The cameras were set up as presented in Table 3 and were synchronized with the load cell to capture the corresponding tensile force for each image. To maximize the accuracy of the results and take advantage of the camera sensor's potential, the two cameras were placed vertically for the D and TopOpt specimens, as shown in Fig. 3. For the TopOpt test, due to its out-of-plane behavior, two identical DIC systems were employed, with one system positioned on each side of the specimen to capture full-field strain data. Both systems, composed of identical cameras and lenses, were independently calibrated and carefully aligned using VIC software to ensure all cameras focused on the same central point of the specimen. A wooden panel was placed in the background to prevent saturation and minimize reflection errors, with a rectangular opening allowing an unobstructed recording by both systems. This configuration was essential for accurately capturing the complex strain fields with the required precision.

The post-processing analysis was performed using MatchID software [42]. For accurate results, it is crucial to carefully choose the correlation algorithm, subset size, and step size when obtaining the experimental displacement fields. In order to select the best DIC settings, a performance analysis is conducted [42]. The goal of this analysis is to find a balance between strain resolution (i.e., the noise floor) and signal reconstruction (e.g., the von Mises strain) [40]. To achieve this, different combinations of DIC settings such as subset size, subset shape functions, interpolation func-

<figure><img src="image_4.png" /><figcaption>**Fig. 2** Speckle pattern applied by spray on the three specimens: (a) Notched, (b) D and (c) TopOpt</figcaption></figure>
<table><caption>**Table 3** Hardware configuration and optimal DIC settings for the analyses</caption><thead><tr><th></th><th>Notched</th><th>D</th><th>TopOpt</th></tr></thead><tbody><tr><td colspan="4"><strong>Hardware specifications</strong></td></tr><tr><td>Camera 0 (noise)</td><td>Basler ac A2440-35um 5MP (0.68%, 1.07%, 1.03%)</td><td></td><td></td></tr><tr><td>Image resolution</td><td>2448 x 2048</td><td>2048 x 2448</td><td>2048 x 2448</td></tr><tr><td>Camera 1 (noise)</td><td>Basler ac A2440-35um 5MP (0.74%, 0.77%, 0.68%)</td><td></td><td></td></tr><tr><td>Image resolution</td><td>2448 x 2048</td><td>2048 x 2448</td><td>2048 x 2448</td></tr><tr><td>Focal length [mm]</td><td>35</td><td></td><td></td></tr><tr><td>Stereo angle [°]</td><td>25</td><td></td><td></td></tr><tr><td>FOV [mm]</td><td>50x80</td><td>70x90</td><td>100x190</td></tr><tr><td>Distance [mm]</td><td>450</td><td>446</td><td>860</td></tr><tr><td colspan="4"><strong>DIC processing parameters</strong></td></tr><tr><td>Correlation algorithm</td><td>ZNSSD</td><td></td><td></td></tr><tr><td>Interpolation</td><td>Local bicubic splines</td><td></td><td></td></tr><tr><td>Subset shape function</td><td>Quadratic</td><td></td><td></td></tr><tr><td>Subset size [px]</td><td>19</td><td>15</td><td>19</td></tr><tr><td>Step size [px]</td><td>6</td><td>6</td><td>8</td></tr><tr><td>Image prefiltering</td><td>Gaussian</td><td></td><td></td></tr><tr><td>Strain window [px]</td><td>7</td><td></td><td></td></tr><tr><td>Strain tensor</td><td>Euler-Almansi</td><td></td><td></td></tr><tr><td>Progress history</td><td>Spatial</td><td>Spatial</td><td>Spatial+Update reference</td></tr><tr><td>Noise floor (U)</td><td>3.9e-4</td><td>3.8e-3</td><td>1.4e-3</td></tr></tbody></table>

tions, step size, and strain window size are used to compute strain fields. The optimal DIC settings are determined by considering the trade-off between noise and signal reconstruction. Based on this selection criterion, the optimal DIC settings for each specimen design are presented in Table 3. Due to the significant differences in the Field of View (FOV) and distances between specimens, the observed noise floor, which represents a threshold value of the displacement below which measurements are indistinguishable from noise, may have different levels, as larger FOVs and greater distances reduce pixel density and increase sensitivity to environmental factors. This study used two different software: VIC-3D for experimental image capture and MatchID for post-processing. This choice was based on software availability and the capability of MatchID to optimize DIC settings through its Performance Analysis feature.

<figure><img src="image_5.png" /><figcaption>**Fig. 3** Experimental setup: (a) An apparatus with one DIC system was similarly mounted for the Notched and D specimens. For the Notched test, the cameras were placed side by side in the horizontal direction. (b) Two identical DIC systems were used for the TopOpt, one on each side of the test. A wood panel is added in the background to prevent saturation of both systems for TopOpt. Only a rectangular opening was cut for recording the images with system 2</figcaption></figure>
<table><caption>**Table 4** Absolute values and relative weights for the adjustment and normalization of the terms of $I_t$ indicator [45]</caption><thead><tr><th colspan="5">Absolute weights (w<sub>ai</sub>)</th><th colspan="5">Relative weights (w<sub>ri</sub>)</th></tr><tr><th>w<sub>a1</sub></th><th>w<sub>a2</sub></th><th>w<sub>a3</sub></th><th>w<sub>a4</sub></th><th>w<sub>a5</sub></th><th>w<sub>r1</sub></th><th>w<sub>r2</sub></th><th>w<sub>r3</sub></th><th>w<sub>r4</sub></th><th>w<sub>r5</sub></th></tr></thead><tbody><tr><td>1</td><td>4</td><td>0.25</td><td>1</td><td>1</td><td>0.3</td><td>0.03</td><td>0.17</td><td>0.4</td><td>0.1</td></tr></tbody></table>

## Mechanical Richness Indicator

In a previous work [20], key performance indicators were proposed to evaluate the potential of heterogeneous mechanical tests. Due to the numerical concept of the work, some metrics were computed based on the stress field. Since stress is not a quantity that can be experimentally measured, only the strain field richness indicator is here computed as in [45]. It can be defined as follows:

$$I_t = w_{r1} \frac{\text{Std}(\varepsilon_2/\varepsilon_1)}{w_{a1}} + w_{r2} \frac{(\varepsilon_2/\varepsilon_1)_R}{w_{a2}} + w_{r3} \frac{\text{Std}(\bar{\varepsilon}^{\text{eq}})}{w_{a3}} + w_{r4} \frac{\bar{\varepsilon}_{\text{max}}^{\text{eq}}}{w_{a4}} + w_{r5} \frac{\text{Av}(\bar{\varepsilon}^{\text{eq}})}{w_{a5}}, \quad (1)$$

where $\varepsilon_1$ and $\varepsilon_2$ are, respectively, the major and minor principal strains in the sheet plane. The ratio between the minor and the major principal strains ($\varepsilon_2/\varepsilon_1$) is used to define the strain state space. In the original formulation of the indicator, the equivalent plastic strain, $\bar{\varepsilon}^P$, is used as an evaluation criterion. However, due to the experimental character of this work, the equivalent strain, $\bar{\varepsilon}^{\text{eq}}$, and its maximum value, $\bar{\varepsilon}_{\text{max}}^{\text{eq}}$, are used instead. This indicator evaluates different features, from left to right in (equation (1)): (i) the strain state standard deviation, (ii) the strain state range, (iii) the standard deviation, (iv) maximum and (v) average values of the equivalent strain. These terms can be divided into two main groups: strain state range and heterogeneity and strain level. The importance of each term is adjusted and normalized using absolute ($w_{ai}$,

with $i \in [1, 5]$) and relative weights ($w_{ri}$, with $i \in [1, 5]$). The values chosen for these parameters, based on the original study performed with a mild steel [45], are given in Table 4.

## Results and Discussion

### Test Repeatability

Analyzing the repeatability of advanced mechanical tests is crucial due to the complexity of their geometry and the sensitivity of the techniques used. These tests provide detailed insights into critical material properties and behavior and, therefore, a thorough investigation of their repeatability is required. Due to that, for instance, a study by Fu et al. [46] adopts a double-notched specimen to run a VFM-based identification and analyzes the repeatability of the load-displacement curves and the temperature distributions of three experimental runs before the identification procedure. While homogeneous tests benefit from consistent conditions and test parameters, which lead to high repeatability by averaging strain over large areas and discarding the influence of material heterogeneity, heterogeneous tests do not offer this advantage. These inherently involve varied strain distributions and stress states across the specimen, which, combined with localized measurements, make the results more susceptible to variations due to material heterogeneity. Each test is unique and modeled with specific local boundary conditions, with full-field measurements as output. However, with the material remaining the same, if the local boundaries are similar, the resulting outputs should exhibit similar trends. Thus, ensuring similar trends across experimental runs confirms the reliability of the material response under comparable conditions. There are not many works discussing the need to

<figure><img src="image_6.png" /><figcaption>**Fig. 4** Displacement evolution of the three virtual extensometers, with the same length, defined at different positions for the (a) Notched, (b) D and (c) TopOpt specimens. Each line represents the displacement of a pair of points with the respective color. The black dashed line represents the average evolution of the three extensometers</figcaption></figure>
analyze the repeatability of heterogeneous tests, but this sen-
sitivity to local variations in material properties highlights
the need for multiple experimental runs to evaluate the con-
sistency and reliability of the test results.

In this work, two experimental runs are performed in the
same conditions with each specimen design. First of all, the
geometries' misalignment during the test is analyzed, which
is likely to appear in this kind of experiment. Due to this,
a non-uniform strain distribution may appear in the speci-
men. For each experiment, the displacement evolution was
recorded during the test and this information allowed the
creation of virtual extensometers that follow specific points
at the specimen surface throughout the test. Figure 4 sets an
example for one test of each specimen design, where the evo-
lution of the deformation of three virtual extensometers along
the Ȳ direction is represented. The three extensometers, each
one with one color, were placed at the same height along the
width of the specimens and the deformation evolution curve
is represented with the same color. An average of the three
evolution curves is represented by the black dashed line.

For the Notched specimen, the displacement evolution of
the three extensometers is rather similar. The three curves
would ideally overlap, however, due to the misalignment of
the specimen, the displacement evolution is not symmetric.
The displacement at rupture strongly depends on the location
of the extensometer and the rupture will occur on one side
of the specimen [47]. Regarding the D specimen, since the
geometry is not vertically symmetric, the behavior of the left
pair of points (orange) is different from the right one (green).
Sliding also occurs at the beginning but at a different level for
the three extensometers. Due to the symmetric design of the
TopOpt specimen, both virtual extensometers on the sides
of the specimen behave similarly to the one at the center
of the specimen. The specimens were held in place using
wedge grips, which are self-tightening. While some sliding
was observed during the tests, this was accounted for by

measuring local displacements with DIC, ensuring that the
sliding did not affect the accuracy of the results.

The information extracted from the virtual extensometers'
deformation makes it possible to plot the load-local displace-
ment curves related to each specimen design in Fig. 5. For
each specimen type, two test runs are analyzed and, from each
one, the average displacement of the virtual extensometers is
used to plot the load evolution. In the y-axis, the load value
is divided by the initial thickness of the sheet specimen so
that the load evolution curve does not account for disper-
sions in the specimens' dimensions. It is worth noting that
the similarity of the load-local displacement curves is a key
indicator of the test's ability to maintain stability and produce
robust results across different experimental runs. Comparing
the two tests with the Notched specimen, it can be noted that
there is a similar behavior between the two curves with a
slight difference in the yielding part. Regarding the D spec-
imen, the load level in Run 2 is higher than in the Run 1
and reaches rupture with a lower displacement. In contrast,
despite the unusual out-of-plane behavior, the load-local dis-
placement curves of the two runs of the TopOpt specimen
are quite similar.

As stated in a previous work [20], the TopOpt specimen
suffers from buckling instability during a uniaxial loading
test, leading to an out-of-plane behavior of the specimen
geometry. This event is represented, for the sake of exam-
ple, for an experimental run, in (Fig. 6(c)). The out-of-plane
displacement is depicted in the transverse direction at several
time steps through the test time until rupture. The grey curve
corresponds to the beginning of the tensile test and the black
one stands for the end of the test, where the displacement of
the grips is 18.43 mm. In order to understand if this behavior
due to a buckling phenomenon can be repeatable, the out-
of-plane displacement maps at the end of the test for the two
experimental runs with the TopOpt specimen are represented
in (Figs. 6(a) and (b)). Overall, both specimens present iden-

<figure><img src="image_7.png" /><figcaption>Fig. 5 Load-local displacement curves for two test runs with each specimen design: (a) Notched, (b) D and (c) TopOpt specimens. The load is divided by the initial thickness of the specimen to account for inaccuracies in the specimens' dimensions</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 6** Analysis of the TopOpt specimen out-of-plane behavior: displacement maps at the end of test for the (a) TopOpt03 and (b) TopOpt04 experimental runs; (c) Evolution of the out-of-plane displacement across the transversal direction through the test time, from the grey (no grips' displacement) to the black line (end of the test)</figcaption></figure>

tical configurations at the end of the test, being the minor differences between the displacement maps in the minimum and maximum values.

The three tests were analyzed across different experimental runs and it is possible to conclude that their behavior is quite similar. Despite the out-of-plane behavior, the TopOpt test presented robust results in the two test runs, meaning a repeatable load behavior and strain evolution. For each specimen type, an experimental run is chosen as a representative one from the two presented previously. Therefore, Run 2 is chosen for all the three tests.

## Mechanical Richness Analysis

This section provides a comprehensive comparison of the three mechanical tests (Notched, D, and TopOpt) taking into account their performance and potential to provide important mechanical information. In the first place, Fig. 7 depicts the principal strain distributions captured at the last frame before fracture as well as the principal strain diagrams associated

with each specimen. Regarding the strain maps, the maximum and minimum strain distributions are represented for the Notched, D, and TopOpt specimens. For the TopOpt, the strain distributions of both surfaces, extracted by DIC systems 1 and 2, are shown. The information from both systems is also presented in two principal strain diagrams.

For the Notched specimen, the highest major strain values are localized in the center part of the specimen and the rest is under low strain values. Due to its similarity to the well-known dogbone specimen, it presents predominantly material points in uniaxial tension. In contrast, the D specimen presents a more dispersed strain field, presenting some areas where the higher values are localized. These are placed near the edges and therefore may be missed during the DIC measurement. Also, due to the small thickness of the thin arm, the extraction of information may not be possible. The peak value is achieved in the thin arm as well as rupture, as represented in (Fig. 8(b)). The D specimen presents points in pure shear besides the points in uniaxial tension filling almost the entire strain state range for small and intermediate magnitude values. For the TopOpt specimen, the strain fields from both
<figure><img src="image_9.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>(c)</figcaption></figure>

<caption>**Fig. 7** Experimental maximum and minimum principal strain fields of the (a) Notched, (b) D, and (c) TopOpt (DIC systems 1 and 2) specimens. Principal strain diagrams for the DP600 with the information exhibited by: (a) Notched, (b) D, and (c) TopOpt specimens</caption>
<figure><img src="image_13.png" /><figcaption>Fig. 8 Specimens after fracture: (a) Notched, (b) D and (c) TopOpt</figcaption></figure>

surfaces are presented, system 1 is related to the surface with the outer side of the bending while System 2 is associated with the surface inside the bent area. Due to the out-of-plane behavior, the strain states are different from one surface to the other. Uniaxial tension is clearly an important strain state on both surfaces, with peak values localized in the center part, where fracture occurs. Plane strain compression is observed with system 2 in the upper and bottom center parts, where the specimen bends, close to the largest hole edge, with $\varepsilon_2$ reaching high values of the order of -0.4 while $\varepsilon_1$ remains close to zero. Plane strain tension is observed with system

1, but without a maximum value near the largest hole edge, with $\varepsilon_1$ remaining below 1. These differences between the two surfaces would require a deeper analysis, with a hybrid experimental-numerical method, which is out of the scope of this paper.

The mechanical richness indicator (cf. Section *Mechanical Richness Indicator*) has been computed for each test design. In Fig. 9, the values of the mechanical indicator for each test configuration are presented as well as each term of $I_t$ (cf. (equation (1))). The experimental data used for the computation of this indicator is taken from the last frame before

<figure><img src="image_14.png" /><figcaption>Fig. 9 Values of the mechanical indicator $I_t$, proposed by Souto et. al. [45], and its terms computed individually for all the test designs</figcaption></figure>
the specimens rupture to capture the largest range of strain and the strain states generated during the test. While for the Notched and D specimens, the load is monotonic and, therefore, there is a progressive and continuous strain evolution until rupture, for the TopOpt test, a non-monotonic behavior is observed due to the onset of buckling. As shown in Fig. 6, buckling occurs relatively rapidly, and once initiated, the strain states stabilize and do not evolve significantly. Thus, even in the presence of buckling, the final frame remains representative of the strain states achieved during the test.

Based on the overall indicator value, the ranking is similar to the one obtained with virtual results in a previous work [20].

The TopOpt presents the highest indicator value followed by the D and the Notched specimens. Regarding the strain state standard deviation, Fig. 10 depicts the data points distribution of the three tests over the strain state range that is chosen to be limited between -15 and 1. The information was extracted from the DIC points after the post-processing. The TopOpt specimen presents a higher value of the standard deviation since the material points are more evenly distributed across the strain state range presented with a higher concentration between uniaxial and plane strain tension. It can be noticed a scarce distribution between uniaxial and plane strain compression for the Notched and D specimens in comparison to the TopOpt. Limits are imposed to the strain state range, explaining the same value for the D and TopOpt specimens for this term. As highlighted in Fig. 9, the standard deviation of the principal strain ratio significantly impacts the calculation of the global indicator. However, this does not diminish the importance of considering all components

in the calculation. The inclusion of multiple terms ensures a comprehensive assessment of the test's performance, capturing different aspects of the strain field behavior. Evaluating the weights assigned to each term or revising the indicator could be addressed, however, it wasn't the primary focus of the present work.

Concerning the standard deviation and the average values of the equivalent strain, the D specimen exceeds the TopOpt one. As shown in Fig. 7, the D specimen presents a more distributed strain field magnitude, leading to a higher average value. The equivalent strain peak values are higher in the TopOpt specimen but are extremely localized. This makes them difficult to be processed and extracted by DIC. With the aid of the principal strain diagrams, it is worth noting that the D specimen has a more important distribution in the tension strain states, explaining its higher term value. For the shear term, although the TopOpt also presents material points in this range, the magnitude value of the D specimen points is higher. The advantage of the TopOpt specimen is the ability to have material points both in tension and compression with interesting strain magnitude values in a single experiment. This happens due to its out-of-plane behavior during the test. Also, it presents material points in biaxial and plane strain tension and compression. A similar analysis was made in a previous work [20], where this mechanical indicator was computed for analyzing the data richness provided by these tests numerically. When compared to this work, the Notched and D specimens provide a similar quantity of information on the same strain states, mostly around uniaxial tension. Regarding the TopOpt specimen, the data on uniaxial tension is identically placed. However, the information extracted

<figure><img src="image_15.png" /><figcaption>Fig. 10 Data points distribution over the strain state range of the (a) Notched, (b) D and (c) TopOpt specimens</figcaption></figure>
experimentally from the surface under compression is placed mostly around plane strain compression while the numerical data pointed to uniaxial compression.

The observed discrepancies between numerical predictions and experimental results for the TopOpt test, particularly in the strain distribution under compression, can be attributed to several factors. Experimental noise, inherent to imaging techniques like digital image correlation, introduces variability affecting strain measurement accuracy. This is particularly significant in regions with high strain gradients or near geometric singularities, due to the low spatial filtering effect of DIC, which typically underestimates these magnitude values in localized strain concentrations areas. Additionally, at this point, the numerical model simplifies certain aspects of the test, such as idealized boundary conditions and chosen material properties, which can lead to deviations from experimental observations. For instance, material anisotropy could alter local strain distributions. Furthermore, imperfections in specimen alignment or gripping conditions during the experiment may introduce additional complexities not accounted for in this numerical setup. These challenges highlight the inherent difficulty in reproducing numerically the experimental real-world behavior of heterogeneous tests, particularly in this case the out-of-plane behavior of this test.

Overall, the results showed that the strain field information provided by the tests remains similar when using data from real experiments. Based on the computation of the mechanical indicator $I_T$, a ranking of the three tests can be made and it can be compared to the one proposed in [20], in which numerical data was used to compute the indicator. Despite the modification in the computation of the indicator regarding the equivalent plastic strain, which cannot be experimentally measured, the ranking remains unchanged, confirming the effectiveness of these tests when using real experimental data.

These findings highlight the robustness of heterogeneous tests in capturing complex material responses, even when using real experimental data. Compared to traditional quasi-homogeneous tests, heterogeneous tests offer significant advantages by providing richer strain field information and enabling the evaluation of diverse stress states within a single setup. This reduces the experimental workload and enhances cost-effectiveness. However, it is acknowledged that the implementation of heterogeneous tests requires advanced methodologies, such as DIC, and specialized post-processing techniques, which may increase complexity. While not a direct replacement for quasi-homogeneous tests, they may serve as a powerful complementary tool for scenarios where traditional tests fall short, particularly in capturing complex material behaviors under non-uniform loading conditions. Standardization of these tests would require robust protocols, extensive validation, and improved accessibility, paving the

way for broader adoption and integration into international standards.

## Test Sensitivity to Different High-strength Steels

In the previous sections, the TopOpt specimen is compared with another two heterogeneous mechanical tests for a DP600 steel. This section is focused on the TopOpt test and its potential is analyzed for three advanced high-strength steels: DP600, DP780 and HF1050. The mechanical information regarding these materials can be found in Table 2. The sensitivity of the results to the different materials is analyzed as well as the robustness of the potential of the test. Therefore, in a similar way to Fig. 5, the load-local displacement curves are presented for two different experimental runs of each material in Fig. 11.

Since the thickness of the metal sheet is different for each material, the load is presented taking into account the initial section of the specimen. It can be noted that a larger load level is required to deform the HF1050 and the DP780 due to their ultimate yield stress and thickness, cf. Table 2. However, it is required a lower displacement of the grips to lead the specimen to rupture. For instance, the TopOpt specimens with DP600, DP780 and HF1050 reach rupture with displacements of 18.43, 10.60, and 9.78 mm, respectively. It is worth noting that the two experimental runs with the DP600 are the most similar, while for the other materials, rupture is achieved earlier for one of the runs.

Figure 12 shows the out-of-plane displacement map of the TopOpt specimen with the three materials, DP600, DP780, and HF1050, just before rupture.

<figure><img src="image_16.png" /><figcaption>**Fig. 11** Load-local displacement curves of two different experimental runs with the TopOpt specimen for each material (DP600, DP780, and HF1050)</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Fig. 12 Out-of-plane displacement map of the TopOpt specimens with (a) DP600, (b) DP780 and (c) HF1050 steels at the end of the test</figcaption></figure>

From both figures, it is worth noting that the TopOpt specimen with the DP600 presents the highest out-of-plane displacement value. This is due to (i) the lowest thickness of the metal sheet of the three materials and (ii) the lowest yield stress of the three. It can be noted that, in the context of the TopOpt test, buckling behavior is largely determined by the sheet thickness, with anisotropic properties playing a less significant role.

Regarding the strain field information, Fig. 13 depicts the principal strain diagrams associated with the TopOpt specimen for each material (DP600, DP780 and HF1050). In the diagram associated with the TopOpt test, it is shown the information from both surfaces extracted by systems 1 and 2.

The strain field information presented in the diagrams is identical for the three materials despite the strain level, demonstrating the specimen's strain state richness, indepen-

dently of the material. Additionally, for each test, there is a difference between the information extracted by both systems in the location where the specimen bends, which has a different behavior in system 1 or 2, being under tension or compression, respectively. This is the main difference that can be noted in the principal strain diagrams between both systems (grey and black colors). While System 2 shows material points under plane strain compression, system 1 presents in contrast points in plane strain tension. However, the majority of the points are common to both and are under uniaxial tension. Regarding the DP780 TopOpt test, there is a significant difference in the strain level between both surfaces. This can be noted in the principal strain diagram, where there are points extracted by system 2 under uniaxial tension with higher strain levels. This happened due to the impossibility of system 1 of extracting data from this time on due to the bad

<figure><img src="image_18.png" /><figcaption>Fig. 13 Principal strain diagrams related to the TopOpt specimen for the three materials: (a) DP600, (b) DP780, and (c) HF1050. Data provided by both systems (1 and 2) is presented</figcaption></figure>
resolution of the images. Both the DP780 and HF1050 tests present more material points under plane strain tension than the DP600. In turn, the DP600 test has material points under equibiaxial tension. This may be due to the higher bending of this test when compared to the others that have a lower out-of-plane displacement (cf. Fig. 12).

To quantify the information given by the principal strain diagrams and the strain field distributions, the strain richness indicator is computed similarly to the previous section. Figure 14 depicts the indicator value as well as its terms individually for the TopOpt with the three different materials.

The final indicator value is quite similar between the three tests with the highest value associated with the DP780 test. Regarding the equivalent strain level, the DP600 TopOpt test presents the best performance, however, the gap to the others, mainly to the DP780 one, is small. The strain state standard deviation is higher in both the DP600 and DP780 tests due to the higher number of points under plane strain tension while the DP600 test presents more points concentrated in the uniaxial tension state. Due to this, since the gap to the other is higher, this test isn't the one with the highest indicator value.

Overall, it is shown that the performance of the TopOpt specimen is quite similar even when different high-strength steels are used. Even though the strain level achieved by

the test when using the HF1050 steel is lower, the strain state heterogeneity remains identical for the three materials. This demonstrates the robustness of the test in maintaining its performance across different materials. While this study focuses on high-strength steels, the proposed methodology is adaptable to other materials. Future work could further explore its applicability to a broader range of materials with varying mechanical properties, enhancing its potential for advanced material behavior characterization.

## Concluding Remarks

This work aims to analyze the potential of three heterogeneous mechanical tests for characterizing sheet metal behavior when using data from real experiments. Heterogeneous tests are able to provide a substantial amount and diversity of mechanical information in a single experiment. However, issues related to the experimental task may arise and impact the results' quality, leading to non-comparable numerical and experimental data. Therefore, it is crucial to verify the potential given to them by numerical models.

First, the repeatability of the three tests is investigated across different experimental runs. The complexity associated with their geometry and the measurement technique

<figure><img src="image_19.png" /><figcaption>**Fig. 14** Values of the mechanical indicator $I_t$, proposed by Souto et. al. [45], and its terms computed individually for the TopOpt specimen with the three materials</figcaption></figure>
may influence the mechanical behavior and, therefore, the
mechanical information extracted. The results showed that,
despite the complex designs, these tests have proven to be
highly repeatable when performed in similar conditions.

The mechanical information richness is analyzed based on
the strain level and strain state range and heterogeneity. Based
on the principal strain diagrams and the quantification of their
information by a mechanical indicator, the results showed
that the TopOpt test provided a larger quantity and diversity
of mechanical data. Due to its out-of-plane behavior during
the test, information was extracted by both surfaces, lead-
ing to the simultaneous existence of tensile and compressive
states in a single experiment. When compared to numerical
data, the Notched and D specimens provide a similar quan-
tity of information on the same strain states. Regarding the
TopOpt specimen, experimental information extracted from
the surface under compression is placed mostly around plane
strain compression while the numerical data pointed to uni-
axial compression.

Finally, the sensitivity of the TopOpt test behavior to dif-
ferent high-strength steels was investigated, resulting in a
similar amount of information provided by each test. Despite
the different strain level achieved by each test, mainly due
to the different mechanical properties, the same strain states
were identified. It is possible to confirm the test's ability to
yield similar mechanical behavior data across three different
high-strength steels highlighting its robustness and reliabil-
ity.

Through the experimental analysis of these heterogeneous
mechanical tests, it is possible to prove that their potential
remains unchanged when real experimental data is used.
By demonstrating their effectiveness in providing relevant
information for material behavior characterization, this work
opens the way for the improvement of model calibration pro-
cedures.

**Acknowledgements** This project has received funding from the Research Fund for Coal and Steel under grant agreement No 888153. This paper was also supported by the projects UIDB/00481/2020 and UIDP/00481/2020 - Fundação para a Ciência e a Tecnologia, DOI 10.54499/UIDB/00481/2020 and DOI 10.54499/UIDP/00481/2020. M. Gonçalves is grateful to the FCT for the Ph.D. grant Ref. UI/BD/151257/2021 and to the ESAFORM association for the mobility grant. The authors would like to acknowledge Anthony Jegat from IRDL for his invaluable expertise and help with the experiments.

**Funding** Open access funding provided by FCTIFCCN (b-on).

Declarations

**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.

**Conflicts of Interest** The authors declare no conflict of interest.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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