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

Ze-Xing Su; Chao-Yang Sun; Ming-Wang Fu; Ling-Yun Qian

doi:10.1007/s40436-018-0243-8

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

Su, Ze-Xing; Sun, Chao-Yang; Fu, Ming-Wang; Qian, Ling-Yun 2019-01-02 00:00:00 Adv. Manuf. (2019) 7:30–41 https://doi.org/10.1007/s40436-018-0243-8 Physical-based constitutive model considering the microstructure evolution during hot working of AZ80 magnesium alloy 1,2 1,2 3 1,2 • • • Ze-Xing Su Chao-Yang Sun Ming-Wang Fu Ling-Yun Qian Received: 23 May 2018 / Accepted: 7 December 2018 / Published online: 2 January 2019 The Author(s) 2018 Abstract A physical-based constitutive model was devel- Keywords AZ80 magnesium alloy Hot deformation oped to model the viscoplastic ﬂow behavior and Constitutive model Microstructure evolution Dynamic microstructure evolution of AZ80 magnesium alloy during recrystallization (DRX) the hot working process. The competing deformation mechanisms, including work hardening, dynamic recovery, and dynamic recrystallization, in an isothermal compres- 1 Introduction sion environment were considered in the model. The internal state variables, including the normalized disloca- Magnesium alloy is one of the most promising lightweight tion density and recrystallized volume fraction, were metallic materials for structural applications in the auto- incorporated into the model to articulate the microstructure motive and aerospace industries owing to its low density, evolution during hot deformation. The kinetic condition high speciﬁc strength, and good recyclability [1, 2]. critical for dynamic recrystallization, considering the However, the slip systems of the alloy are poor at room effects of the deformation temperature and strain rate, was temperature due to its hexagonal close-packed (HCP) obtained by employing both the Poliak-Jonas criterion and structure, which limits its deformation and manufactura- Zener-Hollomon parameter. Microstructure observations bility and thus its application in the industry [3, 4]. To indicate that the recrystallized volume fraction increases improve the workability of the alloy, plastic deformation, with decreasing Z parameter at constant strain, which is such as extrusion and forging, is usually conducted at consistent with the predicted kinetics model. Based on the elevated temperatures, which increases the number of slip developed model, a good correlation was also obtained planes and decreases the critical shear stress of non-basal between the predicted and experimental ﬂow stress. The plane slip [5]. During the hot deformation of magnesium results indicate a good predictability of the model in alloy, dynamic recrystallization (DRX) predominates describing the hot deformation behavior and microstructure because of its low stacking fault energy (60–78 mJ/m ), evolution of AZ80 magnesium alloy. which is an effective way to obtain a reﬁned microstructure and improve the mechanics performance [6]. The under- standing of the ﬂow behavior and metallurgical phenomena during hot deformation is of scientiﬁc signiﬁcance for the & Chao-Yang Sun suncy@ustb.edu.cn optimization of the hot workability and guiding forming process of magnesium alloy. School of Mechanical Engineering, University of Science and A constitutive model is essential for the description of Technology Beijing, Beijing 100083, People’s Republic of the ﬂow characteristics of materials, which could be China combined with a ﬁnite element method to provide an Beijing Key Laboratory of Lightweight Metal Forming, efﬁcient computational platform for the prediction of the Beijing 100083, People’s Republic of China mechanical response. Over the past decades, numerous Department of Mechanical Engineering, The Hong Kong constitutive models have been established such as phe- Polytechnic University, Hung Hom, Hong Kong, People’s nomenological models, e.g., Arrhenius-type model and Republic of China 123 Physical-based constitutive model considering the microstructure evolution during hot working… 31 Johnson-Cook model [7, 8], advanced statistical models, alloy to describe the ﬂow behavior and microstructure e.g., artiﬁcial neural network model [9, 10], and physical- evolution during the hot working process. based internal state variable (ISV) models, e.g., Zerilli- At present, many constitutive models, which are avail- Armstrong model and Kocks-Mecking model [11, 12]. By able for magnesium alloys, are generally phenomenologi- modeling of the deformation process, Liu and Ding [13] cally constructed by introducing a few macroscopic established an Arrhenius-type constitutive model for AZ91 variables to the model and formulated as functions of the magnesium alloy to describe the hot deformation behavior temperature, strain rate and strain. They are basically for a wide range of temperatures and strain rates. Sabokpa experimental and based on classical viscoplastic theory to et al. [14] developed an artiﬁcial neural network (ANN) describe work hardening and thermal softening phenomena model for the representation of the hot compression during the hot working process. However, the models lack behavior of AZ81 magnesium alloy and compared it with an in-depth understanding of the underlying physics. In the Arrhenius-type model. The results indicated that the view of the important role of DRX in the grain reﬁnement trained ANN model was more accurate in predicting the and improvement of mechanical properties during the hot ﬂow stress. Furthermore, Li et al. [15] conducted a com- working of magnesium alloy, the establishment of a con- parative study on Arrhenius-type, ANN, and modiﬁed stitutive model that can reﬂect the evolution of the Zerilli-Armstrong models to predict the hot deformation recrystallized microstructure is urgently needed. In this behavior of T24 steel. The ANN model showed the most study, a physical-based constitutive model was established excellent predictability. However, it was pointed out that by introducing ISVs including the normalized dislocation the ANN model strongly depended on extensive high- density and recrystallized volume fraction. To provide an quality data and characteristic variables did not provide adequate description of the recrystallization kinetics, the physical insights. Poliak-Jonas criterion and the Zener-Hollomon parameter Among the different models, the phenomenological were employed because they represented the microstruc- model is currently the most widely used one due to the few ture evolution, which was simultaneously affected by the parameters and simplicity in determining the parameters by deformation temperature and strain rate. The model was regression analysis. However, the model presents a poor veriﬁed using the experimental ﬂow stress and recrystal- predictability in determining deformation parameters that lized volume fraction and a good agreement was observed. are not in the experimental range. The advanced statistical The proposed constitutive model can be used to analyze the model is the most accurate in predicting the ﬂow behavior hot deformation behavior and microstructure evolution of of a wide range of deformation conditions. However, the AZ80 magnesium alloy. model does not have an intrinsic physics meaning and cannot reﬂect the microstructure evolution during the deformation. Compared with the above-mentioned models, 2 Experimental procedures the physical-based ISV approach has become more important in modeling various ﬂow behaviors and The as-received AZ80 magnesium alloy is an extruded microstructure evolution over the past decades. billet with a diameter of 200 mm and a height of 150 mm. The constitutive ISV model is capable of capturing The chemical composition (mass fraction) of the alloy is: inherent microstructural complexity during the working 8.16% Al, 0.42% Zn, 0.03% Mn, 0.01% Si, 0.005% Fe, process and achieving a reasonable accuracy, which can be 0.001% Cu, 0.001% Ni, and balance Mg. According to the used in computational simulation for process optimization standard of ASTM E209 [20] for compression tests of [16]. Vilamosa et al. [17] established a physical-based metallic materials at elevated temperatures with conven- constitutive model for AA6082 aluminum alloy using the tional or rapid heating rates and strain rates, cylindrical dislocation density as an internal variable without relating specimens with a diameter of 8 mm and height of 12 mm it to microstructure evolution. Austin and Mcdowell [18] were machined from the extruded bar along the extrusion proposed a dislocation-based constitutive model for the direction. Two end surfaces of the specimens were ground viscoplastic deformation of fcc metals at high strain rates to reduce the effect of friction on the hot deformation by introducing path-dependent differential equations for behavior. A uniaxial hot compression test was conducted mobile and immobile dislocation densities. Lin et al. [19] with a Gleeble-1500D thermomechanical simulator; the developed a dislocation-based uniﬁed viscoplastic consti- schematic is illustrated in Fig.1. Five deformation tem- tutive model for C–Mn steel to predict the dislocation peratures (300, 325, 350, 375 and 400 C) and three strain -1 density, recrystallization, and grain size evolution during rates (0.001, 0.01 and 0.1 s ) were used and the height multi-pass hot rolling processing. The aforementioned reduction was 60%, corresponding to a true strain of 0.916. study is a good reference and basis for establishing the Before the compression, the top and bottom surfaces of the physical-based constitutive model for AZ80 magnesium specimens were coated with graphite lubricant and 123 32 Z.-X. Su et al. 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 Fig. 1 Schematic illustration of isothermal compression tests for the decrease in the ﬂow stress. It can be concluded that the AZ80 magnesium alloy dynamic softening stage starts at a lower strain and ﬂow softening more likely occurs at an increased deformation tantalum foil was placed between the specimen and anvil to temperature and decreased strain rate because both the reduce the friction and avoid adhesion. For the measure- abundant energy and sufﬁcient time affect the thermal ment and feedback control of the real-time temperature, activation process including DRV and DRX. K-type thermocouples were welded in the center of spec- imens. The specimens were heated to the test deformation temperature with a heating rate of 5 C/s and then held 4 Constitutive ISV model isothermally for 180 s to eliminate the thermal gradient prior to the deformation. Stress-strain data were automat- 4.1 Flow stress decomposition ically recorded by the testing system during the hot deformation process. After the compression, the specimens According to Hooke’s law, the stress-strain relationship of were immediately water-quenched to retain the deforma- materials can be expressed as [21] tion microstructure. r ¼ E e e ; ð1Þ T p The deformed specimens were sectioned along the compression axis for microstructural analysis using a wire- where r is the overall ﬂow stress; E is the Young’ modulus, electric discharge machine (EDM). The sectioned speci- which is a temperature-dependent parameter; and e and e T p mens were mechanically ground, polished, and then etched represent the total strain and plastic strain, respectively. for 5–12 s using a solution consisting of 5 mL picric acid, Because the material exhibits viscoplastic ﬂow during 5 mL acetic acid, 100 mL ethyl alcohol, and 10 mL dis- the hot working process, the overall ﬂow stress can be tilled water. Microstructure observations were made with a decomposed into three parts: initial yield stress, vis- DM4000M optical microscope. The average grain size and coplastic stress, and hardening stress [22] recrystallized volume fraction were examined by quanti- r ¼ k þ r þ H; ð2Þ tative metallography. where k is the initial yield stress, r the viscoplastic stress, and H the isotropic hardening stress. A hyperbolic sine law 3 Experimental results and analysis is introduced to describe the viscoplastic ﬂow, which is suitable for a wide range of temperatures and strain rates The true compressive stress-strain response of AZ80 [23, 24] magnesium alloy is shown in Fig. 2. Despite the difference e ¼ A sinhðÞ AðÞ r H k ; ð3Þ p 1 2 in the speciﬁc ﬂow stress, the ﬂow response presents a peak stress with subsequent ﬂow softening, which is a typical where e_ is the plastic strain rate; n is the viscoplastic p c DRX characteristic. In general, the variation in the ﬂow exponent, which is temperature-dependent; and A and A 1 2 stress reﬂects the competition between work hardening are material constants. (WH) and dynamic softening during hot deformation. The The hardening stress H in Eq.(3) has a close relationship ﬂow stress ﬁrstly rapidly increases due to dislocation tan- with the reciprocal of the average slip length over which a gling and pileups and dynamic recovery (DRV) simulta- dislocation can run and the mean slip length is determined neously arises through dislocation climbing and gliding, by the inverse square of the dislocation density [25]. The which results in the formation of mobile subgrain bound- isotropic hardening rate can thus be deﬁned as [26, 27] aries and the annihilation of dislocations. As the dislocation 0:5 _ _ H ¼ 0:5Bq q ; ð4Þ density exceeds a threshold, DRX nucleates at low-angle grain boundaries. Therefore, the ﬂow stress decreases 123 Physical-based constitutive model considering the microstructure evolution during hot working… 33 -1 Fig. 2 True stress-strain response obtained from hot compression of AZ80 magnesium alloy under a different temperatures at 0.001 s , and b different strain rates at 300 C 4.3 Dynamic recrystallization kinetics where B is a temperature-dependent parameter; q is the normalized dislocation density formulated as q ¼ 1 q =q The DRX plays an extremely important role in thermo- [28]; q is the initial dislocation density, and q is the dis- mechanical processing, affecting the ﬁnal microstructure location density during the deformation process. The q and mechanical properties of the deformed magnesium value varying from 0 to 1 represents the dislocation that alloys. The critical condition for the initiation of DRX evolves from the initial state to the saturated state. usually depends on the Zener-Hollomon parameter (Z The initial yield stress k in Eq.(3) is temperature-de- parameter) based on empirical equations [29]. The Z pendent and decreases with increasing temperature. It can parameter represents the effects of the temperature and be expressed by an Arrhenius-type equation: strain rate on the deformation behavior and can be k ¼ k expðQ =RTÞ. The parameters k and Q are the 0 k 0 k expressed as Z ¼ e_ expðQ RTÞ. The deformation activation material constant and activation energy of k, respectively. energy Q can be estimated by ﬁtting the peak stress with a hyperbolic sine function [30] 4.2 Dislocation density evolution AðÞ sinhðÞ ar ¼ e_ expðQ=RTÞ; ð6Þ During high temperature deformation, dislocation multi- -1 where e_ is the true strain rate (s ), and A, n, and a are plication and trapping by existing obstacles provide a material constants obtained from linear regression based on driving force for DRV and DRX. Considering the hot the Arrhenius-type equation [31]. Based on the regression deformation mechanisms, the evolution of the dislocation analysis of the lnðÞ sinhðÞ ar versus 1/T diagram under density can be described as [19] different strain rates and Eq. (7), the value of Q is 142.054 d d d 2 3 4 _ kJ/mol q ¼ A ð1 q Þ e_ C q ðA q Þ ðÞ 1 S S; ð5Þ jj 4 P r 3 oðÞ lnðÞ sinhðÞ ar where A , A , d , d , and d are material constants and C is Q ¼ nR : ð7Þ 3 4 2 3 4 r oðÞ 1=T a temperature-dependent parameter. The ﬁrst term repre- sents the accumulation and DRV of the dislocation density. At a constant initial grain size, the peak strain e and The DRV part limits the normalized dislocation density to critical strain e can usually be described with a power-law the saturated state of a dislocation network when q ¼ 1. function of Z (e ¼ KZ )[32], where K and m are material The second term models the static recovery during the constants. The peak strain can be obtained from the ﬂow heating process. The third term expresses the effect of stress curve. To determine the critical strain for DRX, the DRX on the evolution of the dislocation density and S is irreversible thermodynamic principle proposed by Poliak the recrystallized volume fraction. and Jonas [33] was employed o oh ¼ 0; ð8Þ or or 123 34 Z.-X. Su et al. temperature and lower strain rate are more conducive to the where h is the strain hardening rate and is determined by occurrence of DRX. or h ¼ . oe Accordingly, the critical and peak strains as functions of Based on the Poliak-Jonas criterion, the onset of DRX the Z parameter can be expressed by a linear regression in can be identiﬁed from an inﬂection point of the strain logarithmic form (see Fig. 4a) hardening rate h as a function of the ﬂow stress r. Because 3 0:113 e ¼ 2:81 10 Z ; of the following equation, ð10Þ 3 0:114 e ¼ 3:27 10 Z : oh o ln h ¼ : ð9Þ The critical strain was conﬁrmed to be a function of the or oe e_ e_ peak strain (e ﬃ 0:79e ), as shown in Fig. 4b, which is c h The lnh versus e curve should also exhibit an inﬂection consistent with the results for a wide range of materials when DRX takes place. The critical strain of DRX is then [34–36]. mathematically obtained by combining the inﬂection point When the dislocation density increases to a critical criterion of lnh versus e curve with the minimum value value, nucleation and growth of the DRX nucleus will oðÞ lnh criterion of versus e curve. This provides an intu- preferentially occur at high dislocation density zones such oe itionistic description for the analysis of the effect of the as deformation bands and grain boundaries [37]. To deformation conditions on the recrystallized critical strain, describe the evolution of DRX and predict the recrystal- as shown in Fig. 3. Figure 3 shows that the critical strain lized volume fraction, a rate equation is proposed [25] decreases with increasing deformation temperature and S ¼ Q q ðÞ xe eðÞ 1 S ðÞ 1 S ; ð11Þ 0 c decreasing strain rate, which illustrates that a higher oðÞ lnh -1 Fig. 3 Plots of lnh versus e and versus e to determine the critical strain of DRX under a, b different temperatures at 0.001 s ; c, oe d different strain rates at 300 C 123 Physical-based constitutive model considering the microstructure evolution during hot working… 35 Fig. 4 Critical and peak strains a as functions of Z parameter and b the linear ﬁtting relationship between the critical and peak strains E ¼ E expðQ =RTÞ; 0 E where d is a material constant and Q is a temperature- > 1 0 n ¼ n expðQ =RTÞ; c 0 n dependent parameter. The recrystallized volume fraction > B ¼ B expðQ =RTÞ; 0 b S varies from 0 to 1 and its variation is cyclic, depending k ¼ k expðQ =RTÞ; ð14Þ 0 k on the evolution of the dislocation density. > C ¼ C expðQ =RTÞ; r r c It has been proven experimentally that an incubation > Q ¼ Q expðQ =RTÞ; 0 10 Q time is needed for the onset of recrystallization, which can : A ¼ A expðQ =RTÞ; 0 00 A00 be expressed as [38] where R is the universal gas constant (8.314 (JK)/mol) and x ¼ AðÞ 1 x p; ð12Þ T is the absolute temperature (K). The Q in the exponential where x_ is the incubation fraction and A is a temperature- function denotes the activation energy for the correspond- dependent parameter. ing material constants. To determine the material constants within the equations, a genetic algorithm (GA)-based 4.4 Constitutive model development optimization method was developed and programmed by minimizing the residuals of the experimental and calcu- Because the microstructure evolution including hardening, lated stress-strain data. A GA toolbox in Matlab software recovery, and recrystallization mechanisms is considered, a was used to optimize the objective function and determine set of physical-based viscoplastic constitutive ISV equa- the material constants. The details of the optimization tions can be formulated for AZ80 magnesium alloy process are reported in Refs. [39, 40]. The values deter- mined for the material constants are listed in Table 1. e_ ¼ A sinhðÞ AðÞ r H k ; p 1 2 > d > _ S ¼ Q q ðÞ xe e ð1 SÞð1 SÞ ; > 0 c 4.5 Constitutive model validation x_ ¼ A ð1 xÞq ; 0:5 _ _ H ¼ 0:5Bq q ; > . > d d 4 2 d The experimental and predicted stress-strain data for dif- > 3 _ q ¼ A ð1 q Þjj e_ C q ðA q Þ ðÞ 1 S S; 4 P r 3 : ferent deformation conditions are compared in this sec- r ¼ E e e : T p -1 tion. The results for the strain rates of 0.1 and 0.01 s are ð13Þ shown in Fig. 5. Most of the experimental points are close to the predicted results, indicating a preferable prediction Under high-temperature conditions, the deformation is performance of the constitutive model. However, with temperature-dependent due to atom diffusion and disloca- increasing strain rate (see Fig.5a), the deviation increases at tion motion. The material parameters in the constitutive deformation temperatures of 300, 325 and 350 C, espe- model are treated as temperature-dependent parameters, cially in the work hardening-dynamic recovery stage. The which can be expressed as Arrhenius-type equations 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 123 36 Z.-X. Su et al. Table 1 Determined constants of the ISV viscoplastic constitutive model -1 E /MPa n (–) B /MPa k /MPa C (–) Q /(Jmol ) 0 0 0 0 r0 10 440 0.003 2.3 2.4 3.2 30 787.08 -1 -1 -1 -1 -1 -1 Q /(Jmol ) Q /(Jmol ) Q /(Jmol ) Q /(Jmol ) Q /(Jmol ) Q /(Jmol ) E n b k C Q 25 000 37 000 17 800 1000 15 000 56 622.4 -1 Q /(Jmol ) d (–) d (–) d (–) d (–) A (–) A00 1 2 3 4 00 384.56 0.99 0.85 0.8 0.99 10 A (–) A (–) A (–) A (–) 1 2 3 4 -5 7 9 10 0.32 18 14.77 -1 -1 Fig. 5 Comparison of the experimental (symbols) and the predicted (solid curves) stress-strain data at the strain rate of a 0.1 s and b 0.01 s rate is higher than the measured one. It is well known that rate is low, resulting in the deviation between the measured magnesium alloy is a type of low stacking fault energy and predicted ﬂow stresses. alloy; the dynamic recovery rate is low in the alloy because To evaluate the predictability of the proposed model, the of the reduced mobility of the dislocations. A large grain correlation coefﬁcient (R) and average absolute relative boundary migration rate is induced by high local gradients error (e ) are speciﬁed AARE of the dislocation density. Therefore, there is not enough n ðE EÞðP PÞ i i time for the transformation of subgrains to grains, espe- i¼1 -1 R ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ð15Þ cially at higher strain rates, e.g., 0.1 s [41]. On the other hand, the precipitate phase could also play an important ðE EÞ ðE EÞ i i i¼1 role in the hot deformation behavior of AZ80 magnesium alloy. At low deformation temperatures (300, 325 and 1 E P i i 350 C), which are below the temperature of complete e ¼ ; ð16Þ AARE n E i¼1 dissolution of the precipitate phase, the phase can strengthen the alloy due to the pinning effect of the grain where E and P are the experimental and predicted ﬂow i i boundaries and dislocations [42]. An excessive deforma- stress, respectively; E and P are the average values of E tion resistance is therefore induced and the experimental and P ; and n is the number of sampling data. The values of ﬂow stress shows a notable work hardening effect. With R and e were calculated to be 96.0% and 5.9%, AARE increasing temperature, the ﬂow stress becomes insensitive respectively, which indicated a good prediction accuracy of to the deformation temperature due to the dissolution of the the model. precipitate phase [2]. Thus, the actual dynamic recovery 123 Physical-based constitutive model considering the microstructure evolution during hot working… 37 detail, microstructure observations were conducted under 5 Results and discussion various Z values (corresponding to ﬁve speciﬁc deforma- The curves predicted for the dislocation density evolution tion conditions), as shown in Fig. 8. Based on the follow- ing equation, Z ¼ e_ expðQ=RTÞ, a low Z value corresponds exhibit rapid hardening to a single peak at relatively low strain and then ﬂow softening followed by a steady state at to a high temperature and low strain rate, while a high Z value reﬂects a low temperature and high strain rate. An high strain, as shown in Fig. 6. These characteristics are consistent with the variation in the ﬂow stress in Fig. 2.It inhomogeneous microstructure with an average grain size of 35 lm is characteristic for the pre-extruded alloy, as can be concluded that the low strain rate and high defor- mation temperature beneﬁt dislocation annihilation and shown in Fig. 8a. Figure 8b shows that new grains nucleate at the original grain boundaries. Subsequently, nucleation lead to a decreasing dislocation density and increasing occurs at the growing grains, forming a necklace-type mobile dislocations. During the hot deformation, many of structure of grains. In this case, which corresponds to high the dislocations are concentrated on the grain boundaries in Z conditions and low recrystallization kinetics, the nucle- the slip process. With increasing dislocation density near the grain boundaries, DRX takes place in the local area and ation of the recrystallized grains starts at a very high strain, leading to a notably incomplete recrystallized microstruc- then evolves into the interior of the grains. Consequently, both the dislocation density and ﬂow stress are reduced by ture. At a low Z value with high recrystallization kinetics and mobility of grain boundaries, new grains bulge from dislocation rearrangement and annihilation. The curves predicted for the recrystallized volume the preexisting grain boundaries and grow quickly and adequately, forming a completely recrystallized fraction increase with the strain in a sigmoidal manner, as microstructure (see Fig. 8f). shown in Figs. 7a–c. The points marked as (1)–(5) repre- The statistical data of the recrystallized volume fraction sent the experimental data obtained from microstructure under different Z values are presented in Fig. 7d. For a observations in Fig. 8 and are in good agreement with the given strain of 0.916, the recrystallized volume fraction is predicted results. It is notable that the recrystallized vol- less than 90% at high Z values, corresponding to an ume fraction increases with increasing temperature at a given strain rate, while it decreases with increasing strain incomplete recrystallized microstructure (see Figs. 8b and c). On the other hand, full evolution of the recrystallized rate at a constant temperature. This is attributed to the higher mobility at grain boundaries at higher temperature microstructure occurs at low Z values (see Figs. 8d–f); an almost 100% recrystallized volume fraction is shown in and lower strain rate, which beneﬁts the activation of DRX [43]. After the deformation reaches a critical strain, DRX Fig. 7d. Based on the statistical analysis, it can be con- cluded that the recrystallized volume fraction increases occurs and the recrystallized volume fraction increases quickly with increasing strain until a considerable fraction with decreasing Z value for a given true strain, which can be conﬁrmed by the predicted recrystallized kinetics shown of the microstructure is recrystallized. in Figs. 7a–c. Similar features of the DRX microstructure As stated in our previous research [44], there is a close relationship between the recrystallized microstructure during hot deformation were observed for other magnesium alloys [45, 46]. evolution and Z parameter. To explain the DRX kinetics in -1 Fig. 6 Evolution of the normalized dislocation density under a different temperatures at 0.001 s , and b different strain rates at 300 C 123 38 Z.-X. Su et al. -1 -1 -1 Fig. 7 Recrystallized volume fraction at the strain rate of a 0.1 s , b 0.01 s and c 0.001 s (the curves represent the predicted values of the model, and the points marked with (1)–(5) represent the experimental values corresponding to the microstructures in Figs. 8b–f, respectively) d the statistical recrystallized volume fraction and the average grain size as a function of the logarithm of Z parameter Another notable phenomenon under different Z condi- which is plotted as vertical blue dashed line in Fig. 7d. It tions is the statistical result for the grain size distribution in can thus be concluded that the low strain rate contributes to the ﬁnal microstructure (see Fig. 7d). Under high Z condi- high recrystallization kinetics of 99.4%. The grains can tions with a high strain rate effect, work hardening is sig- maintain a small size under the low temperature, which is niﬁcant and the initiation of the recrystallization process is an optimum state for the hot working process. To achieve delayed. Consequently, the stored deformation energy is the desired ﬁnal properties with an optimum microstructure mostly used for the formation of new recrystallized grains (higher volume fraction of DRX grains and lower average and the average grain size becomes small and limited grain size), a predetermined deformation temperature and because there is no sufﬁcient time for grain growth, which strain rate should be considered. results in an inhomogeneous microstructure with an aver- age grain size of 8.2 lm (see Fig. 8c). Under low Z con- ditions with a high temperature effect, some grains evolve 6 Conclusions quickly due to grain growth and grain impingement, which is consistent with an inhomogeneous microstructure with (i) The DRX is the dominant softening mechanism in an average grain size of 15.2 lm (see Fig. 8e). Figure 8d AZ80 magnesium alloy during the hot deformation shows a homogeneous and reﬁned microstructure with an process and its ﬂow response exhibits a single peak average grain size of 7.5 lm. Deformation conditions of followed by steady-state ﬂow. The critical DRX -1 300 C/0.001 s correspond to a low lnZ value of 22.9, strain was determined as a function of the peak strain 123 Physical-based constitutive model considering the microstructure evolution during hot working… 39 Fig. 8 Microstructures of a the pre-extruded state and the hot deformed state with a strain of 0.916 corresponding to the high Z value of -1 -1 -1 -1 -1 b 300 C/0.1 s and c 325 C/0.01 s , the low Z value of d 300 C/0.001 s , e 400 C/0.01 s and f 400 C/0.001 s based on the Poliak-Jonas criterion, which shows a including the normalized dislocation density, recrys- power-law function of the Z parameter. A low tallized volume fraction, and macroscopic variable Z value with high temperature and low strain rate is Z parameter. Consequently, the viscoplastic ﬂow favorable for the occurrence of DRX. behavior and microstructure evolution during the hot (ii) A physical-based constitutive model was developed working process can be characterized. in consideration of work hardening, recovery, and (iii) The stress-strain data predicted by the constitutive DRX. The proposed model was built with ISVs model show a good correlation with the 123 40 Z.-X. Su et al. 11. He A, Xie GL, Zhang HL et al (2014) A modiﬁed Zerilli-Arm- experimental results. Statistical microstructure strong constitutive model to predict hot deformation behavior of observations indicate an increase in the recrystal- 20CrMo alloy steel. Mater Des 56:122–127 lized volume fraction with decreasing Z parameter, 12. Momeni A, Ebrahimi GR, Jahazi M et al (2014) Microstructure which is consistent with the predicted recrystallized evolution at the onset of discontinuous dynamic recrystallization: a physics-based model of subgrain critical size. J Alloy Compd kinetics data. The model shows a good predictability 587:199–210 in describing the hot deformation behavior and 13. Liu LF, Ding HL (2009) Study of the plastic ﬂow behaviors of microstructure evolution of AZ80 magnesium alloy. AZ91 magnesium alloy during thermomechanical processes. J Alloy Compd 484(1–2):949–956 14. Sabokpa O, Zarei-Hanzaki A, Abedi HR et al (2012) Artiﬁcial Acknowledgements The work is ﬁnancially supported by the Natural neural network modeling to predict the high temperature ﬂow Science Foundation of Beijing Municipality (Grant No. 3182025), the behavior of an AZ81 magnesium alloy. Mater Des 39:390–396 National Natural Science Foundation of China (Grant No. 15. Li HY, Wang XF, Wei DD et al (2012) A comparative study on U1730121), the Joint Foundation (general) Project of the Equipment modiﬁed Zerilli-Armstrong, Arrhenius-type and artiﬁcial neural Pre-research of the Ministry of Education (Grant No. 6141A020221) network models to predict high-temperature deformation behav- and the Postdoctoral Science Foundation of China (Grant No. ior in T24 steel. Mater Sci Eng A 536:216–222 2017M620609). 16. Grong Ø, Shercliff HR (2002) Microstructural modelling in metals processing. Prog Mater Sci 47(2):163–282 Open Access This article is distributed under the terms of the 17. Vilamosa V, Clausen AH, Børvik T et al (2016) A physically- Creative Commons Attribution 4.0 International License (http://crea based constitutive model applied to AA6082 aluminium alloy at tivecommons.org/licenses/by/4.0/), which permits unrestricted use, large strains, high strain rates and elevated temperatures. Mater distribution, and reproduction in any medium, provided you give Des 103:391–405 appropriate credit to the original author(s) and the source, provide a 18. Austin RA, Mcdowell DL (2011) A dislocation-based constitu- link to the Creative Commons license, and indicate if changes were tive model for viscoplastic deformation of fcc metals at very high made. strain rates. Int J Plast 27(1):1–24 19. Lin J, Liu Y, Farrugia DCJ et al (2005) Development of dislo- cation-based uniﬁed material model for simulating microstructure References evolution in multipass hot rolling. Philos Mag 85(18):1967–1987 20. Standard practice for compression tests of metallic materials at elevated temperatures with conventional or rapid heating rates 1. Yu ZW, Tang AT, He JJ et al (2018) Effect of high content of and strain rates, astm. https://www.astm.org/DATABASE.CART/ manganese on microstructure, texture and mechanical properties STD_REFERENCE/E9.htm. Accessed 23 May 2018 of magnesium alloy. 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Lin YC, Chen XM, Wen DX et al (2014) A physically-based Dynamic recrystallization in AZ31 magnesium alloy. Mater Sci constitutive model for a typical nickel-based superalloy. Comput Eng A 456(1):52–57 Mater Sci 83(2):282–289 46. Xu Y, Hu LX, Sun Y (2013) Deformation behaviour and dynamic 38. Tang XF, Wang BY, Zhang N et al (2015) Modeling of recrystallization of AZ61 magnesium alloy. J Alloy Compd microstructural evolution and ﬂow behavior of superalloy IN718 580(8):262–269 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Manufacturing Springer Journals http://www.deepdyve.com/lp/springer-journals/physical-based-constitutive-model-considering-the-microstructure-0cp20GE4Az

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Publisher: Springer Journals
Copyright: Copyright © 2018 by The Author(s)
Subject: Engineering; Manufacturing, Machines, Tools, Processes; Control, Robotics, Mechatronics; Nanotechnology and Microengineering
ISSN: 2095-3127
eISSN: 2195-3597
DOI: 10.1007/s40436-018-0243-8
Publisher site: See Article on Publisher Site

Abstract

Adv. Manuf. (2019) 7:30–41 https://doi.org/10.1007/s40436-018-0243-8 Physical-based constitutive model considering the microstructure evolution during hot working of AZ80 magnesium alloy 1,2 1,2 3 1,2 • • • Ze-Xing Su Chao-Yang Sun Ming-Wang Fu Ling-Yun Qian Received: 23 May 2018 / Accepted: 7 December 2018 / Published online: 2 January 2019 The Author(s) 2018 Abstract A physical-based constitutive model was devel- Keywords AZ80 magnesium alloy Hot deformation oped to model the viscoplastic ﬂow behavior and Constitutive model Microstructure evolution Dynamic microstructure evolution of AZ80 magnesium alloy during recrystallization (DRX) the hot working process. The competing deformation mechanisms, including work hardening, dynamic recovery, and dynamic recrystallization, in an isothermal compres- 1 Introduction sion environment were considered in the model. The internal state variables, including the normalized disloca- Magnesium alloy is one of the most promising lightweight tion density and recrystallized volume fraction, were metallic materials for structural applications in the auto- incorporated into the model to articulate the microstructure motive and aerospace industries owing to its low density, evolution during hot deformation. The kinetic condition high speciﬁc strength, and good recyclability [1, 2]. critical for dynamic recrystallization, considering the However, the slip systems of the alloy are poor at room effects of the deformation temperature and strain rate, was temperature due to its hexagonal close-packed (HCP) obtained by employing both the Poliak-Jonas criterion and structure, which limits its deformation and manufactura- Zener-Hollomon parameter. Microstructure observations bility and thus its application in the industry [3, 4]. To indicate that the recrystallized volume fraction increases improve the workability of the alloy, plastic deformation, with decreasing Z parameter at constant strain, which is such as extrusion and forging, is usually conducted at consistent with the predicted kinetics model. Based on the elevated temperatures, which increases the number of slip developed model, a good correlation was also obtained planes and decreases the critical shear stress of non-basal between the predicted and experimental ﬂow stress. The plane slip [5]. During the hot deformation of magnesium results indicate a good predictability of the model in alloy, dynamic recrystallization (DRX) predominates describing the hot deformation behavior and microstructure because of its low stacking fault energy (60–78 mJ/m ), evolution of AZ80 magnesium alloy. which is an effective way to obtain a reﬁned microstructure and improve the mechanics performance [6]. The under- standing of the ﬂow behavior and metallurgical phenomena during hot deformation is of scientiﬁc signiﬁcance for the & Chao-Yang Sun suncy@ustb.edu.cn optimization of the hot workability and guiding forming process of magnesium alloy. School of Mechanical Engineering, University of Science and A constitutive model is essential for the description of Technology Beijing, Beijing 100083, People’s Republic of the ﬂow characteristics of materials, which could be China combined with a ﬁnite element method to provide an Beijing Key Laboratory of Lightweight Metal Forming, efﬁcient computational platform for the prediction of the Beijing 100083, People’s Republic of China mechanical response. Over the past decades, numerous Department of Mechanical Engineering, The Hong Kong constitutive models have been established such as phe- Polytechnic University, Hung Hom, Hong Kong, People’s nomenological models, e.g., Arrhenius-type model and Republic of China 123 Physical-based constitutive model considering the microstructure evolution during hot working… 31 Johnson-Cook model [7, 8], advanced statistical models, alloy to describe the ﬂow behavior and microstructure e.g., artiﬁcial neural network model [9, 10], and physical- evolution during the hot working process. based internal state variable (ISV) models, e.g., Zerilli- At present, many constitutive models, which are avail- Armstrong model and Kocks-Mecking model [11, 12]. By able for magnesium alloys, are generally phenomenologi- modeling of the deformation process, Liu and Ding [13] cally constructed by introducing a few macroscopic established an Arrhenius-type constitutive model for AZ91 variables to the model and formulated as functions of the magnesium alloy to describe the hot deformation behavior temperature, strain rate and strain. They are basically for a wide range of temperatures and strain rates. Sabokpa experimental and based on classical viscoplastic theory to et al. [14] developed an artiﬁcial neural network (ANN) describe work hardening and thermal softening phenomena model for the representation of the hot compression during the hot working process. However, the models lack behavior of AZ81 magnesium alloy and compared it with an in-depth understanding of the underlying physics. In the Arrhenius-type model. The results indicated that the view of the important role of DRX in the grain reﬁnement trained ANN model was more accurate in predicting the and improvement of mechanical properties during the hot ﬂow stress. Furthermore, Li et al. [15] conducted a com- working of magnesium alloy, the establishment of a con- parative study on Arrhenius-type, ANN, and modiﬁed stitutive model that can reﬂect the evolution of the Zerilli-Armstrong models to predict the hot deformation recrystallized microstructure is urgently needed. In this behavior of T24 steel. The ANN model showed the most study, a physical-based constitutive model was established excellent predictability. However, it was pointed out that by introducing ISVs including the normalized dislocation the ANN model strongly depended on extensive high- density and recrystallized volume fraction. To provide an quality data and characteristic variables did not provide adequate description of the recrystallization kinetics, the physical insights. Poliak-Jonas criterion and the Zener-Hollomon parameter Among the different models, the phenomenological were employed because they represented the microstruc- model is currently the most widely used one due to the few ture evolution, which was simultaneously affected by the parameters and simplicity in determining the parameters by deformation temperature and strain rate. The model was regression analysis. However, the model presents a poor veriﬁed using the experimental ﬂow stress and recrystal- predictability in determining deformation parameters that lized volume fraction and a good agreement was observed. are not in the experimental range. The advanced statistical The proposed constitutive model can be used to analyze the model is the most accurate in predicting the ﬂow behavior hot deformation behavior and microstructure evolution of of a wide range of deformation conditions. However, the AZ80 magnesium alloy. model does not have an intrinsic physics meaning and cannot reﬂect the microstructure evolution during the deformation. Compared with the above-mentioned models, 2 Experimental procedures the physical-based ISV approach has become more important in modeling various ﬂow behaviors and The as-received AZ80 magnesium alloy is an extruded microstructure evolution over the past decades. billet with a diameter of 200 mm and a height of 150 mm. The constitutive ISV model is capable of capturing The chemical composition (mass fraction) of the alloy is: inherent microstructural complexity during the working 8.16% Al, 0.42% Zn, 0.03% Mn, 0.01% Si, 0.005% Fe, process and achieving a reasonable accuracy, which can be 0.001% Cu, 0.001% Ni, and balance Mg. According to the used in computational simulation for process optimization standard of ASTM E209 [20] for compression tests of [16]. Vilamosa et al. [17] established a physical-based metallic materials at elevated temperatures with conven- constitutive model for AA6082 aluminum alloy using the tional or rapid heating rates and strain rates, cylindrical dislocation density as an internal variable without relating specimens with a diameter of 8 mm and height of 12 mm it to microstructure evolution. Austin and Mcdowell [18] were machined from the extruded bar along the extrusion proposed a dislocation-based constitutive model for the direction. Two end surfaces of the specimens were ground viscoplastic deformation of fcc metals at high strain rates to reduce the effect of friction on the hot deformation by introducing path-dependent differential equations for behavior. A uniaxial hot compression test was conducted mobile and immobile dislocation densities. Lin et al. [19] with a Gleeble-1500D thermomechanical simulator; the developed a dislocation-based uniﬁed viscoplastic consti- schematic is illustrated in Fig.1. Five deformation tem- tutive model for C–Mn steel to predict the dislocation peratures (300, 325, 350, 375 and 400 C) and three strain -1 density, recrystallization, and grain size evolution during rates (0.001, 0.01 and 0.1 s ) were used and the height multi-pass hot rolling processing. The aforementioned reduction was 60%, corresponding to a true strain of 0.916. study is a good reference and basis for establishing the Before the compression, the top and bottom surfaces of the physical-based constitutive model for AZ80 magnesium specimens were coated with graphite lubricant and 123 32 Z.-X. Su et al. 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 Fig. 1 Schematic illustration of isothermal compression tests for the decrease in the ﬂow stress. It can be concluded that the AZ80 magnesium alloy dynamic softening stage starts at a lower strain and ﬂow softening more likely occurs at an increased deformation tantalum foil was placed between the specimen and anvil to temperature and decreased strain rate because both the reduce the friction and avoid adhesion. For the measure- abundant energy and sufﬁcient time affect the thermal ment and feedback control of the real-time temperature, activation process including DRV and DRX. K-type thermocouples were welded in the center of spec- imens. The specimens were heated to the test deformation temperature with a heating rate of 5 C/s and then held 4 Constitutive ISV model isothermally for 180 s to eliminate the thermal gradient prior to the deformation. Stress-strain data were automat- 4.1 Flow stress decomposition ically recorded by the testing system during the hot deformation process. After the compression, the specimens According to Hooke’s law, the stress-strain relationship of were immediately water-quenched to retain the deforma- materials can be expressed as [21] tion microstructure. r ¼ E e e ; ð1Þ T p The deformed specimens were sectioned along the compression axis for microstructural analysis using a wire- where r is the overall ﬂow stress; E is the Young’ modulus, electric discharge machine (EDM). The sectioned speci- which is a temperature-dependent parameter; and e and e T p mens were mechanically ground, polished, and then etched represent the total strain and plastic strain, respectively. for 5–12 s using a solution consisting of 5 mL picric acid, Because the material exhibits viscoplastic ﬂow during 5 mL acetic acid, 100 mL ethyl alcohol, and 10 mL dis- the hot working process, the overall ﬂow stress can be tilled water. Microstructure observations were made with a decomposed into three parts: initial yield stress, vis- DM4000M optical microscope. The average grain size and coplastic stress, and hardening stress [22] recrystallized volume fraction were examined by quanti- r ¼ k þ r þ H; ð2Þ tative metallography. where k is the initial yield stress, r the viscoplastic stress, and H the isotropic hardening stress. A hyperbolic sine law 3 Experimental results and analysis is introduced to describe the viscoplastic ﬂow, which is suitable for a wide range of temperatures and strain rates The true compressive stress-strain response of AZ80 [23, 24] magnesium alloy is shown in Fig. 2. Despite the difference e ¼ A sinhðÞ AðÞ r H k ; ð3Þ p 1 2 in the speciﬁc ﬂow stress, the ﬂow response presents a peak stress with subsequent ﬂow softening, which is a typical where e_ is the plastic strain rate; n is the viscoplastic p c DRX characteristic. In general, the variation in the ﬂow exponent, which is temperature-dependent; and A and A 1 2 stress reﬂects the competition between work hardening are material constants. (WH) and dynamic softening during hot deformation. The The hardening stress H in Eq.(3) has a close relationship ﬂow stress ﬁrstly rapidly increases due to dislocation tan- with the reciprocal of the average slip length over which a gling and pileups and dynamic recovery (DRV) simulta- dislocation can run and the mean slip length is determined neously arises through dislocation climbing and gliding, by the inverse square of the dislocation density [25]. The which results in the formation of mobile subgrain bound- isotropic hardening rate can thus be deﬁned as [26, 27] aries and the annihilation of dislocations. As the dislocation 0:5 _ _ H ¼ 0:5Bq q ; ð4Þ density exceeds a threshold, DRX nucleates at low-angle grain boundaries. Therefore, the ﬂow stress decreases 123 Physical-based constitutive model considering the microstructure evolution during hot working… 33 -1 Fig. 2 True stress-strain response obtained from hot compression of AZ80 magnesium alloy under a different temperatures at 0.001 s , and b different strain rates at 300 C 4.3 Dynamic recrystallization kinetics where B is a temperature-dependent parameter; q is the normalized dislocation density formulated as q ¼ 1 q =q The DRX plays an extremely important role in thermo- [28]; q is the initial dislocation density, and q is the dis- mechanical processing, affecting the ﬁnal microstructure location density during the deformation process. The q and mechanical properties of the deformed magnesium value varying from 0 to 1 represents the dislocation that alloys. The critical condition for the initiation of DRX evolves from the initial state to the saturated state. usually depends on the Zener-Hollomon parameter (Z The initial yield stress k in Eq.(3) is temperature-de- parameter) based on empirical equations [29]. The Z pendent and decreases with increasing temperature. It can parameter represents the effects of the temperature and be expressed by an Arrhenius-type equation: strain rate on the deformation behavior and can be k ¼ k expðQ =RTÞ. The parameters k and Q are the 0 k 0 k expressed as Z ¼ e_ expðQ RTÞ. The deformation activation material constant and activation energy of k, respectively. energy Q can be estimated by ﬁtting the peak stress with a hyperbolic sine function [30] 4.2 Dislocation density evolution AðÞ sinhðÞ ar ¼ e_ expðQ=RTÞ; ð6Þ During high temperature deformation, dislocation multi- -1 where e_ is the true strain rate (s ), and A, n, and a are plication and trapping by existing obstacles provide a material constants obtained from linear regression based on driving force for DRV and DRX. Considering the hot the Arrhenius-type equation [31]. Based on the regression deformation mechanisms, the evolution of the dislocation analysis of the lnðÞ sinhðÞ ar versus 1/T diagram under density can be described as [19] different strain rates and Eq. (7), the value of Q is 142.054 d d d 2 3 4 _ kJ/mol q ¼ A ð1 q Þ e_ C q ðA q Þ ðÞ 1 S S; ð5Þ jj 4 P r 3 oðÞ lnðÞ sinhðÞ ar where A , A , d , d , and d are material constants and C is Q ¼ nR : ð7Þ 3 4 2 3 4 r oðÞ 1=T a temperature-dependent parameter. The ﬁrst term repre- sents the accumulation and DRV of the dislocation density. At a constant initial grain size, the peak strain e and The DRV part limits the normalized dislocation density to critical strain e can usually be described with a power-law the saturated state of a dislocation network when q ¼ 1. function of Z (e ¼ KZ )[32], where K and m are material The second term models the static recovery during the constants. The peak strain can be obtained from the ﬂow heating process. The third term expresses the effect of stress curve. To determine the critical strain for DRX, the DRX on the evolution of the dislocation density and S is irreversible thermodynamic principle proposed by Poliak the recrystallized volume fraction. and Jonas [33] was employed o oh ¼ 0; ð8Þ or or 123 34 Z.-X. Su et al. temperature and lower strain rate are more conducive to the where h is the strain hardening rate and is determined by occurrence of DRX. or h ¼ . oe Accordingly, the critical and peak strains as functions of Based on the Poliak-Jonas criterion, the onset of DRX the Z parameter can be expressed by a linear regression in can be identiﬁed from an inﬂection point of the strain logarithmic form (see Fig. 4a) hardening rate h as a function of the ﬂow stress r. Because 3 0:113 e ¼ 2:81 10 Z ; of the following equation, ð10Þ 3 0:114 e ¼ 3:27 10 Z : oh o ln h ¼ : ð9Þ The critical strain was conﬁrmed to be a function of the or oe e_ e_ peak strain (e ﬃ 0:79e ), as shown in Fig. 4b, which is c h The lnh versus e curve should also exhibit an inﬂection consistent with the results for a wide range of materials when DRX takes place. The critical strain of DRX is then [34–36]. mathematically obtained by combining the inﬂection point When the dislocation density increases to a critical criterion of lnh versus e curve with the minimum value value, nucleation and growth of the DRX nucleus will oðÞ lnh criterion of versus e curve. This provides an intu- preferentially occur at high dislocation density zones such oe itionistic description for the analysis of the effect of the as deformation bands and grain boundaries [37]. To deformation conditions on the recrystallized critical strain, describe the evolution of DRX and predict the recrystal- as shown in Fig. 3. Figure 3 shows that the critical strain lized volume fraction, a rate equation is proposed [25] decreases with increasing deformation temperature and S ¼ Q q ðÞ xe eðÞ 1 S ðÞ 1 S ; ð11Þ 0 c decreasing strain rate, which illustrates that a higher oðÞ lnh -1 Fig. 3 Plots of lnh versus e and versus e to determine the critical strain of DRX under a, b different temperatures at 0.001 s ; c, oe d different strain rates at 300 C 123 Physical-based constitutive model considering the microstructure evolution during hot working… 35 Fig. 4 Critical and peak strains a as functions of Z parameter and b the linear ﬁtting relationship between the critical and peak strains E ¼ E expðQ =RTÞ; 0 E where d is a material constant and Q is a temperature- > 1 0 n ¼ n expðQ =RTÞ; c 0 n dependent parameter. The recrystallized volume fraction > B ¼ B expðQ =RTÞ; 0 b S varies from 0 to 1 and its variation is cyclic, depending k ¼ k expðQ =RTÞ; ð14Þ 0 k on the evolution of the dislocation density. > C ¼ C expðQ =RTÞ; r r c It has been proven experimentally that an incubation > Q ¼ Q expðQ =RTÞ; 0 10 Q time is needed for the onset of recrystallization, which can : A ¼ A expðQ =RTÞ; 0 00 A00 be expressed as [38] where R is the universal gas constant (8.314 (JK)/mol) and x ¼ AðÞ 1 x p; ð12Þ T is the absolute temperature (K). The Q in the exponential where x_ is the incubation fraction and A is a temperature- function denotes the activation energy for the correspond- dependent parameter. ing material constants. To determine the material constants within the equations, a genetic algorithm (GA)-based 4.4 Constitutive model development optimization method was developed and programmed by minimizing the residuals of the experimental and calcu- Because the microstructure evolution including hardening, lated stress-strain data. A GA toolbox in Matlab software recovery, and recrystallization mechanisms is considered, a was used to optimize the objective function and determine set of physical-based viscoplastic constitutive ISV equa- the material constants. The details of the optimization tions can be formulated for AZ80 magnesium alloy process are reported in Refs. [39, 40]. The values deter- mined for the material constants are listed in Table 1. e_ ¼ A sinhðÞ AðÞ r H k ; p 1 2 > d > _ S ¼ Q q ðÞ xe e ð1 SÞð1 SÞ ; > 0 c 4.5 Constitutive model validation x_ ¼ A ð1 xÞq ; 0:5 _ _ H ¼ 0:5Bq q ; > . > d d 4 2 d The experimental and predicted stress-strain data for dif- > 3 _ q ¼ A ð1 q Þjj e_ C q ðA q Þ ðÞ 1 S S; 4 P r 3 : ferent deformation conditions are compared in this sec- r ¼ E e e : T p -1 tion. The results for the strain rates of 0.1 and 0.01 s are ð13Þ shown in Fig. 5. Most of the experimental points are close to the predicted results, indicating a preferable prediction Under high-temperature conditions, the deformation is performance of the constitutive model. However, with temperature-dependent due to atom diffusion and disloca- increasing strain rate (see Fig.5a), the deviation increases at tion motion. The material parameters in the constitutive deformation temperatures of 300, 325 and 350 C, espe- model are treated as temperature-dependent parameters, cially in the work hardening-dynamic recovery stage. The which can be expressed as Arrhenius-type equations 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 123 36 Z.-X. Su et al. Table 1 Determined constants of the ISV viscoplastic constitutive model -1 E /MPa n (–) B /MPa k /MPa C (–) Q /(Jmol ) 0 0 0 0 r0 10 440 0.003 2.3 2.4 3.2 30 787.08 -1 -1 -1 -1 -1 -1 Q /(Jmol ) Q /(Jmol ) Q /(Jmol ) Q /(Jmol ) Q /(Jmol ) Q /(Jmol ) E n b k C Q 25 000 37 000 17 800 1000 15 000 56 622.4 -1 Q /(Jmol ) d (–) d (–) d (–) d (–) A (–) A00 1 2 3 4 00 384.56 0.99 0.85 0.8 0.99 10 A (–) A (–) A (–) A (–) 1 2 3 4 -5 7 9 10 0.32 18 14.77 -1 -1 Fig. 5 Comparison of the experimental (symbols) and the predicted (solid curves) stress-strain data at the strain rate of a 0.1 s and b 0.01 s rate is higher than the measured one. It is well known that rate is low, resulting in the deviation between the measured magnesium alloy is a type of low stacking fault energy and predicted ﬂow stresses. alloy; the dynamic recovery rate is low in the alloy because To evaluate the predictability of the proposed model, the of the reduced mobility of the dislocations. A large grain correlation coefﬁcient (R) and average absolute relative boundary migration rate is induced by high local gradients error (e ) are speciﬁed AARE of the dislocation density. Therefore, there is not enough n ðE EÞðP PÞ i i time for the transformation of subgrains to grains, espe- i¼1 -1 R ¼ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ð15Þ cially at higher strain rates, e.g., 0.1 s [41]. On the other hand, the precipitate phase could also play an important ðE EÞ ðE EÞ i i i¼1 role in the hot deformation behavior of AZ80 magnesium alloy. At low deformation temperatures (300, 325 and 1 E P i i 350 C), which are below the temperature of complete e ¼ ; ð16Þ AARE n E i¼1 dissolution of the precipitate phase, the phase can strengthen the alloy due to the pinning effect of the grain where E and P are the experimental and predicted ﬂow i i boundaries and dislocations [42]. An excessive deforma- stress, respectively; E and P are the average values of E tion resistance is therefore induced and the experimental and P ; and n is the number of sampling data. The values of ﬂow stress shows a notable work hardening effect. With R and e were calculated to be 96.0% and 5.9%, AARE increasing temperature, the ﬂow stress becomes insensitive respectively, which indicated a good prediction accuracy of to the deformation temperature due to the dissolution of the the model. precipitate phase [2]. Thus, the actual dynamic recovery 123 Physical-based constitutive model considering the microstructure evolution during hot working… 37 detail, microstructure observations were conducted under 5 Results and discussion various Z values (corresponding to ﬁve speciﬁc deforma- The curves predicted for the dislocation density evolution tion conditions), as shown in Fig. 8. Based on the follow- ing equation, Z ¼ e_ expðQ=RTÞ, a low Z value corresponds exhibit rapid hardening to a single peak at relatively low strain and then ﬂow softening followed by a steady state at to a high temperature and low strain rate, while a high Z value reﬂects a low temperature and high strain rate. An high strain, as shown in Fig. 6. These characteristics are consistent with the variation in the ﬂow stress in Fig. 2.It inhomogeneous microstructure with an average grain size of 35 lm is characteristic for the pre-extruded alloy, as can be concluded that the low strain rate and high defor- mation temperature beneﬁt dislocation annihilation and shown in Fig. 8a. Figure 8b shows that new grains nucleate at the original grain boundaries. Subsequently, nucleation lead to a decreasing dislocation density and increasing occurs at the growing grains, forming a necklace-type mobile dislocations. During the hot deformation, many of structure of grains. In this case, which corresponds to high the dislocations are concentrated on the grain boundaries in Z conditions and low recrystallization kinetics, the nucle- the slip process. With increasing dislocation density near the grain boundaries, DRX takes place in the local area and ation of the recrystallized grains starts at a very high strain, leading to a notably incomplete recrystallized microstruc- then evolves into the interior of the grains. Consequently, both the dislocation density and ﬂow stress are reduced by ture. At a low Z value with high recrystallization kinetics and mobility of grain boundaries, new grains bulge from dislocation rearrangement and annihilation. The curves predicted for the recrystallized volume the preexisting grain boundaries and grow quickly and adequately, forming a completely recrystallized fraction increase with the strain in a sigmoidal manner, as microstructure (see Fig. 8f). shown in Figs. 7a–c. The points marked as (1)–(5) repre- The statistical data of the recrystallized volume fraction sent the experimental data obtained from microstructure under different Z values are presented in Fig. 7d. For a observations in Fig. 8 and are in good agreement with the given strain of 0.916, the recrystallized volume fraction is predicted results. It is notable that the recrystallized vol- less than 90% at high Z values, corresponding to an ume fraction increases with increasing temperature at a given strain rate, while it decreases with increasing strain incomplete recrystallized microstructure (see Figs. 8b and c). On the other hand, full evolution of the recrystallized rate at a constant temperature. This is attributed to the higher mobility at grain boundaries at higher temperature microstructure occurs at low Z values (see Figs. 8d–f); an almost 100% recrystallized volume fraction is shown in and lower strain rate, which beneﬁts the activation of DRX [43]. After the deformation reaches a critical strain, DRX Fig. 7d. Based on the statistical analysis, it can be con- cluded that the recrystallized volume fraction increases occurs and the recrystallized volume fraction increases quickly with increasing strain until a considerable fraction with decreasing Z value for a given true strain, which can be conﬁrmed by the predicted recrystallized kinetics shown of the microstructure is recrystallized. in Figs. 7a–c. Similar features of the DRX microstructure As stated in our previous research [44], there is a close relationship between the recrystallized microstructure during hot deformation were observed for other magnesium alloys [45, 46]. evolution and Z parameter. To explain the DRX kinetics in -1 Fig. 6 Evolution of the normalized dislocation density under a different temperatures at 0.001 s , and b different strain rates at 300 C 123 38 Z.-X. Su et al. -1 -1 -1 Fig. 7 Recrystallized volume fraction at the strain rate of a 0.1 s , b 0.01 s and c 0.001 s (the curves represent the predicted values of the model, and the points marked with (1)–(5) represent the experimental values corresponding to the microstructures in Figs. 8b–f, respectively) d the statistical recrystallized volume fraction and the average grain size as a function of the logarithm of Z parameter Another notable phenomenon under different Z condi- which is plotted as vertical blue dashed line in Fig. 7d. It tions is the statistical result for the grain size distribution in can thus be concluded that the low strain rate contributes to the ﬁnal microstructure (see Fig. 7d). Under high Z condi- high recrystallization kinetics of 99.4%. The grains can tions with a high strain rate effect, work hardening is sig- maintain a small size under the low temperature, which is niﬁcant and the initiation of the recrystallization process is an optimum state for the hot working process. To achieve delayed. Consequently, the stored deformation energy is the desired ﬁnal properties with an optimum microstructure mostly used for the formation of new recrystallized grains (higher volume fraction of DRX grains and lower average and the average grain size becomes small and limited grain size), a predetermined deformation temperature and because there is no sufﬁcient time for grain growth, which strain rate should be considered. results in an inhomogeneous microstructure with an aver- age grain size of 8.2 lm (see Fig. 8c). Under low Z con- ditions with a high temperature effect, some grains evolve 6 Conclusions quickly due to grain growth and grain impingement, which is consistent with an inhomogeneous microstructure with (i) The DRX is the dominant softening mechanism in an average grain size of 15.2 lm (see Fig. 8e). Figure 8d AZ80 magnesium alloy during the hot deformation shows a homogeneous and reﬁned microstructure with an process and its ﬂow response exhibits a single peak average grain size of 7.5 lm. Deformation conditions of followed by steady-state ﬂow. The critical DRX -1 300 C/0.001 s correspond to a low lnZ value of 22.9, strain was determined as a function of the peak strain 123 Physical-based constitutive model considering the microstructure evolution during hot working… 39 Fig. 8 Microstructures of a the pre-extruded state and the hot deformed state with a strain of 0.916 corresponding to the high Z value of -1 -1 -1 -1 -1 b 300 C/0.1 s and c 325 C/0.01 s , the low Z value of d 300 C/0.001 s , e 400 C/0.01 s and f 400 C/0.001 s based on the Poliak-Jonas criterion, which shows a including the normalized dislocation density, recrys- power-law function of the Z parameter. A low tallized volume fraction, and macroscopic variable Z value with high temperature and low strain rate is Z parameter. Consequently, the viscoplastic ﬂow favorable for the occurrence of DRX. behavior and microstructure evolution during the hot (ii) A physical-based constitutive model was developed working process can be characterized. in consideration of work hardening, recovery, and (iii) The stress-strain data predicted by the constitutive DRX. The proposed model was built with ISVs model show a good correlation with the 123 40 Z.-X. Su et al. 11. He A, Xie GL, Zhang HL et al (2014) A modiﬁed Zerilli-Arm- experimental results. Statistical microstructure strong constitutive model to predict hot deformation behavior of observations indicate an increase in the recrystal- 20CrMo alloy steel. Mater Des 56:122–127 lized volume fraction with decreasing Z parameter, 12. Momeni A, Ebrahimi GR, Jahazi M et al (2014) Microstructure which is consistent with the predicted recrystallized evolution at the onset of discontinuous dynamic recrystallization: a physics-based model of subgrain critical size. J Alloy Compd kinetics data. The model shows a good predictability 587:199–210 in describing the hot deformation behavior and 13. Liu LF, Ding HL (2009) Study of the plastic ﬂow behaviors of microstructure evolution of AZ80 magnesium alloy. AZ91 magnesium alloy during thermomechanical processes. J Alloy Compd 484(1–2):949–956 14. Sabokpa O, Zarei-Hanzaki A, Abedi HR et al (2012) Artiﬁcial Acknowledgements The work is ﬁnancially supported by the Natural neural network modeling to predict the high temperature ﬂow Science Foundation of Beijing Municipality (Grant No. 3182025), the behavior of an AZ81 magnesium alloy. Mater Des 39:390–396 National Natural Science Foundation of China (Grant No. 15. Li HY, Wang XF, Wei DD et al (2012) A comparative study on U1730121), the Joint Foundation (general) Project of the Equipment modiﬁed Zerilli-Armstrong, Arrhenius-type and artiﬁcial neural Pre-research of the Ministry of Education (Grant No. 6141A020221) network models to predict high-temperature deformation behav- and the Postdoctoral Science Foundation of China (Grant No. ior in T24 steel. Mater Sci Eng A 536:216–222 2017M620609). 16. Grong Ø, Shercliff HR (2002) Microstructural modelling in metals processing. Prog Mater Sci 47(2):163–282 Open Access This article is distributed under the terms of the 17. Vilamosa V, Clausen AH, Børvik T et al (2016) A physically- Creative Commons Attribution 4.0 International License (http://crea based constitutive model applied to AA6082 aluminium alloy at tivecommons.org/licenses/by/4.0/), which permits unrestricted use, large strains, high strain rates and elevated temperatures. 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Physical-based constitutive model considering the microstructure evolution during hot working of AZ80 magnesium alloy

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

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

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

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