Numerical simulation of stress evolutions in 2A14 aluminum alloy components during solution and aging process

ABSTRACT Aluminum alloys are commonly used in the aerospace industry because of their good specific strength and corrosion resistance. However, reducing residual stresses and machining deformation of aluminum alloy components are the focus of researches. In this paper, a material model of 2A14 aluminum alloy during solution treatment and aging process is established with thermal expansion, high temperature tensile and stress relaxation experiments. The stress relaxation during aging is described with the Zener-Wert-Avrami model based on the experimental data. Quenching and aging experiments of 2A14 aluminum alloy C-ring samples were carried out. Temperature changes of C-rings were measured during quenching, residual stresses in the samples after solution, and aging are measured by XRD, respectively. Based on measured temperature variations, surface heat transfer coefficients of a C-ring during quenching were calculated by inverse heat transfer calculation and a novel optimization method. The changes of temperature, stresses, and notch opening distance of a C-ring during solution and aging process were calculated, and calculated results were in good agreement with the measured results.


Introduction
Aluminum alloys are widely used as structural materials in the aerospace industry because of their high specific strength, low density, good manufacturability, and low cost [1]. Heat treatable aluminum alloys need to be strengthened by the solution and aging process before they can meet the product requirements, and 2A14 aluminum alloy is not an exception. During the quenching process, a step of the solution treatment, the temperature of the component changes rapidly and leads to large residual stresses especially for aerospace structural components with thin wall thickness and complex shapes. Residual stresses adversely affect the dimensional stability, corrosion resistance, fatigue strength, and service life of products [2]. Zhang et al. [3] studied the effect of cooling rates on the mechanical properties and residual stresses of 2A14 aluminum alloy through numerical simulation and experiments. The results show that the mechanical properties of 2A14 aluminum alloy are mainly determined by the cooling rate within the quenching sensitive temperature range of 300-400°C. Robinson et al. [4] studied the relationship between residual stress and mechanical properties of eight different aluminum alloys after solution treatment and found that the surface residual stress of the sample changed linearly with hardness and strength.
Aging is the subsequent process of solution treatment, in which the mechanical performance of aluminum alloys increases due to the precipitation of supersaturated solid solutions. In aging processes, the aging temperature and holding time are key parameters affecting the strengthening effect. Alexopoulos et al. [5,6] studied the effects of different aging temperatures and aging time on 2024 aluminum alloy. In the peak range, the strength of 2024 aluminum alloy can be increased by about 50%, while the elongation at break caused by corrosion decreases by only about 11%. In this range, parts with better comprehensive performances can be obtained. During aging processes, variations of internal stresses continue and cause the phenomenon of stress relaxation. Hu [7] studied evolutions of residual stresses of 7050 aluminum alloy plates at different aging temperatures and found that the surface residual stress was released less than 30%. When studying residual stresses and deformation of AA7050 and AA6061 thick plates, Mandy et al. [8] found that the residual stresses of AA7050 plates were not released after T6 heat treatment, and the residual stresses of AA7050 plates were released by 36% after T73 over aging treatment. The residual stresses of AA6061 plates were released by 22% after T6 heat treatment. It can be concluded that the residual stress releasing of aluminum alloy components during the aging process is relatively low. A reasonable explanation is that the strength and elastic limit of aluminum alloys are improving during aging as well as the aging temperature of aluminum alloys is generally low. Moazam et al. [9] calculated the magnitude and distribution of residual stresses after heat treatment by finite element method (FEM), and the simulation results are in good agreement with the residual stresses measured by the contour method. Ran et al. [10] simulated the effect of cold rolling treatment on the residual stress relaxation of AA7050 parts after quenching by FEM and found that the residual stresses in the core of AA7050 parts after cold rolling treatment were greatly released. However, there are a few research works on the thorough process simulation of the solution and aging process of aluminum alloy components.
In this study, the material model and stress relaxation model of 2A14 aluminum alloy is established, and the deformation as well as the stress changes of a C-Type sample during solution and aging process is calculated by FEM. The calculated results are consistent with the experimental results.

Material model of 2A14 aluminum alloy in quenching process
Thermophysical properties of 2A14 aluminum alloy Samples used in this study are all taken from a rolled bar of 2A14 aluminum alloy, and the chemical compositions of this material are shown in Table 1.
At first, samples are heated to 502°C at a heating rate of 10°C/s and held for 2 h. Then samples are cooled down to room temperature at a cooling rate of 20°C/s, 5°C/s, 1°C/s, 0.5°C/s, and 0.05°C/s and the temperatures of samples are measured to get the thermal expansion coefficients shown in Figure 1. Due to the large cooling rate during quenching, the thermal expansion coefficients at the cooling rate of 20°C/s are selected to simulate the quenching process.
Thermophysical parameters of 2A14 aluminum alloy are obtained from the calculation results of JMatPro ® and the data published in literatures [11], as shown in Table 2.

Constitutive model of 2A14 aluminum alloy in quenching process
Flow stresses of aluminum alloys are affected by temperature, strain, and strain rate. The stress-strain curves of 2A14 aluminum alloy are measured through hightemperature tensile tests. The tensile temperatures are set to 100°C, 200°C, 300°C, 400°C and 500°C, and the strain rates are set to 0.05, 0.001 and 0.0005 1/s. The Arrhenius model, the most widely used material model in the study of thermal deformation process, takes the influence of temperature and strain rate on material flow stress into consideration and has high adaptability at high temperature based on deformation activation energy theory. The equations used in this paper are as below: where1 is the strain rate, 1/s, T is the Kelvin temperature, K, s is the yield stress under corresponding conditions, MPa, Q is the deformation activation energy, J/mol, R is the ideal gas constant, 8.314 J/(mol K), A, n, h, b and a(a = b/n) is material parameters. The material parameters used in the Arrhenius model are shown in Figure 2. It can be seen that the values of corresponding parameters of the Arrhenius equation in the high-and low-temperature range are very different. The value of h at high temperatures varies from 3.3 to 3.9, while the value at low temperatures is between 42 and 57, which also shows that 2A14 aluminum alloy has different mechanical behaviors in the high-and low-temperature range.
A backpropagation neural network [12] with two hidden layers is used to fit the constitution model of 2A14 aluminum alloy, and each hidden layer in this BPANN has 15 nodes. Strain, strain rate, and  temperature are input, respectively, to the neural network and flow stresses are output. The function 'trainlm' and Levenberg-Marquardt method are used as the training function and optimization method respectively. After training, the artificial neural network model with good accuracy and reliability is finally obtained, and there are no obvious overfitting problems Table 3. The average error, average error rate, and the maximum absolute error of fitting with the Arrhenius model and BPANN are listed as below: Table 2. Thermophysical parameters of 2A14 aluminum alloy [11].    Comparing the predicted results of the artificial neural network with the experimental data and the fitting results of the commonly used Arrhenius model [13,14], as shown in Figure 3, it can be seen that the predicted results of the artificial neural network are significantly better in the whole temperature range.

Stress relaxation model of 2A14 aluminum alloy
There is an interrelationship between the internal stresses and aging precipitation in the aging process. On the one hand, the precipitation behavior can increase the yield strength and elastic limit of aluminum alloys, so as to make internal stresses of workpieces decrease. On the other hand, variations of internal stresses will influence the aging precipitation. Hu et al. [15] found that the external stress has two opposite effects on the precipitation behavior of an Al-Cu-Cd aluminum alloy. Dislocations caused by applied stresses accelerate the nucleation of precipitates. Meanwhile, the atomic motion of cadmium in the aluminum alloy will be hindered by applied stresses so that the precipitation will be hindered. Now, it is generally believed that the creep theory can explain the release mechanism of residual stresses. Therefore, models commonly used to describe stress relaxation are mostly based on the creep model [7]. The creep theory holds that the elastic deformation in workpieces gradually changes into the creep deformation under constant strain and residual stresses, resulting in the corresponding reduction of elastic stresses [16]. The Zener-Wert-Avrami (ZWA) model is an empirical model to describe the stress relaxation behavior of aluminum alloy based on the principle of thermal activation [17]. Based on the microstructure evolution law and aging strengthening theory [18][19][20], some scholars established the macro-micro coupled constitutive equation of stress relaxation in the aging process. The macro-micro coupled constitutive equation of stress relaxation takes the effect of aging strengthening on stress relaxation into consideration but it is difficult to apply to the numerical simulation due to its multiple parameters. In order to verify the accuracy of different models, the stress relaxation experiment is carried out, and the creep model and the ZWA model are used to fit the experimental curve.

Stress relaxation experiment
The geometry and picture of the specimen used in the stress relaxation experiment are shown in Figure 4. The specimens are transferred to the creep testing machine after solution treatment. The experimental temperatures are set to 150°C, 175°C and 190°C, and the initial applied stresses are set to 90, 120 and 150 MPa.
At the beginning of the experiment, samples are loaded with the axial stress to 0.5 kN at a constant stress loading rate, Then, the samples are heated to the experimental temperature and held for 1 h. After  temperature holding, samples are loaded with the axial stress to the experimental stress at a constant stress loading rate and then a part of the load is unloaded automatically to keep the total deformation unchanged; the relaxation curve can be drawn by measuring the reduction value of stresses with time. The schematic diagram of the test cycle and the equipment used in the experiment are shown in Figure 5. Figure 6 shows the results of the stress relaxation experiment. It can be found that stresses of samples decrease rapidly at the beginning and change very slowly after 8 h. The stresses of the three curves of Figure 6(a) decrease by 91, 69, and 58 MPa, respectively. Figure 6(b,d) shows that there are three stages in the stress relaxation process: initial stage, transition stage and stable stage. In the initial stage, the stress relaxation rate is high and stresses decrease sharply. As to the transition stage, the stress relaxation rate starts to decrease rapidly. Finally, the stress relaxation rate is close to 0 and stresses tend to be steady in the stable stage. It should be noted that in the stress relaxation experiment, the sample is undergoing stress relaxation and precipitation strengthening at the same time. Therefore, the experimental curve can be considered as the combination of stress relaxation and precipitation strengthening of 2A14 aluminum alloy during aging.
The variation of the residual stress in the aging process is usually explained with the creep model. Tjong and Ma [21] proposed a phenomenological model of the creep rate and stress: where A is the material coefficient, s is the stress, n is the material exponent, Q is the activation energy, R is the gas constant, and T is the temperature. The logarithmic relationship between the creep rate and stress can be calculated by using Equation (4), as shown in Figure 7.
There are three distinct linear regions in the figure corresponding to the three stages in Figure 6. The strain rate increases with stresses, which is consistent with classical creep theory. However, the creep model involves both temperature and stress, which will make the simulation complex and increase the computational cost. In contrast, the Zener-Wert-Avrami model is a better choice.
where s RS is the residual stress, s RS 0 is the initial residual stress, t a is the aging time, m is the stress relaxation exponent, A is a material-related coefficient, B is a material constant, k is the Boltzmann constant, T a is the aging temperature and DH is the activation enthalpy of residual stress. Therefore, the residual stress relaxation model of 2A14 aluminum alloy can be expressed as: The relationship between ln ln s s 0 and ln t in Zener-Wert-Avrami model is showed in Figure 8. It is obvious that the ZWA model has better linearity compared with the creep model, which shows that the ZWA model can describe the stress relaxation of 2A14 aluminum alloy at 175°C. It can be seen that the stress relaxation exponent is not constant during aging. There is a great difference in stress relaxation exponent before and after 6 h. Therefore, the stress relaxation during the aging process of 2A14 aluminum alloy is described in the form of a piecewise function.
The parameters used in the stress relaxation model are shown in Table 4. And the fitting result of the stress relaxation curve is shown in Figure 9.

Calculation of heat transfer coefficient
Due to the sharp change of temperature during the quenching process of 2A14 aluminum alloy components, the heat transfer coefficient of the quenching process should be assigned accurately. In this paper, a double hidden layer artificial neural network is used to optimize the heat transfer coefficients [23]. The computational steps of the ANN optimization described above are given as follows: .
Step 1: Random heat transfer coefficients are used for 100 times of FEM calculation to generate temperature curves at measuring points and  corresponding heat transfer coefficients jointly construct the training data set. . Step 2: The ANN model is trained automatically. . Step 3: The experimental temperature data are used to predict the heat transfer coefficient with a trained ANN model to get the predicted heat transfer coefficients. . Step 4: The predicted heat transfer coefficient is used to calculate the temperature field and compare it with the experimental measured temperature. .
Step 5: If the maximum absolute percentage error (MAPE) is greater than the allowable error, the heat transfer coefficient and temperature field will be added to the training data set to continue the calculation until the required accuracy is achieved.
The flowchart of the algorithm is shown in Figure 10.
The temperature measurement experiments are carried out to verify the optimization algorithm. The sample used in the experiment is a C-ring as shown in Figure 11. Holes with diameters of 2.1 mm in different depths are machined on the end face of the C-ring to install thermocouples, so as to get the temperature change curve of each surface of the C-ring sample during heat treatment. The locations of the measuring points are also shown in Figure 11. The   depth of holes 1, 3 and 6 are 6 mm, and the depth of holes 2, 4 and 5 are 11 mm, which basically covers each heat exchange interface and locates as close as possible to the measuring surface. Heat exchange surfaces of the C-ring sample can be divided into three groups. The first group is the outer face where the cooling rate is the fastest; the second group is the inner face where the cooling rate is slower than that of the outer face; the third group is the lower and upper end face where the cooling rate is the slowest. Therefore, three groups of heat transfer coefficients are used to correspond to the above three groups of surfaces.
After 34 iterations, the available heat transfer coefficients of inner surface, outer surface, and end face are obtained. The maximum absolute percentage error is less than 2.5%. The obtained heat transfer coefficient is used to simulate the temperature field of the quenching process. Due to the uneven wall thickness of C-ring and different cooling rates at different locations during quenching, point 1 near the outside face, point 2 near the end face, and point 3 near the inside face are selected for verification. The comparison between the calculated temperatures and the experimental results is shown in Figure 12.

Simulation results
Because the quenching process is the main stage of the generation of deformation and residual stresses, Figure 11. The geometry and photo of the C-ring sample.   only quenching is simulated and the heating process is ignored.
Residual stress distribution after solution and aging treatment.
Residual stresses after quenching and the variation of the residual stress component S11 during the aging process is shown in Figure 13. It can be seen from Figure 13(a,b) that the heating process of aging basically has a slight effect on the distribution of the residual stress S11 of the C-ring sample due to the increase of temperature. It can be seen from Figure 13(c-e) that the distribution of the residual stress S11 basically does not change during aging, and the surface residual compressive stress decreases from -80 MPa to about -50 MPa.
Variation of the opening distance of the notch during the solution and aging process The change of opening distance of the notch is affected by two aspects: one is that the opening distance becomes larger due to the volume shrinkage at the notch and the thick wall area; another is that the overall size is reduced due to the overall volume shrinkage, so the opening distance will also be reduced. In the early and middle stages of quenching, due to the rapid cooling of the thin notch zone and the slow cooling of the bulk zone, the opening distance of the notch is increasing as a whole. However, when the thin notch zone is basically cooled down to the quenchant temperature, the temperature of the bulk zone is still relatively high, so the overall size shrinkage is greater than the impact of the shrinkage of the thick wall area on the opening distance, so the opening size decreases in the later stage of quenching. Finally, the opening distance of the notch is increased by 0.1158 mm. During the heating stage and air cooling stage of the aging process, the opening distance of the notch decreases to the same mechanism. At the temperature holding stage of aging, the opening distance of the notch decreases gradually due to the release of residual stress with the increase of aging time. There is a sudden decrease in the residual stress release rate at about 6 h, resulting in a sudden change in the opening distance at about 6 h. The variation of opening distance during solution and aging process is shown in Figure 14.

Experimental validation
After the sample is heated to 502°C in the muffle furnace, it shall be held at the temperature for 10 h, and then, the sample shall be quickly transferred to the quenching tank for immersion quenching. When entering the water, the end face with thermocouple installed should be upward, and the whole  transfer process should not exceed 5 s. Until the temperature is uniform, residual stress will be measured. After that, the aging treatment is carried out at 175°C, and the aging treatment time varies from 1 h to 16 h. The measurements of residual stress are carried out at 1, 2, 3, 4, 5, 8, 10, 12 and 16 h during the aging process.
The X-350A X-ray stress measuring instrument is used to measure surface residual stresses of C-ring samples. The five locations of residual stresses measurement and measuring direction are shown in Figure 15.
In order to compare with the experimental data, the surface residual stresses along the path shown in Figure 15 are calculated. As shown in Figure 16, the distribution of residual stress from the numerical simulation is consistent with that from experimental measurement. From the opening to the maximum wall thickness of C-ring, the residual stress first increases and then decreases along the outer diameter Figure 16.
The residual stress distribution pattern does not change with the aging time, but the residual stress value decreases with the aging time. Table 5 shows the variation of stress with different aging times.
The non-uniform wall thickness and the notch of the C-ring can produce obvious residual stress during heat treatment. The opening distance of the notch can be used as a characterization of deformation. The comparison between experimental and simulation results of the deformation can be seen in Figure 17.
It can be seen in Figure 17 that errors exist in the simulation results of the thickness variation before and after solution treatment. The reason may be that thermal expansion during heating of aging is not considered. The simulation results of deformation before and after aging are pretty accurate, which shows that the finite element model in this paper is reasonable.

Conclusions
In order to study internal stress evolutions in 2A14 aluminum alloy parts in the whole process of solution treatment and aging, the constitutive model at high temperature during quenching and the stress relaxation model during aging were established. The numerical simulation of residual stresses of a C-ring sample after quenching and the residual stress relaxation of the sample during aging process were carried out; calculated results met experimental results well. Calculated stress evolutions during aging showed that magnitudes of residual stresses obtained during quenching are reduced during aging, and the distribution pattern of residual stresses almost remains.
Numerical simulation of the solution and aging treatment process of aluminum alloys is extremely challenging due to many influence factors. This study can be further improved with more reliable thermophysical parameters and heat transfer boundary conditions. The relationship between the precipitates and stress relaxation during the aging process also needs to be explored.

Author contribution
The contribution of Yang Yuchen to the paper is the determination of heat transfer coefficients during quenching and numerical simulation, Deng Xiaofeng works for the test and modeling of the stress relaxation during aging, and Shi Wei works for material modeling of quenching and aging process.

Disclosure statement
No potential conflict of interest was reported by the author(s).