Constitutive behavior and novel characterization of hot deformation of Al-Zn-Mg-Cu aluminum alloy for lightweight traffic

Isothermal compression tests of 7A21 aluminum alloy were carried out on a Gleeble-3500 thermal simulator, and the stress-strain curves were obtained at temperatures ranging from 350 to 500 °C and strain rates ranging from 0.01 to 10 s−1. The Arrhenius-type constitutive models with/without strain compensation were established to predict hot deformation mechanical behavior of the alloy based on friction and temperature corrected stress-strain curves, respectively. The model with strain compensation shows a higher prediction accuracy by calculating the average absolute relative error and correlation coefficient. The hot processing maps at different strains were constructed based on the dynamic material model (DMM). The safety strain rates map, a new form of processing map which reflects the variation of critical safety strain rates with the deformation temperatures and true strains, was generated to simplify the acquisition of safety zones throughout the whole deformation process.


Introduction
With the development of the society, the requirements for safety, luxury and performance of vehicles are increasing. At the same time, the automobile industry must comply with the increasingly stringent fuel efficiency standards and emission regulations, and the overall vehicle weight must be reduced. The application of aluminum auto parts is considered to be one of the most promising ways to solve this contradiction [1]. Nowadays aluminum alloys have been widely used in automobile panels. For example, 6xxx series aluminum alloy such as A6016, A6111 and A6181A were used in automobile outer panels [2], and 5xxx series aluminum alloy such as AA5052, AA5182 and AA5754 were used in automobile inner panels [3,4]. However, 5xxx and 6xxx aluminum alloy can not meet the strength requirements of some key automotive structures (such as A pillar and B pillar). The application of 7xxx series aluminum alloy with high strength density ratio and excellent mechanical properties in automobile industry gets more and more attentions [5][6][7][8].
In order to optimize the hot-working characters of 7xxx aluminum alloy, improve its mechanical properties and expand its application in the field of automobile, some efforts have been made to study the hot deformation behavior of the material. Lin et al [9] studied the hot deformation behavior of Al-Zn-Mg-Cu alloy under timedependent strain rate and established a constitutive model based on the dislocation density principle and iterative method. S Y Park et al [10] explored a possibility for directly using the as cast 7075 aluminum as a billet for hot working by using hot processing maps. Sun et al [11] developed a continuous dynamic recrystallization model of extruded AA7075 aluminum alloy based on the internal-state-variable (ISV) method to predict the flow stress and microstructure evolution during the deformation process. Mirzadeh [12] contrasted the prediction accuracy of the phenomenological and physical constitutive equations of 7075 aluminum alloy, and the results showed that physical constitutive equations was more suitable for describing the hot deformation behavior of 7075 aluminum alloy. Li et al [13] established an artificial neural network (ANN) model for Al-5.4Zn-2.0Mg-0.35Cu-0.3Mn-0.25Sc-0.10Zr alloy based on back-propagation learning algorithm, and higher accuracy was achieved when compared with Arrhenius-type equation. Li et al [14] analyzed the processing maps Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
of Al-6.2Zn-0.70Mg-0.3Mn-0.17Zr alloy at different strains, and the optimum processing conditions were in deformation temperature range from 703 K to 773 K and strain rate range from 0.03 s −1 to 0.32 s −1 . However, there are few reports on the hot deformation behavior of 7A21 aluminum alloy for automotive lightweight.
In this paper, a single pass isothermal compression test for 7A21 aluminum alloy was carried out, and a thermodynamic constitutive equation describing the relationship between flow stress, deformation temperature, strain rate and strain was established to predict its hot deformation flow behavior. The hot processing map was built to optimize the hot process parameters. An attempt has been made to conduct a new type of hot processing map expressed by strain rates to simplify the acquisition mode of hot working safety zone at any deformation conditions.

Materials and methods
An as cast 7A21 aluminum alloy prepared in laboratory was used in this investigation, and its chemical composition (wt.%) was given in table 1. The experimental material was processed into cylindrical specimens with a diameter of 10 mm and a height of 15 mm for experiments. In order to simulate the mechanical behavior and microstructure evolution during the rolling process, the as cast cylindrical specimens were homogenized at 550°C for 24 h, as shown in figure 1(a). The microstructure of the alloy after homogenization was shown in figure 2.
The isothermal compression tests for these homogenized cylindrical specimens were carried out on a Gleeble-3500 thermal simulator. In order to reduce friction effect and make the specimen deformed uniformly during the thermal deformation process, the tantalum and graphite sheets were added to both polished ends of the specimens separately with high temperature lubricants before the experiments. Four deformation temperatures (350, 400, 450 and 500°C) and four strain rates (0.01, 0.1, 1 and 10 s −1 ) were chosen in the experiments. The specimen was heated to the deformation temperature at a heating rate of 10°C s −1 , and held for 3 min to make the temperature uniform. The height reduction of the specimen was 55%, and the corresponding true strain was 0.8. The specimen was quenched to room temperature immediately after the deformation process, as shown in figure 1(b). Vernier caliper was used to measure the size of the sample after deformation.
3. Correction of the flow stress curves 3.1. The principle of friction correction for the flow stress curves The friction at the die-workpiece interface makes the deformation of the specimen uneven with a phenomenon of the 'waist drum' and affects the accuracy of the measured flow stress. Tantalum sheets and graphite sheets are generally added between the indenter and the specimen to reduce friction, but its adverse effects can not be   figure 3. It can be seen that the phenomenon of 'waist drum' is still very prominent under high strain. Therefore, following equation is used to modify the stress-strain curves obtained by compression test [15]: Where P is the flow stress after friction correction, σ is the flow stress obtained by experiment, R 0 and H 0 are initial radius and height of the specimen respectively, μ is friction coefficient which can be determined by following equation [16]: where H, R, R M and ΔH are theoretical deformation height, theoretical deformation radius, maximum deformation radius and height reduction respectively, as shown in figure 4.  3.2. The principle of temperature correction for the flow stress curves A large amount of deformation heat will be generated inside the specimens during plastic deformation. At low strain rate, these deformation heat spreads to the surrounding environment by heat conduction. At high strain rate, the heat diffusion efficiency is far lower than the formation efficiency which leads to the increase of the temperature of the specimen. Figure 5 shows the actual temperature changes with strain rates at the setting temperature of 350°C. It can be seen that the temperature fluctuation increases with the increase of strain rate. When the strain rate rises to 10 s −1 , the actual maximum temperature exceeds the setting temperature of 16.37°C. Therefore, the temperature rise of the specimens due to deformation heating must be corrected to obtain the accurate flow stress of a specific temperature at high strain rates. The relationship between flow stress before and after temperature correction at a certain strain rate can be expressed as follows [17]: Where σ 1 and σ are flow stress before and after temperature correction respectively, T 1 is setting temperature, T is actual temperature which can be calculated by [18]: Where ρC is heat capacity, q is the adiabatic factor, σ is true stress, ε is true strain and e  is strain rate.

Flow stress behavior
The flow stress curves at different deformation temperatures and different strain rates after the friction and temperature correction are shown in figure 6. It can be seen that the deformation temperature and the strain rate have great influence on hot deformation behavior of 7A21 aluminum alloy, and the flow stress increases significantly with the decrease of deformation temperature and the increase of strain rate. At the early stage of deformation, the flow stress increases rapidly due to dislocation formation and accumulation [7,9,14]. Then, the flow stress reaches a peak value with a declining rate of growth, indicating that the dynamic recovery (DRV) and the dynamic recrystallization (DRX) counteracted part of the work hardening [19]. After that, flow stress tends to maintain or decrease to a steady state, which proves the dynamic equilibrium between work hardening and dynamic softening [20].

Constitutive equation considering peak stress
The hot deformation behavior of material is mainly determined by the deformation parameters such as strain rate and deformation temperature. The relationship between flow stress, temperature and strain rate can be expressed as following equation [21]: Where A, A 1 , A 2 , n, n 1 , β and α are temperature independent material constants, Q is activation energy of hot deformation (J mol −1 ), T is deformation temperature, R is gas constant (8.314 J mol −1 K −1 ), σ is a characteristic stress on the flow stress curve. In this paper, the peak stress is used as the characteristic stress to study the hot deformation behavior of 7A21 aluminum alloy. By substituting equation (8) into equation (7) respectively and taking natural logarithms on both sides, equation (7) are changed into: e s bs as ln ln ln for low stress levels ln for high stress levels ln ln sinh for all stress levels Then n 1 , β, n, and Q can be expressed as: (10) and (11) .87313 KJ/mol, respectively. Thermal activation energy Q of pure aluminum under dynamic mechanism of cross-slip of dislocation is 117 kJ mol −1 . The reason for the high thermal activation energy of the experimental 7A21 aluminum alloy is that the addition of Zn, Mg and other alloying elements significantly reduces the stacking fault energy and restrains the cross slip of dislocation, which increases the energy required for dislocation aggregation before cross slip [22][23][24].

According to equations
The hot deformation behavior of materials at different deformation temperatures and strain rates can also be expressed by Zener-Hollomon parameters [13]: Taking natural logarithms on both sides of equation (14) , following equation can be obtained: By substituting the value of Q into equation (14), the value of Z at different deformation conditions can be obtained. According to equation (20), lnA is the average intercept of lnZ-ln[sinh(ασ p )] plots, as shown in figure 7(e), and the value of A was calculated as 1.15695·10 16 .
When substituting the value of α, n, Q, A into equation (7), the constitutive equation of 7A21 aluminum alloy can be expressed as:

Constitutive equation considering compensation of strain
The constitutive equation established above only consider the effect of deformation temperature and strain rate on the mechanical behavior of materials during hot deformation, and the effect of strain was neglected. However, the study shows that the strain has great influence on the hot deformation activation energy (Q) and other material constants (i.e. α, n, A) [8,21,25]. Therefore, constitutive equations considering compensation of strain should be established in order to predict the flow stress more accurately. Similar to the solution method mentioned above, the value of materials constants α, n, Q and lnA were calculated in the strain range from 0.025 to 0.2 with the interval of 0.025 and strain range from 0.25 to 0.8 with the interval of 0.05. By analyzing the correlation and generalization, an eighth order polynomial was used to represent the influence of strain on materials constants, as shown in equation (19) and figure 8 According to equations (14) and (15), the constitutive equation of 7A21 aluminum alloy considering strain compensation can be written as: , Q (ε) and A (ε) can be calculated by equation (19).
A comparison between the predicted flow stress and the experimental flow stress was carried out as shown in figure 9. It could be observed that the predicted value of flow stress has good agreement with the experimental data, indicating a high prediction accuracy of the constitutive equation with compensation of strain.

Verification of the constitutive equation
The average absolute relative error (AARE) and correlation coefficient (R) are used to further verify the accuracy of the constitutive equations developed above, which can be expressed as [26]: Where Ei and Pi are predicted values and calculated values respectively. Ē and P are the average values of Ei and Pi respectively, N is the total number of data in the investigation. Figure 10(a) shows the comparisons between calculated peak stresses by equation (22) and experimental peak stresses, and the correlation coefficient R and average absolute relative error AARE are 0.9934 and 8.40% respectively. Figure 10(b) shows the comparisons between calculated stresses by equation (24) and experimental stress at different strain, and the correlation coefficient R and average absolute relative error AARE are 0.9963 and 3.75% respectively. It can be seen that the accuracy of the constitutive equation with strain compensation is higher than that of the constitutive equation without strain compensation. Therefore, the constitutive equation with strain compensation is more suitable for predicting the mechanical behavior of 7A21 aluminum alloy during hot deformation.

5.
Processing maps of 7A21 aluminum alloy 5.1. Computation of power dissipation coefficient η and instability coefficient x e (˙) The establishment and analysis of hot processing maps based on dynamic materials model (DMM) is a reliable method to understand the hot deformation behavior of different metal materials and optimize the thermal deformation process [19,[27][28][29][30]. According to the DMM, the hot deformation process of materials can be regarded as an energy dissipation process. The total power (P) absorbed by the workpiece will be consumed in two ways: the first part (G content) is the energy dissipated by plastic deformation; the second part (J co-content) is the energy related to microstructure evolution in the process of deformation, such as dynamic recovery, dynamic recrystallization, phase transition, superplastic rheology and internal defects. The instantaneous total power consumption P can be expressed by flow stress σ and strain rate e  Generally, a larger value of η indicates that the proportion of the energy consumed in microstructure evolution is higher, which shows a better workability [19]. However, structure defects such as local rheology, local shear bands and wedge cracking may also occur at the same time. Based on the extremum principle of the irreversible thermodynamics applied to the large plastic flow body, Ziegler [8,9] proposed the condition of thermal processing instability: Where D is the dynamic metallurgical deformation function, which is equivalent to the J co-content. Substituting J into equation (28), the criterion for the occurrence of flow instability of the material can be expressed as:

Safety strain rates map for thermal processing
The critical strain value of instibility zone at different strain and deformation temperature can be calculated by equation (29). In order to enrich the data points, the critical value of the safety strain rates under different deformation conditions is directly obtained through the unstable region of the processing maps with a temperature interval of 2°C. A three-dimensional diagram is conducted to represent the variation of the safety strain rates with the temperature and strain. The safety zone for hot working is shown in figure 11(a). It can be seen that only a single upper limit and the lower limit are presented in the safety zone. The contour map of the upper and lower limit of the safety strain rates are represented as figures 11(b) and (c) respectively. The safety strain rate map can be constituted by combining figures 11(b) and (c), and then the safety zone for hot working at any deformation condition can be obtained.  In order to ensure the reliability of data acquisition, corresponding acquisition principle is determined. The strain rate value of point A between contour line U 1 and U 2 and the strain value of point B just on contour line U 2 in upper limit map are both equal to the strain rate value corresponding to contour U 1 . The strain rate value of point C between the contour line L 1 and L 2 and the strain value of point D just on contour line L 2 in upper limit map are both equal to the strain rate value corresponding to contour L 2 . The obtained safety strain rate interval of A, B, C, D are shown in figure 11(c). Figure 12 shows the value of critical strain rates obtained at strain 0.4 based on acquisition principle mentioned above, and the shaded region represents instability zone. It can be seen that the obtained value of critical strain rates are all in safety zone. The maximum error of critical strain rates obtained from the safety strain rates map and processing maps is 0.345 which is the gradient interval value of the contour line, indicating that the safety strain rates map and the corresponding acquisition principle is correct and reliable.
Compared with the hot processing maps, one of the advantage of the safety strain rates map is to improve the efficiency of determining the safety zone of the hot working at different strain. For example, in order to get the hot processing safety zone at deformation temperature 450°C and strains ranging from 0.1 to 0.8 with a intercept of 0.1, eight processing maps need to be analyzed. However, only one map is needed to determine the safety zone of hot processing under all the strains by using safety strain rates map, as shown in figure 13. A comparisons are made between strain rates obtained from the safety strain rates map and processing maps, as shown in figure 14. It can be seen that the obtained safety zone is equal to or slightly smaller than the actual safety zone. This is mainly determined by the gradient of the contour line of the safety strain rates map, and the deviation increases with the increase of the gradient interval.
Another advantage of the new map is that it is easy to obtain the safety zone of hot working under any strain rate range. Figures 15(b) and (c) are the hot working safety zones of the experimental materials under the strain rate ranging from 0.01 to 10 s −1 and 0.05 to 1 s −1 respectively, and the boundaries of which are the  corresponding contour lines in the safety strain rates map, as shown in figure 15(a). It can be seen that with the narrowing of the safety strain interval, the area of the safety zone is gradually increasing.
What's more, two strain sensitive regions (region 1 and region 3) and two temperature sensitive regions (region 2 and region 4) can be obtained from figure 15(a). The deformation condition ranges of these regions are as follows: Region 1 and region 2 are strain rate range of 1∼10 s −1 with strain range of 0.13°C∼0.35, temperature range of 350°C∼425°C and strain range of 0.3∼0.8, temperature range of 425°C∼465°C respectively. Region 3 and region 4 are strain range of 0.27∼0.35, temperature range of 350°C∼367°C and strain range of 0.3∼0.8, temperature range of 367°C∼387°C respectively with strain rate interval of 0.01 ∼0.05 s −1 . The contour line in region 1 and region 3 is dense along the direction of strain change, indicating that the occurence of flow instability in which are sensitive to strain. While in region 2 and region 4, the deformation temperature shows decisive influence. The deformation conditions in regions listed above should be avoided in optimizing the hot deformation process especially in industrial production of 7A21 aluminum alloy.
By adding the strain rate dot corresponding to the peak value of the power dissipation coefficient under different deformation conditions to safety strain rates map, a new type of processing map can also be obtained to optimize the hot deformation process, as shown in figure 15(a). The color of the dot represents the strain rate range of the peak region of the power dissipation coefficient, and the number on the dot is the peak power dissipation coefficient. It can be clearly seen from figure 15(a) that the power dissipation stay high at temperature 500°C and strain rate ranging from 0.01 to 0.1 s −1 during hot deformation, indicating it is the optimum deformation parameters. The results agree well with the analysis results of the hot processing maps.
In summary, the safety strain rates map shows good convenience, accuracy and reliability and can be used to guide industrial production.

Summary
Based on friction and temperature modified stress-strain curves, both of the Arrhenius-type constitutive equation with and without strain compensation were obtained to describe the hot flow behaviors. The constitutive equation with strain compensation shows a higher accuracy with the correlation coefficient R and average absolute relative error AARE values of 0.9963 and 3.75% respectively.
Based on instability criterion, a novel type safety strain rates map was developed to analyze the hot deformation behavior of 7A21 aluminum alloy. The safety zone of hot processing at different deformation conditions and the optimum processing windows were obtained. The map shows good convenience, accuracy and reliability and it is of certain significance to guide the industrial production.