1. Introduction
Aluminum alloy is widely used in aerospace, shipbuilding, and automobile manufacturing due to its excellent properties such as low density and high corrosion resistance [
1,
2]. However, aluminum alloy usually exhibits plastic anisotropy after thermomechanical treatments. Therefore, modeling its anisotropy is essential for accurately predicting the mechanical behavior of its sheets in advanced manufacturing engineering. Over the past few decades, plenty of anisotropy yield criteria have been proposed, such as Hill’48 [
3], Yld2000-2D [
4,
5], Yld2004 [
6], and Bron–Besson yield criterion [
7]. Hill’48 is a classical one and is widely used in finite element (FE) simulations of sheet metals due to the simplicity of its parameter identification. While for some advanced anisotropy yield criteria with many parameters, parameter identification is an ongoing need. Currently, the parameter identification procedure can be classified into two main strategies: traditional testing and inverse identification.
The traditional testing strategy is based on some homogeneous deformation tests, such as uniaxial tensile tests, shear tests, etc., to accurately analyze the mechanical properties of sheet metals under well-defined stress states. For the identification of anisotropic yield criterion parameters, the required mechanical properties mainly include the material orientation-dependent yield stresses and
r-values. For example, the parameter identification of the Yld2000-2D yield criterion can be performed using uniaxial tensile tests at 0°, 45°, and 90° to the rolling direction, as well as an equi-biaxial tensile test [
4]. Zang et al. [
8] characterized the yield stresses and
r-values of mild and dual-phase steel sheets by performing shear, uniaxial, and biaxial tensile tests. These results were then used to identify the parameters of the Bron–Besson yield criterion. With a similar identification strategy, Zhang et al. [
9] performed the parameter identification of the Bron–Besson yield criterion at three plastic strain levels to consider the evolution of material anisotropy with plastic strain. Performing equi-biaxial tensile test requires a dedicated biaxial tensile equipment, which is usually expensive and technically complex. An alternative is to substitute it with other tests. Tian et al. [
10] employed a disk compression test to determine the
r-value under the equi-biaxial tension for calibrating the Yld2000-2D yield criterion. Commonly, for the traditional testing strategy, the number of experimental campaigns should not be less than the number of parameters to be identified. However, when the experimental data are insufficient to determine the parameters of the yield criterion, it would be acceptable to compensate the data by reasonable assumptions (e.g., using isotropic parameters) or by numerical predictions based on advanced microstructural models. Khalfallah et al. [
11] identified the parameters of Cazacu and Barlat yield criterion (CB2001) using a reduced set of experimental data combined with some artificially generated data.
The inverse identification strategy is a full-field measurement-based approach that extracts more information from non-standard tests to retrieve anisotropic yield criterion parameters. The full-field measurement usually relies on the Digital Image Correlation (DIC) technique. The non-standard test usually employs a well-designed specimen to generate a heterogeneous deformation field. Due to the heterogeneity, various regions of the specimen experience different stress states and strain paths, and thus, a wealth of information about material anisotropy can be extracted from a single test [
12]. From this perspective, the inverse identification strategy shows a high potential to simplify the experimental campaigns without sacrificing the identification accuracy [
13]. The inverse identification strategy combined with non-standard tests for material parameter identification has been rapidly developed in recent years and is called Materials Testing 2.0 [
14], promising to revamp traditional testing. Avril et al. [
15] presented a complete overview and comparison of several mainstream methods of the inverse identification strategy. Among them, the FEMU [
16] is a more intuitive approach. The principle of the FEMU is to minimize the gap between numerical predictions and experimental measurements by iteratively updating the model parameters. Numerical predictions are obtained from the reproduction of the experiment using an FE model. The experimental measurements can be either full-field measurements or partial measurements of the full-field. The mapping relationship between the prediction and experiment can be established based on displacement, strain, force, etc. The FEMU is a mature technique with high robustness and low sensitivity to measurement noise, and it is capable of modeling complex specimen geometries [
17]. Based on the FEMU method, Pottier et al. [
18] identified the parameters of Hill’48 yield criterion using a well-designed specimen. The specimen was out of plane deformed to generate significant heterogeneous deformation fields with tensile, shear, and expansion strain states. The identified parameters were verified using the deep drawing test, and the results showed that the inverse identification combined with heterogeneous strain field can better predict the actual deformation compared to traditional tests. Wang et al. [
19] used the same specimen for testing at an elevated temperature and identified the parameters of Yld2000-2D criterion for 7B04 aluminum alloy at 200 °C. The biaxial tensile test is a special case of multi-axial loading. By using a well-designed cruciform specimen, a heterogeneous test can also be achieved. Zhang et al. [
20] designed a cruciform specimen with notches to supply experimental data for the inverse procedure. Based on this, the parameters of Bron and Besson yield criterion were accurately identified. Martins et al. [
21] used a biaxial test and the Virtual Fields Method (VFM) to calibrate the parameters of Yld2000-2D criterion and Swift’s hardening law.
The present paper focuses on the parameter identification of Yld2000-2D anisotropy yield criterion of AA5086 sheets using traditional testing and inverse identification strategies. The aim is to provide a detailed insight into the implementation and validation of these two strategies and to provide a comprehensive comparison. The traditional testing method considers three uniaxial tensile tests at different orientations and an equi-biaxial tensile test. The inverse identification strategy relies on the FEMU method coupling with a biaxial tensile test using a dedicated cruciform specimen or the Pottier bulging test. To verify the identified anisotropy parameters, a comparison between the experimental and Yld2000-2D-predicted yield stresses, r-values, and yield loci is performed. In addition, a deep drawing test is carried out. The identified yield criterion is further evaluated in terms of practical forming by comparing the predicted earing height distribution with the experimental results. All specimens in this work are extracted from 2 mm thick AA5086 sheets using laser cutting.
5. Verification of Yield Criterion Parameters
Plastic anisotropy is the main factor leading to earing behavior in the deep drawing test. In order to verify the accuracy of the Yld2000-2D anisotropy yield criterion parameters identified by different methods, a deep drawing test is carried out. The shape of the specimen is circular, with a diameter of 90 mm and a thickness of 2 mm. The schematic diagram of the deep drawing device is shown in
Figure 15a. The punch speed is set as 0.01 mm/s, and a 6 kN blank holder force provided by two nitrogen springs is adopted. Lubrication is applied to the specimen, the punch, and the die. The earing height distribution of formed specimen is measured using a height gauge. The FE model of the deep drawing test is shown in
Figure 15b. The specimen is modeled with four node shell elements, and the punch, the die, and the blank holder are modeled using the discrete rigid bodies. The friction coefficient between the specimen and the punch and the die is set to 0.2. The numerical deep drawing test is then performed with the identified parameter sets of Yld2000-2D yield criterion and stopped at the corresponding experimental drawing depth (65 mm).
The experimental and predicted earing profiles are presented in
Figure 16. The anisotropy of AA5086 sheets leads to four ears, and the peaks of the earing profile appear at about 45°, 135°, 225°, and 315° directions. This distribution is similar to the results presented by Neto et al. [
24]. Considering the test symmetry, here only the earing profiles at 0 ° and 180° are discussed. It is found that the identified Yld2000-2D parameter sets from both the traditional testing and inverse identification strategies accurately predicted the heights and locations of the earing profile peaks. For the valleys of 0° and 180° directions, the three Yld2000-2D parameter sets provide similar predictions of the earing height, but all are about 1.7% below the experimental value. In addition, a significant difference between the three predicted earing profiles is observed around the 90° direction. With
Figure 4,
Figure 10, and
Figure 14, one may notice that there is also a significant difference in the
r90 predicted by the three Yld2000-2D parameter sets. The parameter sets identified using the traditional testing and inverse identification (Pottier bulging test) strategies successfully captured
r90 of AA5086 sheets, and therefore, the prediction of earing height in the 90° direction is accurate. While the
r90 predicted with Yld2000-2D parameters from the inverse identification strategy (biaxial tensile test) is significantly lower than the experimental one, so it provides an underestimated prediction of earing height in the 90° direction. Compared to the heterogeneous deformation field from the biaxial tensile test, the one from the Pottier bulging test provided more information on the inverse identification procedure, especially on the material deformation under shear loading. Therefore, it is revealed that the quality and richness of the information provided by the heterogeneous tests could lead to better identification of Yld2000-2D parameters.
In summary, the parameters of Yld2000-2D yield criterion identified based on the traditional testing strategy can accurately describe the anisotropy behaviors of AA5086 sheets as expected. The inverse identification strategy based on the FEMU method and heterogeneous tests is also an effective alternative to identify the Yld2000-2D parameters with satisfactory accuracy. In addition, the capacity to predict the practical forming process is improved when test heterogeneity increases. The inverse identification strategy eliminates the need for a large number of experiments as required by the traditional testing strategy, and a single heterogeneous test allows the simultaneous identification of eight parameters of Yld2000-2D yield criterion.