Physical-based constitutive model considering the microstructure evolution during hot working of AZ80 magnesium alloy

A physical-based constitutive model was developed to model the viscoplastic flow behavior and microstructure evolution of AZ80 magnesium alloy during the hot working process. The competing deformation mechanisms, including work hardening, dynamic recovery, and dynamic recrystallization, in an isothermal compression environment were considered in the model. The internal state variables, including the normalized dislocation density and recrystallized volume fraction, were incorporated into the model to articulate the microstructure evolution during hot deformation. The kinetic condition critical for dynamic recrystallization, considering the effects of the deformation temperature and strain rate, was obtained by employing both the Poliak-Jonas criterion and Zener-Hollomon parameter. Microstructure observations indicate that the recrystallized volume fraction increases with decreasing Z parameter at constant strain, which is consistent with the predicted kinetics model. Based on the developed model, a good correlation was also obtained between the predicted and experimental flow stress. The results indicate a good predictability of the model in describing the hot deformation behavior and microstructure evolution of AZ80 magnesium alloy.


Introduction
Magnesium alloy is one of the most promising lightweight metallic materials for structural applications in the automotive and aerospace industries owing to its low density, high specific strength, and good recyclability [1,2]. However, the slip systems of the alloy are poor at room temperature due to its hexagonal close-packed (HCP) structure, which limits its deformation and manufacturability and thus its application in the industry [3,4]. To improve the workability of the alloy, plastic deformation, such as extrusion and forging, is usually conducted at elevated temperatures, which increases the number of slip planes and decreases the critical shear stress of non-basal plane slip [5]. During the hot deformation of magnesium alloy, dynamic recrystallization (DRX) predominates because of its low stacking fault energy (60-78 mJ/m 2 ), which is an effective way to obtain a refined microstructure and improve the mechanics performance [6]. The understanding of the flow behavior and metallurgical phenomena during hot deformation is of scientific significance for the optimization of the hot workability and guiding forming process of magnesium alloy.
A constitutive model is essential for the description of the flow characteristics of materials, which could be combined with a finite element method to provide an efficient computational platform for the prediction of the mechanical response. Over the past decades, numerous constitutive models have been established such as phenomenological models, e.g., Arrhenius-type model and Johnson-Cook model [7,8], advanced statistical models, e.g., artificial neural network model [9,10], and physicalbased internal state variable (ISV) models, e.g., Zerilli-Armstrong model and Kocks-Mecking model [11,12]. By modeling of the deformation process, Liu and Ding [13] established an Arrhenius-type constitutive model for AZ91 magnesium alloy to describe the hot deformation behavior for a wide range of temperatures and strain rates. Sabokpa et al. [14] developed an artificial neural network (ANN) model for the representation of the hot compression behavior of AZ81 magnesium alloy and compared it with the Arrhenius-type model. The results indicated that the trained ANN model was more accurate in predicting the flow stress. Furthermore, Li et al. [15] conducted a comparative study on Arrhenius-type, ANN, and modified Zerilli-Armstrong models to predict the hot deformation behavior of T24 steel. The ANN model showed the most excellent predictability. However, it was pointed out that the ANN model strongly depended on extensive highquality data and characteristic variables did not provide physical insights.
Among the different models, the phenomenological model is currently the most widely used one due to the few parameters and simplicity in determining the parameters by regression analysis. However, the model presents a poor predictability in determining deformation parameters that are not in the experimental range. The advanced statistical model is the most accurate in predicting the flow behavior of a wide range of deformation conditions. However, the model does not have an intrinsic physics meaning and cannot reflect the microstructure evolution during the deformation. Compared with the above-mentioned models, the physical-based ISV approach has become more important in modeling various flow behaviors and microstructure evolution over the past decades. The constitutive ISV model is capable of capturing inherent microstructural complexity during the working process and achieving a reasonable accuracy, which can be used in computational simulation for process optimization [16]. Vilamosa et al. [17] established a physical-based constitutive model for AA6082 aluminum alloy using the dislocation density as an internal variable without relating it to microstructure evolution. Austin and Mcdowell [18] proposed a dislocation-based constitutive model for the viscoplastic deformation of fcc metals at high strain rates by introducing path-dependent differential equations for mobile and immobile dislocation densities. Lin et al. [19] developed a dislocation-based unified viscoplastic constitutive model for C-Mn steel to predict the dislocation density, recrystallization, and grain size evolution during multi-pass hot rolling processing. The aforementioned study is a good reference and basis for establishing the physical-based constitutive model for AZ80 magnesium alloy to describe the flow behavior and microstructure evolution during the hot working process.
At present, many constitutive models, which are available for magnesium alloys, are generally phenomenologically constructed by introducing a few macroscopic variables to the model and formulated as functions of the temperature, strain rate and strain. They are basically experimental and based on classical viscoplastic theory to describe work hardening and thermal softening phenomena during the hot working process. However, the models lack an in-depth understanding of the underlying physics. In view of the important role of DRX in the grain refinement and improvement of mechanical properties during the hot working of magnesium alloy, the establishment of a constitutive model that can reflect the evolution of the recrystallized microstructure is urgently needed. In this study, a physical-based constitutive model was established by introducing ISVs including the normalized dislocation density and recrystallized volume fraction. To provide an adequate description of the recrystallization kinetics, the Poliak-Jonas criterion and the Zener-Hollomon parameter were employed because they represented the microstructure evolution, which was simultaneously affected by the deformation temperature and strain rate. The model was verified using the experimental flow stress and recrystallized volume fraction and a good agreement was observed. The proposed constitutive model can be used to analyze the hot deformation behavior and microstructure evolution of AZ80 magnesium alloy.

Experimental procedures
The as-received AZ80 magnesium alloy is an extruded billet with a diameter of 200 mm and a height of 150 mm. The chemical composition (mass fraction) of the alloy is: 8.16% Al, 0.42% Zn, 0.03% Mn, 0.01% Si, 0.005% Fe, 0.001% Cu, 0.001% Ni, and balance Mg. According to the standard of ASTM E209 [20] for compression tests of metallic materials at elevated temperatures with conventional or rapid heating rates and strain rates, cylindrical specimens with a diameter of 8 mm and height of 12 mm were machined from the extruded bar along the extrusion direction. Two end surfaces of the specimens were ground to reduce the effect of friction on the hot deformation behavior. A uniaxial hot compression test was conducted with a Gleeble-1500D thermomechanical simulator; the schematic is illustrated in Fig.1. Five deformation temperatures (300, 325, 350, 375 and 400°C) and three strain rates (0.001, 0.01 and 0.1 s -1 ) were used and the height reduction was 60%, corresponding to a true strain of 0.916. Before the compression, the top and bottom surfaces of the specimens were coated with graphite lubricant and Physical-based constitutive model considering the microstructure evolution during hot working… 31 tantalum foil was placed between the specimen and anvil to reduce the friction and avoid adhesion. For the measurement and feedback control of the real-time temperature, K-type thermocouples were welded in the center of specimens. The specimens were heated to the test deformation temperature with a heating rate of 5°C/s and then held isothermally for 180 s to eliminate the thermal gradient prior to the deformation. Stress-strain data were automatically recorded by the testing system during the hot deformation process. After the compression, the specimens were immediately water-quenched to retain the deformation microstructure. The deformed specimens were sectioned along the compression axis for microstructural analysis using a wireelectric discharge machine (EDM). The sectioned specimens were mechanically ground, polished, and then etched for 5-12 s using a solution consisting of 5 mL picric acid, 5 mL acetic acid, 100 mL ethyl alcohol, and 10 mL distilled water. Microstructure observations were made with a DM4000M optical microscope. The average grain size and recrystallized volume fraction were examined by quantitative metallography.

Experimental results and analysis
The true compressive stress-strain response of AZ80 magnesium alloy is shown in Fig. 2. Despite the difference in the specific flow stress, the flow response presents a peak stress with subsequent flow softening, which is a typical DRX characteristic. In general, the variation in the flow stress reflects the competition between work hardening (WH) and dynamic softening during hot deformation. The flow stress firstly rapidly increases due to dislocation tangling and pileups and dynamic recovery (DRV) simultaneously arises through dislocation climbing and gliding, which results in the formation of mobile subgrain boundaries and the annihilation of dislocations. As the dislocation density exceeds a threshold, DRX nucleates at low-angle grain boundaries. Therefore, the flow stress decreases dramatically by further dislocation annihilation. When the dislocation proliferation and annihilation achieve a dynamic equilibrium, the flow characteristics affected by strain hardening and dynamic softening maintain a steady state.
Note that the flow stress has a negative sensitivity to the temperature and a positive sensitivity to the strain rate. High temperature leads to a much higher driving energy for dislocation movement, while a low strain rate provides sufficient time for DRV and DRX, which both contribute to the decrease in the flow stress. It can be concluded that the dynamic softening stage starts at a lower strain and flow softening more likely occurs at an increased deformation temperature and decreased strain rate because both the abundant energy and sufficient time affect the thermal activation process including DRV and DRX.

Constitutive ISV model 4.1 Flow stress decomposition
According to Hooke's law, the stress-strain relationship of materials can be expressed as [21] r where r is the overall flow stress; E is the Young' modulus, which is a temperature-dependent parameter; and e T and e p represent the total strain and plastic strain, respectively. Because the material exhibits viscoplastic flow during the hot working process, the overall flow stress can be decomposed into three parts: initial yield stress, viscoplastic stress, and hardening stress [22] r where k is the initial yield stress, r v the viscoplastic stress, and H the isotropic hardening stress. A hyperbolic sine law is introduced to describe the viscoplastic flow, which is suitable for a wide range of temperatures and strain rates [23,24] where _ e p is the plastic strain rate; n c is the viscoplastic exponent, which is temperature-dependent; and A 1 and A 2 are material constants.
The hardening stress H in Eq.
(3) has a close relationship with the reciprocal of the average slip length over which a dislocation can run and the mean slip length is determined by the inverse square of the dislocation density [25]. The isotropic hardening rate can thus be defined as [26,27] _ where B is a temperature-dependent parameter; q is the normalized dislocation density formulated as q ¼ 1 À q 0 =q [28]; q 0 is the initial dislocation density, and q is the dislocation density during the deformation process. The q value varying from 0 to 1 represents the dislocation that evolves from the initial state to the saturated state.
The initial yield stress k in Eq. (3) is temperature-dependent and decreases with increasing temperature. It can be expressed by an Arrhenius-type equation: k ¼ k 0 expðQ k =RTÞ. The parameters k 0 and Q k are the material constant and activation energy of k, respectively.

Dislocation density evolution
During high temperature deformation, dislocation multiplication and trapping by existing obstacles provide a driving force for DRV and DRX. Considering the hot deformation mechanisms, the evolution of the dislocation density can be described as [19] where A 3 , A 4 , d 2 , d 3 , and d 4 are material constants and C r is a temperature-dependent parameter. The first term represents the accumulation and DRV of the dislocation density. The DRV part limits the normalized dislocation density to the saturated state of a dislocation network when q ¼ 1. The second term models the static recovery during the heating process. The third term expresses the effect of DRX on the evolution of the dislocation density and S is the recrystallized volume fraction.

Dynamic recrystallization kinetics
The DRX plays an extremely important role in thermomechanical processing, affecting the final microstructure and mechanical properties of the deformed magnesium alloys. The critical condition for the initiation of DRX usually depends on the Zener-Hollomon parameter (Z parameter) based on empirical equations [29]. The Z parameter represents the effects of the temperature and strain rate on the deformation behavior and can be expressed as Z ¼ _ e expðQ=RTÞ. The deformation activation energy Q can be estimated by fitting the peak stress with a hyperbolic sine function [30] A sinh ar ð Þ ð where _ e is the true strain rate (s -1 ), and A, n, and a are material constants obtained from linear regression based on the Arrhenius-type equation [31]. Based on the regression analysis of the ln sinh ar ð Þ ð Þ versus 1/T diagram under different strain rates and Eq. (7), the value of Q is 142.054 kJ/mol At a constant initial grain size, the peak strain e h and critical strain e c can usually be described with a power-law function of Z (e ¼ KZ m ) [32], where K and m are material constants. The peak strain can be obtained from the flow stress curve. To determine the critical strain for DRX, the irreversible thermodynamic principle proposed by Poliak and Jonas [33] was employed Physical-based constitutive model considering the microstructure evolution during hot working… 33 where h is the strain hardening rate and is determined by h ¼ or oe . Based on the Poliak-Jonas criterion, the onset of DRX can be identified from an inflection point of the strain hardening rate h as a function of the flow stress r. Because of the following equation, The lnh versus e curve should also exhibit an inflection when DRX takes place. The critical strain of DRX is then mathematically obtained by combining the inflection point criterion of lnh versus e curve with the minimum value criterion of À o lnh ð Þ oe versus e curve. This provides an intuitionistic description for the analysis of the effect of the deformation conditions on the recrystallized critical strain, as shown in Fig. 3. Figure 3 shows that the critical strain decreases with increasing deformation temperature and decreasing strain rate, which illustrates that a higher temperature and lower strain rate are more conducive to the occurrence of DRX. Accordingly, the critical and peak strains as functions of the Z parameter can be expressed by a linear regression in logarithmic form (see Fig. 4a) The critical strain was confirmed to be a function of the peak strain (e c ffi 0:79e h ), as shown in Fig. 4b, which is consistent with the results for a wide range of materials [34][35][36].
When the dislocation density increases to a critical value, nucleation and growth of the DRX nucleus will preferentially occur at high dislocation density zones such as deformation bands and grain boundaries [37]. To describe the evolution of DRX and predict the recrystallized volume fraction, a rate equation is proposed [25]  where d 1 is a material constant and Q 0 is a temperaturedependent parameter. The recrystallized volume fraction S varies from 0 to 1 and its variation is cyclic, depending on the evolution of the dislocation density.
It has been proven experimentally that an incubation time is needed for the onset of recrystallization, which can be expressed as [38] where _ x is the incubation fraction and A 0 is a temperaturedependent parameter.

Constitutive model development
Because the microstructure evolution including hardening, recovery, and recrystallization mechanisms is considered, a set of physical-based viscoplastic constitutive ISV equations can be formulated for AZ80 magnesium alloy r ¼ E e T À e p À Á : Under high-temperature conditions, the deformation is temperature-dependent due to atom diffusion and dislocation motion. The material parameters in the constitutive model are treated as temperature-dependent parameters, which can be expressed as Arrhenius-type equations where R is the universal gas constant (8.314 (JÁK)/mol) and T is the absolute temperature (K). The Q in the exponential function denotes the activation energy for the corresponding material constants. To determine the material constants within the equations, a genetic algorithm (GA)-based optimization method was developed and programmed by minimizing the residuals of the experimental and calculated stress-strain data. A GA toolbox in Matlab software was used to optimize the objective function and determine the material constants. The details of the optimization process are reported in Refs. [39,40]. The values determined for the material constants are listed in Table 1.

Constitutive model validation
The experimental and predicted stress-strain data for different deformation conditions are compared in this section. The results for the strain rates of 0.1 and 0.01 s -1 are shown in Fig. 5. Most of the experimental points are close to the predicted results, indicating a preferable prediction performance of the constitutive model. However, with increasing strain rate (see Fig.5a), the deviation increases at deformation temperatures of 300, 325 and 350°C, especially in the work hardening-dynamic recovery stage. The predicted flow stresses agree well with the measured ones at higher deformation temperatures. The main reason for this phenomenon is that the predicted dynamic recovery Physical-based constitutive model considering the microstructure evolution during hot working… 35 rate is higher than the measured one. It is well known that magnesium alloy is a type of low stacking fault energy alloy; the dynamic recovery rate is low in the alloy because of the reduced mobility of the dislocations. A large grain boundary migration rate is induced by high local gradients of the dislocation density. Therefore, there is not enough time for the transformation of subgrains to grains, especially at higher strain rates, e.g., 0.1 s -1 [41]. On the other hand, the precipitate phase could also play an important role in the hot deformation behavior of AZ80 magnesium alloy. At low deformation temperatures (300, 325 and 350°C), which are below the temperature of complete dissolution of the precipitate phase, the phase can strengthen the alloy due to the pinning effect of the grain boundaries and dislocations [42]. An excessive deformation resistance is therefore induced and the experimental flow stress shows a notable work hardening effect. With increasing temperature, the flow stress becomes insensitive to the deformation temperature due to the dissolution of the precipitate phase [2]. Thus, the actual dynamic recovery rate is low, resulting in the deviation between the measured and predicted flow stresses.
To evaluate the predictability of the proposed model, the correlation coefficient (R) and average absolute relative error (e AARE ) are specified where E i and P i are the experimental and predicted flow stress, respectively; E and P are the average values of E i and P i ; and n is the number of sampling data. The values of R and e AARE were calculated to be 96.0% and 5.9%, respectively, which indicated a good prediction accuracy of the model.  000 37 000 17 800 1000 15 000 56 622.4 A 00 (-) 384.56 0.99 0.85 0.8 0.99 10 A 4 (-) 7 9 10 -5 0.32 18 14.77

Results and discussion
The curves predicted for the dislocation density evolution exhibit rapid hardening to a single peak at relatively low strain and then flow softening followed by a steady state at high strain, as shown in Fig. 6. These characteristics are consistent with the variation in the flow stress in Fig. 2. It can be concluded that the low strain rate and high deformation temperature benefit dislocation annihilation and lead to a decreasing dislocation density and increasing mobile dislocations. During the hot deformation, many of the dislocations are concentrated on the grain boundaries in the slip process. With increasing dislocation density near the grain boundaries, DRX takes place in the local area and then evolves into the interior of the grains. Consequently, both the dislocation density and flow stress are reduced by dislocation rearrangement and annihilation. The curves predicted for the recrystallized volume fraction increase with the strain in a sigmoidal manner, as shown in Figs. 7a-c. The points marked as (1)-(5) represent the experimental data obtained from microstructure observations in Fig. 8 and are in good agreement with the predicted results. It is notable that the recrystallized volume fraction increases with increasing temperature at a given strain rate, while it decreases with increasing strain rate at a constant temperature. This is attributed to the higher mobility at grain boundaries at higher temperature and lower strain rate, which benefits the activation of DRX [43]. After the deformation reaches a critical strain, DRX occurs and the recrystallized volume fraction increases quickly with increasing strain until a considerable fraction of the microstructure is recrystallized.
As stated in our previous research [44], there is a close relationship between the recrystallized microstructure evolution and Z parameter. To explain the DRX kinetics in detail, microstructure observations were conducted under various Z values (corresponding to five specific deformation conditions), as shown in Fig. 8. Based on the following equation, Z ¼ _ e expðQ=RTÞ, a low Z value corresponds to a high temperature and low strain rate, while a high Z value reflects a low temperature and high strain rate. An inhomogeneous microstructure with an average grain size of 35 lm is characteristic for the pre-extruded alloy, as shown in Fig. 8a. Figure 8b shows that new grains nucleate at the original grain boundaries. Subsequently, nucleation occurs at the growing grains, forming a necklace-type structure of grains. In this case, which corresponds to high Z conditions and low recrystallization kinetics, the nucleation of the recrystallized grains starts at a very high strain, leading to a notably incomplete recrystallized microstructure. At a low Z value with high recrystallization kinetics and mobility of grain boundaries, new grains bulge from the preexisting grain boundaries and grow quickly and adequately, forming a completely recrystallized microstructure (see Fig. 8f).
The statistical data of the recrystallized volume fraction under different Z values are presented in Fig. 7d. For a given strain of 0.916, the recrystallized volume fraction is less than 90% at high Z values, corresponding to an incomplete recrystallized microstructure (see Figs. 8b and c). On the other hand, full evolution of the recrystallized microstructure occurs at low Z values (see Figs. 8d-f); an almost 100% recrystallized volume fraction is shown in Fig. 7d. Based on the statistical analysis, it can be concluded that the recrystallized volume fraction increases with decreasing Z value for a given true strain, which can be confirmed by the predicted recrystallized kinetics shown in Figs. 7a-c. Similar features of the DRX microstructure during hot deformation were observed for other magnesium alloys [45,46]. Another notable phenomenon under different Z conditions is the statistical result for the grain size distribution in the final microstructure (see Fig. 7d). Under high Z conditions with a high strain rate effect, work hardening is significant and the initiation of the recrystallization process is delayed. Consequently, the stored deformation energy is mostly used for the formation of new recrystallized grains and the average grain size becomes small and limited because there is no sufficient time for grain growth, which results in an inhomogeneous microstructure with an average grain size of 8.2 lm (see Fig. 8c). Under low Z conditions with a high temperature effect, some grains evolve quickly due to grain growth and grain impingement, which is consistent with an inhomogeneous microstructure with an average grain size of 15.2 lm (see Fig. 8e). Figure 8d shows a homogeneous and refined microstructure with an average grain size of 7.5 lm. Deformation conditions of 300°C/0.001 s -1 correspond to a low lnZ value of 22.9, which is plotted as vertical blue dashed line in Fig. 7d. It can thus be concluded that the low strain rate contributes to high recrystallization kinetics of 99.4%. The grains can maintain a small size under the low temperature, which is an optimum state for the hot working process. To achieve the desired final properties with an optimum microstructure (higher volume fraction of DRX grains and lower average grain size), a predetermined deformation temperature and strain rate should be considered.

Conclusions
(i) The DRX is the dominant softening mechanism in AZ80 magnesium alloy during the hot deformation process and its flow response exhibits a single peak followed by steady-state flow. The critical DRX strain was determined as a function of the peak strain  Physical-based constitutive model considering the microstructure evolution during hot working… 39 experimental results. Statistical microstructure observations indicate an increase in the recrystallized volume fraction with decreasing Z parameter, which is consistent with the predicted recrystallized kinetics data. The model shows a good predictability in describing the hot deformation behavior and microstructure evolution of AZ80 magnesium alloy.