# Material Parameter Identification Based on Heterogeneous Testing: Validation in case of a forming process

Lisa Germain, Mafalda Gonçalves, Antonio Andrade-Campos, Sandrine Thuillier

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Lisa Germain, Mafalda Gonçalves, Antonio Andrade-Campos, Sandrine Thuillier. Material Parameter Identification Based on Heterogeneous Testing: Validation in case of a forming process. IDDRG 2025, Jun 2025, Lisbon, Portugal. pp.01075, 10.1051/matecconf/202540801075. hal-05102080

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# Material Parameter Identification Based on Heterogeneous Testing: validation in case of a forming process

Lisa Germain<fnref n="1" />*, Mafalda Gonçalves<fnref n="2" />, Antonio Andrade-Campos<fnref n="2" />, and Sandrine Thuillier<fnref n="1" />

**Abstract.** The mechanical characterization of sheet metals requires a series of quasi-homogeneous mechanical tests, requesting the use of multiple specimens to reproduce the large range of strain states characteristics of forming processes. Recent studies have shown that heterogeneous testing can provide enough data in a single test to cover a large range of strain states. The design and choice of the heterogeneous test is still a matter of investigation and validation of the model calibration in virtual forming is needed. The aim of this study is to use a mechanical model for 0.8 mm thick DP600 Dual Phase steel calibrated from heterogeneous test data as input to the numerical simulation of a forming process. The material parameters of an anisotropic plasticity model were determined with the finite element model updating method applied to two heterogeneous tests. The study explores the practical relevance of these parameters in the case of the deep drawing of a cylindrical cup. Numerical simulations of the forming process are conducted with ABAQUS software and instrumented forming experiments are carried out to validate the model calibration. The comparison between experimental and numerical data is performed using the forming load, the deformed shape and the strain field.

**Keywords:** Heterogeneous test; Anisotropic plasticity; Deep drawing; High strength steel.

## 1 Introduction

Though numerical simulation of sheet metal forming may appear nowadays a well-established high added value technique, it has not more than 60 years old [1]. Its development relies on numerical techniques (mathematical formulation, contact, tool representation) and the material mechanical behavior modelling. This last topic has arisen a huge interest in the scientific community, as the reliability of the numerical results highly depends on it. Sheet metals have an anisotropic behavior, sustain large irreversible deformation, exhibit viscosity, Bauschinger and cross-loading effect, hysteresis and a strain-dependent effect during unloading, all of which may influence the strain and stress states of the deformed part. The representation of the mechanical behavior has progressively evolved from rigid plastic, elasto-plastic with isotropic hardening, normal anisotropy to advanced models, especially for the initial anisotropy and the strain-hardening, involving more and more material parameters [2]. Determining the values of these parameters for a given material is still a challenge.

Indeed, the underlying idea, within a phenomenological model framework, is to reach as many as possible similar strain and stress states in the calibration tests and in the forming process. However, advanced model calibration requires a series of quasi-homogeneous mechanical tests, involving the use of multiple specimens to replicate the full range of strain states characteristic of forming processes [3]. A rather recent alternative approach is the use of a reduced number of heterogeneous tests [4], coupled with an inverse identification strategy like finite element model updating (FEMU). This last method is still under development, the focus being more on the test design then its validation. A previous study focused on calibrating a mechanical model for a DP600 steel using both quasi-homogeneous (uniaxial and biaxial tensions) and heterogeneous tests. Two geometries, i.e., Notched and Dshape, subject to a uniaxial load, were selected and tested experimentally, to identify the material parameters of Yld2000-2d yield criterion coupled with a linear combination of Swift-Voce hardening law, leading to several material parameter sets [5]. A comparison of the strain and stress states obtained with these parameter sets, in the case of a deep drawing of cylindrical cup, was performed.

<footnote n="1">¹Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56100 Lorient, France</footnote>

<footnote n="2">²Center for Mechanical Technology and Automation (TEMA), LASI – Intelligent Systems Associate Laboratory, Department of Mechanical Engineering, University of Aveiro, Portugal</footnote>

<footnote>* Corresponding author: lisa.germain@univ-ubs.fr</footnote>
Although no experimental validation was performed, it seemed that the strain and stress states predicted in forming with the Dshape parameter set present the highest overlapping with the calibration test. However, the hardening curve and the strain distribution in forming are better predicted using the Notched specimen.

The aim of this study is the experimental validation of the parameter sets identified from heterogeneous tests, in the case of the deep drawing of cylindrical cups. The paper is organized in 2 parts: the first one is dedicated to the presentation of the mechanical model and material parameter sets identified for a DP600 steel, as well as of the forming set-up and its numerical counterpart. And the second part is focused on results and discussion.

## 2 Material and methods

A dual phase DP600, provided in sheets of thickness 0.8 mm by OCAS, was used in this study. The yield stress at 0.2% of plastic strain is 354.5 MPa, the ultimate tensile stress is 648.8 MPa. The normal anisotropy coefficient is 0.999 and the planar anisotropy coefficient is 0.085. The material exhibits a slight anisotropy in strain, although it is almost negligible on the stress level, which may have an influence on the strain states predictions. Therefore, the anisotropic yield criterion Yld2000-2d is selected [6], coupled with isotropic hardening, to describe the mechanical behavior.

### 2.1 Mechanical model

The equivalent stress is defined as the sum of 3 terms, each depending on the eigenvalues of 2 tensors obtained by linear transformations of the Cauchy stress tensor. These transformations depend on material parameters $\alpha_i$, $i = 1, ..., 8$. The exponent is fixed equal to 6, as advised for body centred cubic materials. Isotropic hardening is considered, and a linear combination of Swift's power law and Voce's saturation equation is used to describe the hardening law:

$$ \bar{\sigma}(\bar{\epsilon}^p) = \beta K(\epsilon_0 + \bar{\epsilon}^p)^n + (1-\beta)(Q_1 + Q_2(1-\exp(-Q_3\bar{\epsilon}^p))), \quad (1) $$

where $\bar{\sigma}$, $\bar{\epsilon}^p$ are respectively the equivalent stress and plastic strain and $\beta$, $K$, $\epsilon_0$, $n$, $Q_1$, $Q_2$, $Q_3$ are the material parameters.

The identification of the material parameters was performed using experimental data from heterogeneous tests, either Notched [7] or Dshape [8] specimen cut such that the uniaxial loading direction is inclined at 45° to the rolling direction, using the finite element model updating method. Moreover, out of comparison's sake, experimental data of a tensile

test in the rolling direction (maximum homogeneous strain of 0.15) and hydraulic bulge test (maximum homogeneous strain of 0.6) were used to calibrate the material parameters of Eq. 1 using a curve fitting algorithm in Python. The 7 parameters of the hardening law are further referred as Iso; in this case, von Mises yield criterion is used and the initial anisotropy is neglected.

The hardening parameters are given in Table 1 and the parameters of Yld2000-2d, identified from Notched and Dshape specimen only, in Table 2.

<table><caption>**Table 1.** Material parameters for the hardening law (Eq. 1) identified from homogeneous (Iso) or heterogeneous tests (Notched and Dshape)</caption><thead><tr><th></th><th>Iso</th><th>Notched</th><th>Dshape</th></tr></thead><tbody><tr><td>β</td><td>0.62</td><td>0.62</td><td>0.62</td></tr><tr><td>K (MPa)</td><td>1115.15</td><td>1338.18</td><td>1338.18</td></tr><tr><td>ε<sub>0</sub></td><td>0.0013</td><td>0.0010</td><td>0.0015</td></tr><tr><td>n</td><td>0.31</td><td>0.34</td><td>0.37</td></tr><tr><td>Q<sub>1</sub> (MPa)</td><td>582.92</td><td>466.40</td><td>466.40</td></tr><tr><td>Q<sub>2</sub> (MPa)</td><td>467.38</td><td>403.04</td><td>559.98</td></tr><tr><td>Q<sub>3</sub></td><td>13.06</td><td>15.55</td><td>15.72</td></tr></tbody></table>

<table><caption>**Table 2.** Material parameters of Yld2000-2d identified from heterogeneous tests (Notched and Dshape)</caption><thead><tr><th></th><th>α<sub>1</sub></th><th>α<sub>2</sub></th><th>α<sub>3</sub></th><th>α<sub>4</sub></th></tr></thead><tbody><tr><td>Notched</td><td>0.858</td><td>0.895</td><td>1.037</td><td>1.044</td></tr><tr><td>Dshape</td><td>1.054</td><td>1.092</td><td>0.987</td><td>0.965</td></tr><tr><td></td><td>α<sub>5</sub></td><td>α<sub>6</sub></td><td>α<sub>7</sub></td><td>α<sub>8</sub></td></tr><tr><td>Notched</td><td>1.017</td><td>0.978</td><td>0.800</td><td>1.195</td></tr><tr><td>Dshape</td><td>1.032</td><td>1.046</td><td>1.199</td><td>0.851</td></tr></tbody></table>

It can be seen that the value of the $\beta$ parameter, that weights the Swift and Voce contributions, is the same for the 3 sets, although the identification strategies are different. This result may depend on the initial chosen value; however, an increase of respectively 3% and 12% from the initial guess for Notched and Dshape was obtained after optimization, highlighting a convergence towards the same value for the 3 methods used here. The initial value of the equivalent stress is similar for Notched and Dshape, respectively 257.1 MPa and 254.1 MPa but higher for Iso, i.e. 309.6 MPa. Fig. 1 shows a comparison of the 3 predicted hardening laws with experimental data, highlighting similar trends. The comparison of the projection of the initial yield surface, obtained either with Notched parameter set or with Dshape one, is also presented in Fig. 1. The biaxial stress state yield point is similar whereas the difference is large in the shear stress state.
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Comparison of hardening laws (Eq. 1) with 3 different parameter sets (Table 1) with experimental data obtained in uniaxial tension in the rolling direction (UT) and in bulge test. Below: projection of the initial yield surface, in the normalized $\sigma_1-\sigma_2$ plane, obtained either with Notched parameter set or with Dshape one. $\sigma_0$ is the initial value of the equivalent stress and Ref. corresponds to von Mises yield criterion coupled with Iso parameter set.</figcaption></figure>

## 2.2 Deep drawing of cylindrical cups

The cup drawing test involves stretching, bending, unbending and friction, which occur during drawing [2]. Experiments were conducted on DP600 circular blanks, punched out to the diameter $Ø_s$. The set-up provided by Zwick was used as a reference for the blank and tools dimensions, cf. Fig. 2. A cylindrical punch of diameter 33 mm is used to draw a cup from a circular blank held between a blank holder and a die. The gap between the punch and the die is 1.15 mm, leading to a clearance (after removal of the blank thickness) of 0.35 mm.

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** 2D scheme of the set-up used for the deep drawing of cylindrical cups: $Ø_B = 33.6$ mm, $Ø_P = 33$ mm, $r_P = 5$ mm, $Ø_D = 35.3$ mm, $r_D = 5$ mm, $h_D = 8.75$ mm, $Ø_S = 60$ mm.</figcaption></figure>

Lubricant was set on the punch and the blank holder, in contact with the inner surface of the blank. A black dot random pattern, of size 5 pixel, over a white layer, was printed on the outer blank surface, to be used for digital image correlation (DIC), cf. Fig. 3. Two layers of baking paper were put between the die and the blank, to protect the painted surface. VIC3D system was used to record the digital images and to perform the calculation; DIC parameters are given in Table 3. The blank holder force was set initially to 6 kN but was observed to decrease slightly during the drawing test by around 3%. The punch stroke was set so that to obtain a fully drawn cup. The punch speed is set to 1 mm.s$^{-1}$.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Initial outer surface of the blank, printed with a uniform white layer and a printed random black pattern. And position of the deformed part. Characteristic directions are indicated.</figcaption></figure>

The punch stroke, load and the blank holder force are recorded during the test. Moreover, the strain field on the cup outer surface was calculated by DIC after the tool removal, for the two punch strokes. Due to the friction and the deformation, the painted pattern was damaged in some areas, making it difficult to calculate the strain field. Moreover, the earning profile is measured by contact with a 3D measuring machine.

<table><caption>**Table 3.** DIC parameters – Two digital cameras Basler acA2440</caption><tr><td>Camera image resolution</td><td>2448 px * 2048 px</td></tr><tr><td>Image scale [pixel.mm<sup>-1</sup>]</td><td>12.4</td></tr><tr><td>Lens focal distance [mm]</td><td>35</td></tr><tr><td>Stand-off distance [mm]</td><td>470</td></tr><tr><td>Image filtering</td><td>Gaussian</td></tr><tr><td>Subset size [px]</td><td>39</td></tr><tr><td>Step size [px]</td><td>6</td></tr><tr><td>Strain window</td><td>5</td></tr><tr><td>Virtual strain gauge</td><td>63</td></tr><tr><td>Strain formulation</td><td>Hencky</td></tr></table>

## 2.3 Numerical simulation of the forming process

The numerical model is developed within an implicit framework with Abaqus. One fourth of the blank is modelled, due to the material and process symmetries.
The blank is meshed with shell elements with reduced integration and 5 integration points along the thickness. The mesh size is set to 0.3 mm in the radial direction, except in the center part, where it is coarser. In the circumferential direction, one element is set every 2°. The tools (punch, die and blank holder) are meshed as discrete rigid bodies. The blank holder force is applied using the recorded experimental value.

Contact with friction is considered between the blank and the tools, with a Coulomb's law and a constant friction coefficient. The influence of the friction coefficient is sensitive on the maximum load, the width of the punch load-stroke curve and the plateau level of the last part of the punch load evolution, corresponding to just sliding of the blank along the die [9]. To capture well the last two features, a non-zero friction coefficient with the punch, blank-holder and the horizontal part and radius of the die is considered, whereas a frictionless condition with the vertical part of the die is set. Moreover, 2 different values for the friction coefficient are used, 0.07 and 0.1, and numerical results are presented for each value, labelled respectively as "f0.07" and "f0.1". The material model presented above is implemented via a user subroutine and 3 material parameter sets are considered, depending on the test used for the calibration: Notched, Dshape and Iso, with values of the material parameters given in Tables 1 and 2.

## 3 Results and discussion

The first step is a comparison of the predicted punch force evolution with the stroke, as well as the deformed shape and the strain distribution at the end of the process.

### 3.1 Punch load-stroke evolution

Comparison of experimental and numerical evolution of the punch load with the stroke is shown in Fig. 4. The experimental data exhibited a very good repeatability, illustrated here by 3 tests. The average maximum load over these 3 tests is 40.8 +/- 0.09 kN reached at a stroke of 11.8 mm. The numerical prediction depends strongly on the material parameter set and the friction coefficient, with a maximum load value equal to, for the lowest value of the friction coefficient, respectively 34.5 kN for Dshape, 37.7 kN for Iso and 40.9 kN for Notched. A slight increase of around 4% is obtained for a friction coefficient of 0.1, leading to respectively 36.0 kN for Dshape, 39.3 kN for Iso and 42.8 kN for Notched.

The stroke value at the maximum load depends slightly on the parameter set and the friction

coefficient. The smallest predicted value is 11.4 mm (Notched/f0.07) and the highest one is 11.9 mm (Notched/f0.1). Generally speaking, the predicted stroke value is closer to the experimental one when the friction coefficient is higher, except for Dshape parameter set.

The global shape of the punch load-stroke curve is very well captured by the Notched parameter set, with the experimental data in-between the prediction for "f0.07" and "f0.1". The load is underestimated by the 2 other parameter sets, i.e., Iso and Dshape. Though the equivalent stress for Dshape is higher than the one for Notched (cf. Fig. 1), the maximum predicted load is much lower, highlighting the influence of the anisotropy. The influence of the friction coefficient could be taken in account by introducing a pressure-dependent model [10].

<figure><img src="image_5.png" /><figcaption>Fig. 4. Punch stroke-load: comparison of experimental values (3 test data are plotted with black lines, they are nearly all superimposed) with numerical ones.</figcaption></figure>

### 3.2 Deformed shape and strain field

The earing profile is shown in Fig. 5. It should be highlighted that only the variation with respect to the average cup height ($h_{avg}$) is presented, for both experimental and numerical data. It is defined as $\Delta h = h - h_{avg}$, with h the cup height. The numerical height variation is calculated in the range 0°-90° and plotted over the whole angular range by using symmetries. The slight experimental height variation, with a decrease at +/- 45° and +/-135° to the rolling direction, is clearly shown. A slight dispersion is highlighted over 3 tests, as well as a higher value in the rolling direction than at 90° to the rolling direction. It should be highlighted that the anisotropy effect is slight, as expected from the planar anisotropy coefficient, leading to a maximum+/- 45° and +/-135° to the rolling direction. Indeed, the material exhibits a normal isotropy but a sensible planar anisotropy, the plastic anisotropy coefficients are such that $r_{45} < 1 < r_0 < r_{90}$ [5]. The average height of the drawn cup is 19.9 mm.
The numerical prediction depends on the material parameter set and, at a lesser scale, on the friction coefficient, with an average height equal to, for the lowest value of the friction coefficient, respectively 18.9 mm for Notched, 19.2 mm for Iso and 20.9 mm for Dshape. A slight increase, below 1%, is obtained for a friction coefficient of 0.1. The height variation is rather well captured by the Notched parameter set, with a slight overprediction in the rolling direction, with a maximum value slightly sensitive to the friction coefficient, higher for “f0.1” than for “f0.07”. As already discussed [5], the predicted earing profile with the Dshape parameter does not match the experimental variation, with a positive variation predicted at +/- 45° and +/-135° to the rolling direction, whereas negative values are obtained experimentally.

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Earing profile: comparison of experimental values (3 tests are represented with black dots) with numerical ones.</figcaption></figure>

Fig. 6 shows the major strain distribution, in the sheet plane. The maximum major strain is measured in the rolling and transverse directions, above the cup radius, with an average value of 0.46. A slightly lower value is reached at 45° to the rolling direction, around 0.28. Moreover, the principal strain in the cup radius is 0.09, which is consistent with the process parameters. The pattern is submitted to sliding on the baking paper and high strains, which lead to areas for which the correlation was not possible.

The comparison is only qualitative at this stage, as a quantitative comparison requires a post-treatment of DIC data, as the subset size and step impose a filtering operation different from the finite element one [11]. The numerical predictions are rather different from the experimental distribution. Indeed, the strain localisation obtained at the bottom of the cup wall is not predicted. The Iso parameter set succeeds to predict the major strain magnitude in the cup wall, with a similar value of 0.28.

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Major strain distribution for a punch stroke of 30 mm. The same scale is used for the 4 results.</figcaption></figure>

### 3.3 Discussion

The selection of a relevant heterogeneous test for model calibration test is still a matter of investigation and should be validated against the application in metal forming, here the drawing of cylindrical cups. A numerical comparison of the strain and stress states was performed with the states reached for a punch stroke of 15 mm [5], as the stress levels reached during forming were much higher than the ones reached during the calibration tests. The strain and stress states predicted in forming with the Dshape parameter set presented the highest overlapping with the calibration test. However, for the cup drawing test, the prediction of the maximum punch load and the deformed shape are not correctly predicted with the Dshape parameter set and a better prediction is obtained with Notched.

Moreover, the identifiability of the material parameters with the Dshape and Notched specimen was evaluated previously, though with a simpler equation for hardening (only Swift law) [12]. The main idea is to check that the numerical strain field is sufficiently sensitive to individual changes in each material parameter and that changes in the model output due to a variation of a single parameter cannot be compensated by changes in other parameters. Two identifiability metrics can be used: the collinearity index $\gamma_K$, where K refers to the subset of parameters related to anisotropy or hardening and the identifiability index $I_K$. A value lower than 20 for the first one qualifies the test to accurately identify the material parameters; a value below 2 for the second one corresponds to a good identifiability or a moderate one when it is in-between 2 and 3. The detailed analysis is presented in [12] and the values are recalled in Table 4. The corresponding indicator value was larger for Dshape specimen, considering the yield
criterion, than for the Notched specimen. The rather
poor prediction with the Dshape parameter set could
be related to identifiability issues.

<table><caption>**Table 4.** Identifiability metrics calculated for Notched and Dshape specimen [12]. The numerical values given in the first column correspond to threshold values for having a good identifiability. Values highlighted in grey are above these thresholds.</caption><thead><tr><th>Identifiability metrics</th><th>K</th><th>Notched</th><th>Dshape</th></tr></thead><tbody><tr><td>γ<sub>K</sub> (&lt; 20)</td><td>Hardening</td><td>4.99</td><td>2.32</td></tr><tr><td>γ<sub>K</sub> (&lt; 20)</td><td>Anisotropy</td><td>16.98</td><td>94.83</td></tr><tr><td>I<sub>K</sub> (2 &lt; . &lt; 3)</td><td>Hardening</td><td>1.84</td><td>0.99</td></tr><tr><td>I<sub>K</sub> (2 &lt; . &lt; 3)</td><td>Anisotropy</td><td>3.19</td><td>4.56</td></tr></tbody></table>

The calibration with quasi-homogeneous tests involved 2 types of tests, uniaxial and biaxial tensions, and tensile tests at 45° and 90° to the rolling direction should be added to quantify the anisotropy, leading to a minimum of 4 quasi-homogeneous tests. One single heterogeneous test (Notched) leads to a rather good prediction of the forming process, which highlights the strength of heterogeneous testing.

## 4 Conclusions

The aim of this study is to use a mechanical model for 0.8 mm thick DP600 Dual Phase steel calibrated from heterogeneous test data as input to the numerical simulation of a forming process. The material parameters of an anisotropic plasticity model were determined with the finite element model updating method applied to 2 heterogeneous tests. The study explores the practical relevance of these parameters in the case of the deep drawing of cylindrical cups. Numerical simulations of the forming process are conducted with ABAQUS software and instrumented forming experiments are carried out to validate the model calibration. The comparison between experimental and numerical data shows that the Dshape parameters set does not predict a realistic deformed shape, whereas the numerical simulation with the Notched parameter set captures rather well several features of the forming process.

## Acknowledgments

This project has received funding from the Research Fund for Coal and Steel under grant agreement No 888153.

## Disclaimer

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

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