# A Physically Based Constitutive Model and Continuous Dynamic Recrystallization Behavior Analysis of 2219 Aluminum Alloy during Hot Deformation Process

^{1}

^{2}

^{*}

## Abstract

**:**

^{−1}, respectively, and the deformed microstructures were observed. The flow curves of the 2219 Al alloy obtained show that flow stress decreases with the increase in temperature and/or the decrease in strain rate. The physically based constitutive model is applied to describe the flow behavior during hot deformation. In this model, Young’s modulus and lattice diffusion coefficient are temperature-dependent, and the creep exponent is regarded as a variable. The predicted values calculated by the constitutive model are in good agreement with the experimental results. In addition, it is confirmed that the main softening mechanism of the 2219 Al alloy during hot deformation is dynamic recovery and incomplete continuous dynamic recrystallization (CDRX) by the analysis of electron backscattered diffraction (EBSD) micrographs. Moreover, CDRX can readily occur under the condition of high temperatures, low strain rates, and large strains. Meanwhile, the recrystallization grain size will also be larger.

## 1. Introduction

## 2. Materials and Methods

_{m}was adopted in this paper. Four different compression temperatures (773 K, 723 K, 673 K, 623 K) and four different strain rates (0.01 s

^{−1}, 0.1 s

^{−1}, 1 s

^{−1}, 10 s

^{−1}) were applied in the tests; the final true strain is up to 0.9. Each temperature was tested only once under four different strain rates, respectively. Tantalum foil and boron nitride were used to reduce the friction between anvils and the specimen. Each sample was heated to the specified forming temperature at a heating rate of 5 K/s, and then held for 3 min to eliminate the temperature gradient. The true stress-strain data were automatically collected from the computer system during the compression process. After the tests, all the samples were immediately water quenched to preserve the deformed microstructure. Then, the compressed samples were sectioned parallel to the compression axis for subsequent microstructural analysis by the electron backscattered diffraction (EBSD) technique. After the samples were grinded, they were then electropolished in a solution of 30% nitric acid and 70% methanol at 18 V for 50 s. HKL Channel5 software was used to analyze the EBSD data. In all the EBSD images, the LAGBs (orientation angle: 2°–15°) were marked by thin white lines, and the HAGBs (orientation angle: >15°) were marked by the thick black lines. Figure 1 shows the EBSD microstructure of 2219 Al alloy before compression. Coarse equiaxed recrystallized grains and deformed grains with many substructures were both detected.

## 3. Results and Discussion

#### 3.1. Flow Behaviors

^{−1}and 10 s

^{−1}were modified by temperature compensation according to Humphreys F.J [40]. It can be clearly seen from Figure 2 that the flow stress increased rapidly to a peak value and then slightly fell to a relatively steady value, which shows an obvious characteristic of a single-peak type. Thus, each flow stress curve can be divided into three parts by strains: work-hardening part, softening part, and steady part [41,42]. The mechanism of each part is mainly controlled by the competition between work hardening and DRV and/or DRX during the hot deformation process [43]. For the work-hardening part, the rapid accumulation and proliferation of dislocations leads to obvious work hardening, but DRV develops too slowly. The synthesis result is the rapid increase of flow stress at this stage. For the softening part, high dislocation density largely promotes the development of DRV and the formation of DRX, which has exceeded the effect of work hardening. That is to say, the flow stress will slightly decrease. For the steady part, the multiplication and annihilation of dislocations approximately reached an offsetting state due to the effect of continuous deformation and DRV and/or DRX, respectively, thus, the flow stresses almost remained relatively stable values.

^{−1}, 773 K/0.1 s

^{−1}, 723 K/0.01 s

^{−1}. This is because hot compression is a thermal-activation process [5]. Therefore, there will be a larger driving force for the development of DRV and/or DRX due to the easier movement and migration of dislocation and grain boundary [44] at higher temperatures. On the other hand, it will develop more sufficiently to consume more dislocation density at lower strain rate. Thus, either of higher forming temperature and lower strain rate can promote the development of DRV and/or DRX, and finally result in the decrease of the flow stress.

#### 3.2. Physically Based Constitutive Modeling

^{−1}) and $\sigma $ (MPa) are the strain rate and the flow stress, respectively, B and $\alpha $ are the material constants, and the index 5 represents the ideal value of the creep exponent. However, microstructural evolution (i.e., DRX) generally has a certain influence on the value of creep exponent, so it is reasonable to be considered as a variable (i.e., n) [39].

_{0}is the pre-exponent coefficient of the lattice diffusion (1.7 × 10

^{−4}m

^{2}/s), Q

_{sd}is the activation energy of lattice diffusion (142 KJ/mol), R is the universal gas constant (8.31 J·mol

^{−1}K

^{−1}), and T is the forming temperature (K). $\upsilon $ is Poisson’s ratio (0.33), and G is the shear modulus; its relation with temperature can be expressed as [46]:

_{0}is the shear modulus at 300 K (2.54 × 10

^{4}MPa), T

_{M}is the melting temperature of 2219 Al alloy (916 K), and $\eta $ indicates the temperature dependence of shear modulus (–0.50). According to the parameters above, D(T) and E(T) can be expressed as follows:

_{1}, B

_{2}, n

_{1}are material parameters and $\alpha =\beta /{n}_{1}$.

_{1}and $\beta $ can be obtained from the gradient of the graph of $\mathrm{ln}\left[\dot{\epsilon}/D\left(T\right)\right]$ against $\mathrm{ln}\left[\sigma /E\left(T\right)\right]$ and $\mathrm{ln}\left[\dot{\epsilon}/D\left(T\right)\right]$ against $\sigma /E\left(T\right)$. These plots are shown in Figure 3 (i.e., taking the strain of 0.5 as an example).

^{2}), which is expressed as follows [39]:

_{i}and P

_{i}are the experimental and predicted values of each point, respectively, E and P are the average experimental and the predicted values, and N is the total number of the data sample used. According to the data analysis, $\alpha $ has very good fitting with correlation coefficient R

^{2}= 0.972; therefore, $\alpha $ can be mathematically expressed as follows:

^{2}= 0.992; therefore, lnB is expressed mathematically as follows:

^{2}= 0.999. Therefore, n is expressed mathematically as follows:

^{−1}to 10

^{−1}, and the strain range of 0~0.8 can be predicted.

#### 3.3. Verification of the Model

^{2}) and average absolute relative error (AARE) were determined; the AARE is expressed as follows [39]:

_{i}and P

_{i}are the experimental and predicted values of each point, and N is the total number of the data sample used.

^{2}= 0.985 and the AARE = 3.88%. Conclusively, the physically based constitutive model can be used to predict the flow stress of 2219 Al alloy during hot deformation with high accuracy.

#### 3.4. Microstructural Evolution

#### 3.4.1. The Formation of CDRX

^{−1}. Comparing with the samples before compression (Figure 1), it is observed that the grains were flattened in the direction of compression and a large amount of substructures with LAGBs were generated in the original grains. The deformed microstructure showed obvious characteristics of DRV. The SFE of the metals and alloys had a significant effect on the softening mechanisms (DRV and/or DRX). For a 2219 Al alloy with high SFE, it is more difficult to dissociate the perfect dislocation into two partials, but perfect dislocation glide, climb, and cross slip can occur easily. During hot deformation, rapid DRV takes place readily, and the stored energy is decreased by rearrangement and annihilation of dislocations, which generally retards the development of DDRX. However, it can be seen from Figure 9a that many fine grains with a size less than 10 μm appeared near the grain boundaries and a few incomplete fine grains (marked with black dotted circles) composed of HAGBs; partial LAGBs were also observed. It is reasonable to deduce that these incomplete fine grains will develop to whole recrystallized grains with continued deformation; that is to say, these partial LAGBs will transform into HAGBs [50]. Therefore, the special formation mechanism of new recrystallized grains is proved to be CDRX rather than the traditional DDRX on 2219 Al alloy during hot deformation [51,52].

#### 3.4.2. The Influence of TMP on CDRX

^{−1}, and the strains of 0.2, 0.5, and 0.9, respectively. It can be seen clearly that, as the deformation continued, the original grains were gradually elongated, grain boundaries became more and more jagged, and the amount of small, recrystallized grains near the grain boundaries increased progressively. Thus, the increase in the strain can promote the formation of CDRX. However, the percentage of recrystallized grains is still very small; even at the strain of 0.9, the original grains and their internal substructures occupied the majority, which also proves that the predominant softening mechanism of the 2219 Al alloy during hot deformation was DRV.

^{−1}and strain of 0.9 at different temperatures of 623 K, 673 K, and 773 K, respectively. It can be seen that, as the temperature increased, the amount of recrystallized grains near the grain boundaries gradually increased, and also the grain size gradually increased. This is due to the fact that the mobility of the grain boundaries will increase with the increase in temperature. Moreover, DRX is a thermal-activation process; therefore, there will be a greater driving force to promote the growth of the new grains at high temperature. So, the increase of forming temperature can promote the development of CDRX and growth of new grains. Finally, Figure 10f–h show the microstructures deformed at the temperature of 773 K, strain of 0.9, and different strain rates of 1 s

^{−1}, 0.1 s

^{−1}, and 0.01 s

^{−1}, respectively. It can also be observed that as the strain rate increased, the amount of recrystallized grains decreased, and the growth of new fine grains was greatly restricted. This is because the reduction of forming time restrains the growth of recrystallized grains at high strain rate. Therefore, the increase in strain rates limits the development of CDRX and the growth of its recrystallized grains. The results of the influence of TMP on CDRX obtained are consistent with the study of Wang [51] on AA7050 aluminum alloy.

## 4. Conclusions

- The flow stress of the 2219 Al alloy is very sensitive to temperatures and strain rates, and its value decreases with the increase in temperatures and/or the decrease in strain rates.
- The physically based constitutive model of 2219 Al alloy established is proved to have good predictive performance, which can be used to accurately describe the flow behavior of the 2219 Al alloy in the temperature range of 623 K to 773 K, strain rate range of 0.01 s
^{−1}to 10^{−1}, and the strain range of 0~0.8. - It has been proved that the main microstructure evolution of 2219 Al alloy under hot deformation is DRV and incomplete CDRX. Moreover, CDRX can occur readily at high temperatures, low strain rates and high strains; meanwhile, the recrystallized grains size will also be larger.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The electron backscattered diffraction (EBSD) micrograph of 2219 Al alloy before deformation.

**Figure 2.**The true strain-stress curves obtained at various strain rates and temperatures of (

**a**) 0.01 s

^{−1}; (

**b**) 0.1 s

^{−1}; (

**c**) 1 s

^{−1}; (

**d**) 10 s

^{−1}.

**Figure 3.**Relationships between material parameters: (

**a**) $\mathrm{ln}\left[\dot{\epsilon}/D\left(T\right)\right]$ against $\mathrm{ln}\left[\sigma /E\left(T\right)\right]$; (

**b**) $\mathrm{ln}\left[\dot{\epsilon}/D\left(T\right)\right]$ against $\sigma /E\left(T\right)$.

**Figure 4.**The variation of $\alpha $: (

**a**) at different strains and temperatures; (

**b**) 3D illustration and its nonlinear surface fitting.

**Figure 5.**Relationship between $\mathrm{ln}\left[\dot{\epsilon}/D\left(T\right)\right]$ against $\mathrm{ln}\left\{sinh\left[\alpha \sigma /E\left(T\right)\right]\right\}$ for different parameters.

**Figure 6.**The variation of lnB: (

**a**) at different strains and temperatures; (

**b**) 3D illustration and its nonlinear surface fitting.

**Figure 8.**Correlations between the experimental and predicted flow stresses of the physically based constitutive model: (

**a**) under modeling set; (

**b**) under prediction set; (

**c**) by linear fit.

**Figure 9.**EBSD measurement showing continuous dynamic recrystallization (CDRX) in 2219Al alloy. (

**a**) Micrograph of the deformed sample under the condition of 673 K and 10 s

^{−1}; (

**b**) the cumulative misorientation along the dotted line from A and B to the grain boundary.

**Figure 10.**The EBSD micrographs of the deformed samples under conditions of: (

**a**) 673 K and 0.1 s

^{−1}with the strain of 0.2; (

**b**) 673 K and 0.1 s

^{−1}with the strain of 0.5; (

**c**) 673 K and 0.1 s

^{−1}with the strain of 0.9; (

**d**) 623 K and 1 s

^{−1}with the strain of 0.9; (

**e**) 673 K and 1 s

^{−1}with the strain of 0.9; (

**f**) 773 K and 1 s

^{−1}with the strain of 0.9; (

**g**) 773 K and 0.1 s

^{−1}with the strain of 0.9; (

**h**) 773 K and 0.01 s

^{−1}with the strain of 0.9.

Si | Fe | Cu | Mn | Mg | Zn | V | Ti | Zr | Al |
---|---|---|---|---|---|---|---|---|---|

0.20 | 0.30 | 5.8~6.8 | 0.20~0.40 | 0.02 | 0.10 | 0.05~0.15 | 0.02~0.10 | 0.10~0.25 | Bal |

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Liu, L.; Wu, Y.; Gong, H.; Li, S.; Ahmad, A.S.
A Physically Based Constitutive Model and Continuous Dynamic Recrystallization Behavior Analysis of 2219 Aluminum Alloy during Hot Deformation Process. *Materials* **2018**, *11*, 1443.
https://doi.org/10.3390/ma11081443

**AMA Style**

Liu L, Wu Y, Gong H, Li S, Ahmad AS.
A Physically Based Constitutive Model and Continuous Dynamic Recrystallization Behavior Analysis of 2219 Aluminum Alloy during Hot Deformation Process. *Materials*. 2018; 11(8):1443.
https://doi.org/10.3390/ma11081443

**Chicago/Turabian Style**

Liu, Lei, Yunxin Wu, Hai Gong, Shuang Li, and A. S. Ahmad.
2018. "A Physically Based Constitutive Model and Continuous Dynamic Recrystallization Behavior Analysis of 2219 Aluminum Alloy during Hot Deformation Process" *Materials* 11, no. 8: 1443.
https://doi.org/10.3390/ma11081443