Coupled CA-FE simulation for dynamic recrystallization of magnesium alloy during hot extrusion

Yu-Ying He,Sheng-Wen Bai,Gang Fang

State Key Lab of Tribology,Department of Mechanical Engineering,Tsinghua University,100084 Beijing,China

Abstract In the present research,the dynamic recrystallization (DRX) behavior of a newly-developed Mg-Al-Zn-RE alloy with abundant secondphase particles during hot extrusion is investigated by coupling finit element (FE) and cellular automaton (CA) models.A two-dimensional CA model is developed to quantitatively and topologically evaluate the DRX process during deformation with constant forming conditions.Considering the fact that second-phase particles with various sizes extensively exist in the studied Mg-Al-Zn-RE magnesium alloy,models of DRX nucleation and grain growth velocity are modified The coefficient of the modifie CA model are calibrated by isothermal compression experiments of the magnesium alloy.Subsequently,the CA model is coupled with FE analysis to investigate the DRX behavior during the hot extrusions of the Mg-Al-Zn-RE alloy.The DRX behavior of the magnesium alloy at different stages and positions of extruded plates is simulated by the established model.Finer grains near the edge than in the inner of the plates result from higher strain and strain rate.The influenc of extrusion conditions on microstructural evolution is explored.Under the employed forming conditions,average grain size decreases 28-62 times from as-cast and solution-treated to as-extruded state due to grain refinemen by DRX.With increasing initial billet temperature or extrusion speed,average grain size increases.The fines grains are obtained at the initial billet temperature of 623K and the extrusion speed of 7.83mm/s.Low initial billet temperature or high extrusion speed benefit homogeneous grain distribution.The simulated results are in good agreement with experimental ones.

Keywords: Cellular automaton;Finite element;Dynamic recrystallization;Extrusion;Magnesium alloy.

1.Introduction

It is vital to control microstructure formed during hot working,which influence the mechanical properties of metallic materials largely.In general,wrought magnesium alloys possess poor workability at room temperature,so hot working becomes an important way.Because magnesium alloys have relatively low-to-medium stacking fault energy,dynamic recrystallization(DRX)is their principal softening mechanism during hot working and leads to severe microstructure evolution.Therefore,an in-depth understanding of DRX behavior during hot working is the prerequisite of optimum microstructures and improved mechanical performances of magnesium alloys.

In the last decades,various phenomenological and physically-based models have been proposed to describe plastic fl w behavior [1,2],DRX kinetics [3,4],and microstructural characters [5,6] of materials during hot working.Though predicting the average behavior of DRX,these models fail to predict locally and topologically microstructural evolutions,which are also difficul to be observed in experiments.

To explore the evolution of microstructural morphology more comprehensively and accurately,some mesoscale models such as Monte Carlo (MC) [7],cellular automaton (CA)[8],and phase-fiel (PF) methods [9] are employed.Owing to the superiority of real-time and real-space scale as well as high efficien y and fl xibility,the CA method is most widely adopted to model the DRX of metals and their alloys [10].First introduced into DRX simulation by Goetz and Seetharaman [8],the CA model reproduced DRX characters such as the necklace structure and the single to multi-peak transition of fl w stress with initial grain size increasing,yet the effects of thermomechanical parameters were not taken into account.To associate DRX behavior with deformation conditions (temperature and strain rate),Ding and Guo [11] developed a CA model coupled with fundamental metallurgical principles of DRX.On this basis,with the aim of more accurate simulation or more suitability for certain materials,various modification were applied to the model of Ding and Guo.As for the authenticity of initial microstructures in the simulation,instead of using a normal grain growth algorithm,Chen et al.[12] instituted the state transition rules for initial grain growth based on physical metallurgical principles.Wang et al.[13] guaranteed the distribution of initial grain size by a CA model with multi-step nucleation.Considering topological changes of grains during compression,Xiao et al.[14] realized the topology deformation by vector operation.While Chen et al.[12] established an unchangeable cellular coordinate system and a changeable material coordinate system synchronously to remain recrystallized grains (R-grains)equiaxed when other grains were compressed.Metallurgical theories of dislocation density,nucleation,and grain growth are critical to CA models.Models of inhomogeneous distribution of dislocation density inside grains [15] and dislocationdependent nucleation [16] were employed to make simulation closer to experimental processes.

For highly-alloyed materials,the influenc of alloying elements on DRX cannot be ignored.Lee et al.[17] introduced a new coefficien into the equation of grain boundary mobility to characterize the solute drag effect.Goetz [18] randomly set coarse second-phase particles in the interior of grains and treated them as potential nucleation sites.Concerning rodlikeδphase in Inconel 718,Zhang et al.[19] placed ellipsoid particles in the initial microstructure and terms considering particle-stimulated nucleation (PSN) and pinning effect of second-phase particles were added to expressions of nucleation and grain boundary migration.The extruded magnesium alloy studied in the present work is newly developed by adding rare earth (RE) element La and Gd to commercial AZ80 magnesium alloy to improve its strength [20].It is found that bulky second-phase particles,including needlelike or lamellar phase Al11La3as well as block-shaped phases Al8Mn5and Al2Gd,are widespread in the Mg-Al-Zn-RE alloy,promoting PSN for DRX,consequently accelerating DRX process and refinin R-grains.In addition,fin globular precipitates Mg17Al12generating at boundaries hindered nucleation and grain growth of DRX,resulting in a suppression of recrystallization and a small R-grain size.Thus,models of nucleation and grain growth velocity are modifie for the new Mg-Al-Zn-RE magnesium alloy in this work to integrate the effects of second-phase particles in the CA model.

To make the simulation using CA models more consistent with the reality,the finit element method (FEM)is employed to provide the temporal and spatial variation of working con-ditions macroscopically.Wu et al.[2] deemed that strain rate and temperature during isothermal hot compression on the thermo-mechanical simulator were not constant in reality and a FEM incorporating a physically-based constitutive model was combined with a CA model to trace these changes.Coupled CA-FE model is a powerful method to predict DRX behavior during practical hot forming processes.Lee et al.[17] investigated the DRX during non-isothermal hot compression of pure copper and gear hot forging by a CA model and FEM,but the simulated results for hot forging were not verifie by experiments.Asadi et al.[21] and Akbari et al.[22] applied the CA-FE method to friction stir welding.DRX behavior in stir zone with severe deformation and intense frictional heat was studied,while grain crushing under severe plastic deformation (SPD) was not considered.Li et al.[23] simulated microstructures of an Mg-Al-Ca-Sr magnesium alloy after hot extrusion by the combination of CA and FE models,but microstructural evolution was not analyzed.Hot extrusion is one of the main processes to product profile of magnesium alloys.DRX is the principal mechanism to refin grains during hot extrusions of magnesium alloy,so the microstructural evolution during DRX is necessary to investigate and control to enhance the mechanical properties of products.

In the present work,a CA model is proposed to simulate the microstructural evolution of an Mg-Al-Zn-RE magnesium alloy during isothermal hot compression with constant deformation conditions.The CA model considers the pinning effect of fin second-phase particles on grain boundary mobility,as well as the effects of PSN and fin precipitates on nucleation.Coupled with FE simulation,the CA model is further employed to the hot extrusion of the Mg-Al-Zn-RE magnesium alloy,during which deformation temperatures and strain rates vary with stroke.

2.Material and experiments

2.1.Material

A newly-developed Mg-Al-Zn-RE extruded alloy was investigated and its chemical compositions are shown in Table 1.To homogenize initial microstructures and improve the workability of the magnesium alloy,solution treatment at 693K for 24h followed by water quenching was implemented to the as-cast alloy and the corresponding microstructure is shown in Fig.1.The grains with an average size of 372 μm were coarse.With abundant alloying elements,there existed large quantities of second-phase particles Al11La3,Al2Gd,etc.in the magnesium alloy [20,24].

Table 1 Chemical composition of the Mg-Al-Zn-RE alloy (wt.%) [24].

Fig.1.Microstructure of as-cast and solution-treated Mg-Al-Zn-RE alloy.

2.2.Experiments

A series of isothermal hot compression tests of the Mg-Al-Zn-RE alloy were performed to calibrate the CA model,determine the constitutive model used for FE simulations,and validate the reliability of the established CA model.As a typical forming process of material,hot extrusions were conducted to validate the coupled CA-FE model in this work.

2.2.1.Compressions

Cylindrical specimens of the Mg-Al-Zn-RE alloy with a size of ?8×12mm were machined from the as-cast and solution-treated ingot.Hot compression tests were carried out on a thermo-mechanical simulator (DSI,Gleeble 1500D) at different temperatures (573,603,633,663 and 693K) and strain rates (0.001 s-1,0.01 s-1,0.1 s-1,1 s-1and 10 s-1).Graphite foils were placed between the specimen ends and platens to reduce the friction.The specimens were heated to the predefine temperature at a rate of 5K/s and held for 180s to obtain a homogeneous temperature field When the true equivalent strain reached 0.693,the compressions were ceased and the specimens were immediately water quenched to retain the microstructures after hot compressions.

The measured fl w stress-strain curves during hot compression are shown in Fig.2,which are found to be sensitive to temperature and strain rate.

2.2.2.Extrusions

Hot direct extrusions of Mg-Al-Zn-RE alloy plates were conducted on a hydraulic press with a load capacity of 2000kN.The extrusion die setup of magnesium alloy plates was designed and its cutaway schematic is shown in Fig.3.The extrusion conditions are listed in Table 2.Billets with a rectangular cross-section of 50mm×30mm and a length of 20mm were extruded to plates with a cross-section of 30mm×3mm at an extrusion ratio of 16.67.Two initial billet temperatures 623 and 673K and two extrusion speeds 7.83 and 81.68mm/s were employed in the extrusion tests.The extrusion speeds 7.83 and 81.68mm/s were calculated as the product of the corresponding extrusion stem speeds(0.47mm/s and 4.9mm/s,respectively) and the extrusion ratio 16.67.

Table 2 Extrusion conditions.

Table 3 Material constants in the CA model.

Magnesium alloy billets and the extrusion die were heated to the preset temperatures by four heating rods at a rate of 10K/s and held for 600s.During the extrusions,once extruded out of the bearing,the magnesium alloys were water quenched to maintain the as-extruded microstructure.When the extrusion stroke attained 10mm,the extrusion test was finished Temperatures around the extrusion die bearing,extrusion stem speeds,and extrusion forces were measured by thermocouples and sensors.

2.2.3.Microstructure observations

Compressed specimens were longitudinally split in half for metallographic analysis.Extruded plates were cut along the extruded direction,in which the microstructures were observed (Fig.4).

All samples were mechanical ground,polished,and then chemically etched by a solution of alcohol (60ml)+acetic acid (20ml) +water (15ml) +hydrogen nitrate (5ml) for 6-12s.An optical microscope (Zeiss,Axio Scope.A1) was used for metallographic observation and an image analysis software (Media Cybernetics,Image-Pro-Plus) was applied to measure average grain size and DRX fractions.

3.Coupled CA-FE modeling

3.1.Dislocation density

During hot deformation,dislocation density evolves under the competitive effects of working hardening (WH) and dynamic softening.The Kocks-Mecking model (KM model)[25],considering the combined actions of WH and dynamic recovery (DRV),is widely adapted to describe the variation of dislocation densityρwith the strainε.

where two terms of the right side represent the dislocation accumulation concerning WH and dislocation annihilation associated with DRV,respectively.WH coefficienk1is a constant while softening coefficienk2varies with deformation conditions (temperature and strain rate).

Fig.2.Flow stress-strain curves of Mg-Al-Zn-RE alloy at different temperatures and strain rates of (a) 0.001 s-1,(b) 0.01 s-1,(c) 0.1 s-1,(d) 1 s-1,(e)10 s-1.

Coefficientk1andk2in Eq.(1) are derived from experimental stress-strain data since plastic fl w behavior is the macroscopic reflectio of dislocation movement.

In the KM model,it is assumed that fl w stressσis merely determined by average dislocation densityρa(bǔ)s given in the following equation.

where the dislocation interaction parameterαis taken as 0.5 for most metals;μis the shear modulus;bis the Burger’s vector.Average dislocation densityis calculated by

wherenis the total number of cells in the CA model andρi,jis the dislocation density of a single cell (i,j).

A typical fl w stress-strain curve with the occurrence of DRX is illustrated in Fig.5.According to softening mechanisms,the evolution of the fl w stress comprises two stages.In Stage Ⅰ,the material is only softened by DRV and the fl w stress increases with the strain monotonously due to the relatively weak softening effect of DRV for metals with low-to-medium stacking-fault-energy.Stage Ⅱappears when the fl w stress increases to the critical valueσcfor DRX.In this stage,DRV and DRX,both as the softening mechanisms,occur simultaneously.Flow stress increases slowly and then reduces rapidly after reaching peak stressσpdue to the enhanced softening effect of DRX through the formation of undistorted R-grains.Finally,a steady-state of fl w stress(σss) attains,meaning that a dynamic balance between WH,DRV,and DRX reaches.

The following method is adopted to acquire the values ofk1andk2from the fl w stress-strain curve.Firstly,the differential equation Eq.(1) is integrated and combined with Eq.(2) to derive the dependence of fl w stressσon plastic strainε.

Fig.3.Cutaway schematic of extrusion die setup of magnesium alloy plates:(a) the front view and (b) the side view of a half die schematic.

Fig.4.Schematic of sections of (a) a compressed specimen and (b) an asextruded plate for metallographic analysis.

Fig.5.Typical fl w stress-strain curve with the occurrence of DRX.

where saturation stressσsatand initial stressσ0are boundary conditions corresponding to plastic strainε=+∞andε=0,respectively.The value ofσ0is considered as the nominal yield stress with a true strain of 0.002 andσsatcan be obtained by extrapolating the curve of WH rateθ(θ=?σ/?ε)againstσ(Fig.6a).Critical stressσcfor triggering DRX in Fig.6a is regarded as the stress inflectio inθ-σcurve,i.e.?2θ/?σ2=0 [26].Then taking logarithm on both sides of Eq.(4-a).

where the value ofk2is determined by fittin the line ln(σsat-σ) -ε(Fig.6b).It should be noted that since the KM model describes dislocation density evolution induced by WH and DRV,only the experimental stress-strain data among Stage Ⅰ(WH+DRV) are employed to evaluate the values ofk2.Then according to Eq.(4-b),work hardening coefficienk1is calculated based on saturation stressσsatand softening coefficienk2,i.e.k1=k2?(σsat/αμb).

In this work,the values ofσsatandk2under different deformation conditions are derived from the fl w stress-strain curves during hot compressions (Fig.2).To enable the application of the dislocation density model at any temperature and strain rate,the functional expressions ofσsatandk2are established.The relationship between saturation stressσsatand deformation conditions obeys the Sellars-Tegart model [27],in which the values of the parameters are obtained by regression analysis described in Ref.[1].

whereZis the Zener-Hollomon parameter;is the strain rate;Qactis the deformation activation energy;Ris the universal gas constant;Tis the absolute temperature.

The dependence of the softening coefficienk2onZfollows a power law.

whereBkandpkare material constants.The values of material parameters in Eq.(7) are obtained by linear regression and then the numeric expression of softening coefficienk2of the studied magnesium alloy is written as the following equation.

3.2.Nucleation and growth of grain

In hot worked metals,the dislocation density of grain evolves according to the rule shown in Eq.(1).Then new nuclei are formed on high-angle boundaries (HABs) by the bulging mechanism when dislocation density,i.e.stored energy,reaches a critical valueρc.A criterion for the initiation of DRX proposed by Roberts and Ahlblom [28] is adopted to calculateρc.

Fig.6.Determination of (a) saturation stress σsat and (b) softening coefficien k2.

whereγiandMiare the grain boundary energy and the grain boundary mobility of theith R-grain,respectively;lis the dislocation mean free path (DMFP);τ=0.5μb2is the dislocation line energy.Grain boundary energyγiis expressed by the Read-Shockley relationship [29] to describe its dependence on grain boundary misorientation.

whereθiis the grain boundary misorientation between theith R-grain and its adjacent grains;θmis the minimum misorientation of HABs (15 °);γmis the grain boundary energy of HABs calculated by

whereνis the Poisson’s ratio.Grain boundary mobilityMiis calculated by Eq.(12) [30].

whereMHABis the grain boundary mobility of HABs given by [31].

whereδis the characteristic grain boundary thickness;D0bis the grain boundary self-diffusion coefficien at 0K;kis the Boltzmann’s constant;Qbis the activation energy of grain boundary diffusion.The DMFPlis taken as the subgrain size and can be calculated by Eq.(14) [32].

whereKis a constant whose value is about 10 for most metals.

According to the model proposed by Ding and Guo [11],the nucleation rate ˙nfor DRX applied to calculate the amount of the nuclei generated per cellular automata step (CAS) has a quantitative relationship with temperature and strain rate:

whereCis a constant and the indexmis set to 1.

R-grains grow through the grain boundary migration under the driving force.The growth velocityviof theith R-grain is obtained by the product of the mobilityMiand the driving force per unit areafi.

The driving force in Eq.(16) comes from the energy changes caused by R-grain growth.Assuming that the shape of R-grain is spherical,the energy changes dEican be represented by the following formula:

Recently,some studies [17,19] have pointed out that for certain alloys,the influenc of the pinning effect of fin second-phase particles on grain boundary mobility cannot be neglected.In the present work,a correction factorkmsynthesizing the effects of second phases on the grain growth velocity is introduced.

where the values ofkmcan be obtained by inverse analysis[16].

Fig.7.Microstructure with PSN of Mg-Al-Zn-RE alloy.

Because the values of the coefficientCin the nucleation rate model andkmin the grain growth model are difficul to acquire from the experimental results,the method calibrating parameters by minimizing the deviation between the predicted results with the experimental ones is adopted [16].It is found thatCandkmdepend on temperature and strain rate,which is also proved by other studies [19,33].In other words,the coefficienCin Eq.(15) is no longer a material constant.A reasonable explanation is the effects of second phase particles on nucleation.There exist a considerable fraction of secondphase particles and particle clusters with sizes over 1 μm in the studied Mg-Al-Zn-RE alloy (Fig.7),causing PSN (denoted by red arrows in Fig.7) due to the large misorientation gradients and the high dislocation density around these coarse particles and particle clusters.Besides,the past research of the studied magnesium alloy showed that fin particles suppressed nucleation of R-grains by preventing the formation of lattice curvature necessary for nucleation [20].As the shape,size,volume fraction,and distribution of particles are not considered in the present work,the comprehensive influenc of second-phase particles on the nucleation rate is reflecte by the variation of theCvalue and the model of nucleation rate is modified

In the present work,the functional equations ofCandkmare considered as Eq.(20).

whereC0,mc,Qc,k0,mk,andQkare material constants evaluated by the nonlinear fitting ThenCandkmare quantitatively expressed as:

Modeling the influenc of shape,size,volume fraction,distribution and dynamic precipitation of the second-phase particles in the Mg-Al-Zn-RE magnesium alloy on nucleation and grain growth in the CA method is left for a further investigation as an improvement of the current model.

3.3.CA modeling for DRX in hot compression

To establish a practical and efficien CA model,the following assumptions are proposed.The initial dislocation density in all primary grains is uniform and identical.The dislocation density in each grain is uniform and increases withcontinuous deformation.Corresponding to the grain boundary bulging nucleation mechanism,DRX nuclei are only formed on the grain boundaries (including primary grain boundaries and R-grain boundaries).The initial dislocation density in the newly-formed R-grains is set to zero.

For fully characterizing the recrystallized process,each cell in the present CA model has fi e state variables including:

(1)The grain numberdistinguishes different grains and their grain boundaries.

(2)The orientationcalculates the grain boundary energy and mobility.

(3)The dislocation densityrepresents the stored energy.

(4)The migration distancedetermines the grain growth.

(5)The DRX rounddenotes the DRX times.

Considering the computational efficien y and simulation accuracy,the lattice scale of 1000×1000 and each cell size representing 1×1μm2in the real space are adopted in the present two-dimensional CA model.Hence the simulation area corresponds to 1×1 mm2in the real material.The von Neumann neighboring rule is selected.The periodic boundary condition is used to weaken the boundary effect.The values of the material constants input into the current CA model are given in Table 3.

The simulation procedure of the CA model for isothermal compression tests is summarized in Fig.8.The initial microstructure is created by a normal grain growth model.The orientations of grains in the matrix are randomly selected and the dislocation density of all cells is primitively set asρ0(ρ0=(σ0/αμb)2).According to the deformation conditions(temperature and strain rate),the values of CA model parameters are input.Dislocation density evolves with increasing strain.As critical dislocation densityρcis reached,nucleation and grain growth occur constantly by transferring the state variables of boundary cells following the specifi rules of evolution.

When the critical condition of nucleation is reached,i.e.the dislocation density of cells located on the boundaries of initial grains or R-grains exceedsρc,a certain number of cells belonging to these grain boundaries are selected randomly as the new nuclei of R-grains.The nucleation probabilityPnof each critical cell is:

whereΔtCAis the simulation time step in a CAS andd0is the side length of a cell.The maximum migration distance of cells should be less thand0in a CAS,and thus simulation time stepΔtCAis define as:

Fig.8.Flowchart of the DRX simulation using CA model for isothermal hot compression.

wherevmaxis the highest migration velocity.Radom numbers are assigned to each critical cell and then cells with smaller random numbers than the nucleation probability are considered as the new nuclei in the present CAS.Each nucleus consists of one cell [12].As described in the above assumption,the initial density of a new nucleus is set to zero.Meanwhile,a random orientation between 1 and 90 as well as a new grain number is assigned.The value of migration distance returns to zero and the DRX round increases by one.

The migration of grain boundaries in the CA model is realized by converting the state variables of cells belonging to the shrink grain boundaries to those of the adjacent expanded grains.To weaken the influenc of the discrete model on the accuracy,the following probabilistic rules are accepted to determine the transformation of cell states:

(1) For the boundary cell (i,j) at time stept+ΔtCA,its adjacent cells with positive growth velocities to the cell (i,j)and bigger DRX rounds are picked out.The selected cells are referred to as Candidate Cells Ⅰbelow.

(2) The sum of the swept area by Candidate Cells Ⅰis computed as

(3) If the swept area by Candidate Cells Ⅰ∑A(i,j)is larger than the area of a cell,the boundary cell (i,j) will change its state,i.e.it will be gobbled up by one of its neighboring grains.Otherwise,the state of the cell (i,j) remains unchanged in this CAS and the steps (4)-(6) are skipped.

(4) Threshold values are assigned to the Candidate Cells Ⅰin the form of following equation.

(5) Each cell in Candidate Cells Ⅰis given a random number in the range of 0-1 as the transformation probability and is added to Candidate Cells Ⅱon the condition that its random number is less than its threshold value.

(6) The state of the cell (i,j) is transformed to that of the cell with the maximum transformation probability among Candidate Cells Ⅱ.

Evolution of fl w stress,average grain size,DRX fractions,and microstructural topology with strain throughout simulation processes are recorded and output as the predicted results.The DRX fraction in the CA model is represented as:

wherenDRXis the number of recrystallized cells.To weaken the influenc of the huge difference between initial grain size and R-grain size on the authenticity of the calculated average grain sizedavg,the method incorporated a weight factor of grain area is employed as Eq.(27) [34].

whereNgis the total number of grains;Aiis the area of theith grain;Ais the total area of the investigated region (1 mm2);niis the number of cells within theith grain.

Fig.9.Simulation scheme of the coupled CA-FE model.

3.4.Coupled CA-FE modeling for hot extrusion

During the practical working process,the deformation is non-isothermal and heterogeneous.That is to say,temperature,strain,and strain rate shift severely with time and space,which directly influence the microstructural evolution of the workpiece.Since the traditional CA model excludes thermomechanical analysis,finit element (FE) analysis is utilized to capture the varying strain field strain rate field and temperature field

The simulation scheme of the coupled CA-FE model is described in Fig.9.The changes in process variables (strain,strain rate,and temperature) with time at local regions of workpieces are extracted from the results of FE analysis.Strain,strain rate,and temperature are assumed constant within each FE step (FES).The time step in an FESΔtFEis determined by the time increment and step interval for recording results presented before FE simulation.Generally,the time step in a CASΔtCAcalculated by Eq.(23) is much smaller thanΔtFE.Thus the CA model established above is adopted to predict the DRX behavior in each FES.As the simulation of microstructural evolution in theith FES is fin ished,state variables of cells in the CA model are transmitted to the (i+1)th FES and model parameters as well asΔtCAare recalculated according to the new temperature and strain rate.

In the present study,the coupled CA-FE model was applied in the hot extrusion of the Mg-Al-Zn-RE alloy plate.A three-dimensional (3D) FE model for the hot extrusion of magnesium alloy plate (Fig.10) was built using the commercial software DEFORM-3D.The geometry of the billet,extrusion stem and die was set as one-quarter of the experimental ones due to their symmetry.The billet and tooling were discretized with tetrahedral elements and refine meshes were employed for the billet and die bearing.The extrusion die and stem were considered as rigid bodies without deformation but with heat transferring.The billet of magnesium alloy was considered as rigid-plastic and the thermal-mechanical coupling effect was involved.

Fig.10.Quarter symmetrical FE model for the hot extrusion of the plate.

The constitutive data of the studied magnesium alloy was define by directly inputting the fl w stress-strain data of isothermal hot compression tests under different temperatures and strain rates (Fig.2).The constant heat transfer coeffi cient of 11N/(s·mm· °C) and a constant shear friction model with the friction coefficien of 1 were adopted [24].The thermal properties of the tooling and billet were reported in Ref.[24].Thermal and mechanical behavior during hot extrusion of magnesium alloy under the same process conditions as the experimental ones was simulated by the established FE model.

Fig.11.Compressive (a) fl w stress and (b-e) microstructure of Mg-Al-Zn-RE alloy at a temperature of 663K and a strain rate of 0.1 s-1.

Fig.12.Simulated and experimental microstructures of Mg-Al-Zn-RE alloy after hot compression at a strain rate of 0.1 s-1 and temperatures of (a,d) 633K,(b,e) 663K and (c,f) 693K.

4.Results and discussion

4.1.Microstructures formed in compressions

The simulated microstructural evolution of the Mg-Al-Zn-RE alloy during the compression at a temperature of 663K and a strain rate of 0.1 s-1together with the corresponding fl w stress curves are shown in Fig.11.The simulated microstructures as shown in Fig.11b-e,in which white,colored,and black regions represent the matrix,the R-grains with different orientations and grain boundaries respectively,indicate the evolution process of the microstructure during the compression.After a short induction period from the loading,new grain nuclei began to appear rapidly on the grain boundaries when the critical stressσcwas reached (Fig.11b).Then the nuclei grew up into equiaxed recrystallized grains through consuming the deformed matrix,leading to the formation of a typical necklace structure (Fig.11c).Meanwhile,new rounds of DRX were triggered once the pre-generated R-grains met the recrystallization criterion (Fig.11d),in which stage the volume fraction of R-grainsXDRXincreased sharply and thus the alloy was continuously softened.The fina microstructure was dominated by fin R-grains as fl w stress and DRX fraction tended to a steady-state,even though there still existed a few regions occupied by bulk matrix grains (Fig.11e).

The fl w stress in Fig.11 is slightly overestimated by simulation as strain is greater than 0.2.The reason for the discrepancy may be that the temperature rising which was induced by deformation heat and could result in fl w stress declining[13,35] was neglected.Besides,the additional nucleation sites induced by second-phase particles were not integrated into the simulation,causing a lower degree of DRX and higher fl w stress.

As revealed by the experimental results (Fig.2),deformation temperature and strain rate have significan impacts on the fl w behavior of the studied magnesium alloy.That is to say,the DRX process varies with deformation conditions.

Fig.12 illustrates the CA predicted and experimental grain topology at a strain rate of 0.1 s-1and different temperatures(633K,663K,and 693K).Both R-grain size and volume fraction of recrystallized grains became larger and larger as the temperature increased.It is known that R-grain size and DRX fraction are determined by nucleation rate and grain growth velocity.Under higher temperature,both nucleation rate and grain boundary mobility are increased with the rise of the thermally-activated motion of atoms and molecules in alloys.The influenc of temperature on grain growth velocity is greater than that on the nucleation rate.Thus,the average R-grain size and DRX fraction increased with temperature.

Fig.13.Simulated and experimental microstructures of Mg-Al-Zn-RE alloy after hot compression at a temperature of 663K and strain rates of (a,d) 0.01 s-1,(b,e) 0.1 s-1 and (c,f) 1 s-1.

Fig.14.Comparison of simulated and experimental (a) average R-grain size and (b) volume fraction of DRX during the compression of Mg-Al-Zn-RE alloy.

Fig.13 shows experimentally observed and CA simulated microstructures of magnesium alloy at a temperature of 663K and strain rates of 0.01 s-1,0.1 s-1,and 1 s-1.As the strain rate increased,the nucleation rate of DRX was largely enhanced and thus R-grains became fine DRX fraction decreased owing to higherρcfor nucleation (Eq.(9)) and less time for grain growth.

Fig.14 displays the histograms of average R-grain sizesdavgand DRX fractions,which were measured from experimental metallographs and simulated by the CA model.High consistency between the experimental and simulated results in Fig.14 proves the reliability of the established CA model.It should be pointed out that the average R-grain size of the Mg-Al-Zn-RE magnesium alloy calculated by the modifie CA model in this work was much smaller compared with the predicted results of AZ80 magnesium alloy under similar deformation conditions reported by Li et al.[36].As mentioned earlier,the new Mg-Al-Zn-RE magnesium alloy was developed by adding La and Gd to AZ80 magnesium alloy.Large quantities of second-phase particles appeared in the new magnesium alloy,which refine R-grains to a great extent.Therefore,it is concluded that the modifie CA model reproduced the grain refinemen promoted by second-phase particles successfully.

Fig.14 also demonstrates that the size of R-grains is very sensitive to deformation conditions.It is generally known thatdavgfollows the relationship as Eq.(28) [5],where the working conditions are expressed using the Zener-Hollomon parameterZ.

whereBdandpdare material constants.

The linear regression of the experimental and simulated average R-grain sizes with parameterZis plotted in Fig.15.It could be conducted that R-grains became fine as the value of the parameterZincreased.Quantitative expressions ofdavg(μm) for experiment and simulation are separately written as Eqs.(29-a) and (29-b).

Fig.15.Linear regression of experimental and simulated ln davg with ln Z.

In conclusion,the simulation implemented by the present CA model successfully reproduced the characteristic microstructural evolution of DRX observed in the experiments such as the formation of a necklace structure [37,38].The simulated fl w stress curve,DRX kinetics,and microstructure morphology agreed well with the experimental ones,indicating that the established CA model quantitatively predicted DRX behavior of the Mg-Al-Zn-RE alloy with high accuracy.

4.2.Microstructures formed in extrusions

4.2.1.Microstructural distribution during hot extrusion

Fig.16 illustrates the FE-predicted equivalent strain field equivalent strain rate fiel and temperature fiel of the extruded magnesium alloy at the initial billet temperature of 673K and the steady-state extrusion speed of 7.83mm/s,where the longitudinal cross-section of the extruded plate is shown.All strains and strain rates mentioned below refer to equivalent plastic strain and equivalent plastic strain rates.

Fig.16.Distributions of equivalent strain,equivalent strain rate,and temperature as well as the corresponding four zones of the Mg-Al-Zn-RE alloy near the die orifice extruded at the initial billet temperature of 673K and the extrusion speed of 7.83mm/s.

Fig.17.Positions of sampling points in (a) the undeformed billet and (b)extruded billet and plate for microstructure analysis.

According to the distribution of strain and strain rate as well as material fl w paths during extrusion [39,40],the magnesium near the die orific is roughly and qualitatively divided into four typical zones of direct extrusion.Material before passing through the outlet of the die bearing is divided into Zones I,II and III.Zone I (dead metal zone,DMZ) is around the corner of the container and characterized by low strain and strain rate.The material in this zone undergoes high hydrostatic pressure and almost stays still during extrusion.Therefore,a contour of strain is regarded as the boundary between Zone I and Zone II as shown in Fig.16.Zone II (shear intensive zone,SIZ) is the transition between Zone I with minor material fl w and Zone III with severe material fl w.Consequently,the material fl wing in Zone II undergoes large shear deformation,inducing high strain and strain rate.After being extruded out of the die bearing,the material in SIZ fl ws to the edge of the plate.Zone III,where the material directly fl ws to the die orifice is called the main deformation zone(MDZ).In Zone III,the dominant deformation is elongation,and strain,as well as strain rate,is smaller than those in SIZ.The material fl wing out of MDZ finall becomes the inner area of the plate.The boundary between Zone II and Zone III in Fig.16 is drawn based on a strain contour and material fl w paths.After passing through the die bearing,the material fl ws to Zone IV (exiting profil zone,EPZ).In this zone,the strain level is high while the strain rate decreases to zero.The schematic line denoting the location of the outlet of the die bearing is treated as the boundary between Zone IV and other zones in Fig.16.

Fig.18.Microstructural evolution of fi e points from the billet to the extruded plate.

Microstructural distribution in the four zones during extrusion was predicted by the coupled CA-FE model.By point tracking of FE simulation results,the strain,strain rate,and temperature of the fi e points (Points A,B,C,D,and E),whose initial positions and fl w history are demonstrated in Fig.17,were extracted and then were input into the CA model to simulate the microstructural evolutions during hot extrusion.With extrusion time increasing to 10.3,15.1,20.5,22.8,23.2 and 25.5s,the material at Point X0 (X refers to A,B,C,D or E) successively fl wed to Points X1-X5 in Fig.17.The positions of the four fl wing points (Points B,C,D and E) at the same extrusion time are connected by blue dotted lines.As extrusion reached a steady-state,microstructural evolution from Points X1 to X5 also represented the microstructural distribution in the four zones.

The microstructural evolution of Points A,C,and E are indicated in Fig.18a.The R-grains in Zone I (Point A) were much coarser than ones in other zones.This is because Point A was nearly motionless and thus the strain rate was kept at quite small values (less than 0.015 s-1),which induced a low nucleation rate and more time for grain growth.It is worth noting that,in Fig.18a,the DRX fraction at Point A1 is larger than that at Points C1 and E1.The reason is that the DRX process started earlier for Point A than that for other points.At the beginning of extrusion,material in the billet didn’t arrive at the die orifice and the main deformation concentrated in Zone I where the material fille the corner of the die.During this period,strain at Point A increased to the critical value for DRX and nucleation and subsequent growth of R-grains occurred,while strain at other points almost remain zero and DRX was not triggered.

Similar microstructural evolutions of Points C and E were found.Before arriving at the die orifice material gradually fl wed from Point C1(E1) to Point C3(E3) and was continually deformed at a relatively low strain rate.During this period,the recrystallization was triggered and the recrystallized fraction increased with strain until the material was fully recrystallized.As material fl wed to the die bearing (Points C4 and E4),the strain rate increased drastically and an evident temperature enhancement was observed due to severe deformation,leading to grain refinemen to some extent.Under the large strain rate and high temperature,both nucleation and grain growth was promoted.The reduction of R-grain size is explained as the effect of the higher nucleation rate on Rgrain refinin prevailed the effect of the faster grain growth on R-grains coarsening.At the end of the deformation at the bearing (Points C5 and E5),the strain rate rapidly dropped to a very low value while the temperature maintained high,which promoted grain growth and caused an R-grain size increase.Once extruded out of the die bearing,the material was water quenched and the microstructure entirely composed of fin R-grains was reserved (Points C6 and E6).

Although microstructural evolutions in Zones II (Point C)and III (Point E) were similar,R-grain size varied with strain rates and temperatures in the two zones.To further distinguish the microstructural distribution in Zones II,III,and IV,the microstructures of Points B,C,D,and E at extrusion time of 10.3s and 25,5s are given in Fig.18b.As revealed by the microstructures at Points B1,C1,D1,and E1,the DRX fraction was larger at points in Zone II (Points D1 and E1)than that at points in Zone III (Points B1 and C1).Lee et al.[17] pointed out that strain is one of the principal parameters to determine the microstructure at the partially recrystallized area.Larger strain results in more grains satisfying the critical criterion for DRX and thus a larger DRX fraction.Due to the higher strain level in Zone II,the recrystallized area was bigger and the DRX process was faster.

Fig.19.Comparison of average R-grain size between experimental measurement and simulated results at different points.

The microstructure in Zone IV depended on the recrystallization history of the material.The average grain sizes at Points B6,C6,D6 and E6 were 9.83μm,11.34μm,12.47μm and 12.97μm,respectively.In the inner of the extruded plate (Points D6 and E6),the microstructure was homogeneous due to uniform deformation in Zone III.While R-grain size near the edge of the plate (Points B6 and C6) decreased significantl because of the high strain rate and low deformation temperature in Zone II.

The simulated and experimental average grain size in Zone IV is compared in Fig.19.From experimental and CA-FE coupled simulations,the grain size varying with the position shows the same tendency.The essentially good agreement between the predicted and experimental results proved the capacity of the established method to reproduce the microstructural evolution during hot extrusion and verifie the coupled CA-FE model.

4.2.2.Effects of extrusion process parameters on DRX

The microstructural evolution under the same extrusion conditions as ones of Extrusion tests 1,2,and 3 were simulated by the coupled CA-FE model.The experimental and predicted microstructures after extrusion on the symmetrical axis of the as-extruded pates are displayed in Fig.20.Under all investigated extrusion conditions,due to DRX processes during hot forming,grains were refine intensely after extrusion compared with ones in solution-treated state (Fig.1).Higher extrusion temperature or speed resulted in coarser grains.For the alloy extruded under high extrusion speed,though larger strain rate benefitte grain refinement temperature enhancement induced by severe deformation promoted the grain growth to a great extent.

Fig.20.Simulated and experimental microstructures after extrusion under conditions of (a,d) 623K &7.83mm/s,(b,e) 673K &7.83mm/s,(c,f) 623K &81.67mm/s.

Fig.21.Results of grain size from experiments and simulations.(a) Average grain size at different extrusion conditions;grain size distribution under extrusion conditions of (b) 623K &7.83mm/s,(c) 673K &7.83mm/s and (d) 623K &81.67mm/s.

Fig.21 compares the experimental and simulated average grain size as well as grain size distribution under various process parameters.Both experimental and simulated results showed that the microstructures at lower temperature or higher extrusion speed were more homogeneous.With temperature rising or strain rate decreasing,grains grew more rapidly and the small grains generated later scattered on the boundaries of the coarse grains that have grown up.It was found that though the average grain sizes of experimental and simulated results were almost identical,there were some discrepancies for the grain size distribution.The number of grains with small sizes was underestimated and the microstructures were more even for simulation,which is because the CA parameters were calibrated by average grain size rather than grain size distribution.In general,the coupled CA-FE model was a powerful method to predict DRX behavior during the practical forming process within an acceptable error range.

5.Conclusions

The DRX behavior of an Mg-Al-Zn-RE alloy during hot extrusion is studied by the coupling simulations of CA and FE.The good agreement between the simulated and experimental results reveals the established method is reliable to predict the extruded microstructure of magnesium alloy.The following conclusions are drawn.

(1) The modifie CA model considers the effects of second phase particles on DRX nucleation and grain growth.Grain refinemen of the Mg-Al-Zn-RE alloy by second-phase particles is successfully simulated.The CA model parameters including the softening coeffi cient,the saturation stress,the coefficien of nucleation rate,and the correction factor of grain growth velocity are calibrated by isothermal compression experiments.

(2) The established CA model is coupled with finit element model to simulate microstructural evolution during practical forming process with varying deformation conditions.The coupled CA-FE model is applied in the extrusion of magnesium alloy plates.The simulated grain morphology,grain size,and grain distribution are validated by the extrusion experiments.

(3) During the plate extrusion,due to the higher strain and strain rate,the DRX process is faster and the R-grains are fine in the shear intensive zone (SIZ) compared with the main deformation zone (MDZ).Materials in MDZ and SIZ finall fl w to the inner and edge of the exiting profil zone (EPZ),resulting in average grain size in EPZ decreasing from about 13μm at the core to about 10μm near the edge at the initial billet temperature of 673K and the extrusion speed of 7.83mm/s.

(4) The billet’s grains with an average size of 372μm are dramatically refine to 6-13μm in extruded plates,depending on the employed processing conditions.High temperature or extrusion speed leads to coarse grains of the extruded plate of the magnesium alloy.At the initial billet temperature of 623K and the extrusion speed of 7.83mm/s,the fines grain size (about 6μm) is obtained.With decreasing temperature or increasing extrusion speed,the plate with more homogeneous microstructures are extruded.

Acknowledgments

The authors greatly acknowledge the financia support of the National Key Research and Development Program of China (No.2017YFB0701801) and the National Natural Science Foundation of China (No.51675300).