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01.12.2020 | Original Article | Ausgabe 1/2020 Open Access

# Investigation on Yield Behavior of 7075-T6 Aluminum Alloy at Elevated Temperatures

Zeitschrift:
Chinese Journal of Mechanical Engineering > Ausgabe 1/2020
Autoren:
Jianping Lin, Xingyu Bao, Yong Hou, Junying Min, Xinlei Qu, Zhimin Tao, Jiajie Chen

## 1 Introduction

Lightweight materials, e.g., advanced high strength steels and aluminum alloys, have been identified as a key priority for improving fuel efficiency of automotive [ 1, 2]. 7xxx-series aluminum alloys (AlZnMgCu) offer a potential to replace a portion of high strength steels [ 35]; however, formability of AA7xxx at ambient temperatures is limited. Warm forming technology is a promising approach to improve the formability of peak-aged 7075 aluminum alloys (AA7075-T6) [ 69], where the AA7075-T6 blank is heated to target temperatures followed by warm forming with heated or cold dies. The formability of aluminum alloys under warm conditions can be evaluated by forming limit curves (FLCs) [ 10, 11]. Sotirov et al. [ 12] obtained temperature-dependent forming limits of EN AW-7075-T6 using the Nakazima tests and found that the major strain of FLC raised from 0.15 to 0.3 at plane strain condition as the temperature increased from room temperature to 230 °C. Grohmann et al. [ 13] compared the formability of AA7075-T6 at various temperatures with a typical automotive aluminum alloy AA6016 in T4 temper. It is concluded that at 170 °C, the FLC of AA7075-T6 is similar to that of AA6016-T4 at room temperature.
To reduce the cost, finite element analysis has become an obligatory step to simulate the deformation process of sheet metals. The accuracy of numerical simulation is dependent on the constitutive model which describes the plastic deformation behavior of sheet metals. Studies have been carried out on the influence of constitutive models on the prediction accuracy of finite element analysis (FEA) for the sheet metal forming processes. Significant conclusion was drawn by Hou et al. [ 14] that usage of the Yoshida-Uemori kinematic hardening model [ 15] increases the accuracy of springback prediction for the U-shaped specimens of a 980 MPa grade multiphase steel. For materials that exhibit little Bauschinger effect but pronounced texture anisotropy (e.g., AA6022-T4), the use of a yield criterion that accounts for anisotropy, e.g., Yld2000-2d, is more important for improving the accuracy of springback prediction [ 16]. Pilthammar et al. [ 17] found that the most reliable prediction of strains and forces in the simulations of Nakajima testing can be achieved by the BBC05 yield criterion with a non-integer exponent, demonstrating the need of yield criterion with high flexibility. In addition, there are some investigations on the yield behavior of sheet metals at elevated temperatures. Based on the results from uniaxial tensile test results, Zhou et al. [ 18] calculated the theoretical yield loci from the von Mises, Hill48 and Barlat89 yield criterion of AA7075-T6 at five temperatures (50, 100, 150, 200 and 250 °C). Sheet metals are usually subjected to multiple deformation paths during forming processes, and the multiaxial formability of sheets cannot be merely investigated through the uniaxial tests. Therefore, it is of importance to investigate the mechanical properties of sheet metals under multiaxial loading conditions, e.g., biaxial tension conditions. To date, methods for biaxial tensile testing at elevated temperatures have been used in some literatures. Terriault et al. [ 19] conducted the equal-biaxial tensile tests and obtained the experimental yield surface of a Ti-Ni alloy using a designed pantograph and a conventional tensile testing machine. Hamasaki et al. [ 20] equipped the biaxial testing machine with a thermostatic bath system, and investigated the influence of temperature on the yield surface of AA5182-O. However, there are few works focused on the yield behavior of AA7075-T6 under biaxial loading condition at warm temperatures, and the applicability of various constitutive models requires further investigation.
In this paper, the deformation behavior of AA7075-T6 within the temperature range from 25 °C to 230 °C was studied through uniaxial and biaxial tensile testing with the aid of induction heating system and digital image correlation techniques. Theoretical yield loci calculated by the von Mises, Hill48 and Yld2000-2d yield criteria were compared with experimental ones at various temperatures. In addition, the influence of the exponent in the Yld2000-2d criterion on yield loci was investigated to improve the prediction accuracy.

## 2 Experimental Details

### 2.1 Uniaxial Tensile Tests at Elevated Temperatures

The material investigated in this work is a commercial peak aged (T6 tempered) aluminum alloy sheet, AA7075-T6, with a thickness of 2.0 mm. Uniaxial tensile specimens were modified following the ASTM Standard E8/E8M [ 21] and were cut by wire electro discharge machining (WEDM) at 0° (rolling direction, RD), 45° (diagonal direction, DD) and 90° (transverse direction, TD) to the rolling direction. In order to reduce the heat transfer from heated specimens to the hydraulic grips of the testing machine, the length of the gripper ends in specimen was extended, as shown in Figure  1. Uniaxial tensile tests were performed on the universal testing machine MTS E45.105 with the aid of induction heating at four temperatures, namely 25, 130, 180 and 230 °C, at a nominal strain rate of 0.001/s. The variations of tensile rates are detailed in Table  1. Digital image correlation (DIC) techniques were used to measure strains during the tests. To prepare the speckle pattern for DIC measurements, the following standard procedure was applied. The specimen surface was cleaned with alcohol, and a layer of white paint was sprayed onto the cleaned surface and then black paint dots with sizes of ~0.3 mm were applied onto the dried white paint layer. The DIC system was set on one side of specimen, and the induction heating coil and infrared thermal imager were set on the other side, as illustrated in Figure  2. A graphite paint layer was applied to the measured area of specimen and the infrared thermal imager provided real-time monitoring and closed-loop control of the temperatures of heated specimens. The temperature of the hot spot in the measured area was used as the feedback temperature in the induction heating system. Figure  3 presents the temperature contours of a modified specimen at 230 °C. It is clear that the temperature decreases from the central area of the gage section to the gripper ends, and a uniform temperature distribution section with an approximate length of 8 mm was noted as depicted by the dashed square in Figure  3, where the temperature difference was controlled within 5 °C below the testing temperature. The specimens were heated to the testing temperatures followed by a holding time of 2 min to ensure a homogeneous temperature distribution on specimens, and then the specimens were pulled until localized necking or fracture happened. It should be noted that the strain data from DIC calculation within the dashed square in Figure  3 were used in the post-processing with the Vic-3D software to acquire stress vs. strain curves.
Table 1
Tensile rates of specimens at different temperatures
Temp. (°C)
25
130
180
230
Tensile rate (mm/min)
3
1
1
0.4

### 2.2 Biaxial Tensile Tests at Elevated Temperatures

To obtain the yield loci of AA7075-T6 at elevated temperatures, biaxial tensile tests of cruciform specimens at 25, 130, 180 and 230 °C were carried out on the testing machine MTS BIA5105 equipped with an induction heating system. The dimensions of cruciform specimen are illustrated in Figure  4, where the central gage section was designed following Hanabusa et al. [ 22] and ISO 16842 [ 23]. Specimens were cut by laser, and the same operation was implemented for strain measurements using DIC techniques. The horizontal and vertical directions were defined as the x and y-axes, respectively. The RD of specimens was along x-axis, and the force ratios ( $$F_{x} :F_{y}$$) were set to 1:2, 1:1, and 2:1. The DIC system was set up in front of the tested specimen, and the induction heating coil was placed 10 mm from the specimen on the other side, as shown in Figure  5. In order to ensure the feasibility of applying induction heating to biaxial tensile testing, the temperature contours of a cruciform specimen were measured by an infrared camera at 230 °C in Figure  6(a). A graphite paint layer was applied to the measured area of specimen. It can be found that the temperature decreased from the central area of the specimen to the surrounding arms gradually, while the temperature contours in the central square gage area (45 mm by 45 mm) was relatively uniform. The temperature profiles extracted along the x-axis and y-axis are presented in Figure  6(b), where the origin of the coordinate system located in the cruciform center. Within the range of 45 mm along the x-axis and the y-axis, the maximum temperature was 234 °C and the lowest temperature was 222 °C. Hence, the temperature fluctuation is smaller than ± 8 °C. During the tensile testing, a thermocouple was taped to the center of the specimen using a high temperature adhesive, and the temperature at this center was monitored and used in the feedback control of the induction heating system. The heating rate was set to ~ 150 °C/min.
The biaxial true strain components were extracted and averaged over the gage area (45 mm by 45 mm) of cruciforms in the DIC software. True stress components and plastic strain components of the deformed gage regions were calculated based upon the Hooke’s law and the law of volume constancy in plastic deformation. The detailed processing of stress vs. strain data in the biaxial tensile testing refers to Min et al. [ 24].

## 3 Results and Discussion

### 3.1 Flow Curves

Figure  7 shows the true stress ( $$\sigma$$) vs. equivalent plastic strain ( $$\overline{\varepsilon }_{\text{p}}$$) curves obtained from uniaxial tensile tests along different directions at the temperatures of 25 °C, 130 °C, 180 °C and 230 °C. The equivalent plastic strain ( $$\overline{\varepsilon }_{\text{p}}$$) is defined by Eq. ( 1) to evaluate plastic deformation and impose plastic work equivalency between various stress states,
$$\overline{\varepsilon }_{\text{p}} = \int {\text{d}}\overline{\varepsilon }_{\text{p}} = \int \frac{{\dot{w}_{\text{p}} }}{{\sigma_{0} }} = \int \frac{{\varvec{\sigma}:\dot{\varvec{\varepsilon }}^{\text{p}} }}{{\sigma_{0} }} = \int \frac{{\sigma_{\theta } {\text{d}}\varepsilon_{\theta }^{\text{p}} }}{{\sigma_{0} }}$$
(1)
where $$\sigma_{0}$$ is the yield stress under uniaxial tension along the RD, and $$\theta$$ is the angle between the length direction of specimens and the RD. The initial yield stress decreases with an increase of temperature. Work hardening was observed at temperatures from 25 to 180 °C, while $$\sigma$$ vs. $$\overline{\varepsilon }_{\text{p}}$$ curves are almost flat at 230 °C and softening is dominated at this temperature. The softening effect is due to the increasing dislocation mobility caused by dynamic recovery at warm temperatures [ 25].

### 3.2 Experimental Yield Loci

As shown in Figure  8(a), fracture occurred on the arm of cruciform specimens at 25 °C. The plastic deformation in the gage area of cruciform specimens was limited, and only the initial yield locus (with $$\overline{\varepsilon }_{\text{p}}$$ of 0.002) can be obtained at 25 °C. To enlarge the plastic deformation in the gage region of cruciform, scholars modified the cruciform by reducing thickness of the gage or by strengthening the arms. Liu et al. [ 26] developed an optimal shape with a flat circular thickness reduction in the gauge region, which permits to reach an experimental biaxial stress vs. strain curve for AA5086 up to 0.30 of equivalent plastic strain. An arm strengthening method via laser deposition using materials compatible with the test material was developed by Hou et al. [ 27]. This method was verified using various materials to significantly enlarge the obtainable maximum plastic strain in the gage of cruciforms of aluminum alloy AA5754-O [ 28], dual-phase steels (e.g., DP590, DP780 and DP980) [ 29] and quenching and partitioning steel QP980 [ 30, 31].
Fracture occurred at the end of slits close to the central gage area at elevated temperatures, as shown in Figure  8(b), which led to relatively larger plastic deformation. The maximum equivalent plastic strains achieved in the gage area of specimen at the four investigated temperatures and three loading ratios are listed in Table  2. According to the data processing procedure proposed by Min et al. [ 24], experimental yield loci of AA7075-T6 associated with $$\overline{\varepsilon }_{\text{p}}$$ = 0.002 are presented in Figure  9. Note that the data associated with stress ratios ( $$\sigma_{22} :\sigma_{11}$$) of 0:1 and 1:0 are from the uniaxial tensile tests of specimens along RD and TD, respectively. It can be found from Figure  9 that the experimental yield locus at initial yielding decreases with increasing temperature.
Table 2
Maximum equivalent plastic strains achieved in the gage area of cruciform specimens at various temperatures
Temp. (°C)
$$F_{x} :F_{y}$$ = 1:2
$$F_{x} :F_{y}$$ = 1:1
$$F_{x} :F_{y}$$ = 2:1
25
0.002
0.002
0.002
130
0.006
0.010
0.006
180
0.008
0.020
0.008
230
0.030
0.035
0.015
To compare the shape of yield loci at various temperatures as well as their evolving behavior as a function of $$\overline{\varepsilon }_{\text{p}}$$, the yield loci are normalized by the yield stress ( $$\sigma_{0}$$) under uniaxial tension along the RD. Figure  10 presents the normalized yield loci associated with $$\overline{\varepsilon }_{\text{p}} = 0.002$$ and $$\overline{\varepsilon }_{\text{p}} = 0.01$$ at elevated temperatures. It can be found that the shapes of yield loci are almost temperature-independent at both $$\overline{\varepsilon }_{\text{p}} = 0.002$$ and $$\overline{\varepsilon }_{\text{p}} = 0.01$$.
The normalized yield loci associated with different $$\overline{\varepsilon }_{\text{p}}$$ at 130, 180 and 230 °C are presented in Figure  11. At the same temperature, it is clear that the yield locus expands with an increase of $$\overline{\varepsilon }_{\text{p}}$$, especially under the equi-biaxial tension. This is attributed to the fact that the work hardening rate under biaxial tension is higher than that under uniaxial tension, e.g., as shown in Figure  12 when the testing temperature was 180 °C. It is also observed that the evolving characteristics of yield loci is more pronounced at 180 and 230 °C.

### 3.3 Modeling the Yield Behavior of AA7075-T6

#### 3.3.1 Evaluation of Constitutive Models

The von Mises, Hill48, and Yld2000-2d yield criteria are three classical ones and widely used in finite element simulation. A brief introduction to the three yield criteria is provided below.
Von Mises yield criterion is isotropic and expressed by:
$$\overline{\sigma } = \frac{1}{\sqrt 2 }\sqrt {\left( {\sigma_{1} - \sigma_{2} } \right)^{2} + \left( {\sigma_{2} - \sigma_{3} } \right)^{2} + \left( {\sigma_{3} - \sigma_{1} } \right)^{2} }$$
(2)
where $$\overline{\sigma }$$ is the effective stress.
Under plane stress condition, Hill48 yield criterion [ 32] which considers in-plane anisotropy can be defined as:
$$\left( {G + H} \right)\sigma_{11}^{2} - 2H\sigma_{11} \sigma_{22} + \left( {H + F} \right)\sigma_{22}^{2} + 2N\sigma_{12}^{2} = \overline{\sigma }^{2}$$
(3)
where F, G, H and N are four anisotropic parameters, which can be identified from three r-values from uniaxial tensile tests using Eq. ( 4):
$$F = \frac{{r_{0} }}{{r_{90} \left( {r_{0} + 1} \right)}}, G = \frac{1}{{r_{0} + 1}}, H = \frac{{r_{0} }}{{r_{0} + 1}}, N = \frac{{\left( {r_{0} + r_{90} } \right)\left( {1 + 2r_{45} } \right)}}{{2r_{90} \left( {r_{0} + 1} \right)}} .$$
(4)
Unlike the quadratic von Mises and Hill48 yield criteria, Yld2000-2d yield criterion is non-quadratic and is expressed as [ 33]
$$\varPhi = \varPhi^{\prime}\left( {\varvec{X^{\prime}}} \right) + \varPhi^{\prime\prime}\left( {\varvec{X^{\prime\prime}}} \right) = 2\overline{\sigma }^{m} ,$$
(5)
where
$$\varPhi^{\prime} = \left| {\varvec{X}_{1}^{ '} - \varvec{X}_{2}^{ '} } \right|^{m} ,$$
(6a)
$$\varPhi^{\prime\prime} = \left| {2\varvec{X^{\prime\prime}}_{2} + \varvec{X^{\prime\prime}}_{1} } \right|^{m} + \left| {2\varvec{X^{\prime\prime}}_{1} + \varvec{X^{\prime\prime}}_{2} } \right|^{m} .$$
(6b)
The exponent $$m$$ is a material coefficient associated with crystal structure and is suggested to be 8 for materials with face centered cubic (FCC), e.g., aluminum alloys. $$\varvec{X}_{1}$$ and $$\varvec{X}_{2}$$ are the principal values of matrix $$\varvec{X^{\prime}}$$ and $$\varvec{X^{\prime\prime}}$$.
$$\varvec{X}_{1} = \frac{1}{2}\left( {X_{11} + X_{22} + \sqrt {\left( {X_{11} - X_{22} } \right)^{2} + 4X_{12}^{2} } } \right),$$
(7a)
$$\varvec{X}_{2} = \frac{1}{2}\left( {X_{11} + X_{22} - \sqrt {\left( {X_{11} - X_{22} } \right)^{2} + 4X_{12}^{2} } } \right). .$$
(7b)
Components $$\varvec{X^{\prime}}$$ and $$\varvec{X^{\prime\prime}}$$ are obtained from the following linear transformations of the Cauchy stress.
$$\varvec{X^{\prime}} = \varvec{L^{\prime}} \cdot\varvec{\sigma},$$
(8a)
$$\varvec{X^{\prime\prime}} = \varvec{L^{\prime\prime}} \cdot\varvec{\sigma},$$
(8b)
where
$$\left[ {\begin{array}{*{20}c} {L_{11}^{'} } \\ {\begin{array}{*{20}c} {L_{12}^{'} } \\ {\begin{array}{*{20}c} {L_{21}^{'} } \\ {\begin{array}{*{20}c} {L_{22}^{'} } \\ {L_{66}^{'} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {2/3} \hfill & 0 \hfill & 0 \hfill \\ { - 1/3} \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & { - 1/3} \hfill & 0 \hfill \\ 0 \hfill & {2/3} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\alpha_{1} } \\ {\alpha_{2} } \\ {\alpha_{7} } \\ \end{array} } \right],$$
(9a)
$$\left[ {\begin{array}{*{20}c} {L_{11}^{''} } \\ {\begin{array}{*{20}c} {L_{12}^{''} } \\ {\begin{array}{*{20}c} {L_{21}^{''} } \\ {\begin{array}{*{20}c} {L_{22}^{''} } \\ {L_{66}^{''} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} { - 2} \hfill & 2 \hfill & 8 \hfill & { - 2} \hfill & 0 \hfill \\ 1 \hfill & { - 4} \hfill & { - 4} \hfill & 4 \hfill & 0 \hfill \\ 4 \hfill & { - 4} \hfill & { - 4} \hfill & 1 \hfill & 0 \hfill \\ { - 2} \hfill & 8 \hfill & 2 \hfill & { - 2} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 1 \hfill \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\alpha_{3} } \\ {\begin{array}{*{20}c} {\alpha_{4} } \\ {\alpha_{5} } \\ {\alpha_{6} } \\ \end{array} } \\ {\alpha_{8} } \\ \end{array} } \right]$$
(9b)
In Eqs. (8)–(9), $$\varvec{\sigma}$$ is Cauchy stress and the material parameters $$\alpha_{1}$$$$\alpha_{8}$$ are eight anisotropy coefficients calibrated from $$\sigma_{0}$$, $$\sigma_{45}$$, $$\sigma_{90}$$, $$r_{0}$$, $$r_{45}$$, $$r_{90}$$, $$\sigma_{\text{b}}$$ and $$r_{\text{b}}$$, where $$\sigma_{\theta }$$ and $$r_{\theta }$$ are the uniaxial yield stress and r-value (Lankford coefficient) measured along the direction having an angle of $$\theta$$ to the RD, respectively, and $$\sigma_{\text{b}}$$ and $$r_{\text{b}}$$ are the equi-biaxial yield stress and the ratio of plastic strain rate under equi-biaxial tension, respectively [ 34].
The yield stresses and r-values of AA7075-T6 with $$\overline{\varepsilon }_{\text{p}} = 0.002$$ at various temperatures are summarized in Table  3. It is found that $$r_{90}$$ decreases as the temperature increases from 180 to 230°C. In addition, the r-value obtained from uniaxial tension remains nearly constant despite the temperature variation and $$r_{\text{b}}$$ decreases with increasing temperature gradually. The fitted parameters of the Hill48 yield function in Eq. ( 3) and the Yld2000-2d yield function in Eq. (9) are summarized in Tables  4 and 5, respectively.
Table 3
Flow stresses and r-values of AA7075-T6 with $$\overline{\varepsilon }_{\text{p}} = 0.002$$
Temp.
(°C)
$$\sigma_{0}$$
$$\sigma_{45}$$
$$\sigma_{90}$$
$$\sigma_{\text{b}}$$
$$r_{0}$$
$$r_{45}$$
$$r_{90}$$
$$r_{\text{b}}$$
25
536
540
542
527
0.63
0.81
1.1
0.89
130
408
407
413
410
0.63
0.8
1.11
0.71
180
305
313
309
300
0.65
0.84
1.07
0.67
230
198
198
197
194
0.67
0.81
0.85
0.58
Table 4
Identified parameters in the Hill48 yield function
Temp.
(°C)
F
G
H
N
25
0.3491
0.6146
0.3854
1.2644
130
0.3492
0.6131
0.3869
1.2481
180
0.3678
0.6064
0.3936
1.3045
230
0.4639
0.6039
0.3961
1.403
Table 5
Identified parameters in the Yld2000-2d yield function with an exponent of 8
Temp.
(°C)
$$\alpha_{1}$$
$$\alpha_{2}$$
$$\alpha_{3}$$
$$\alpha_{4}$$
$$\alpha_{5}$$
$$\alpha_{6}$$
$$\alpha_{7}$$
$$\alpha_{8}$$
$$m$$
25
0.9129
1.0522
1.0654
0.9937
1.0241
1.0014
0.981
0.989
8
130
0.9212
1.0378
0.974
0.9755
1.022
0.9943
0.9835
1.044
180
0.9457
1.0201
0.9957
0.9827
1.0329
1.0581
0.9681
0.9463
230
0.9688
0.997
0.9118
0.9934
1.0395
1.1013
0.9871
1.0094
The yield loci calculated by the von Mises, Hill48 and Yld2000-2d yield criteria are presented in Figure  13. Compared with the other two yield functions, Yld2000-2d gives prediction in better agreement with the experimental data.
Figure  14 shows the predicted uniaxial yield stresses and r-values along various angles to the RD using three yield functions, and these predictions are compared with the experimental results at 130 °C. Uniaxial stresses predicted by the Hill48 criterion are larger than the experimental stresses, and the von Mises and Yld2000-2d yield functions give accurate prediction of uniaxial yield stresses, as shown in Figure  12(a). In Figure  12(b), large deviations between r-values predicted by von Mises and experimental results are observed. The Hill48 and Yld2000-2d yield criteria give nearly the same prediction in r-value along different angles to the RD, and the results are in good agreement with experimental data. In summary, the Yld2000-2d criterion, namely, the non-quadratic anisotropic yield equation, provides the most accurate prediction in the yield behavior of AA7075-T6 at elevated temperatures, among the three investigated yield criteria.

#### 3.3.2 Influence of the Exponent Value in the Yld2000-2d Criterion on Calculated Yield Loci

In order to investigate the influence of the exponent m in the Yld2000-2d criterion on the predicted yield surface, initial yield loci from Yld000-2d with exponent varying from 6 to 26 are shown in Figure  15. It can be observed from Figure  15 that the exponent has a significant impact on the curvature of the predicted yield surface. As the exponent m increases, the curve between the uniaxial tension and equi-biaxial tends to be straight. To evaluate the errors when applying different exponent values to predicting yield loci, the root mean square error (RMSE) is used,
$${\text{RMSE}} = \sqrt {\frac{1}{N}\mathop \sum \limits_{i}^{N} \left( {\frac{{\sigma_{\text{f}} }}{{\sigma_{\text{e}} }} - 1} \right)^{2} } ,$$
(10)
where $$\sigma_{\text{e}}$$ is the experimental value, $$\sigma_{\text{f}}$$ is the predicted value, and N is the number of experimental data sets (or stress ratios), which is 5 in this work. Figure  16 shows the calculated RMSEs by the Yld2000-2d criterion with different $$m$$ values at various temperatures. At 25 °C, the RMSE decreases as $$m$$ increases. At other temperatures, the RMSE decreases at first and then increases with an increase of $$m$$ value. The averaged RMSE is indicated by the solid line in Figure  16. It can be seen that the averaged RMSE decreases as $$m$$ increases from 6 to 14, and increases slightly as $$m$$ increases from 14 to 26. Therefore, the Yld2000-2d yield function with $$m$$ = 14 gives the lowest RMSE when predicting yield loci of AA7075-T6. The experimental yield loci associated with $$\overline{\varepsilon }_{\text{p}} = 0.01$$ are compared with the calculated ones by the Yld2000-2d yield criterion with $$m$$ = 14 in Figure  17. Results show that the predicted yield loci are in good agreement with experimental data points, which indicates that the Yld2000-2d yield criterion with $$m = 14$$ accurately predicts yield loci of AA7075-T6 at elevated temperature not only at the initial yielding but also during the subsequent yielding. The identified parameters of the Yld2000-2d yield function with $$m$$ = 14 when $$\overline{\varepsilon }_{\text{p}}$$ is equal to 0.002 and 0.01 are listed in Tables  6 and 7, respectively.
Table 6
Identified parameters of Yld2000-2d yield function with $$m=14$$ ( $$\overline{\varepsilon }_{\text{p}} = 0.002$$)
Temp.
(°C)
$$\alpha_{1}$$
$$\alpha_{2}$$
$$\alpha_{3}$$
$$\alpha_{4}$$
$$\alpha_{5}$$
$$\alpha_{6}$$
$$\alpha_{7}$$
$$\alpha_{8}$$
$$m$$
25
0.9666
1.0098
1.0479
0.9984
1.0172
1.0233
0.9864
0.9795
14
130
0.9673
1.0048
0.9825
0.983
1.0108
0.9954
0.9924
1.0284
180
0.9778
0.9976
1.0174
0.9915
1.0216
1.0476
0.9711
0.9399
230
0.9737
1.0111
0.9759
1.0022
1.0262
1.0698
0.9933
0.995
Table 7
Identified parameters of Yld2000-2d yield function with $$m=14$$ ( $$\overline{\varepsilon }_{\text{p}} = 0.01$$)
Temp.
(°C)
$$\alpha_{1}$$
$$\alpha_{2}$$
$$\alpha_{3}$$
$$\alpha_{4}$$
$$\alpha_{5}$$
$$\alpha_{6}$$
$$\alpha_{7}$$
$$\alpha_{8}$$
$$m$$
130
0.9701
0.9823
0.9063
0.9581
0.989
0.8982
0.969
1.0341
14
180
0.9783
0.9733
0.904
0.9577
0.99
0.9063
0.9581
0.9966
230
0.9505
1.0397
0.8822
0.9928
1.0073
0.9819
1.0029
1.074

## 4 Conclusions

In this study, the yield loci of AA7075-T6 at elevated temperatures from 25 to 230 °C were obtained through uniaxial and biaxial tensile testing with the aids of induction heating and DIC techniques, and the yield behavior at various temperatures have been investigated. The following conclusions can be drawn.
(1)
The von Mises, Hill48 and Yld2000-2d yield functions are calibrated using uniaxial yield stresses and r-values at 0°, 45°, 90° to the RD, the equi-biaxial yield stress and the plastic strain ratio under equi-biaxial tension at various temperatures.

(2)
Through comparison of experimental results with three yield functions, Yld2000-2d yield function, namely, the non-quadratic anisotropic yield equation, is verified to give a better prediction to the yield loci of AA7075-T6 within the investigated temperature range.

(3)
An exponent of 14 in the Yld2000-2d yield function is for the yield loci of AA7075-T6 at elevated temperatures.

(4)
Hardening of AA7075-T6 at elevated temperatures is found stress-state-dependent.

## Acknowledgements

Not applicable.

### Competing interests

The authors declare no competing financial interests.
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