Mechanical Properties and Formability of Heat-Treated 7000-Series High-Strength Aluminum Alloy: Experiments and Finite Element Modeling

Abstract

The feasibility of room-temperature (cold) forming of peak-aged 7075 high-strength aluminum alloy (7075-T6) sheets after solution heat treatment followed by water quenching was investigated. The anisotropic plastic properties of the heat-treated aluminum sheet (7075-WT) were experimentally characterized by performing uniaxial tensile tests (along different loading directions) and a balanced biaxial tension test. Limit dome height (LDH) and cylindrical cup drawing experiments were conducted to evaluate the formability at room temperature. With the obtained anisotropic mechanical properties, different yield functions, including Yld2000-2d, Hill 1948, and von-Mises, were determined and implemented in the finite element simulations to numerically predict the formability at room temperature. The formability of the 7075-WT sheet at room temperature could be significantly enhanced owing to the prior heat treatment, and the finite element simulations performed using the anisotropic yield functions and Voce hardening law were in good agreement with the measured LDH. The LDH of the 7075-WT sheet in the plane strain and biaxial stress modes and the earing in the circular cup drawing tests could be better predicted non-quadratic Yld2000-2d function than those by isotropic and Hill 1948 quadratic yield functions.

Graphic abstract

1 Introduction

Because of the weight reduction requirement for vehicles in the automotive industry, the use of aluminum alloy sheets has dramatically increased over the past few decades. In particular, high-strength aluminum alloy sheets, such as 5000–7000 series alloys, are in contention to replace conventional high-strength steel sheets owing to their higher specific strength. For example, the tensile strength of 7000-series aluminum alloy sheets is over 500 MPa, making them a strong candidate for the structural parts of automobiles. However, despite the high strength, the inferior formability and fracture properties at room temperature limit their application to automotive part forming [1].

As one of the methods for improving the formability of aluminum sheets, the warm forming process has been widely used for many years. In this method, the typical forming temperature is in the range of 150–300 °C [2, 3], and the springback of the formed parts is reduced (or equivalently, increased shape accuracy) because of the lower forming stress [4]. However, several limitations of the warm forming process in addition to the increased cost have been reported, such as low strength because of the heat effect during forming [5] and the need for a special lubricant to reduce the friction between the tool and the sheet [6]. Another method of increasing the ductility of aluminum alloy sheets involves applying a hot stamping process at even higher temperatures than that used in the warm forming process [7]. In the hot stamping process, the sheet metal is heated to the solution heat treatment (SHT) temperature, transferred to a forming tool, and quenched in cooled dies. Optimizing the process parameters, such as the temperature and transfer time, could help improve the capacity of the sheets in enduring large plastic deformation without failure [8]. However, more investigations on the complex material flow behavior in terms of the temperature, strain, and strain rate at elevated temperatures need to be performed to ensure good quality of the products following the hot stamping operation [9].

Studies on the cold (room-temperature) forming of 7000-series aluminum alloy sheets are lacking, because of their low ductility in this temperature range [10, 11]. To obtain better formability, heat treatments, such as W and O tempers, were applied to 7000-series alloy sheets. Consequently, they exhibited higher formability but lower strength than the conventional T6 tempered sheets because of the absence of hardening Guinier–Preston (GP) zone and η′ precipitates [12]. With the improved formability realized via W and O tempers, automotive parts could be successfully manufactured [10, 11], and the strength of the sheets could be increased to that of the T6-tempered sheets by applying additional heat treatments after forming. Although experimental studies are many, studies on the plastic deformation behavior of high-strength aluminum alloys following heat treatment and corresponding numerical analyses are lacking.

The plastic deformation of aluminum alloys can be accurately described using appropriate anisotropic yield functions and hardening models. Various yield functions have been proposed to reproduce the anisotropic material behaviors under different stress states [13,14,15,16,17,18]. Non-quadratic yield models were proposed to account for the crystal structure, thus obtaining better agreement with the measured plastic deformation behavior after appropriately determining the non-quadratic exponent of the yield functions [14, 15]. In addition, numerical simulations for predicting the plastic behavior of aluminum alloys were performed with the non-quadratic anisotropic yield functions based on linear transformations [19,20,21].

In this study, finite element modeling was performed to evaluate the effectiveness of the anisotropic constitutive models in predicting the plastic forming process of high-strength aluminum alloy 7075-T6 sheets after SHT followed by water quenching. In Sect. 2, experiments for determining the mechanical properties of the sheets, such as the uniaxial tension, in-plane biaxial tension, and limit dome height (LDH), are described, including cylindrical cup drawing tests. The finite element models with the corresponding anisotropic yield functions and hardening law are presented in Sect. 3. In Sect. 4, different plasticity models are evaluated by comparing the simulated results with the mechanical behaviors obtained from the LDH and cylindrical cup drawing tests. Finally, the conclusions of this study are presented in Sect. 5.

2 Experiments

2.1 Material and Heat Treatment

A commercial 7000-series high-strength aluminum alloy under peak-aged condition, i.e., 7075-T6 sheet, was investigated in this study. The sheet was obtained from Kaiser Aluminum Corporation. The thickness of the sheet is 1.0 mm. Table 1 lists the chemical compositions in wt%, in which the additions of Zn and Cu are presented for improving strength compared to other grades of aluminum alloys.

Table 1 Chemical composition of the aluminum alloy 7075-T6 sheet (in wt%)

As the ductility of the sheet under the as-received T6 temper condition is known to be limited for effective forming at room temperature, a special heat treatment was performed in this study. The T6 sheet material was heated to 470 °C at a heating rate of 1.5 °C/s and held for 30 min for SHT. The heated sheet was then water quenched down to room temperature. Figure 1 shows the schematic of the heat treatment procedure. Following the SHT, the high-strength aluminum alloy sheet exhibited lower strength and higher ductility than the as-received material. This process is often called the W-temper (WT) condition. Although the mechanical properties of the solution heat-treated sheet can change depending on the holding time, a constant holding time of 30 min was applied to all the specimens. Moreover, the mechanical tests are conducted within 30 min after quenching, as shown in Fig. 1.

Fig. 1

Temperature profile of the W-temper condition and subsequent mechanical testing

2.2 Uniaxial Tension Test

Uniaxial tension tests were performed on the ASTM E8 M sub-size standard specimens using a universal testing machine (UTM) at a strain rate of 10⁻³ /s. To evaluate the anisotropy of the W-tempered sheet, the tensile specimens were cut along 0, 45, and 90° from the rolling direction (RD). In addition to obtaining the stress–strain curves of the test specimens, the strains along the longitudinal (or length) direction and transverse (or width) direction were measured using the ARAMIS 3D digital image correlation (DIC), from which the r-value (${\text{r = }}\frac{{{\text{d}}\upvarepsilon_{\text{w}} }}{{{\text{d}}\upvarepsilon_{\text{t}} }}$) was obtained. Here, dε_w and dε_t denote the strain increments along the width and thickness directions, respectively.

2.3 In-Plane Biaxial Tension Test

In-plane balanced biaxial tension tests [22, 23] were conducted on a cruciform biaxial specimen. From this test, the yield stress in the equi-biaxial stress state and the ratio of strain increment in the transverse direction to that in the RD (r_b = dε_TD/dε_RD) were obtained. Figure 2a shows the biaxial tensile specimen placed on the testing machine. Figure 2b shows the detailed dimensions of the cruciform specimen. The testing machine has two servo-controlled hydraulic actuators located at right angles to each other. The actuators were used to maintain a constant load ratio of 1:1. Four strain gauges are attached onto the surface of the specimen, as shown in Fig. 2b, to measure the strains along the two loading directions. The strain rate was maintained at approximately 10⁻³/s in all the tests. At least three tests were conducted to ensure the reproducibility of the measured data.

Fig. 2

a In-plane biaxial testing machine and b cruciform specimen with dimensions

2.4 Limit Dome Height (LDH) Test

Figure 3a shows the schematic of the LDH test set up along with the detailed dimensions of the forming tools. The LDH test was conducted using a universal sheet metal testing machine (manufactured by Erichsen™). A hemispherical punch with a radius of 50 mm was used, and the punch speed was 0.5 mm/min. A lock bead with a depth of 10 mm was used to constrain the drawing of the sheet during the experiment. A blank holding force of 50 kN was applied to the holder, and Teflon^® sheet with a thickness of 0.1 mm was used to minimize the influence of friction.

Fig. 3

a Tool dimensions for LDH test and b specimen dimensions

Figure 3b shows the specimen dimensions for the LDH tests. The specimens were prepared in accordance with the ISO standard ISO 12004-2. Five different sample widths, i.e., 40, 60, 100, 140, and 200 mm, were selected to realize various deformation modes in the tests. Finally, the punch load–displacement relationships were recorded during the test.

2.5 Cylindrical Cup Drawing Test

Cylindrical cup drawing tests were performed to evaluate the formability and anisotropy of the heat-treated aluminum alloy sheet. Figure 4 shows the schematic of the tool set up for the tests with detailed tool dimensions. The same test machine used for the LDH test was used. The diameter of the circular sheet sample was 87 mm. The punch speed was set to 0.5 mm/s to ensure that all the tests are quasi-static, and a holding force of 30 kN was applied. Tallow was applied to both the specimen and tool surfaces to reduce the friction. Like the LDH tests, at least three tests were conducted.

Fig. 4

Schematic of cup drawing test tools

3 Constitutive Laws and Finite Element Modeling

3.1 Strain Hardening Law

The engineering stress–strain curves along the RD obtained from the uniaxial tension tests were used to determine the strain hardening law for the investigated sheet. The true stress–strain curves were transformed from their engineering curves only up to the point of uniform elongation, whereas the flow stress curves beyond the ultimate tensile stress were fitted as per the Voce hardening law using Eq. (1).

$${\bar{\upsigma}} = {\text{A}} + {\text{B}} \cdot \left( {1.0 - e^{{ - {\text{C}} \cdot {\bar{\upvarepsilon }}}} } \right).$$

(1)

where A, B, and C are material constants.

3.2 Forming Limit Diagram Using Marciniak–Kuczyinski (M–K) Theory

Marciniak and Kuczyinski [24] proposed a theoretical approach to describe the sheet formability. The M–K model assumes a pre-existing imperfection oriented at angle φ° to the direction of the principal minor stress direction (y-axis) as shown in Fig. 5. The x and y axes correspond the principal stress directions in the homogeneous part <A>, ${\vec{\mathbf{t}}}$ and ${\vec{\mathbf{n}}}$ are the parallel and perpendicular directions to the imperfection region <B>, respectively. In the figure, $t_{0}^{\text{A}}$ and $t_{0}^{\text{B}}$ are the initial material thicknesses of homogeneous and grooved parts, respectively. The initial imperfection is represented by the ratio of two initial thicknesses (${\text{f}}_{0} = \frac{{t_{0}^{\text{B}} }}{{t_{0}^{\text{A}} }}$), and the imperfection promotes the localized necking. The initial value of the imperfection is assumed to be 0.996 [25].

Fig. 5

Schematic description of Marciniak–Kuczynski model

In the model, it is assumed that the stress ratio, α = $\frac{{\upsigma_{yy}^{\text{A}} }}{{\upsigma_{xx}^{\text{A}} }}$, in homogeneous region A at a specific angle φ° is constant during plastic deformation. By applying the force equilibrium and compatibility conditions, the stress and strain responses inside the grooved region are obtained by using Newton–Raphson algorithm. The failure occurs when the ratio of the strain increment in homogeneous region along x-axis to the corresponding strain increment in grooved region ($\frac{{\Delta\upvarepsilon_{xx}^{\text{A}} }}{{\Delta\upvarepsilon_{xx}^{\text{B}} }}$) becomes less than the specific critical tolerance set as 0.1. The calculations are continued from angle φ = 0° to angle φ = 90°. And the forming limit strains are the minimum strains in the homogeneous region from calculations.

Note that quadratic or non-quadratic anisotropic yield models (See Appendix A) were associated with M–K theory to predict the forming limit curve in this study.

3.3 Finite Element Modeling: Limit Dome Height (LDH) Test

The finite element simulations were performed using the implicit FE software ABAQUS/Standard 2018 via user-defined material subroutine (UMAT) [26]. Considering the symmetry of the model and boundary conditions, only a quarter of the specimen was modeled. Figure 6 shows the symmetric boundary conditions. To reduce the computation time, the blank draw-in in the bead region was completely constrained. A minimum element size of 1 mm by 0.2 mm and a maximum element size of 1 mm by 1 mm were used with four-node shell elements with reduced integration (S4R). Nine integration points along the thickness direction were used. The tools were modeled with analytical rigid body elements. The Coulomb friction coefficient between the tools and the blank sheet was set to 0.05 considering the lubrication provided in the experiments [27, 28].

Fig. 6

Finite element model for the LDH tests

The isotropic hardening law and the associated flow rule were implemented to describe the plastic deformation behavior of the sheet [29]. In addition, a linear isotropic elasticity was assumed with a Young’s modulus (E) of 72 GPa and a Poisson’s ratio (ν) of 0.33.

3.4 Finite Element Modeling: Cylindrical Cup Drawing Test

The cylindrical cup drawing simulations were performed using ABAQUS/Explicit 2018 via the user-defined material subroutine VUMAT [26]. The punch velocity was carefully chosen to ensure negligible dynamic effect in the quasi-static forming process. The FE model includes a die system and an aluminum alloy sheet blank, as shown in Fig. 4. In addition, the shell elements S4R with nine integration points along the thickness direction are used for the blank. The maximum element size is 1 mm by 0.5 mm, and the number of elements is 5411. As shown in Fig. 7, only 1/4th of the model is used for the sheet blank owing to model symmetry. The tools were modeled using analytical rigid body elements. A Coulomb friction coefficient of 0.1 was employed [30, 31] though more rigorous model exists [32]. The same elastoplastic laws as in the LDH simulations were used.

Fig. 7

Finite element model for the discretized blank

4 Results and Discussion

4.1 Experimental Results

Figure 8 shows the engineering stress–strain curves of both the as-received (7075-T6) and W-tempered (7075-WT) sheets. For the 7075-WT sheets, the stress–strain curves along the three directions (RD, TD, and 45° from RD) are compared. Compared to the 7075-T6 sheets, the 7075-WT sheets exhibit lower tensile strength (~ 580 MPa vs. ~ 350 MPa) but significantly improved uniform elongation (~ 11 vs. ~ 21%). Table 2 lists the basic mechanical properties of the 7075-WT sheets, obtained from the uniaxial tension tests. In terms of the anisotropy, the differences in the flow stresses along the three loading directions are not noticeable. However, the deformation anisotropy (i.e., r-values) exhibits significant deviations for the three specimen orientations. Because of the strong anisotropy in the r-values, anisotropic yield functions are required for the finite element simulations.

Fig. 8

Engineering stress–strain curves of the 7075-T6 and 7075-WT sheets along the three loading directions

Table 2 Summary of the mechanical properties

From the measured uniaxial tensile and balanced biaxial tensile properties, the plastic constitutive models were identified. Table 3 lists the identified model parameters for the Voce hardening law, Hill 1948 yield function, and Yld2000-2d yield function (See Appendix A). For the hardening law and Yld2000-2d yield function, the least squares method is used to obtain the best fitting parameters. In contrast, the four anisotropic coefficients of the Hill 1948 yield function can be analytically obtained from the measured r-values, which exhibit significantly higher anisotropy than the yield stresses. Figure 9a shows the obtained yield criteria for the different yield functions. For comparison, the isotropic von Mises yield function is also presented in the figure. The figure shows that the yield stress in the biaxial tensile region predicted using the Hill 1948 yield function is lower than that predicted using the isotropic yield function, and the non-quadratic nature of the Yld2000-2d yield function is reflected in the sharp stress drop from the balanced biaxial state to the plane strain state. In the plane strain tension state, the stress predicted using the Yld2000-2d model is the lowest, whereas the stress predicted using the isotropic model is the highest. Figure 9b, c show the variations in the r-values and normalized yield stresses along the loading directions (or equivalently, sheet orientation) from RD. The variations in the yield stresses and r-values are better described using the Yld2000-2d model, whereas only the r-values could be well fitted to the experimental results.

Table 3 Material constants for the constitutive models

Fig. 9

Comparison between different yield models in terms of: a Yield loci, b stress, and c r-value

The effect of yield models on the predicted localized necking or formability was presented in Fig. 10a. The predicted forming limit diagrams (FLDs) clearly showed very high sensitivity of the limiting strains on the choice of the yield models. Particularly, in biaxial deformation regions, high limiting strains were predicted when von-Mises plastic yielding model was used. As for the comparison, the FLD of 7075-T6 was constructed by using M-K theory incorporated with Yld2000-2d model. As shown in Fig. 10b, the predicted formability of 7075-WT sheets is much better than that of 7075-T6 sheets.

Fig. 10

a Forming limit diagrams (FLDs) predicted by M–K approach using various yield models, and b comparison of predicted formability of 7075-WT and 7075-T6 sheets

4.2 Numerical Results of LDH Test

Figure 11 shows the FE-calculated punch load–displacement curves of the 7075-WT sheet, obtained from the LDH tests using the different constitutive models. The experimentally measured results are also included for comparison. For the 40 mm-width specimen, the LDH result of the 7075-T6 sheet was also included. Following are the conclusions drawn from the results:

Fig. 11

Comparison between the punch load–displacement curves obtained using experimental and numerical analyses

• The punch load–displacement curves predicted using the Voce hardening law are in good agreement with the measured curves with a reasonable accuracy. The curves of the specimens with sheet widths of 40 and 60 mm are in good agreement with the experimental results for all the three yield functions. Figure 11a shows that the formability of the 7075-WT sheet is significantly greater than that of the 7076-T6 sheet, i.e., an improvement of 10 mm in the punch displacement before fracture. Figure 12a shows the major strain distribution of the 40 mm-width specimen at the onset of necking, in which case the strain is greater than the uniform elongation observed under uniaxial tension. This result helps validate the use of the Voce hardening law for large strain simulations.

  Fig. 12

  

  a Strain distribution of the specimen with a width of 40 mm at the onset of necking, and b major–minor strains corresponding to specimen widths of 100 and 140 mm

• For the specimens with sheet widths of 100 and 140 mm, the simulations with the Yld2000-2d model are in good agreement with the measured load–displacement curves and punch displacements at localized failure (Fig. 11c, d). Figure 12b shows the major–minor strain curves corresponding to sheet widths of 100 and 140 mm, obtained from the experiments. Here, the deformation states correspond to plane strain deformation, in which case the Yld2000-2d model is more suitable owing to its non-quadratic characteristic. The early localization corresponds to the sharper shape in the plane strain region, whereas the bulged-out shape obtained when using the von Mises and Hill 1948 functions delays the localization.

• The punch load–displacement curves corresponding to a sheet width of 200 mm show that only the Yld2000-2d model could predict the maximum load and displacement at localized failure (Fig. 11e). Both the Hill 1948 and von Mises yield functions failed to predict the failure even when the punch displacement is greater than 50 mm. This could be because the yield function model parameter identification was done considering a balanced biaxial state only in the Yld2000-2d model.

4.3 Numerical Results of Cylindrical Cup Drawing Test

Figure 13a, c show the results of the cylindrical cup drawing tests for the 7075-T6 and 7075-WT sheets. A holding force of 30 kN was applied in both the tests. The figures show that the formability obtained following the WT heat treatment is greater than that obtained under the T6 condition, in which case an early fracture is observed in the specimens under the same forming condition. For the formed part with the 7075-WT sheet, four ears are observed after complete forming without fracture. This test result demonstrates the potential of room-temperature forming of WT heat-treated 7000-series high-strength aluminum alloys. Additionally, the corresponding simulation results incorporating M–K FLDs for the 7075-T6 and 7075-WT sheets are shown in Fig. 13b, d. The simulated result of the 7075-T6 sheet shows that the failure in the cup wall is successfully predicted in quantitative way. Both location of failed position (7.4 mm from the top) and punch displacement at failure (12 mm) from FE simulation shown in Fig. 13b gave good agreement with experimental values of 7.2 mm and 11.1 mm shown in Fig. 13a, respectively. As for 7075-WT, FE simulation predicts successfully drawn cup, which is consistent with the experimental observation.

Fig. 13

Experimental and corresponding simulated results of cylindrically drawn cups with a b 7075-T6 and c d 7075-WT test specimens

As a result of numerical modeling, Fig. 14 compares the load–punch displacement curves predicted from the cylindrical cup drawing test with the measured curve. Because of the higher strength of the 7075-T6 sheet, its punch load is significantly higher than that of the 7075-WT sheet. The FE simulation results obtained using the three yield functions are compared with the experimental results. Note that all the three models could effectively predict the load–displacement curves, unlike that in the LDH test case. This result shows that the investigated yield functions could represent the stress state of the WT-tempered sheet in the drawing mode.

Fig. 14

Comparison between punch load–displacement curves obtained using experimental and numerical analyses

To investigate the effect of deformation anisotropy, the earing profiles were simulated and compared with the experimentally measured profiles. Figure 15 shows that the normalized cup heights (the height normalized using the cup height along RD) at intervals of 15° from the RD strongly depend on the yield functions. Evidently, no ears could be predicted using the isotropic von-Mises yield function. In contrast, the two anisotropic yield functions, i.e., Hill 1948 and Yld2000-2d, could accurately predict the number of ears. The Yld2000-2d yield function gave a slightly better prediction of the ear height, as the anisotropy in the yield stresses and r-values could be simultaneously considered in this model. Figure 16 shows the equivalent stresses after forming, obtained using the different constitutive models. The stress distribution is uniform when using the isotropic von-Mises yield function, whereas a non-uniform stress distribution is observed when using the two anisotropic yield functions. This non-uniform stress distribution can be attributed to the earing profile.

Fig. 15

Comparison between earing profiles obtained using experimental and numerical analyses

Fig. 16

Equivalent stresses obtained using a von-Mises b Hill 1948 and c Yld2000-2d yield functions

Although the earing profiles obtained using the two anisotropic yield functions are different, the absolute magnitude is very low, i.e., only a 4% difference, as shown in Fig. 15. This is interesting because the Hill 1948 yield function cannot predict the directional change in the yield stress, whereas the Yld2000-2d yield function can. It is reported that the formation of ears in the cylindrical cup drawing process depends on both the stress and r-value along the loading directions. From Fig. 8, which compares the stress and r-value with respect to the sheet orientation along RD obtained using the different yield functions, it is found that the relative difference in the yield stresses obtained using the Hill 1948 and Yld2000-2d yield functions is only approximately 5%, which is close to the difference in the predicted earing profiles. The similar predictions of the r-values using the Hill 1948 and Yld2000-2d yield functions may be one of the reasons for the similar earing profiles predicted using the two anisotropic yield functions.

4.4 Effect of Paint-Bake After Plastic Deformation

In the previous sections, the enhanced formability and the validity of the anisotropic hardening model in predicting the LDH and cylindrical cup drawing formability of the 7075-WT sheet were presented. However, the WT heat treatment significantly reduced the strength of the sheet compared to that under the as-received T6 condition. For actual automotive components in engineering practice, the strength is recovered using the bake-hardening process.

Therefore, the final strength after bake-hardening was investigated by applying a heat treatment analogous to the industrial bake-hardening process used for 7075-WT sheet. The heat treatment performed in this study involves heating the sheet up to 180 °C for 30 min followed by air-cooling, as shown in Fig. 17. This heat treatment is known as artificial ageing of heat treatable aluminum alloys. To consider the pre-strain effect on the mechanical properties of the heat-treated 7075-WT sheet, additional uniaxial tension tests were conducted after a tensile strain of 15%.

Fig. 17

Temperature profile of heat treatment

Figure 18 shows the engineering stress–strain curves of the 7075-WT sheet subjected to prior tensile deformation and subsequent paint-bake heat treatment. The heat-treated specimen exhibits increased stress up to 450 MPa, which is approximately 90% of the ultimate tensile stress of T6 sheet. Another interesting observation is that the elongation at fracture is improved (up to 25%) compared to the WT sheet. The results obtained in this study confirm that the forming of 7075-WT aluminum alloys can be a potential application alternative to the conventional cold forming of high-strength aluminum alloys (which is characterized by limited formability at room temperature).

Fig. 18

Experimental results of the engineering stress–strain relationship for the 7075-WT sheet subjected to paint-bake cycle

5 Conclusion

In this study, the mechanical properties of heat-treatable 7000-series aluminum alloy, namely 7075-WT sheet, following solution heat treatment and subsequent water quenching were characterized by performing uniaxial and balanced biaxial tensile tests. Moreover, the formability of the investigated material was evaluated by conducting limit dome height and cylindrical cup drawing tests at room temperature. The plasticity laws based on the anisotropic yield functions were identified from the measured anisotropic mechanical properties. Both quadratic Hill 1948 yield function and non-quadratic Yld2000-2d yield function were considered and implemented in the finite element simulations to predict the formability of the 7075-WT sheet. Following are the conclusions drawn from the combined experimental and numerical analyses.

• The investigated 7075-WT sheet exhibited strong anisotropy in terms of the r-values, whereas the anisotropy was not noticeable in terms of the flow stress. The anisotropic mechanical properties of the 7075-WT sheet, including the r-values and flow stresses, along various sheet orientations could be well reproduced using the non-quadratic anisotropic yield function Yld2000-2d. The Hill 1948 model identified from the r-values along the RD, 45° from RD, and TD could predict only the r-values along the sheet orientations, but failed to predict the anisotropy in the stresses.

• The finite element simulations for the LDH tests showed that the Yld2000-2d model with the Voce hardening law could predict the load–punch displacement curves with a higher accuracy than the Hill 1948 model. The discrepancy between the experimentally measured data and the simulated results obtained using the Hill 1948 model was more pronounced in the plane strain state. The shape of the yield surface with a sharp corner in the biaxial tensile region, obtained using the Yld2000-2d model, helped increase its predictive accuracy compared to the Hill 1948 model, in which case a bulged-out shape was observed in the same stress state.

• Regarding the cylindrical cup drawing tests, the load–punch displacement curves predicted using the von Mises, Hill 1948, and Yld2000-2d yield functions were in good agreement with the experimental results. This shows that the drawing dominant deformation mode in the 7075-WT sheet can be accurately modeled using both the quadratic and non-quadratic yield functions. However, the Yld2000-2d model gave better predictions of the earing profile of the investigated material than the Hill 1948 model.

• To verify the application prospect of the WT-tempered high-strength aluminum alloy sheet to cold sheet metal forming, the strength recovery was validated by conducting a tensile test following paint-bake heat treatment on a pre-strained 7075-WT sheet. Finally, the second heat treatment applied to the pre-strained 7075-WT sheet was found to enhance its ductility. This will be further investigated as a future study.

Appendix A. Anisotropic Yield Models

To consider the material anisotropy of the investigated aluminum alloy, the non-quadratic anisotropic Yld2000-2d yield function was used. This yield function is formulated using two linear transformations, and a brief summary of the model is given below [18]:

$$\phi \left( {\varvec{\upsigma}} \right) = \left( {\frac{{\phi^{\prime } + \phi^{\prime \prime } }}{2}} \right)^{{{1 \mathord{\left/ {\vphantom {1 {\text{m}}}} \right. \kern-0pt} {\text{m}}}}} = {\bar{\sigma }} ,$$

(2)

$$\phi^{\prime } = \left| {{\text{S}}_{1}^{(1)} - {\text{S}}_{2}^{(1)} } \right|^{\text{m}} \quad {\text{and}}\quad \phi^{\prime \prime } = \left| {2{\text{S}}_{2}^{(2)} + {\text{S}}_{1}^{(2)} } \right|^{\text{m}} + \left| {2{\text{S}}_{1}^{(2)} + {\text{S}}_{2}^{(2)} } \right|^{\text{m}} ,$$

(3)

where m is an exponent, the recommended value of which is 8 for face-centered cubic metals such as aluminum alloys. ${\text{S}}_{1,2}^{{ ( { }i{ )}}}$ (i = 1, 2) denotes the principal stress obtained from the linearly transformed stress tensor ${\mathbf{S}}^{{ ( { }i{ )}}}$ (i = 1, 2). The two linear transformations are applied from the Cauchy stress σ to ${\mathbf{S}}^{{ ( { }i{ )}}}$ (i = 1, 2) as follows.

$${\mathbf{S}}^{{ ( { }i{ )}}} = {\mathbf{C}}^{{ ( { }i{ )}}} \cdot {\mathbf{s}} = {\mathbf{C}}^{{ ( { }i{ )}}} \cdot {\mathbf{T}} \cdot {\varvec{\upsigma}} = {\mathbf{L}}^{{ ( { }i{ )}}} \cdot {\varvec{\upsigma}}\quad {\text{with}}\quad i = \, 1, \, 2.$$

(4)

The tensor T transforms the Cauchy stress σ into the deviatoric stress tensor s, and the anisotropic property of metals can be represented using the transformation tensors C⁽¹⁾ and C⁽²⁾. The products of the two transformations are represented as L⁽¹⁾ and L⁽²⁾, respectively, which can be expressed in matrix forms as follows.

$$\left[ {\begin{array}{*{20}c} {{\text{S}}_{xx}^{(i)} } \\ {{\text{S}}_{yy}^{(i)} } \\ {{\text{S}}_{xy}^{(i)} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\varvec{L}_{11}^{(i)} } & {\varvec{L}_{12}^{(i)} } & 0 \\ {\varvec{L}_{21}^{(i)} } & {\varvec{L}_{22}^{(i)} } & 0 \\ 0 & 0 & {\varvec{L}_{66}^{(i)} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\upsigma_{xx} } \\ {\upsigma_{yy} } \\ {\upsigma_{xy} } \\ \end{array} } \right]\quad {\text{with}}\quad i = \, 1, \, 2$$

(5)

$$\left[ {\begin{array}{*{20}c} {\varvec{L}_{{\varvec{11}}}^{(1)} } \\ {\varvec{L}_{{\varvec{12}}}^{(1)} } \\ {\varvec{L}_{{\varvec{21}}}^{(1)} } \\ {\varvec{L}_{22}^{(1)} } \\ {\varvec{L}_{66}^{(1)} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {2/3} & 0 & 0 \\ { - 1/3} & 0 & 0 \\ 0 & { - 1/3} & 0 \\ 0 & {2/3} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\upalpha_{1} } \\ {\upalpha_{2} } \\ {\upalpha_{7} } \\ \end{array} } \right]\quad {\text{and}}\quad \left[ {\begin{array}{*{20}c} {\varvec{L}_{{\varvec{11}}}^{(2)} } \\ {\varvec{L}_{{\varvec{12}}}^{(2)} } \\ {\varvec{L}_{{\varvec{21}}}^{(2)} } \\ {\varvec{L}_{22}^{(2)} } \\ {\varvec{L}_{66}^{(2)} } \\ \end{array} } \right] = \frac{1}{9}\left[ {\begin{array}{*{20}c} { - 2} & 2 & 8 & { - 2} & 0 \\ 1 & { - 4} & { - 4} & 4 & 0 \\ 4 & { - 4} & { - 4} & 1 & 0 \\ { - 2} & 8 & 2 & { - 2} & 0 \\ 0 & 0 & 0 & 0 & 9 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\upalpha_{3} } \\ {\upalpha_{4} } \\ {\upalpha_{5} } \\ {\upalpha_{6} } \\ {\upalpha_{8} } \\ \end{array} } \right]$$

(6)

The material constants α₁–α₈ can be obtained from the experimental data of the uniaxial tension tests (at intervals of 45°) and balanced biaxial tension tests, i.e., from each test, the yield stress and r-value are obtained. Thus, a total of eight test variables are obtained to identify the eight parameters in Eq. (6).

For comparison, the following quadratic anisotropic yield function introduced by Hill [13] was also used.

$$\varPhi \left( {\varvec{\upsigma}} \right) = \left( {\text{G + H}} \right)\upsigma_{ 1 1}^{ 2} {\text{ + H}}\upsigma_{ 1 1}\upsigma_{ 2 2} { + }\left( {\text{F + H}} \right)\upsigma_{ 2 2}^{ 2} {\text{ + 2N}}\upsigma_{ 1 2}^{ 2} = {\bar{\sigma }}^{ 2} ,$$

(7)

where ${\bar{\upsigma }}$ is the effective stress, F, G, H, and N are the four anisotropic coefficients, and σ_ij denotes the Cauchy stress component under the plane stress condition. The anisotropic yield function helps recover the well-known isotropic von-Mises yield function when G = H = 0.5 and N = 1.5. The four coefficients of the yield function can be determined using either the yield stresses or the r-values obtained from the uniaxial tensile tests along the RD, diagonal direction (45° from RD), and TD.