# Virtual forming based on model calibration from heterogeneous tests

Lisa Germain, Mafalda Gonçalves, Thibault Barret, Antonio Andrade-Campos, S. Thuillier

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Lisa Germain, Mafalda Gonçalves, Thibault Barret, Antonio Andrade-Campos, S. Thuillier. Virtual forming based on model calibration from heterogeneous tests. ESAFORM 2025, May 2025, Paestum, Italy. pp.1568-1576, 10.21741/9781644903599-169. hal-05062344

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# Virtual forming based on model calibration from heterogeneous tests

Lisa Germain¹, a*, Mafalda Gonçalves², b, Thibault Barret¹, c,
Antonio Andrade-Campos², d, and Sandrine Thuillier¹, e

¹Univ. Bretagne Sud, UMR CNRS, IRDL, F-56100 Lorient, France

²Centre for Mechanical Technology and Automation (TEMA), LASI, Department of Mechanical Engineering University of Aveiro, Portugal

a.lisa.germain@univ-ubs.fr, b.mafalda.goncalves@ua.pt, c.thibault.barret@univ-ubs.fr,

d.gilac@ua.pt, e.sandrine.thuillier@univ-ubs.fr

**Keywords:** Heterogeneous Test, Anisotropic Plasticity, Deep Drawing, High Strength Steel

**Abstract.** This study focuses on calibrating a mechanical model for a DP600 steel using both quasi-homogeneous and heterogeneous tests. Several sample geometries subject to a uniaxial load were selected and tested experimentally, and a finite element model updating method was used to identify the material parameters of an anisotropic plasticity model, leading to several material parameter sets. Then, the virtual forming of cylindrical cup is considered, using these different parameter sets. The strain and stress states of the mechanical tests used for the model calibration are compared with those of the forming process, to analyse their relevance. The main goal is to check the whole chain from the indicator-based design of heterogeneous tests up to a numerical case study of the forming of cylindrical cups.

**Introduction**

The mechanical characterization of materials, such as advanced high-strength steels (AHSS), to derive input data for sheet metal forming, 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 [1]. However, recent studies have shown that heterogeneous tests can provide sufficient data in a single heterogeneous test [2], or a reduced number of these tests [3], to calibrate anisotropic plasticity models. The mechanical design of heterogeneous tests is still a matter of investigation, and there is no established best choice nor standardization. Methods for the mechanical design of heterogeneous tests were categorized, pointing out design by intuition, by strain state, by identification quality and by full identification simulation [4]. A method based on topology optimization has also been used to design specimen shapes [5], e.g., the TopOpt geometry [6], which highlights the possibility to cut a specimen into the desired complex shape and to perform experiments [7], leading to another category for the design of heterogeneous tests. Selecting the relevant heterogenous test is still a matter of investigation. Recently, an indicator based on a combination of metrics related to strain and stress fields, identifiability analysis and synthetic images for Digital Image Correlation quality, was suggested, leading to a ranking of specimen shapes subject to a uniaxial loading [8]. In particular, the D-shape specimen [9] shows a better capability for isotropic hardening model coupled with an anisotropic yield criterion than the Notched specimen [10], based on this indicator. The aim of this study is to compare the predictions of virtual forming with material parameter sets identified from these 2 heterogeneous tests.

The design of specimen shape is an essential step, which must be followed by a validation step. Indeed, the samples designed for heterogeneous testing of metallic sheets are optimized with regards to the strain and stress states reached during the forming processes, encompassing a large diversity for the strain path ratio, stress triaxiality and Lode parameter. The underlying idea, within

<figure><img src="image_2.png" /></figure>
a phenomenological model framework is to reach as many as possible similar strain and stress states in the heterogeneous test and in the forming process. The calibrated model should then be tested against the forming process, to check the whole chain. To the authors' knowledge, this step was rarely performed up to now [11], the focus being more on the test design then its validation. This study specifically aims to calibrate a mechanical model for a DP600 steel sheet using data from 2 heterogeneous tests, as well as data from quasi-homogeneous tests. It also explores the practical relevance of the identified parameters for forming applications, such as the deep drawing of cylindrical cups.

The paper is organized in 3 parts, starting with model calibration with either quasi-homogeneous or heterogeneous tests, then the presentation of the numerical model for the forming of cylindrical cups and finally a presentation of the results, in terms of comparison of punch-stroke curves and strain and stress fields for several parameter sets, and a discussion.

## Model calibration

Material. A dual phase DP600, provided in sheets of thickness 0.8 mm by OCAS, was used in this study. Tensile tests on dog-bone samples following the standards ISO 6892-1, with a rectangular section of 12.5x75 mm², were performed at several orientations to the rolling direction. The yield stress at 0.2% of plastic strain is 354.5 MPa (R<sub>p0.2</sub>), the ultimate tensile stress is 648.8 MPa. The plastic anisotropy coefficients for 3 directions relative to the rolling direction are given in Table 1, the normal anisotropy coefficient is r̄ = 0.999 and the planar anisotropy coefficient is Δr = 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 [12], coupled with isotropic hardening, to describe the mechanical behavior. This approach requires at least 4 quasi-homogeneous tests, 3 tensile tests at different orientations to the rolling direction and hydraulic bulge test.

<table><caption>Table 1 – plastic anisotropy coefficients at 0°, 45° and 90° to the rolling direction, calculated from experimental data obtained in tension.</caption><thead><tr><th>r<sub>0</sub></th><th>r<sub>45</sub></th><th>r<sub>90</sub></th></tr></thead><tbody><tr><td>1.012</td><td>0.889</td><td>1.187</td></tr></tbody></table>

Constitutive equations. The equivalent stress σ̄ is defined by Yld2000-2d anisotropic yield criterion for plane stress states, by:

$$2\bar{\sigma}^a = |X'_1 - X'_2|^a + |2X''_2 + X''_1|^a + |2X''_1 + X''_2|^a, \quad (1)$$

with X'_i and X''_i, i = 1,2, the eigenvalues of tensors X' and X'' obtained after linear transformations of the Cauchy stress tensor σ, i.e., X'=L'σ and X''=L''σ, which depend on the material parameters α_i, i = 1,..., 8:

$$L' = \begin{bmatrix} L'_{11} \\ L'_{12} \\ L'_{21} \\ L'_{22} \\ L'_{66} \end{bmatrix} = \begin{bmatrix} 2/3 & 0 & 0 \\ -1/3 & 0 & 0 \\ 0 & -1/3 & 0 \\ 0 & 2/3 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \alpha_1 \\ \alpha_2 \\ \alpha_7 \end{bmatrix}, \quad (2)$$
$$L'' = \begin{bmatrix} L''_{11} \\ L''_{12} \\ L''_{21} \\ L''_{22} \\ L''_{66} \end{bmatrix} = \frac{1}{9} \begin{bmatrix} -2 & 2 & 8 & -2 & 0 \\ 1 & -4 & -4 & 4 & 0 \\ 4 & -4 & -4 & 1 & 0 \\ -2 & 8 & 2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 9 \end{bmatrix} \begin{bmatrix} \alpha_3 \\ \alpha_4 \\ \alpha_5 \\ \alpha_6 \\ \alpha_8 \end{bmatrix} \quad (3)$$

Isotropic hardening is considered, and a combination of Swift's power law and Voce's saturation equation is used to describe the hardening law:

$$\bar{\sigma}(\bar{\varepsilon}^p) = \beta K(\varepsilon_0 + \bar{\varepsilon}^p)^n + (1-\beta)(Q_1 + Q_2(1-\exp(-Q_3\bar{\varepsilon}^p))), \quad (4)$$

where $\bar{\varepsilon}^p$ is the equivalent plastic strain and $\beta$, $K$, $\varepsilon_0$, $n$, $Q_1$, $Q_2$, $Q_3$ are material parameters to be identified.

Calibration with quasi-homogeneous tests. Experimental data from the 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. 4 using a curve fitting algorithm in Python. A viscoplastic part was also added but this contribution is negligible for the small strain rates used under quasi-static conditions of this study. The 7 parameters of the hardening law are further referred as Iso.

Calibration with heterogeneous tests. Two heterogeneous tests subject to uniaxial loading, inclined at 45° to the rolling direction, were chosen and experiments were carried out with the sample geometry given in Fig. 1. They were cut by water jet, and some characteristics dimensions were checked after cutting. Rectangular parts at the two extremities are set under the grips. The strain field on one surface was recorded continuously using Digital Image Correlation, as well as the load in the tensile direction. Repeatability was investigated, by plotting the load evolution with a local displacement as well as the strain spatial evolution at some specific areas and checked over 3 tests for each geometry. The global displacement speed of the grips was set to respectively 1 and 3 mm.min⁻¹ for Notched and D-shape specimen. Experimental details are given in [13].

Material parameter identification was performed by the Finite Element Model Updating (FEMU) method. A numerical model for both geometries was performed with Abaqus software within an implicit scheme, with shell elements and using local measured displacement fields at the two extremities, to take account of the experimental boundary conditions. The material model was implemented via a user subroutine. The cost function is made up of an addition of four normalized terms, corresponding to the 3 planar strain components and the load; the normalization factor is equal to the maximum experimental strain and load respectively. Moreover, shear and plane strain constraints were added, to force the anisotropic yield criterion to represent correctly these stress states [14]. A multi-starting approach for the initial parameter set was used to avoid local minima and to overcome the dependence of the results to the initial set, which involves performing 5 optimization runs. More details are given in [15].

<figure><img src="image_3.png" /><figcaption>Figure 1 – specimen dimensions in mm, for (a) Notched and (b) D-shape specimen. The Y-direction corresponds to the tensile direction and is inclined by 45° to the rolling direction.</figcaption></figure>
All the parameters of the anisotropic yield criterion and the hardening law were identified by the FEMU method, using either the Notched or the D-shape specimen data, leading to 2 sets of material parameters, further referred to as Notched and Dshape.

## Virtual forming process

A cup drawing process is considered in this study, as representative of more complex industrial forming processes. The set-up provided by Zwick was used as a reference for the blank and tools dimensions, e.g. [1] and Fig. 2a. 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, with a blank holder force of 1.3 kN. 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.

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 tools (punch, die and blank holder) are discrete rigid bodies. Contact with friction is considered between the blank and the tools, with a Coulomb's law and a pressure dependent friction coefficient, i.e., it varies from 0.14 (pressure null) down to 0.085 for a large contact pressure of 40 MPa [16]. These values are consistent with a drawing process in lubricated conditions. 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. Moreover, out of comparison's sake, Hill's 48 yield criterion, with parameters identified from the anisotropic plastic coefficients (cf. Table 1) coupled with Iso, is also used as a reference case.

<figure><img src="image_4.png" /><figcaption>Figure 2 – (a) 2D scheme of the set-up used for the deep drawing of cylindrical cups [1]:</figcaption></figure>

$\Phi_B = 33.6 \text{ mm}, \Phi_P = 33 \text{ mm}, r_P = 5 \text{ mm}, \Phi_D = 35.3 \text{ mm}, r_D = 5 \text{ mm}, h_D = 8.75 \text{ mm}, \Phi_S = 60 \text{ mm}$ (b) finite element model for the blank, meshed with shell elements.

## Results and discussion

Model calibration. Table 2 gives the hardening law parameters optimized from the 3 experimental tests and Table 3 the parameters for the anisotropic yield criterion determined with Notched and D-shape specimens.

<table><caption>Table 2 – values of the 7 parameters for the hardening law (Eq. 4) identified from quasi-homogeneous tests (Iso) and from the 2 heterogeneous tests, Notched and Dshape.</caption><thead><tr><th></th><th>β</th><th>K [MPa]</th><th>ε<sub>0</sub></th><th>n</th><th>Q<sub>1</sub> [MPa]</th><th>Q<sub>2</sub> [MPa]</th><th>Q<sub>3</sub></th></tr></thead><tbody><tr><td>Iso</td><td>0.62</td><td>1115.15</td><td>0.0013</td><td>0.31</td><td>582.92</td><td>467.38</td><td>13.06</td></tr><tr><td>Notched</td><td>0.62</td><td>1338.18</td><td>0.0010</td><td>0.34</td><td>466.40</td><td>403.04</td><td>15.55</td></tr><tr><td>Dshape</td><td>0.62</td><td>1338.18</td><td>0.0015</td><td>0.37</td><td>466.40</td><td>559.98</td><td>15.72</td></tr></tbody></table>

It can be seen in Table 2 that the value of the β 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 yield stress, equal to $\sigma_0 = \bar{\sigma}(\bar{\varepsilon}^p=0)=\beta K\varepsilon_0^n+(1-\beta)Q_1$, is similar for Notched and Dshape, respectively 257.1 MPa and 254.1 MPa but higher for Iso, i.e. 309.6 MPa, though all the values are significantly lower than $R_{p0.2}$. This certainly comes from the identification strategy, where the plastic strain is calculated for the homogeneous test approach whereas the whole elasto-plastic information is used for heterogeneous tests. However, the predicted load level for both Notched and Dshape geometries is slightly underestimated near the elasto-plastic transition. Finally, Fig. 3 shows a comparison of the 3 predicted hardening laws with experimental data, highlighting similar trends.

<figure><img src="image_5.png" /><figcaption>Figure 3 – comparison of hardening laws (Eq. 4) with 3 different parameter sets, i.e., Iso, Notched and Dshape with experimental data obtained in uniaxial tension in the rolling direction (UT) and in bulge test.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4 – 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 defined above and Ref. corresponds to von Mises.</figcaption></figure>

The optimized parameters of Yld2000-2d yield criterion are given in Table 3. The comparison of the projection of the initial yield surface in the normalized $\sigma_1-\sigma_2$ plane, obtained either with Notched parameter set or with Dshape one, is shown in Fig. 4. The biaxial stress state yield point is similar whereas the difference is large in the shear stress state. It should be highlighted that the anisotropic yield loci do not go through the point (1,0), as the initial yield stress in the rolling direction, predicted by Yld2000-2d, is not equal to $\sigma_0$.

<table><caption>Table 3 – values of the 8 parameters for Yld2000-2d yield criterion (Eqs. 1, 2 and 3) identified from the 2 heterogeneous tests, Notched and Dshape. The exponent $\alpha$ (Eq. 1) was fixed and equal to 6.</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><th>α<sub>5</sub></th><th>α<sub>6</sub></th><th>α<sub>7</sub></th><th>α<sub>8</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><td>1.017</td><td>0.978</td><td>0.800</td><td>1.195</td></tr><tr><td>Dshape</td><td>1.054</td><td>1.092</td><td>0.987</td><td>0.965</td><td>1.032</td><td>1.046</td><td>1.199</td><td>0.851</td></tr></tbody></table>

Virtual forming. The first step is a comparison of the predicted punch force evolution with the stroke, as well as the equivalent plastic strain distribution nearly at the end of the process, cf. Fig. 5. The simulations with the user subroutine (Notched and Dshape) encountered some convergence issues at the end of the process, related to the contact management with only the cup edge sliding along the die; therefore, the discussion is based on the strain state for the same punch stroke of 24 mm. At this value, the cup is fully drawn, just sliding along the die length. Another simulation with Iso parameter set, coupled with Hill’s 1948 yield criterion identified from the plastic anisotropy coefficients, labelled IsoH, is added. Indeed, the deformed shape of the cup and the resulting strain states are influenced by the anisotropy (cup earing) and IsoH can be considered as
a reference for the strain states, in the absence of experimental results. The maximum load reached during the forming process depends on the parameter set, ranging for a minimum value of 34 kN for Dshape up to a maximum value of 41 kN for Notched; and a smaller value of 38 kN for Iso. When comparing the stress levels of the hardening laws, for an equivalent plastic strain of respectively 0.2 and 0.6, the highest relative gap is 4% and 9%, taking Iso as a reference. The relative gap on the maximum load is of the order of 10%, but not in the same trend as the stress levels of Fig. 1. Therefore, anisotropy should influence the maximum load prediction. Fig. 5 shows that the maximum equivalent plastic strain reached during the process is close to 0.6 and the earning profile is similar for IsoH and Notched. Though not shown here, the earning profile predicted by Dshape is in the reverse direction at 45° to the rolling direction, which seems not realistic.

Though all the states reached during the calibration tests or the forming process are relevant, plotting all of them makes the diagram difficult to read. The comparison of the strain and stress states was attempted first with the states reached at the end of the process. However, the stress levels reached during forming were much higher than the ones reached during the calibration tests and the comparison presented in the following is performed for a punch stroke of 15 mm.

For this value, the cup is not fully drawn and the maximum equivalent plastic strain reached is around 0.4. Fig. 6 shows this comparison, for each parameter set. The calibration with quasi-homogeneous tests involved 2 types of tests, uniaxial and biaxial tensions, highlighted as straight lines in Fig. 6a. The strain states with a low equivalent plastic strain correspond to a plane strain state and are close to uniaxial tension and shear for higher levels of equivalent plastic strain. The use of Hill's 1948 yield criterion leads to more dispersed strain states near the shear and compression states. The strain states in forming predicted by the 4 parameter sets are rather similar, with differences mainly in the largest equivalent plastic strain area (around 0.4). However, the spatial location of the material points corresponding to the highest equivalent plastic strain is not the same, being at 0° and 90° to the rolling direction for IsoH and Notched and at 45° to the rolling direction for Dshape. Moreover, the envelop of the strain states used for model calibration, reached just before rupture, are represented as black lines. There is no large overlapping of the strain states in forming and in the calibration test, neither for quasi-homogeneous tests nor for heterogeneous ones. However, a certain match can be seen for Dshape.

<figure><img src="image_7.png" /><figcaption>Figure 5 – (a) punch load evolution with the stroke for the different parameter sets Iso, IsoH, Notched, Dshape (b,c) distribution of the equivalent plastic strain on the outer cup surface for a punch stroke of 24 mm for IsoH and Notched parameter sets respectively. PEEQ and SDV1 stand for the equivalent plastic strain.</figcaption></figure>

The stress diagrams in Fig. 6 also share a similar distribution whatever the parameter set. The contours with a black line in Figs. 6e and 6f were obtained for a DP600, but with slightly different mechanical properties, modeled with Swift's hardening law and Yld2000-2d yield criterion [8]. Some stress states in tension and towards biaxial tension are obtained both in the calibration test
and in forming for the Notched parameter set; however, a larger overlapping is obtained for the Dshape parameters set (Fig. 6f), though the global stress level reached during forming is higher.

<figure><img src="image_8.png" /><figcaption>Figure 6 – representation of the in plane major-minor strain ($ε_1 - ε_2$) (a,b,c) or stress ($σ_1 - σ_2$) (d,e,f) states, for a punch stroke of 15 mm. The colored points correspond to the forming process, with the color scale related to the magnitude of the equivalent plastic strain. The contours in black lines correspond to the states related to the test used for the model calibration.</figcaption></figure>

It seems for this comparison 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 seem better predicted using the Notched specimen. This point requires an experimental validation.

It should be emphasized that using 4 quasi-homogeneous tests or one single heterogeneous test (Notched) leads to similar results, which highlights the strength of heterogeneous testing. The identifiability was evaluated previously, though with a simpler equation for hardening (only Swift law) [8]. The corresponding indicator values were larger than the threshold ones for Dshape specimen, considering the yield criterion, but not for the Notched specimen. The results presented in this study seem in agreement with these previous results, though further analysis is requested, as the issue about uniqueness of the optimized parameter sets and local minima is always a concern with FEMU.
## Conclusions

This study focuses 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 specimen, subject to a uniaxial load, were selected and tested experimentally, and a finite element model updating method was used 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. Then, the virtual forming of a cylindrical cup is considered, using these different parameter sets. Although no experimental validation was performed at this stage, it seems 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.

## 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.

## References

[1] N. Souto, A. G. Andrade-Campos and S. Thuillier, Material parameter identification within an integrated methodology considering anisotropy, hardening and rupture, J Mater Process Technol 220 (2015) 157-172. http://dx.doi.org/10.1016/j.jmatprotec.2015.01.017

[2] S. Zhang, L. Léotoing, D. Guines, S. Thuillier, S.L. Zang, Calibration of anisotropic yield criterion with conventional tests or biaxial test, Int J Mech Sci 85 (2014) 142-152
http://dx.doi.org/10.1016/j.ijmecsci.2014.05.020

[3] Y. Zhang, A. Yamanaka, S. Cooreman, T. Kuwabara, S. Coppieters, Inverse identification of plastic anisotropy through multiple non-conventional mechanical experiments, Int J Solids Structures 285 (2023) 112534. https://doi.org/10.1016/j.ijsolstr.2023.112534

[4] F. Pierron, M. Grédiac, Towards Materials Testing 2.0. A review of test design for identification of constitutive parameters from full-field measurements, Strain 57 (2021) e12370.
https://doi.org/10.1111/str.12370

[5] M. Gonçalves, A. Andrade-Campos, B. Barroqueiro, On the design of mechanical heterogeneous specimens using multilevel topology optimization, Advances in Engineering Software 175 (2023) 103314. https://doi.org/10.1016/j.advengsoft.2022.103314

[6] M. Gonçalves, A. Andrade-Campos, S. Thuillier, On the topology design of a mechanical heterogeneous specimen using geometric and material nonlinearities, IOP Conf. Series: Materials Sciences and Engineering 1328 (2022) 012055. doi:10.1088/1757-899X/1238/1/012055

[7] M. Gonçalves, B. Guegan, S. Thuillier, A. Andrade-Campos, Experimental investigation of heterogeneous mechanical tests Materials Research Proceedings, Vol. 41, pp 1742-1751, 2024.
https://doi.org/10.21741/9781644903131-193

[8] M. Gonçalves, M.G. Oliveira, S. Thuillier, A. Andrade-Campos. Key performance indicators for heterogeneous mechanical tests. Int J Mech Sci 264 (2024) 108821.
https://doi.org/10.1016/j.ijmecsci.2023.108821
[9] E.M. Jones, J.D. Carroll, K.N. Karlson, S.L.B. Kramer, R.B. Lehoucq, P.L. Reu PL, D.Z. Turner, Parameter covariance and non-uniqueness in material model calibration using the Virtual Fields Method. Comput Mater Sci 152 (2018) 268-90.
http://dx.doi.org/10.1016/j.commatsci.2018.05.037

[10] M. Rossi, F. Pierron, M. Stamborská. Application of the virtual fields method to large strain anisotropic plasticity. Int J Solids Struct (2016) 97-98:322-35.
http://dx.doi.org/10.1016/j.ijsolstr.2016.07.015.

[11] M. Conde, Recommendation engine and design techniques for the efficient calibration and selection of mechanical constitutive models, PhD thesis, Universidad de Aveiro, 2024

[12] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E.Dick, D.J. Lege, F. Pourboghrat, S.H. Choi, E. Chu. Plane stress yield function for aluminum alloy sheets - Part 1: Theory, Int J Plast 19 (2003) 1297-319. http://dx.doi.org/10.1016/S0749-6419(02)00019-0

[13] M. Gonçalves, S. Thuillier, A. Andrade-Campos, Assessing the potential of heterogeneous mechanical tests for sheet metals through experimentally measured full-fields, submitted for publication in Experimental Mechanics (2024)

[14] A. Abedini, C. Butcher, T. Rahmaan, M.J Worswick, Evaluation and calibration of anisotropic yield criteria in shear loading: constraints to eliminate numerical artefacts, Int J Solids Structures 151 (2018) 118-134. http://dx.doi.org/10.1016/j.ijsolstr.2017.06.029

[15] M. Gonçalves, Design of a novel heterogeneous mechanical test using topology optimization for sheet metal characterization, PhD thesis, Universidad de Aveiro, 2024

[16] A. Shrivastava, D.R. Kumar, Simulation of deep drawing of dual phase steel sheets incorporating tribological behaviour predicted by advanced friction modelling, IOP Conf. See.: Mater.Sci. Eng. 1284 (2023) 012047. http://dx.doi.org/10.1088/1757-899X/1284/1/012047