# Determination of a machine learning constitutive model from biaxial tensile test: application to thermal forming limits prediction of AA6061 sheets from shear to equi-biaxial tension

Zhihao Wang, Dominique Guines, Lionel Leotoing

► To cite this version:

Zhihao Wang, Dominique Guines, Lionel Leotoing. Determination of a machine learning constitutive model from biaxial tensile test: application to thermal forming limits prediction of AA6061 sheets from shear to equi-biaxial tension. ESAFORM 2025, May 2025, Paestum, Italy. pp.1891 - 1900, 10.21741/9781644903599-203. hal-05115130

HAL Id: hal-05115130

https://hal.science/hal-05115130v1

Submitted on 16 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Determination of a machine learning constitutive model from biaxial tensile test: application to thermal forming limits prediction of AA6061 sheets from shear to equi-biaxial tension

Zhihao WANG¹,ª, Dominique GUINES¹,ᵇ and Lionel LEOTOING¹,ᶜ

¹Univ Rennes, INSA Rennes, LGCGM (Laboratoire de Génie Civil et Génie Mécanique), F-35000 Rennes, France

ªzhihao.wang@insa-rennes.fr, ᵇdominique.guines@insa-rennes.fr,

ᶜlionel.leotoing@insa-rennes.fr

**Keywords:** Biaxial Tensile Test, Thermal Formability, Constitutive Modeling, Machine Learning, Fracture Forming Limit

**Abstract.** Characterization and prediction of forming failure behaviors of aluminum alloys are crucial for achieving lightweight design. In this work, we investigated thermal forming limits of AA6061-T4 sheets from shear to equi-biaxial stretching using biaxial tensile test. The two primary failure modes, localized necking and fracture, were studied. A machine learning constitutive model of AA6061-T4 sheets, including temperature and strain rate dependent flow behaviors, was determined and implemented in simulations. Failure behaviors of the material under various proportional loadings were predicted using a machine learning constitutive model and an uncoupled fracture criterion.

**Introduction**

In many fields, lightweighting is critical to improving performance and reducing costs. Accurate characterization of thermal forming limits of sheet metals is directly related to achieving lightweighting goals. Hot forming is a key technique for manufacturing components with complex geometries, as it effectively reduces forming force and enhances the formability of materials. However, during hot forming processes, the mechanical properties of sheet metals often undergo complex changes, including plastic flow behaviors, microstructural evolution, and damage accumulation, all of which critically influence formability. Accurate characterization of thermal forming limits contributes to the optimization of process parameters, preventing cracking or excessive deformation, and improves the quality and reliability of final products. Additionally, it facilitates the application of novel materials, taping their forming potential.

The forming limit diagram (FLD) is a widely used tool for describing the formability of materials. It includes a set of forming limit curves based on necking (FLCN), which depict the boundary between uniform deformation and the onset of localized necking failure. Some studies have considered cases where materials, such as aluminum alloys and steels, fail without necking under specific loading conditions. Forming limit curves based on fracture (FLCF) is introduced to the FLD as a supplementary tool for comprehensive evaluation of sheet metal formability. Two formability tests have been standardized to characterize FLD of sheet metals under ambient temperature, which are Nakazima test (out-of-plane forming) and Marciniak test (in-plane forming). These tests have also been applied to characterize forming limits at elevated temperatures. Wang et al. [1] utilized a heating furnace to heat the Nakazima tooling, obtaining FLCNs for AA7075-W under temperatures ranging from 200 to 480 °C and strain rates from 0.1 to 10 s⁻¹. The results showed that the effect of temperature on formability was non-monotonic: forming limits increased within the lower temperature range of 200 to 400 °C but decreased at higher temperatures. As the temperature exceeded 280°C, the forming limits decreased with increasing strain rates. Tang et al. [2] performed hot Nakazima tests to determine FLCN and FLCF

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

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under license by Materials Research Forum LLC.
of 22MnB5 at 600°C, and the results were further employed to validate the performance of a modified Lemaitre-based damage law. Naka et al. [3] carried out thermal Marciniak tests to determine FLD of 5083 Al-Mg alloys under temperatures ranging from 20 to 300 °C and test velocities from 0.2 to 200 mm/min. The results showed that material formability exhibits more significant strain rate sensitivity with increasing temperature, and lower formability at higher strain rates. Chu et al. [4] carried out thermal Marciniak tests by installing heating rods inside blank holders and punch to characterize the FLC of AA5086 from 20 to 200°C. The positive effect of the temperature and the negative effect of the strain rate on the formability are clearly observed.

Some studies have discussed the limitations of standard formability tests. For the Nakazima test, a hemispherical punch is used, and good lubrication between the punch and the specimen is critical to ensure that initial localization and fracture occur near the punch dome. However, under hot forming conditions, the lubrication system could fail. Mahalle et al. [5] performed Nakazima tests to characterize FLCN and FLCF of Inconel 718 superalloy over a temperature range from 20 to 700 °C, and MOLYKOTE lubricant was used. They reported that several specimens exhibited fracture locations slightly offset from the punch apex due to lubrication failure during hot forming. The Marciniak test, on the other hand, employs a cylindrical punch and requires the use of carrier blanks with lubrication measures to minimize contact and friction between the punch and the specimen. However, as reported by Hsu et al. [6], initial localization and fracture often occur near the rounded corners of the cylindrical punch, leading to invalid results and poor repeatability. Noder and Butcher [7] emphasized that the shape of the carrier blank is a critical factor in ensuring that the initial fracture occurs within the specimen's center, requiring iterative adjustments to accommodate different test materials. Customizing carrier blanks for testing across various temperatures, however, poses a significant challenge due to the complexity and workload involved.

Recognizing the limitations of traditional formability tests, some studies have proposed biaxial tensile tests to characterize the formability of sheet metals. The key advantage of biaxial tensile test lies in the ability to determine formability under in-plane forming conditions without friction or contact. Strain paths can be directly controlled by adjusting the biaxial loading ratio, enabling a single specimen geometry to cover multiple strain paths. Through cruciform specimen design, typically with a local thickness reduction, necking and initial fracture are always localized at the specimen center. Zhang et al. [8] summarized the geometries and applications of existing cruciform specimens and proposed a new design to ensure proportional equi-biaxial strain paths at the location of fracture initiation. Subsequently, they used this specimen to characterize the FLCN and FLCF of boron steel sheets at temperatures ranging from 750 to 925 °C. Leotoing et al. [9] and Song et al., [10] developed several cruciform specimen designs for formability characterization, successfully determining the FLCNs of AA5086 and DP600 sheets at room temperature, with validation performed through Nakazima and Marciniak tests. Wang et al. [11] used cruciform specimens to characterize the forming limits of DP590 sheets at temperatures ranging from 300 to 500 °C. The strain paths that can be covered by the above-mentioned cruciform specimens range from equi-biaxial tension to nearly uniaxial tension. Wang et al. [12] proposed an asymmetric cruciform specimen to cover strain paths from shear to plane strain. Using two distinct cruciform specimens, Wang et al. [13] characterized the FLCN of AA6061 sheets from shear to equi-biaxial tension. In the present study, the two distinct cruciform specimens were further adopted to characterize the FLCN and FLCF of AA6061 sheets at 240 °C.

The prediction of failure behaviors in hot forming processes relies on accurate constitutive modeling. At elevated temperatures, the constitutive behavior of metallic materials is highly nonlinear, posing significant challenges to traditional modeling approaches. Recent years, machine learning (ML)-based constitutive modeling has emerged as a promising alternative. ML offers powerful function approximation capabilities and remarkable efficiency in capturing material behaviors, particularly for new materials with mechanical properties and microstructural
mechanisms that are not yet fully understood. As a data-driven approach, ML models depend heavily on high-quality datasets. Uniaxial tensile (UT) tests are limited in providing comprehensive observations of material constitutive relations, especially for large strain levels, which are critical for predicting failure stages. The biaxial tensile test has proven to be an effective method for characterizing the constitutive behavior of sheet metals. In a prior study [14], biaxial tensile test with a dedicated cruciform specimen was proposed to identify the stress-strain response of AA5086 sheets at room temperature, achieving an equivalent strain level of up to 30%. This dedicated specimen, through geometric design and thickness reduction, allows the observation of plastic flow behaviors under large strain levels. Moreover, the specimen main deformation zone exhibits non-uniform deformation features, involving material deformations at different strain levels and strain rates; and presents heterogeneous stress/strain state distributions. Due to the nature of non-uniform deformation, the specimen allows for exhibiting of material responses under diverse strain paths and stress states in a single equi-biaxial test, providing rich features related to material constitutive behaviors. In this study, this cruciform specimen was employed to perform biaxial tensile tests under different temperatures and strain rates, generating datasets for training an artificial neural network (ANN) model for constitutive modeling. An unsupervised learning approach was introduced, integrating ANN with the finite element method (FEM) to learn constitutive relations from biaxial tensile tests. This approach eliminates the need for direct flow stress computation, instead relying on partial strain and global force measurements, offering an efficient and robust framework for constitutive modeling. Based on the ANN model and an uncoupled fracture criterion, failure behaviors of the AA6061 sheets at 240°C under various proportional loadings were predicted.

## FLD characterization using thermal biaxial tensile tests

The adopted two cruciform specimen shapes are shown in Fig. 1. The specimen I was designed by Song et al. [10] and the specimen II was designed by Wang et al. [12]. Specimens were fabricated from a 2 mm thick AA6061-T4 sheet, with thickness reductions achieved using CNC machining. Biaxial tensile tests were conducted using a servo-hydraulic testing machine developed at LGCGM (Laboratoire de Génie Civil et Génie Mécanique). Six different tensile ratios (as given in Table 1) between the two loading axes were conducted to cover strain paths from shear to equi-biaxial tension. The strain rates for material deformation from the uniform deformation stage to the necking stage range from approximately 0.45 s⁻¹ to 1.1 s⁻¹, which falls in the typical strain rate range for sheet forming. An induction heating device was utilized for thermal tests. As shown in Fig. 1, a heat-conducting plate was placed parallel beneath the specimen with a gap of 1 mm. The plate, heated by an induction coil beneath it, transfers heat to the specimen via conduction and radiation. A DIC (Digital Image Correlation) system, operating at a frame rate of 2000 frames per second, was utilized to capture the strain fields of the specimens during the tests. To mitigate thermal disturbances, a gentle horizontal airflow was introduced between the specimen and the DIC camera.

<table><caption>Table 1 – Biaxial tensile ratios to cover strain paths ranging from shear to equi-biaxial.</caption><thead><tr><th rowspan="2">Specimen</th><th colspan="3">Cruciform specimen I</th><th colspan="3">Cruciform specimen II</th></tr><tr><th>v<sub>x</sub> (mm/s)</th><th>1</th><th>1</th><th>1</th><th>1</th><th>1</th><th>1</th></tr></thead><tbody><tr><th>v<sub>y</sub> (mm/s)</th><td>1</td><td>0.7</td><td>0</td><td>0.5</td><td>0</td><td>free</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>Figure 1 – Servo-hydraulic biaxial testing machine, schematic diagram of the induction heating device, and the two cruciform specimen shapes.</figcaption></figure>

Fig. 2 shows the major strain distributions of cruciform specimens under equi-biaxial tension and shear loadings prior to fracture, as well as the corresponding temperature distributions at the target temperature of 240°C. The specimen temperature gradually decreases from the center towards the arms, while the temperature within the main deformation zone remains uniform. Fig. 3 shows the FLCN and FLCF for AA6061-T4 sheets at 240°C, and compared with results at ambient temperature. The onset of localized necking was determined based on the thickness variation method [13]. The fracture strain was extracted from the last DIC image before visible crack appeared. At 240°C, the material formability was significantly improved. For necking-based forming limits, the improvement under plane strain state, which corresponds to the lowest point on the FLCN, was approximately 48%. For fracture-based forming limits, the improvement under equi-biaxial tension state, which corresponds to the lowest point on the FLCF, was about 46%.
<figure><img src="image_4.png" /><figcaption>Figure 2 – Major strain and temperature fields for (a) equi-biaxial tension using cruciform specimen I and (b) shear using cruciform specimen II at the target forming temperature of 240°C.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3 – (a) FLCN and (b) FLCF of AA6061-T4 sheets determined using biaxial tensile tests.</figcaption></figure>

## ANN constitutive model training using thermal biaxial tensile tests

An unsupervised training approach combining ANN with the FEM for ANN model to learn constitutive relations from biaxial tensile tests was adopted. Global forces and partial strain from fifteen biaxial tensile tests at different temperatures (from 21 to 360°C) and strain rates (from 0.001 to 100s⁻¹) were measured by the authors and used as training dataset. A well-validated cruciform specimen shape [14] was employed. Fig. 4 shows the equivalent strain distribution in the main deformation zone of the specimen. Under equi-biaxial tension, the specimen allows observation of plastic flow behaviors up to large strain levels. In addition, the main deformation zone exhibits non-uniform deformation features, involving material deformations at different strain and strain rate levels, as well as different strain states. Feature selection was performed within the strain field to identify the most representative and compact feature path. The path from the specimen center to the necking zone was defined as the feature path to extract equivalent strains for ANN training.

A fully connected neural network with 3-15-15-15-1 architecture was used, which has three inputs (strain, strain rate, and temperature) and one output (stress). The architecture was determined based on basic hyperparameters tuning. As illustrated in Fig. 4, an ANN-FEM framework was developed to train the ANN model. In detail, each iteration of the training process includes the following steps: generation of constitutive relations by the ANN model; conducting FE jobs to predict material deformations; calculating the loss function and performing gradient descent scheme. The process is iterated until the gap between the predictions and experimental
measurements is minimized. Convergence was assumed when the discrepancy between two iterations is less than 1%. The training strategy was inspired by the finite element model updating (FEMU) method [15] used for inverse analysis, but the ANN model was used instead of the traditional explicit constitutive equations. Fig. 5 shows the hardening curves (plotted in colored lines) predicted by the ANN model after training, along with experimentally determined hardening curves (plotted in symbols) from UT tests. The predictions are in good agreement with the experimental results and also provide reasonable extrapolations for large strain levels.

<figure><img src="image_6.png" /><figcaption>Figure 4 – Flow chart of the ANN model training procedure.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 5 – Predicted hardening curves by the ANN model at different strain rates: (a) 0.002, (b) 0.2, and (c) 4 s⁻¹, in comparison with experimental results from UT tests.</figcaption></figure>

## Prediction of failure behaviors in thermal biaxial tensile tests

The ANN model was employed to predict material deformations under thermal biaxial tensile tests at 240°C, and the DF2015 fracture criterion [16] was calibrated to reproduce fracture behaviors. The integral form of DF2015 criterion is represented below (Eqs. 1 and 2):
$$ \int_{0}^{\bar{\varepsilon}_{f}^{p}} \left( \frac{2}{\sqrt{L^{2} + 3}} \right)^{C_{1}} + \left( \left\langle \frac{f(\eta, L, C)}{f(1/3, -1, C)} \right\rangle \right)^{C_{2}} d\bar{\varepsilon}^{p} = C_{3}, \quad \langle x \rangle = \begin{cases} x & x \ge 0 \\ 0 & x < 0 \end{cases} \qquad (1) $$

$$ f(\eta, L, C) = \eta + C_4 \frac{3-L}{3\sqrt{L^2+3}} + C \qquad (2) $$

where the Lode parameter is defined as $L=(2\sigma_2 - \sigma_1 - \sigma_3)/(\sigma_1 - \sigma_3)$. $\sigma_1$, $\sigma_2$, and $\sigma_3$ are the principal stresses, $\eta$ is the stress triaxiality. $C_1$, $C_2$, $C_3$, $C_4$ and $C$ are material constants. A linear damage accumulation function was used (Eq. 3):

$$ dD = \frac{1}{\bar{\varepsilon}_{f}^{p}} d\bar{\varepsilon}^{p} \qquad (3) $$

The damage index $D = 1$ indicates that the material is fully damaged.

A hybrid experimental-numerical approach was adopted to identify the material constants of the DF2015 criterion. The thermal biaxial tensile tests were simulated first to obtain the corresponding stress triaxiality and Lode parameter. Then, the MSE function between the DF2015 analytically determined fracture strain and the experimental fracture strain was minimized by optimizing the material constants in DF2015. The calibrated material constants of DF2015 are listed in Table 2. Fig. 6 shows the comparison between predictions and experimental results for equi-biaxial tension and shear tests at 240°C. It can be seen that the simulations accurately predict the equivalent strain distributions under different loading conditions. Under shear loading, elongated shear bands were observed, with strain levels at the specimen center reaching approximately 100%. The predicted axial forces were in good agreement with the experimental results, indicating that the ANN model provides accurate predictions of constitutive behavior under large strain conditions. Fig. 7 presents the predicted FLCF from numerical biaxial tensile tests. The DF2015 fracture criterion demonstrated notable flexibility and accuracy in predicting fracture behavior across a wide range of linear strain paths, from shear to equi-biaxial tension, for AA6061-T4 sheets at 240°C.

<table><caption>Table 2 – Calibrated material constants for DF2015 fracture criterion.</caption><thead><tr><th>Temperature</th><th>C<sub>1</sub></th><th>C<sub>2</sub></th><th>C<sub>3</sub></th><th>C<sub>4</sub></th><th>C</th></tr></thead><tbody><tr><td>240°C</td><td>0.8349</td><td>2.3307</td><td>0.7630</td><td>0.7038</td><td>0.0211</td></tr></tbody></table>
<figure><img src="image_8.png" /><figcaption>Figure 6 – Comparison between predictions and experimental results for equi-biaxial tension and shear tests at 240°C, including the equivalent strain fields prior to fracture, force curves and strain evolutions (fracture strains are marked by red stars).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 7 – Comparison between experimental FLCF and predicted results using DF2015 fracture criterion at 240°C.</figcaption></figure>

## Conclusions

In this work, FLCF and FLCF of AA6061-T4 sheets at 240°C were characterized using thermal biaxial tensile tests. Failure behaviors of the material under various proportional loadings were predicted using an ANN constitutive model and DF2015 fracture criterion. The following findings were obtained:

1. Thermal biaxial tensile test is an effective testing approach for temperature-dependent FLD determination. Using two distinct cruciform specimen enables covering strain states from equi-biaxial tension to shear.

2. AA6061-T4 sheets exhibit a significant improvement in formability at 240 °C compared to that at ambient temperature.

3. The ANN-FEM training framework enables accurate constitutive modeling of materials using experimental measurements from biaxial tensile tests.

4. The ANN model and DF2015 fracture criterion accurately predict material failure behaviors under various proportional loadings. The FLCF predicted through numerical biaxial tensile tests agrees well with experimental observations.
## References

[1] N. Wang, A. Ilinich, M. Chen, G. Luckey, G. D'Amours, A comparison study on forming limit prediction methods for hot stamping of 7075 aluminum sheet, Int. J. Mech. Sci. 151 (2019) 444-460. https://doi.org/10.1016/j.ijmecsci.2018.12.002

[2] B.T. Tang, S. Bruschi, A. Ghiotti, P.F. Bariani, An improved damage evolution model to predict fracture of steel sheet at elevated temperature, J. Mater. Process. Tech. 228 (2016) 76-87. https://doi.org/10.1016/j.jmatprotec.2015.08.007

[3] T. Naka, G. Torikai, R. Hino, F. Yoshida, The effects of temperature and forming speed on the forming limit diagram for type 5083 aluminum-magnesium alloy sheet, J. Mater. Process. Tech. 113 (2001) 648-653. https://doi.org/10.1016/S0924-0136(01)00650-1

[4] X. Chu, L. Leotoing, D. Guines, E. Ragneau, Temperature and strain rate influence on AA5086 Forming Limit Curves: Experimental results and discussion on the validity of the M-K model. Int. J. Mech. Sci. 78 (2014) 27-34. https://doi.org/10.1016/j.ijmecsci.2013.11.002

[5] G. Mahalle, A. Morchhale, N. Kotkunde, A.K. Gupta, Y.C. Lin, Forming and fracture limits of IN718 alloy at elevated temperatures: Experimental and theoretical investigation. J. Manuf. Process. 56 (2020) 482-499. https://doi.org/10.1016/j.jmapro.2020.04.070

[6] E. Hsu, J.E. Carsley, Verma R., Development of forming limit diagrams of aluminum and magnesium sheet alloys at elevated temperatures. J. Mater. Eng. Perform. 17 (2008) 288-296. https://doi.org/10.1007/s11665-007-9196-y

[7] J. Noder, C. Butcher, A comparative investigation into the influence of the constitutive model on the prediction of in-plane formability for Nakazima and Marciniak tests. Int. J. Mech. Sci. 163 (2019) 105138. https://doi.org/10.1016/j.ijmecsci.2019.105138

[8] R. Zhang, Z. Shao, Z. Shi, T.A. Dean, J. Lin, Effect of cruciform specimen design on strain paths and fracture location in equi-biaxial tension. J. Mater. Process. Tech. 289 (2021) 116932. https://doi.org/10.1016/j.jmatprotec.2020.116932

[9] L. Leotoing, D. Guines, I. Zidane, E. Ragneau, Cruciform shape benefits for experimental and numerical evaluation of sheet metal formability. J. Mater. Process. Tech. 213 (2013) 856-863. https://doi.org/10.1016/j.jmatprotec.2012.12.013

[10] X. Song, L. Leotoing, D. Guines, E. Ragneau, Characterization of forming limits at fracture with an optimized cruciform specimen: Application to DP600 steel sheets. Int. J. Mech. Sci. 126 (2017) 35-43. https://doi.org/10.1016/j.ijmecsci.2017.03.023

[11] Z. Wang, X. Chu, Z. Yue, L. Leotoing, J. Gao, A novel mechanical attachment for biaxial tensile test: Application to formability evaluation for DP590 at different temperatures. J. Mater. Res. Tech. 28(2024) 4025-4039. https://doi.org/10.1016/j.jmrt.2024.01.023

[12] Z. Wang, D. Guines, X. Chu, L. Leotoing, Characterization of forming limits at fracture from shear to plane strain with a dedicated cruciform specimen. Int. J. Mater. Form. 15 (2022) 7. https://doi.org/10.1007/s12289-022-01658-8

[13] Z. Wang, D. Guines, X. Chu, L. Leotoing, Prediction and characterization of forming limits at necking from shear to equi-biaxial loading using the biaxial tensile testing method: Feasibility study and application to AA6061-T4. Mech. Mater. 179 (2023) 104598. https://doi.org/10.1016/j.mechmat.2023.104598