Temperature rises and constitutive equation of homogenized 6063 aluminum alloy for extrusion

The effect of temperature rises on the flow stress and constitutive equation of homogenized 6063 aluminum alloy was studied by theoretical analysis, numerical simulation, and experiments. The results showed that the temperature rises increased with the decreases of deformation temperatures and the increases of strain rates, which caused the flow softening. The modification of the flow stresses of 6063 aluminum alloy was carried out. A strain compensation constitutive equation based on the Arrhenius equation and Zener-Hollomon parameter was introduced to predict the flow behavior of 6063 aluminum alloy. The calculated flow stresses were consistent with the experimental results, and its average absolute relative error was only 3.25%. Finally, the established constitutive equation was substituted into the Deform-3D software. The corresponding extrusion experiments were carried out. The maximum extrusion pressures and maximum exit temperatures in the numerical simulation were in good agreement with those in the experiments, which confirmed the accuracy and reliability of the established constitutive equation.


Introduction
6063 aluminum alloy has been widely used in the construction industry, transportation, and aerospace, since it can be processed into complex shapes and anodized [1][2][3].Hot extrusion is one of the main processing methods for 6063 aluminum alloy.Extrusion temperature, strain rate, and strain significantly affect the microstructure evolution and properties of 6063 aluminum alloy.Therefore, a reliable constitutive equation coupled with extrusion temperature, strain rate, and strain is very important, because it can be used to forecast the flow behavior, microstructure evolution, and mechanical properties [4][5][6].In addition, numerical simulation has been widely applied in industrial profile extrusion with the development of computer simulation technology.In the simulation process, the constitutive equations are usually used as input code, and the reliability and accuracy of embedding constitutive equations largely determine the reliability of the simulation results.
To optimize the process parameters and material properties, scholars have carried out a lot of research.On this basis, a series of constitutive equations have been proposed and modified.Chen et al [7] employed four different constitutive models to predict the influence of microstructure evolution on the flow stress of the 6026 aluminum alloy and found that the initial Johnson-Cook model was not sufficient to describe the flow behavior of the 6026 aluminum alloy.Hallberg et al [8] proposed a finite strain elasto-viscoplastic model considering dislocation and grain size evolution.Rudnytskyj et al [9] investigated the isothermal flow behavior of 6061 aluminum alloy and found that the Garofaro Arenius model provided the highest accuracy in any strain range compared to other models.Based on a substructure-based constitutive equation, Eivani et al [10] analyzed the flow behavior and microstructure evolution of AA7020 aluminum alloy at a high temperature.REZAEI ASHTIANI et al [11] investigated the hot deformation of AA1070 aluminum alloy according to the phenomenological and empirically-based constitutive equation.
The Z-parameter is often introduced into the constitutive equation to study the effects of temperature and velocity on the deformation behavior of aluminum alloys [12].As a consequence, the flow stress can be expressed by the mathematical relationship between temperature and strain rate [13][14][15].The Arrhenius model with strain compensation and Zener-Hollomon parameter effectively improved the prediction accuracy of flow characteristics.Wang et al [16] analyzed the flow behavior of 2219-O aluminum alloy and proposed an Arrhenius constitutive model with a simple function structure.DAI et al [17] used a strain-compensated Arrhenius-type equation to predict the flow behavior of AA5083 aluminum alloy.Zhou et al [18] investigated the flow behavior and mechanical properties of aged AA6082 aluminum alloy using an optimized Arrhenius constitutive model.Cao et al [19] studied the flow behavior and microstructure evolution of an Al-Cu-Li-Mg-Ag Alloy under different temperatures and strain rates.
Consequently, due to its few material constants and ease of establishment and calibration, the Arrhenius equation has been widely applied in scientific research and engineering.In this study, a series of thermal compression tests were carried out on AA6063 aluminum alloy at the temperatures from 573 K to 773 K with the interval of 50 K with the strain rates from 0.01 s −1 to 10 s −1 with the increase of 10x relationship.The temperature rises during plastic deformation were calculated, and the flow stresses were modified.Based on the Arrhenius-type equation, an optimized constitutive equation with Zener-Hollomon parameter and strain compensation was proposed.Finally, the extrusion experiments were conducted to verify the accuracy of the numerical simulations and constitutive equation.The results obtained in this study could provide significant theoretical support and basic data for further study in the field.

Materials and experimental procedures
The chemical composition of the ingot used in thermal simulation is Al-0.85Mg-0.47Si-0.2Fe-0.21Cu-0.01Mn-0.13Cr-0.25Zn-0.15Ti(wt%).Firstly, the ingot was homogenized at 813 K for 24 h in a box-type resistance furnace and then cooled to room temperature in the air.After homogenization treatment, the cylindrical samples (a diameter of 10 mm and a height of 15 mm) were processed through wire-electrode cutting and sanding.The wire-cutting position was at the center of the ingot.Secondly, the cylindrical samples were compressed on a Gleeble-1500 thermal-mechanical simulation machine.In order to reduce the friction, the graphite foil was placed between the sample and the indenter before the start of thermal compression.The samples were heated by a K-type thermocouple at the rate of 10 K s −1 .The K-type thermocouple wire was welded to the middle surface position of the sample.Before exerting compressive loading, the samples were held for 120 s at the preset temperature to ensure the consistency of the sample temperature.The preset temperatures were set from 573 K to 773 K with an interval of 50 K.The preset strain rates were set from 0.01 s −1 to 10 s −1 with the increase of 10x relationship.Finally, each sample was compressed to a height of 6.1 mm with the preset temperature and strain.Twenty groups of thermal compression experiments were conducted and each group of tests was repeated three times.All tests were carried out in a nitrogen atmosphere.

Experimental results and discussions
3.1.Temperature rises Under dynamic load, the heat generation rate exceeds the heat loss rate, resulting in an increase in temperature.Temperature rises could be calculated according to references [20].The highest temperature rises were 44 K, 35 K, 25 K, 18 K, and 13 K at the five different preset temperatures with a strain rate of 10 s −1 , respectively, as shown in figure 1.It can be seen that temperature rises increased with the decrease of preset temperatures and increase of strain rates.This was because the thermal emissivity of the aluminum alloy increases with the increase in temperature.At low strain rates, there was sufficient time for heat conduction and radiation, and the generated energies were almost lost to the indenter and the environment.The temperature rises were nearly zero, such as at the strain rate of 0.01 s −1 .However, at high strain rates, the rate of deformation energy generation was greater than that of heat loss, and there was insufficient time for heat conduction and radiation.The temperatures of specimens increased due to the high adiabatic correction coefficients.
Figure 2 shows the temperature rises at the strain rate of 1 s −1 and 10 s −1 with different preset temperatures.It can be seen that temperature rises increased with the increase of strain and strain rate and the decrease of deformation temperature.There was a linear relationship between temperature rise and strain under a constant deformation temperature with the strain rate, and similar behaviors were also observed in other materials [21,22].The calculated temperature rises were consistent with those of the measured, which indicated the reliability of the calculation results.

Isothermal stresses of 6063 aluminum alloy
The isothermal stresses ( c s ) during thermal simulation deformation could be expressed as [23]: swas subject to a 2nd-order polynomial fitting with a fitting degree of 0.99.Therefore, the isothermal flow stresses of 6063 aluminum alloy could be determined according to equation (1).
The corrected and uncorrected flow stresses are shown in figure 4. It was found that the temperature rises had a significant impact on the flow stresses of 6063 aluminum alloy.There were great differences between the corrected and uncorrected curves, especially in a low temperature and a high strain rate.This was due to the high flow stresses and not enough time for heat dissipation.Consequently, enough mechanical energy was converted into heat, resulting in the temperature rise.Under the conditions of this experiment, the maximum softening flow stress value was 40 MPa.

Strain compensated Arrhenius model
The constitutive equation can be used to predict the flow behavior of materials during extrusion.At present, researchers have established several constitutive models through theoretical analysis and experimental    verification, and applied them to practical production.However, it has been proven that the Arrhenius equation can accurately and comprehensively describe the relationship of various parameters during hot deformation [24]: where e  indicates the strain rate, Q stands for the activation energy, R stands for the universal gas constant, T indicates the absolute temperature, A, n′, , b a and n stand for the material constants.And: According to equation (2), the flow stresses of 6063 aluminum alloy can be represented by: Under a constant preset temperature, substituting equation (2) into equation (3) and taking the natural logarithm of the results, and subsequently taking the differentiation of both sides, respectively: According to equation (6), the value of n′ can be determined by the slops of linear regression of the ln ln e s - curve under a constant temperature.Then, the average value of n′ can be calculated under different temperatures for a constant strain level (figures 5(a) and (b)).The same method can be used to obtain the average value of b according to equation (7).Finally, the average values of n′ and b were substituted into equation (4) to obtain the value of α.
Zener-Holloman parameter (Z) has been introduced to describe the impact of pre-set temperature and strain rate on specimen deformation behavior (equation ( 2)).According to mathematical calculation methods and then tanking the differentiation of both sides of equation (2), the value of Q can be expressed as: plot under the preset temperature with a constant strain level.Repeating the above process, the value of n could be obtained by   6) under a constant strain rate.Then, the value of Q could be obtained by substituting the above two calculation results into formula (8) at the strain of 0.6.
According to the value of n, the According to mathematical calculation methods and then tanking the natural logarithm of both sides of equation (2), the result was shown as follows: The 2 # order polynomial fit results of the parameters were shown in figure 8 and the corresponding equations were displayed in equation (10).From figure 8(a), it can be seen the a value increased steadily from 0.0131 to 0.0157 MPa −1 with the strain increasing to 0.9.Similar behaviors were also observed in other materials [19,25,26], in which the values of a changed with different materials composition and deformation conditions.According to figure 8(b), the value of n value decreased steadily from 10.83 to 6.45 with the increase of strain, which reflected that the workability of the specimen enhanced with the increasing strain.The trend of the n value decreased with increasing strain, which was the same with that for the 2099 Al-Li alloy investigated by Zhang et al [27] and the AA7050 aluminum alloy studied by Wang et al [28].This may be related to the addition  According to equations (5) and (10), the impacts of stain, strain rate, and deformation temperature on the flow stresses could be simultaneously considered by introducing the Z ener-Hollomon parameter.Therefore, the constitutive equation for predicting flow stresses of the 6063 alloy can be described as: Verification of the established model According to the above results, the calculated flow stresses are shown in figure 9.It was found that the calculated flow stresses were consistent with the measurements.In addition, the parameters of R and AARE were introduced to validate the reliability of the established model [25][26][27]:

Extrusion simulation and its experimental verification
To further validate the accuracy and reliability of the model, the established constitutive equation was substituted into the Deform-3D software.A numerical simulation model was established to simulate the extrusion process of the 6063 aluminum alloy.And the corresponding extrusion experiments were carried out to verify the accuracy of the established constitutive equation and numerical simulation model.Before the extrusion experiment, the 6063 aluminum alloy ingot was homogenized at 813 K for 24 h.After that, the ingot was cooled to room temperature in the air and then reheated to the preset extrusion temperature at a speed of 10 K min −1 .The heat preservation time was 0.5 h.The details of the extrusion process parameters are shown in table 1. Due to the symmetry of the square, a 1/8 numerical model was used.The total calculation time could be reduced and much finer meshes could be generated.Therefore, the reliability and accuracy of the simulation could be ensured.In addition, the materials and geometric dimensions of the extrusion tools and the extrusion process parameters in the simulation were consistent with those in the extrusion experiments.The schematic diagram of the acquisition device for real temperature and pressure data is shown in the figure 11.The maximum extrusion pressure and maximum exit temperature were measured by GEFRAN KS-E-E-Z-B4C-M-V pressure sensor and AE-3000 type high-temperature sensor during extrusion, respectively.
The maximum extrusion pressures and maximum exit temperatures are shown in figures 12 and 13, respectively.Results showed that the maximum extrusion pressures and maximum exit temperatures increased with the increase of ram speeds.This was because the material underwent two competing processes: work hardening and dynamic softening, during the extrusion process.At the beginning, work hardening dominated.As the extrusion speed increased, the extrusion temperature increased, and dynamic softening gradually dominated, resulting in a decrease in maximum extrusion pressures.The predicted maximum extrusion pressures and maximum exit temperatures in the finite element simulation were consistent with those in the extrusion experiment, which verified the accuracy and reliability of the constitutive model.

Conclusion
In this study, the hot deformation behaviors of 6063 aluminum alloy were studied by theoretical analysis, numerical simulation, and experiments.The temperature rises during hot deformation were calculated.On this basis, the flow stresses were modified and an optimized constitutive equation with Zener-Hollomon parameter and strain compensation was introduced.The main conclusions were as follows: 1.During thermal compression, the temperature increased significantly and its impact on flow stress could not be ignored.Under the conditions of this experiment, the highest temperature rise reached 44 K during compression, which led to a flow stress softening of 45 MPa.
2. Based on the Arrhenius-type equation, an optimized constitutive equation with Zener-Hollomon parameter and strain compensation was proposed with an AARE value of 3.25%.
3. The predicted maximum extrusion pressures and maximum exit temperatures in the finite element simulation were consistent with those of the extrusion experiment, which indicated the results obtained in this study could provide significant theoretical support during extrusion.
Where u s indicates the measured flow stress, c s stands for the corrected flow stress, of the isothermal stress-temperature curve at the temperature of T 0 and indicates the stress reduction caused by temperature rise.Based on equation (1), the T c i sdata could be calculated by the data of T u 0 sunder the given temperature and strain.The relationships between T c i sand T u 0 sunder different strain rates are showed in figure 3 at a strain level of 0.6.It can be seen the relationship between T c i sand T u 0

Figure 3 s
Figure 3. Relationships between


indicates the parameter n in equation (2) under a constant deformation temperature.It is determined by the slope of linear fitting of [

Figure 5 .
Figure 5.The curves of (a) ln ln ; e s - (b) ln ln .e s - in figure7and the value of A ln could be determined by the intercept of the [ constant strain level.Repeating this process, the values of material constants (a, Q and A ln ) in the range of 0.05-0.9with an interval of 0.05 could be determined.

Figure 8 .
Figure 8. Relationships between Material constants and strains: (a) ; a (b)n; (c)Q and (d) A ln .

Figure 10 .
Figure 10.Correlation between the corrected and calculated flow stresses with strain compensation.

Figure 11 .
Figure 11.The measurement of the temperature and pressure data during extrusion.

Figure 12 .
Figure 12.Maximum extrusion pressures at different ram speeds.

Table 1 .
Billet dimensions and process parameters.