Zexing Su, Chaoyang Sun,d,?, Mingjia Wang, Lingyun Qian,?, Xintong Li

b Beijing Key Laboratory of Lightweight Metal Forming, Beijing 100083, China

c College of Nuclear Equipment and Nuclear Engineering, Yantai University, Yantai 264005, China

d Shunde Graduate School of University of Science and Technology Beijing, Foshan, 528000, China

Abstract In this paper, a unified internal state variable (ISV) model for predicting microstructure evolution during hot working process of AZ80 magnesium alloy was developed.A novel aspect of the proposed model is that the interactive effects of material hardening, recovery and dynamic recrystallization(DRX)on the characteristic deformation behavior were considered by incorporating the evolution laws of viscoplastic flow, dislocation activities, DRX nucleation and boundary migration in a coupled manner.The model parameters were calibrated based on the experimental data analysis and genetic algorithm (GA) based objective optimization.The predicted flow stress, DRX fraction and average grain size match well with experimental results.The proposed model was embedded in the finite element(FE)software DEFORM-3D via user defined subroutine to simulate the hot compression and equal channel angular extrusion (ECAE) processes.The heterogeneous microstructure distributions at different deformation zones and the dislocation density evolution with competitive deformation mechanisms were captured.This study can provide a theoretical solution for the hot working problems of magnesium alloy.

Keywords: AZ80 magnesium alloy; Internal state variable model; Microstructure evolution; Dynamic recrystallization; Hot working process; Finite element simulation.

1.Introduction

Magnesium alloys have exhibited considerable application prospects in the aerospace and automobile lightweight fields owing to the high specific strength, excellent damping characteristics and favorable electromagnetic shielding capability[1,2].However, due to the few slip systems and poor formability, extensive applications of these alloys are restricted to die-cast components [3,4].Bulk formation methods including extrusion and forging are carried out at elevated temperature to activate more slip systems and improve workability [51].During hot working process, the dynamic recrystallization (DRX) mechanism is beneficial to material softening and grain refinement [5][52].Unfortunately, the microstructure evolution is sensitive to deformation parameters [6][53],which increases the control difficulties in mechanical properties and forming quality.It is critical to clarify the microstructure evolution affected by processing parameters, which is the premise for the workability optimization and the production of high-quality magnesium alloy parts.

At present, plentiful constitutive modeling research has been conducted to investigate the material deformation characteristics.For the understanding of flow behavior,the empirical approaches including Johnson-Cook and Arrhenius models show simplicity in the determination of material constants,which have been widely used [7-9].However, the nonlin-ear relationship among material parameters brings difficult to the prediction of hot deformation characteristic beyond the experimental conditions.In contrast, the advanced statistical method such as neural network model shows considerable expertise in self-learning, which can accurately predict the flow stresses within wide deformation conditions [10,11].It is a pity that the prediction capability of neural network model largely depends on the characteristic variables and high quality data.It should be noted that the empirical and advanced statistical methods mainly select strain rate, temperature and plastic strain as macroscopic state variables, which offers less physical insight in metallurgical phenomena.To understand the microstructure evolution of alloys with low stacking fault energy (SFE) including magnesium alloy [12], titanium alloy [13]and nickel-based alloy [14], the Avrami equation is used and contains the DRX fraction as a microstructural variable [15].However, the separating application of macroscopic and microcosmic models ignores the relationship between microstructure evolution and mechanical response.Therefore, a physically-based constitutive model for hot working process,which can relate the deformation behavior to microstructure characteristic, is urgently needed.

Compared with phenomenological modeling approaches,physically-based model involves internal state variables(ISVs) including dislocation density, recrystallized fraction and average grain size (AGS), which can describe the hot deformation characteristic coupled with different microstructure mechanisms.The evolution of physical variables at different scales has a clear correlation with time and strain, which can easily achieve the interactivity and integration with finite element (FE) platform [16].Lin et al.studied the grain evolution with recrystallization effect during the rolling processes of C-Mn steel through a dislocation density based ISV model [17].Li et al.set up a physically-based model for titanium alloy, which involves the physical variables including dislocation density, damage and globularization [18].Wang et al.established the constitutive equations for Inconel 740 superalloy, which includes the grain evolution, critical strain and volume fraction of DRX behavior during hot deformation process [19].Mohamed et al.developed a viscoplastic constitutive framework coupling dislocation density and damage to describe the deformation and failure characteristics in hot stamping of aluminum alloy [20].

At present, most constitutive models for magnesium alloys use classical viscoplastic equations based on work hardening (WH) and dynamic softening mechanisms, which lacks an in-depth insight of microstructure evolution.Although the flow behavior coupling microstructure mechanisms has been widely studied via the unified ISV approach among different materials such as titanium, aluminum and nickel-based alloys.An accurate ISV model for hot working of magnesium alloys is rarely reported, and is urgently needed for the material processing applications.Xu et al.put forward a unified ISV model of Mg-RE alloy by incorporating dislocation density, recrystallized grain size, un-recrystallized grain size and DRX fraction [21], which provides theory guidance for the hot working process of magnesium alloy.However, the characteristic flow behavior of magnesium alloy was not revealed, especially the dynamic softening with the DRX sensitivity at different deformation conditions.In our previous research, a physically-based model involving dislocation density and DRX fraction in hot working of AZ80 magnesium alloy was developed [22].It is a pity that the mechanical response with the interactions of different deformation mechanisms and grain size evolution has not been involved.And the modeling applicability for an actual processing problem combined with experimental validation is not considered.

Fig.1.A typical flow stress curve coupling various microstructural mechanisms in hot compression process of AZ80 magnesium alloy.

In the current study,firstly a unified ISV model for predicting microstructure evolution during hot working process of AZ80 magnesium alloy was established.Especially, the flow stress characteristic with microstructural effect was reasonably described by incorporating the evolution laws of viscoplastic flow, dislocation activities, DRX nucleation and boundary migration.The experimental flow stress, DRX fraction and AGS matched well with calculated results.Secondly, the FE applicability for hot compression process based on the secondary development technology was analyzed.The heterogeneous distribution of ISVs in the deformed specimen and the dislocation density evolution with competitive deformation mechanisms were well captured.Finally, the ISV modeling approach was used to simulate equal channel angular extrusion (ECAE) process.The microstructure evolution under complex thermomechanical boundary condition at characteristic zones was revealed.This study can provide a solution for the hot working problems of magnesium alloy.

2.The unified ISV model

2.1.Modeling viscoplastic flow coupling microstructural mechanisms

Fig.1 shows a typical flow stress curve coupling various microstructural mechanisms in hot compression process of AZ80 magnesium alloy.The overall stressσis composedof the viscoplastic flowσv, isotropic hardeningHand initial yieldk[23].The general flow characteristic shows an initial WH to the peak and a subsequent softening, which is dominated by the occurrence of DRX [24].Generally, DRX process needs to achieve a critical strainεc[25].Dynamic recovery (DRV) appears in the whole deformation stage, and static recovery (SRV) occurs at elevated temperature in the heating and insulation stages, both of which play a competitive role to DRX in the consumption process of stored energy[26].At the grain level, the grain growth in atomic diffusion process and the grain refinement in DRX process result in the evolution of AGS, which can also affect the material behavior.Therefore,the microstructural effects including SRV,WH,DRV, DRX and grain size should be considered in modeling flow behavior.

The stress-strain relation is given by Hooke’s law [27]:

whereEdenotes the Young’s modulus, and its values at different temperatures can be calculated by JmatPro software;εTandεpdenote the total strain and plastic strain, respectively.

To describe the characteristic metal flow within wide deformation conditions and relate it to microstructure evolution,a hyperbolic-sine type viscoplastic flow equation is introduced[21]:

The evolution rate of isotropic hardening stress in Eq.(2)is defined as [28]:

whereBdenotes the material hardening constant, which is temperature-dependent.denotes a normalized concept for dislocation density [29]:

whereρ0andρrepresent the initial and deformed dislocation densities, respectively.With the development of hot working process from the initial state to saturated state,the normalized dislocation density varies from 0 to 1.

The evolution rate of normalized dislocation density is expressed as [30]:

whereA4,A3,δ1,δ2,δ3andδ4are material constants.The first term represents the storage and DRV of dislocations,and(d/d0)δ1describes the grain size sensitivity on the dislocation multiplication.The second term denotes the SRV of dislocations, andCdenotes the SRV coefficient, which is temperature-dependent.The third term represents the DRX,andSdenotes the DRX fraction relevant to the evolution of new grains, which is clarified in the next section.

2.2.Modeling DRX nucleation and grain boundary migration

During the hot deformation of low SFE metals, the formation and migration of high angle grain boundaries are driven by the stored energy.Then the deformed microstructure is replaced by the recrystallized grains and results in a decrease of dislocation density.The DRX nucleation rate is sensitive to temperature and strain rate, which can be designated as[31,32]:

whereN0andbare material constants;Qndenotes the recrystallization activation energy.

Based on the DRX kinetics equation in the literature [33],by coupling the nucleation rate and critical strain of DRX process with normalized dislocation density, the evolution of recrystallized fraction is designated as:

whereNis a parameter related to temperature;λ1is a material constant.εcdenotes the DRX critical strain relevant to deformation conditions, which can be determined by Zener-Hollomon (Z) parameter [34]:

xin Eq.(7) represents the incubation period of DRX initiating, which is given by [35]:

whereA0is a material parameter related to temperature.

The grain boundary migration in hot working process contains the grain growth with thermal effect and the grain refinement with DRX, which affects the variation of AGS [36].The grain growth associated with grain boundary migration is expressed as [37]:

whereσsurfandMdenote the energy density and mobility of grain boundaries, respectively.TheMσsurfterm is affected by temperature, thus Eq.(10) can be designated as:

whereG1is a material constant relevant to temperature;φ1is a material constant.

The grain refining rate coupling DRX fraction is expressed as [38]:

whereλ2andφ2are material constants.G2is a material parameter related to temperature.

By taking the relative grain size ˉd=d/d0into account and combining Eq.(11) with Eq.(12), the AGS evolves as:

The two terms represent the grain growth and grain refinement, respectively.

3.Model calibration and validation

3.1.Calibration of model parameters

The developed constitutive equations can be formulated as:

B, k, C, N, A0,G1andG2are the temperature-dependent variables with thermal activation effect, which can be formulated as Arrhenius-type relations [19,26]:

whereB0,k0,C0,N0,A00,G10andG20are material constants;QB, Qk, QC, QN, QA0,QG1andQG2are the corresponding activation energies.

For the calibration of model parameters, firstly the experimental verifiability and physical meaning need to be considered in the corresponding equations.Especially, the critical strain for DRX initiation in Eq.(7) was obtained according to the criterion of irreversible thermodynamics proposed by Poliak and Jonas [39,40].The inflection points in the lnθ-εcurves (θdenotes the strain hardening rate,θ= dσ/dε) indicate the DRX initiation.On the basis of the literature [21],the function betweenεcandZparameter in Eq.(8) was determined by the linear regression analysis.The initial values of AGSd0, initial yield stressk0and Young’s modulusE0were obtained from the experimental data.

Fig.2.Cubic polynomial fitting between SRV coefficient (C0) and temperature (T).

For the material parameters that are difficult to determine by the experimental data, the initial values were selected based on the reference papers [33,41], and then the lower and upper limits of 25%?200% were defined for the initial values.Since many ISVs in the equations(such as normalized dislocation density, AGS and DRX fraction) are expressed in differential forms.An optimization solution based on genetic algorithm (GA) [42]was adopted to solve the nonlinear relationship among different material constants.The MATLAB GA toolbox was applied to the minimization of residuals between calculated and experimental flow stress, DRX fraction and AGS.The detailed optimization process has been reported elsewhere [38,43].It should be noted from Fig.2 that the stress softening effect at low temperature is much more significant compared with that at high temperature.The stress softening differences at various temperatures can be attributed to the SRV effect in the heating and insulation process, which further affects the microstructure evolution as well as flow characteristics during the hot working process.Taking this physical meaning into account, the SRV coefficientC0at a fixed temperature was firstly determined.Then a cubic polynomial fitting betweenC0andTwas employed,as shown in Fig.2.The empirical equation was given by:C0= ?157.818 + 1.569T?0.0053T2+ 6.08 × 10?6T3,and the determination coefficient(R-square)was calculated as 0.98841, which indicates a good fitting effectiveness.The calibrated model parameters with the physical significance and feasibility are listed in Table 1.

Table 1The calibrated model parameters with the physical significance and feasibility.

3.2.Validation of flow stress

Fig.3(a)-(c) show the experimental (symbols) and calculated (lines) true stresses during hot compression process at 0.1 s?1, 0.01 s?1and 0.001 s?1, respectively.The flow characteristics including WH and dynamic softening can be well reflected, and the calculated results match well with experimental data.

To quantitatively evaluate the modeling accuracy, the root mean square error (RMSE), average absolute relative error(AARE) and correlation coefficient (R) are introduced as:

Fig.3.Comparison between experimental (symbols) and calculated (lines) true stresses during hot compression process: (a) 0.1 s?1, (b) 0.01 s?1, (c) 0.001 s?1 and (d) correlation analysis.

whereNdenotes number of sampling point;EiandCirepresent experimental and calculated true stress, respectively; ˉEand ˉCdenote the averages ofEiandCi,respectively.Fig.3(d)shows the correlation analysis between experimental and calculated true stress.Most of stress data locate near the best fitting line,while a few stress data deviate slightly from the best fitting line at high stress level.The calculatedRMSE, AAREandRare 2.38 MPa, 3.23% and 0.994, respectively, indicating a low deviation and high correlation between experimental and calculated true stresses.Therefore, the characteristic flow behavior with the WH and dynamic softening effects can be well presented by the unified constitutive model.

3.3.Validation of microstructure evolution

To verify the microstructure evolution under different deformation conditions, the microstructure characterization was conducted at the center zone of specimen by electron backscatter diffraction (EBSD) and optical microscopy(OM) tests.The as-extruded billet was annealed before the hot compression process.The initial microstructure with near-equiaxial static recrystallized grains can be observed in Fig.4(a).The AGS was determined to be 20.8 μm.The deformed microstructures at 300 °C and 0.001 s?1with true strains of 0.1, 0.3 and 0.916 are shown in Fig.4(b)-(d), respectively.At lower true strains, the necklace structure composed of DRX grains appears at initial grain boundaries, as shown in Fig.4(b) and (c).As true strain increases to 0.916,DRX fraction increases and AGS decreases, resulting in a refined microstructure with full DRX grains, as shown in Fig.4(d).Increasing deformation degree can accumulate deformation stored energy and provide driving force for grain boundary migration [44].Fig.4(e) shows the comparisonsbetween experimental and calculated results of DRX fraction and AGS.As deformation degree reaches DRX critical strainεc, the AGS curve significantly decreases with the increasing DRX fraction curve.The experimental DRX fraction and AGS match well with the corresponding calculated curves.

Fig.4.The microstructure evolution at different deformation conditions: initial state (a) and deformed state (300 °C and 0.001 s?1) with true strains of 0.1(b), 0.3 (c) and 0.916 (d); (e) Comparisons between experimental (symbols) and calculated (lines) results of DRX fraction and AGS.

To verify the AGS under different strain rates and temperatures, the deformed specimens with true strain of 0.916 were characterized.Fig.5(a) shows an inhomogeneous microstructure composed of small and large grains at 300 °C and 0.01 s?1.As strain rate decreases to 0.001 s?1, a refined and homogeneous microstructure containing complete DRX grains can be observed in Fig.5(b).Low strain rate provides sufficient time for DRX initiating, and low temperature can inhibit the grain growth.As temperature increases to 400 °C,grains grow significantly with enhanced mobility, as shown in Fig.5(c).In Fig.5(d), a necklace structure composed of numerous DRX grains appears at grain boundaries at 400 °C and 0.1 s?1, which is due to the incomplete recrystallization kinetics at high strain rate.Fig.5(e) shows the comparison between experimental and calculated AGS.With the help of Image Pro-Plus software 6.0, three representative locations were selected, and the linear intercept method was used to measure the AGS of each location.Then the standard deviation range and the average value of the experimental AGS were obtained.The calculated AGS locates in the range of experimental results with relatively low standard deviation.The proposed constitutive model can capture the microstructureevolutions including DRX fraction and AGS under various deformation conditions.

Fig.5.The microstructure evolution with a true strain of 0.916: (a) 300 °C, 0.01 s?1, (b) 300 °C, 0.001 s?1, (c) 400 °C, 0.001 s?1, (d) 400 °C, 0.1 s?1 and(e) Comparison between experimental (black symbol) and calculated (red symbol) AGS.

4.Results and discussion

4.1.FE simulation of hot compression process

Based on the secondary development technique,the constitutive equations were embedded in FE software DEFORM-3D via the user defined subroutine to simulate the hot compression experiment.The cylindrical specimen adopted the dimension ofΦ8 × 12 mm, and the tetrahedron element mesh was used.The shear friction mode was applied between anvil and specimen, and a friction coefficient of 0.3 was adopted.The deformation conditions were set as 300 °C and 0.001 s?1,with a true strain of 0.916 (corresponding to a reduction of 7.2 mm).

Fig.6(a) and (b) show the experimental and simulated compressed specimens, respectively.The obvious bulging phenomenon can be observed due to the effect of friction.The experimental and simulated section sizes at different zones were compared, as shown in Table 2.The experimental section sizes at the end and center zones match well with the simulated results.The maximum relative error was calculated to be 1.30%.The established FE model can present the hot compression process of AZ80 magnesium alloy.

Table 2Experimental and simulated section sizes at different locations.

Table 3The decomposition of dislocation density evolution.

The comparison of the experimental and simulated loads during hot compression process at 300 °C and 0.001 s?1is shown in Fig.7.The experimental load shows an initial increasing stage and remains stable at medium strain level,followed by a continuous increasing stage with the increase of deformation degree.The experimental load accords with the simulated result, while a few experimental points deviate slightly at high strain level, which can be attributed to the increasing friction effect at the specimen-anvil interface.

Fig.6.Experimental and simulated results of compressed specimen.

Fig.7.Experimental and simulated loads during hot compression process at 300 °C and 0.001 s?1.

4.2.Heterogeneous ISV distribution

Fig.8 shows the simulated distributions of various ISVs in compressed specimen.In Fig.8(a), (b), effective strain locally accumulates at center zone P1, and then decreases from bulging zone P2 to end zone P3.With the increase of stroke to 7.2 mm, effective strains at different deformation zones reach 1.4,0.7 and 0.3,respectively.The inhomogeneous distribution of effective strain is caused by the different stress states with friction at specimen-anvil interface [45].The distributions of DRX fraction and AGS are shown in Fig.8(c),(d)and(e),(f),respectively, which accords with the distribution of effective strain.The zone with larger deformation exhibits higher DRX fraction and lower AGS.Maximum DRX fraction of 0.98 and minimum AGS of 3.9 μm appear at center zone P1, corre-sponding to a maximum effective strain.High plastic strain provides much stored energy and promotes the activation of DRX process, which results in refined microstructure with high DRX fraction [46].The evolving forms of microstructural ISVs simulated by FE model at P1 and P2 accord with the results obtained by MATLAB in Fig.4.An interesting phenomenon is that the state variable evolution at end zone P3 differs from the other two zones.As the stroke increases to～5 mm, the state variables at P3 accelerate to rise.This can be attributed to the increasing friction effect on deformation and microstructure evolution at high strain level.

To verify the FE model during hot compression process,the EBSD characterization was supplemented at the bulging and end zones, as shown in Fig.9.A wide observation area was adopted to obtain the statistical accuracy of heterogeneous microstructure with relatively low DRX degree.Combined with the microstructure at the center zone in Fig.4, the statistical AGS and DRX fraction at different zones were plotted in Fig.8.In the zones of P1, P2 and P3, the AGSs were 3.9, 4.9 and 6.3 μm, respectively; the corresponding DRX fractions were 0.98, 0.88 and 0.68, respectively.The heterogeneous microstructure distributions at different deformation zones are in good agreement with the predicted results.

4.3.Dislocation activities with microstructural mechanisms

The normalized dislocation density provides a physical insight for microstructure evolution in hot working.In this section, the dislocation density evolution is discussed based on the interactions among various deformation mechanisms.The evolution rate of normalized dislocation density in Eq.(5) is divided into three parts, including the WH as well as DRV,SRV and DRX.The decomposition of dislocation density evolution is shown in Table 3.

Based on the FE simulation of hot compression, the dislocation density evolution at 350 °C and 0.1 s?1are obtained,as shown in Fig.10.The contributed proportions of different microstructure mechanisms are calculated in percentage form.The normalized dislocation density ˉρrises rapidly to a peak and decreases gradually,which is in consistence with the variation of true stress in Fig.3.During the initial deformation stage(ε＜εc),dislocations proliferate rapidly,and DRV occurs to annihilate the entangled dislocation network, which results in an initial decreasing proportion of WH+DRV term.The increasing normalized dislocation density ˉρa(bǔ)t this stage provides the deformation storage energy for the subsequent DRX process.As the deformation reaches the DRX critical strainεc, the DRX initiates and then evolves sufficiently, which leads to the annihilation of abundant dislocations.Thus the proportion of WH+DRV term further decreases significantly,and the proportion of DRX term increases rapidly.When the evolution rate of DRX term exceeds WH+DRV term, the normalized dislocation density continuously decreases.By involving the effects of different deformation mechanisms including WH+DRV and DRX, the proportion of SRV term shows a similar evolution characteristic to the normalized dislocation density.

Fig.10.The simulated dislocation density evolution with contributed proportions of different microstructure mechanisms including WH+DRV, SRV and DRX at 350 °C and 0.1 s?1.

Fig.8.The simulated distributions of effective strain (a) (b), DRX fraction (c) (d) and AGS (e) (f) in compressed specimen at 300 °C and 0.001 s?1.

Fig.9.Microstructures in bulging zone P2 (a) and end zone P3 (b) at 300 °C and 0.001 s?1.

Fig.11 shows the simulated dislocation density evolution with contributed proportions of different microstructure mechanisms at different deformation conditions.As temperature decreases and strain rate increases, normalized dislocation density and the contributed proportion of WH+DRV both increase, as shown in Fig.11(a) and (b), respectively.High strain rate and low temperature contribute to increasing critical shear stress and decreasing fraction of mobile dislocations,which is beneficial to the multiplication and entanglement of dislocations.In Fig.11(c), the contributed proportion of SRV increases with increasing temperatures and decreasing strain rates.High temperature and low strain rate increase SFE and provide a stronger thermal fluctuation for activation, which decreases the energy barrier of dislocation annihilation [47].With deformation storage energy consumed by recovery at high temperature and low strain rate, the contributed proportion of DRX decreases, as shown in Fig.11(d).At one giventemperature (such as 350 °C), the decrease of strain rate results in decreasing DRX critical strain and contributes to the activation of DRX process, which leads to the intersections among DRX curves with various strain rates.The competition between recovery and DRX process with effects of various deformation conditions can also be intuitively explained via true stress curves in Fig.3.The small difference between steady stress and peak stress exists at high temperature and low strain rate (such as 400 °C and 0.001 s?1), which is similar to a recovery type curve.

Fig.11.The simulated dislocation density evolution (a) with the contributed proportions of WH+DRV (b), SRV (c) and DRX (d) at various deformation conditions.

Fig.12.Experimental setup (a) and FE model (b) of ECAE process.

5.Application in ECAE process

5.1.Experimental and numerical scheme

During the complicated hot working processes of magnesium alloy such as forging and extrusion, the material behavior and microstructure evolution are sensitive to the nonuniform deformation, stress states and thermal conditions.Thus the reliability of the proposed constitutive model need to be validated via the complex hot working problems.In this section, the FE modeling approach was used to simulate the ECAE process.Meanwhile, the extrusion experiment was carried out for validation.

Fig.12(a) shows the experimental setup of ECAE process.The ECAE moulds including the top and bottom dies were assembled and then fixed on a 600 ?N hydraulic press.The channel in the bottom die bents through an angular of 120°The channel diameter is 15 mm, and the dimension of workpiece isΦ15 × 40 mm.The inlet and outlet diameters of bottom die were determined to be 20 mm to facilitate the entrance and exit of the workpiece.Before the ECAE process,the die-sets were heated by the heating coil and held for 2 h,and the insulation was used for the reduction of heat dissipation.The workpiece was heated by the heating furnace and held for 30 min.The mold cavity and workpiece were lubricated using molybdenum disulfide (MoS2).The workpiece was placed in the cavity and held for ～10 min to maintain stable temperature distribution.The extrusion parameters were set to be 380 °C and 3 mm/s.The stroke of top die was set to be 40 mm, and the ECAE process lasted for ～40 s.After ECAE process, the workpiece was water-quenched.

Fig.13.Experimental and simulated loads during ECAE process.

Fig.12(b) shows the FE model of ECAE process.The dies and workpiece were set as rigid bodies and plastic body,respectively.The die material adopted H-13 steel.The boundary conditions of heat transfer (including convection and radiation) among the workpiece, dies and ambient atmosphere were considered.A heat transfer coefficient of 11 N/s/mm/°C was set between workpiece and dies, while the heat transfer coefficient to atmosphere was 0.02 N/s/mm/ °C.The workpiece was meshed via 20,000 tetrahedron elements.A shear friction coefficient of 0.2 was applied between dies and workpiece.

5.2.Deformation and microstructure evolution analysis

The comparison of the experimental and simulated loads during ECAE process is shown in Fig.13.The experimental load increases to the peak, followed by a relatively stable stage and decreases ultimately.The variation trend of sim-ulated load matches well with the experimental result.The experimental load is slightly higher than the predicted curve,which is due to the increasing deformation resistance of alloy with heat dissipation effect.The established FE model based on the unified ISV approach can reflect the plastic deformation feature of AZ80 magnesium alloy during ECAE process.

Fig.14.Simulated distributions of normalized dislocation density (a), DRX fraction (b) and AGS (c).

The simulated distributions of microstructural ISVs including normalized dislocation density, DRX fraction and AGS are shown in Fig.14(a)-(c), respectively.The shear band at the channel angular can be observed by the distribution of normalized dislocation density in Fig.14(a).As workpiece entering the shear deformation zone, the proliferation and entanglement of dislocations result in high dislocation density, which provides storage energy for the subsequent DRX [48].When the workpiece flows out of the shear deformation zone, the entangled dislocation network annihilatesby the DRX effect, and the dislocation density significantly decreases.Fig.14(b) and (c) show the similar distribution characteristics of DRX fraction and AGS, respectively.With the shear deformation effect at the channel angular, the DRX fraction increases and AGS decreases.At the outlet of extrusion channel, the distributions of DRX fraction and AGS are different to the distributions at the stable deformation zone,which can attributed to the uneven metal flow with the variation of channel diameter.

Fig.16.Microstructures in different deformation zones at A1 (a), A2 (b) and A3 (c).

Fig.15 shows the evolutions of normalized dislocation density, DRX fraction and AGS with the variation of deformation pattern.The distance at horizontal coordinate is defined as the center axis started at the upper surface of channel in Fig.14(c).With the development of ECAE process, the normalized dislocation density reaches a peak and then decreases gradually, which indicates the hot deformation mechanisms including WH and DRX softening.The AGS significantly decreases with the increase of DRX fraction.

To validate the reliability of FE model in predicting microstructure evolution, three representative locations in the center of deformed workpiece in Fig.14(c) were selected.The microstructures in different deformation zones of A1, A2 and A3 are shown in Fig.16.In Fig.16(a), the workpiece undergoes initial upsetting deformation with large extrusion force before entering the shearing zone, which results in the equiaxed grains with a few DRX grains nucleating at grain boundaries.In Fig.16(b), with the effect of shear deformation at the channel angular, the recrystallization process is developed sufficiently, and the AGS decreases rapidly.As the shearing deformation process finished, the proportion of DRX grains increases, and the microstructure is further refined, as shown in Fig.16(c).The statistical AGS and DRX fraction in different deformation zones were plotted in Fig.15.The experimental microstructure data matches well with the predicted curves.The ECAE process contributes to the refinement of microstructure and the improvement of comprehensive mechanical properties, which has been proved by many literatures [49,50].It can be concluded that the ISV modeling approach can describe the microstructure evolution coupling deformation mechanisms during the ECAE process of AZ80 magnesium alloy.

Fig.15.The evolutions of normalized dislocation density, DRX fraction and AGS during ECAE process.

6.Conclusions

In this study, a unified ISV model for predicting microstructure evolution during hot working process of AZ80 magnesium alloy was established.The macroscopic and microscopic kinetics including viscoplastic flow, dislocation activities, DRX nucleation and boundary migration were introduced.The microstructure evolution features during hot compression and ECAE processes were investigated.The following conclusions can be summarized:

(1) The characteristic deformation behavior including work hardening and dynamic softening was dominated by the interactive effects among dislocation multiplication, recovery, recrystallization and grain size.The recrystallized microstructure evolution was modeled by incorporating the evolution laws of critical strain, nucleation rate and boundary migration rate.The model parameters were calibrated based on experimental data analysis and GA-based objective optimization.Predicted flow stress, DRX fraction and AGS match well with experimental results.

(2) The set of constitutive equations was integrated into FE model via the user defined subroutine to simulate hot compression process.The friction at specimenanvil interface leads to the heterogeneous microstructure distribution.Maximum DRX fraction and minimum AGS locate at the center zone of deformed specimen.The experimental microstructure distributions at center,bulging and end zones agree well with the simulated results.The effect of deformation conditions on dislocation density evolution was explained by competitive processes between different microstructural mechanisms.

(3) Using the calibrated FE modeling approach, the heterogeneous microstructure features during ECAE process were captured.With the shear deformation effect, high dislocation density accumulates at the channel angular.DRX fraction increases and AGS decreases with the development of ECAE process.The unified ISV model can provide scientific understanding and reasonable prediction for the hot working process of magnesium alloy.

Declaration of Competing Interest

None.

Acknowledgments

Authors acknowledge the funding supported by National Natural Science Foundation of China(No.52175285),Beijing Municipal Natural Science Foundation (No.3182025), National Defense Science and Technology Rapid support Project(No.61409230113), Scientific and Technological Innovation Foundation of Shunde Graduate School, USTB and Fundamental Research Funds for the Central Universities(No.FRFBD-20-08A, FRF-TP-20-009A2).