materials Article
Effects of Annealing on the Residual Stress in γ-TiAl Alloy by Molecular Dynamics Simulation Ruicheng Feng 1,2 and Longlong Li 1 1
2
*
ID
, Wenyuan Song 1 , Haiyan Li 1,2, *, Yongnian Qi 1 , Haiyang Qiao 1
School of Mechanical and Electronical Engineering, Lanzhou University of Technology, Lanzhou 730050, China;
[email protected] (R.F.);
[email protected] (W.S.);
[email protected] (Y.Q.);
[email protected] (H.Q.);
[email protected] (L.L.) Key Laboratory of Digital Manufacturing Technology and Application, Ministry of Education, Lanzhou University of Technology, Lanzhou 730050, China Correspondence:
[email protected]; Tel.: +86-150-0251-2021
Received: 21 May 2018; Accepted: 13 June 2018; Published: 15 June 2018
Abstract: In this paper, molecular dynamics simulations are performed to study the annealing process of γ-TiAl alloy with different parameters after introducing residual stress into prepressing. By mainly focusing on the dynamic evolution process of microdefects during annealing and the distribution of residual stress, the relationship between microstructure and residual stress is investigated. The results show that there is no phase transition during annealing, but atom distortion occurs with the change of temperature, and the average grain size slightly increases after annealing. There are some atom clusters in the grains, with a few point defects, and the point defect concentration increases with the rise in temperature, and vice versa; the higher the annealing temperature, the fewer the point defects in the grain after annealing. Due to the grain boundary volume shrinkage and and an increase in the plastic deformation of the grain boundaries during cooling, stress is released, and the average residual stress along Y and Z directions after annealing is less than the average residual stress after prepressing. Keywords: residual stress; molecular dynamics; γ-TiAl alloy; anneal
1. Introduction Due to its low density, high specific strength, excellent high-temperature properties, good oxidation resistance and creep resistance, γ-TiAl alloy is considered to be the high-temperature structural material with the most potential. It has wide application prospects in many fields, such as aerospace and so on [1–4]. However, poor ductility at room temperature greatly restricts the application of γ-TiAl alloy, and introduces residual stress during its forming and processing; the main problems of γ-TiAl alloy are surface deformation, higher surface roughness and residual stress, defects that will become initial crack-extension points, leading to workpiece failure [5,6]. Residual stress is also the root cause of workpiece deformation during the manufacture–service process. The magnitude and distribution of residual stresses have played a major role in the stability of the workpiece, which directly affects its service life and mechanical properties [7]. Lots of work has been done to obtain the ideal residual stress distribution and improve the service life of the workpiece, especially on the relationship among process parameters and the generation mechanism of residual stress and the method of eliminating that stress. Pawade et al. [8] proved the effects of cutting speed, cutting depth, feed rate and tool geometry parameters on residual stress by an experimental method; a residual compressive stress could be produced on a machined surface by adjusting cutting parameters and tool shape properly. Cheng et al. [9] combined the cutting Materials 2018, 11, 1025; doi:10.3390/ma11061025
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experiment results with finite element analysis methods, and found that the combination of cutting force and cutting heat produced high temperature and high pressure when acting on the rake face of the cutting tool, and inhomogeneous plastic deformation were introduced which led to residual stress. The effect of cutting force and temperature on residual stress in the milling process has been investigated by Jiang et al. [10], and the results show that the cutting force plays a leading role in the residual stress, and the tangential residual stress was mainly caused by the tangential force and temperature. Szczepan et al. [11] has studied the residual stresses in a hot-rolled steel strip during cooling in coils, and found that the phase transformations have a significant influence on the level of residual stress. Xiao et al. [12] analyzed the thermal residual stress in glass/glass laser bonding, and the results show that the scale of the temperature field control is closely related to the residual stress, and that by combining the point, line and surface-scaled control, the thermal residual stress can be reduced. A new multistage aging method has been exploited in reducing residual stresses in the quenching process by Sun et al. [13]. Laser shock peening of austenitic stainless steel has been researched by Prabhakaran et al. [14], who found that the residual compressive stress of the material can be increased by adjusting the pulse parameters. R Sola et al. [15,16] used the experimental methods to study the application of the cryogenic treatment and post-tempering cryogenic treatment of AISI M2, AISI D2, X105CrCoMo18 steels. The results show that the precipitation of carbides that occurs during heating from the cryogenic treatment temperature is responsible for the residual stress relaxation, and the precipitation of more hard carbides in the cryogenically treated samples can reduce residual stresses and also enhance the steel fracture toughness. Shao H et al. [17] has investigated joule-induced microstructure evolution and residual stress in a Ti–Al–4V U-shaped screw by an experimental method, and the results show that dislocation density decreases with increasing heated time; joule heating at 900 ◦ C is sufficiently high to enhance the dislocation mobility and the rearrangement, causing the recrystallization of the alpha phase and the change of residual stress. The effect of annealing on the microstructure and residual stress of zirconium has been researched by Zhang C H et al. [18], who found that grain size increases after annealing, and that the decrease of dislocation density and the rearrangement of dislocation after annealing are closely related to the release of residual stress. Through a large number of experiments, it has been found that the generation and elimination of residual stress in a workpiece, and the mechanical properties of workpiece material, are closely related to the microstructure of the material. Yang et al. [19] puts forward the concept of “making materials plain” that considers that the material properties can be improved by regulating the defects of materials at different scales. Wawszczak et al. [20] has studied the evolution of microstructure and residual stress during annealing at different temperatures. The results show that the stress in the samples would be relaxed subjected to heat treatment, and the stress in the samples was correlated with the progress of recovery process which depends on the value of stacking fault energy. Marzbanrad et al. [21] has found grain refinement on the substrate surface results in higher residual compressive stress during a cold spray process. The evolution of microstructure and residual stress during rapid thermal annealing is observed by Hsiao et al. [22]. It is considered that the tensile stress of the film originates from the annihilation of L10 grain boundaries in single-layered FePt films. Zhao et al. [23], has found that the lattice strain caused by thermal expansion mismatch between perovskite and substrate is an important factor affecting the stability of perovskite solar cells. This residual strain is caused by the annealing process in the preparation of perovskite thin films. Nakano et al. [24] has explained the relationship between dislocation density and residual stress in a GaN single crystal during the cooling process by numerical analysis, with the results showing that the residual stress increases with the rise of dislocation density during cooling. As a powerful supplement to the experiment and the verification of theoretical model, molecular dynamics (MD) methods are used to simulate the relationship among microdefects’ evolution, processing parameters and the mechanical properties of the processed materials [25–28]. Although ample research has been conducted on the evolution of microstructure and residual stress, there is little research on the distribution of residual stress in annealing of γ-TiAl alloys. In this
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paper, we seek to better understand: the effects of annealing on residual stress and improve the microdefects evolution during annealing, and the relationship between microstructure and residual annealing process of γ-TiAl alloy; and investigate the distribution of residual stress after annealing, stress with MD simulations. In Section 2, the simulation model and details will be introduced. In theMaterials microdefects 2018, 11,evolution x FOR PEERduring REVIEWannealing, and the relationship between microstructure and residual 3 of 13 Section 3, the simulation results will be obtained and the corresponding analysis undertaken. Finally, stress with MD simulations. In Section 2, the simulation model and details will be introduced. some conclusionsevolution will be drawn in Section 4.and the relationship between microstructure and residual microdefects In Section 3, the simulationduring resultsannealing, will be obtained and the corresponding analysis undertaken. Finally, stress with MD will simulations. some conclusions be drawnIninSection Section2,4.the simulation model and details will be introduced. In 2. Simulation Details Section 3, the simulation results will be obtained and the corresponding analysis undertaken. Finally, 2. Simulation Details some conclusions will be drawn in Section 4. 2.1. Interatomic Potential 2.1.2.Interatomic Potential Simulation The effects ofDetails vacancy concentration and temperature on mechanical properties of single-crystal γ-TiAlThe have beenof carried outconcentration by using the Large Scale Atomic/Molecular Massively Parallel Simulator effects vacancy and temperature on mechanical properties of single-crystal 2.1. Interatomic Potential (Sandia National Laboratories, Albuquerque, NM, USA) (LAMMPS) [29]. It is widely believed that γ-TiAl have been carried out by using the Large Scale Atomic/Molecular Massively Parallel Simulator effectsLaboratories, of vacancy concentration and temperature on mechanical properties of single-crystal interatomic potentials are important for MD simulations, the selection ofisthem critically affects (SandiaThe National Albuquerque, NM, USA) and (LAMMPS) [29]. It widely believed that γ-TiAl have carried out by using Scale Atomic/Molecular Massively Parallel the accuracy of been the MD simulation. For example, the embedded method (EAM) hasSimulator been used interatomic potentials are important forthe MDLarge simulations, and the atom selection of them critically affects the (Sandia National Laboratories, Albuquerque, NM, USA) (LAMMPS) [29]. It is widely believed that to study phase of Ti–Al alloy andthe theembedded defects and theirmethod evolution on crack accuracy of thetransformation MD simulation. For example, atom (EAM) has propagation been used to interatomic potentials important for MD and selection of them critically affects behavior [30,31]. In this are paper, EAMalloy is employed to describe the interaction atoms between study phase transformation of Ti–Al andsimulations, the defects and the their evolution onof crack propagation the accuracy of the MD simulation. For example, the embedded atom method (EAM) has been used materials. behavior [30,31]. In this paper, EAM is employed to describe the interaction of atoms between materials. to study phase transformation of Ti–Al alloy and the defects and their evolution on crack propagation In (MD) this paper, 2.2. Molecular[30,31]. Dynamics (MD) ModelEAM is employed to describe the interaction of atoms between 2.2. behavior Molecular Dynamics Model materials. Thecrystal crystalstructure structureof ofγ-TiAl γ-TiAl alloy alloy is is L1 L100[32,33] [32,33]which whichisisshown shownin inFigure Figure1;1;the thelattice latticeconstants constants The area2.2. a==4.001, 4.001,bb==4.001, 4.001,cc==(MD) 4.181, respectively. are 4.181, respectively. Molecular Dynamics Model The crystal structure of γ-TiAl alloy [001]is L10 [32,33] which is shown in Figure 1; the lattice constants are a = 4.001, b = 4.001, c = 4.181, respectively. [001]
Ti Al [010] 1 / 2[110]
Ti Al
[010] 1 / 2[110]
[100]
Figure 1. The crystal structure of γ-TiAl alloy. Figure 1. The crystal structure of γ-TiAl alloy. [100] The ATOMSK crystal structure of γ-TiAl alloy. of a Voronoi tesselation to The initial model is createdFigure using1. the package by means The initial model is created using the ATOMSK package by means of the a Voronoi tesselation to construct polycrystals, which is shown in Figure 2, where the green area is grain and the white construct polycrystals, which is shown in Figure 2, where the green area is the grain and the white The initial model is created using the ATOMSK package by means of a Voronoi tesselation to area is the grain boundary. construct polycrystals, which is shown in Figure 2, where the green area is the grain and the white area is the grain boundary. area is the grain boundary.
Z Y
Z
X
Y X Figure 2. The simulation model. Figure2.2.The Thesimulation simulation model. model. Figure
The model contains 239,084 atoms and has a size of 200a × 200b × 100c which contains eight unit The model contains atoms hasdirections a size of 200a × 200b × 100c which conditions. contains eight unit cells with different random 239,084 directions; alland three are periodic boundary cells with different random directions; all three directions are periodic boundary conditions.
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The model contains 239,084 atoms and has a size of 200a × 200b × 100c which contains eight unit cells with different random directions; all three directions are periodic boundary conditions.4 of 13 Materials 2018, 11, x FOR PEER REVIEW 2.3. 2.3.Definition DefinitionofofResidual Residual Stress Stress The the model modelisiscompletely completelydefined definedbyby nine stresses, and be expressed Thestress stressat atany any point point in in the nine stresses, and cancan be expressed asasa asecond-order tensor as follows [34]: second-order tensor as follows [34]: σσ1111 σσ1212 σσ1313 σ21 σ22 σ23 σσij == , (1) (1) ij σσ21 σσ22 σσ23 , 31 32 33
σ 31
σ 32
σ 33
InInthis paper, the stress after fully relaxing is defined as residual stress which is calculated by this paper, the stress after fully relaxing is defined as residual stress which is calculated by virial virialtheory theory[35]. [35].Moreover, Moreover,the theresidual residualstress stressisisobtained obtainedfor foreach eachatom atombybyaveraging averagingthe theoutput outputdata ofdata the of standard LAMMPS command “compute stress/atom” over the region within a range the standard LAMMPS command “compute stress/atom” over the region within a range ofof22 Å along the directions. Define σxrs , σσyrs σyrs stressstress in theinX,the Y, ZX,directions. zrs, are Å along corresponding the corresponding directions. Define xrs,, σ σzrs the are residual the residual Y, Z The residualThe tensile stress was stress introduced into the model a 2%by prepressing deformation in the Z directions. residual tensile was introduced into theby model a 2% prepressing deformation direction annealing. The distribution of σxrs , σof , xrs σzrs is shown in Figure in the Z before direction before annealing. The distribution , σafter yrs, σzrsprepressing after prepressing is shown in 3. yrs σ The maximum residual tensile stress in the X, Y,inZthe directions 443.75 MPa, 447.25 MPa, 489.875 MPa, Figure 3. The maximum residual tensile stress X, Y, Z is directions is 443.75 MPa, 447.25 MPa, respectively, and the corresponding depths are 54depths Å, 28 Å, 489.875 MPa, respectively, and the corresponding are4054Å. Å , 28 Å , 40 Å . xrs
500
yrs zrs
Residual stress(MPa)
400
300
200
100
0 0
50
100
150
200
depth(Å)
-100
Figure 3. The after prepressing. Figure The residual residualstress stressininthe theX,X,Y,Y,ZZdirections directions after prepressing.
TheMD MD using a constant-pressure,constant-temperature ensemble (NPT ensemble), firstly The using a constant-pressure, constant-temperature ensemble (NPT ensemble), firstly relaxes relaxes at 30 ps, and the annealing simulation is carried out after reaching equilibrium, using the at 30 ps, and the annealing simulation is carried out after reaching equilibrium, using the Nose–Hoover Nose–Hoover thermostat for the temperature control. The annealing processes are divided into four thermostat for the temperature control. The annealing processes are divided into four cases, and the cases, and the annealing parameters are shown in Table 1. The timestep is 0.001 ps; kinetic energy, annealing parameters are shown in Table 1. The timestep is 0.001 ps; kinetic energy, potential energy potential energy and total energy are recorded every 500 steps during annealing.
and total energy are recorded every 500 steps during annealing.
Table 1. Annealing parameters. Table 1. Annealing parameters.
Case Annealing Temperature (K) Annealing Case
1 2 3 4
Temperature (K)
700
1 2 3 4
700700
700 900900 1100 1100
Heating Rate Heating Rate (K/ps) (K/ps)
2
22 2 22 22
Cooling Rate Holding Temperature Cooling Holding (K/ps) Rate (K) (K/ps)
2
12 1 22 22
Temperature (K)
700
700 700 700
700, 900 700, 900 700, 1100 700, 1100
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3. Results 3. Results
Fcc Other
80
Atomic number of structure(%)
Atomic number of structure(%)
3.1. Structural Evolution Evolution in in Annealing 3.1. Structural Annealing Taking as an an example, example,the theatomic atomicnumber numberofofthe thestructure structureduring during annealing is shown Taking Case Case 11 as annealing is shown in in Figure 4. In this paper, common neighbor analysis (CNA) is employed to analyze the atoms’ Figure 4. In this paper, common neighbor analysis (CNA) is employed to analyze the atoms’ distortion. The γ-TiAl γ-TiAl alloy alloy atoms atoms with with L1 -type structure structure belong belong to to the the recognizable distortion. The L100-type recognizable FCC FCC lattice lattice structure, structure, while while the the lattice lattice structures structures corresponding corresponding to to the the atoms atoms at at the the grain grain boundary boundary are are atypical atypical other In order order to to avoid of surface other lattice lattice structures. structures. In avoid the the interference interference of surface atoms, atoms, the the surface surface atoms atoms were were selected and then eliminated by centrosymmetric parameters. The number of FCC atoms decreases selected and then eliminated by centrosymmetric parameters. The number of FCC atoms decreases gradually increase during means that that the the grain grain boundary boundary gradually while while the the other other type type atoms atoms increase during heating, heating, which which means volume expands when the temperature rises, and the grains are compressed. By observing the output volume expands when the temperature rises, and the grains are compressed. By observing the output of shown inin Figure 5, of the the internal internal energy energy diagram diagramand andthe theradial radialdistribution distributionfunctions functions(RDF) (RDF)curve curveasas shown Figure it can found that the internal energies persistently and smoothly increase with the rise of temperature. 5, it can found that the internal energies persistently and smoothly increase with the rise of The RDF curves thatillustrate the structure classic crystal state shape elevated temperature. Theillustrate RDF curves that is thea structure is a classic crystalatstate shapetemperature, at elevated there are fourthere independent peaks, and the value ofthe curves peaks is zero, which indicates temperature, are four independent peaks, and valuebetween of curves between peaks is zero, which that the lattice structure of the grain has good long-term order. There is a certain phenomenon of indicates that the lattice structure of the grain has good long-term order. There is a certain broadening of of thebroadening peaks during heating, means that which the crystal atoms exactly in the phenomenon of the peakswhich during heating, means thatare thenot crystal atoms areideal not lattice position, and the lattice order is reduced. The FCC atoms increase and the other atoms decrease exactly in the ideal lattice position, and the lattice order is reduced. The FCC atoms increase and the when cooling; also, thewhen internal energies the dropdecrease of temperature, RDF other atoms decrease cooling; also,persistently the internaldecrease energieswith persistently with thethe drop of curve peak width decreases gradually, and the lattice order increases gradually when the temperature temperature, the RDF curve peak width decreases gradually, and the lattice order increases gradually drops. After cooling down to 300 K and fully relaxing, theKaverage grain size slightly increased when the temperature drops. After cooling down to 300 and fully relaxing, the average grainfrom size 4.639 nm to 4.643 nm. In combination with the variation of internal energy, the RDF curve and the slightly increased from 4.639 nm to 4.643 nm. In combination with the variation of internal energy, atomic number of the it is found thatstructure, there is noitphase transition during but the the RDF curve and thestructure, atomic number of the is found that there is noannealing, phase transition atoms’ distortion occurs with the change of temperature. during annealing, but the atoms’ distortion occurs with the change of temperature.
70 60 50 40 30 20 300
400 500 600 Temperature(K)
700
Fcc Other
80 70 60 50 40 30 20 700
600
500
Temperature(K)
(a)
400
300
(b)
Figure 4. 4. The (a) the Figure The atomic atomic number number of of the the structure structure during during annealing: annealing: (a) the atomic atomic number number of of the the structure structure during heating; (b) the atomic number of the structure during cooling. during heating; (b) the atomic number of the structure during cooling. Internal energy
6
-1.044x10
3.2. Microdefects Evolution during the Annealing 300K
Internal energy(eV)
6 400K 8 -1.046x10 500K To observe the evolution of the microdefects, their distribution after prepressing is shown in 6 600K -1.048x10 Figure6 6, with use of the CNA analysis to700K identify the defect atoms and then delete the non-defective 6 -1.050x10
g(r)
normal atoms. After prepressing plastic deformation, there are 6some atom clusters in the crystal grains, -1.052x10 some 4atom clusters are neatly arranged in a ring, some atom clusters are gathered together disorderly 6 -1.054x10 with few dislocations, and there are a few point defects in the crystal grains. There are different types 6 -1.056x10 of dislocations at the grain boundaries after prepressing. 2 6 -1.058x10
6
0
-1.060x10
2
3
r(Å)
4
(a)
5
6
300
400 500 600 Temperature(K)
(b)
700
At
20
20 300
400 500 600 Temperature(K)
700
700
600
(a)
500
Temperature(K)
400
300
(b)
Figure 4.11, The atomic number of the structure during annealing: (a) the atomic number of the structure 6 of 12 Materials 2018, 1025 during heating; (b) the atomic number of the structure during cooling.
g(r)
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-1.044x10
Internal energy
6
-1.046x10
6
Internal energy(eV)
8
300K 400K 500K 600K 700K
-1.048x10
6
-1.050x10
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6
-1.052x10
Figure 5. The radial distribution functions (RDF) curve during annealing: (a) the RDF curve during 4 6 -1.054x10 heating; (b) the internal energy curve during heating. 2
3.2. Microdefects Evolution during the Annealing
6
-1.056x10
6
-1.058x10
6
-1.060x10
To0observe the3 evolution of5 the microdefects, their distribution after prepressing is shown in 2 4 6 300 400 500 600 700 ) Figure 6, with use of ther(Å CNA analysis to identify the defect atoms and then Temperature(K) delete the non-defective (a) (b)clusters in the crystal normal atoms. After prepressing plastic deformation, there are some atom grains, some atom clusters are neatly arranged in a ring, some atom clusters are gathered together Figure 5. The distribution (RDF) curve during annealing: the RDF curveThere duringare disorderly with fewradial dislocations, andfunctions there are a few point defects in the (a) crystal grains. heating; (b) the internal energy curve during heating. different types of dislocations at the grain boundaries after prepressing.
Figure 6. Microdefects in the initial model after prepressing: (a) (a) the the atom clusters in the clusters’ Figure 6. Microdefects in the initial model after prepressing: atom clusters in the clusters’ grains; (b) a Ti anti-site point defect (blue atoms are the Ti atoms, red atoms are the Al atoms); (c) thethe grains; (b) a Ti anti-site point defect (blue atoms are the Ti atoms, red atoms are the Al atoms); (c) dislocations at the atom clusters; (d) the dislocations at the grain boundaries. dislocations at the atom clusters; (d) the dislocations at the grain boundaries.
The change of temperature has a great influence on the point defect concentration of Ti–Al alloy The change of temperature has a great influence on the point defect concentration of Ti–Al [36]. In general the energy of the point defect formation is based on the Arrhenius equation which alloy [36]. In general the energy of the point defect formation is based on the Arrhenius equation can be given as follows: which can be given as follows: Q Q , C = Aexp − C = Aexp − (2) (2) RTRT , where C is point defect concentration, A is the equilibrium constant, Q is the point defect formation where C is point defect A is the constant,According Q is the point defect formation activation energy, R is concentration, the molar constant, andequilibrium T is the temperature. to Equation (2), it can be activation energy, R is the molar constant, and T is the temperature. According to Equation (2), it can derived that the point defect concentration increases with the rise of temperature. The variation trend be of derived thatdefect the point defect concentration increases with the rise ofwith temperature. The variation the point concentration in the simulation results is consistent the theoretical calculation. trend of the that point in the simulation results consistent We found thedefect grain concentration boundaries precipitate atoms to form the is atom clusterswith whenthe thetheoretical temperature calculation. We found that the grain boundaries precipitate atoms to form the atom clusters when thethe rises by tracing the atomic trajectory, as shown in Figure 7a. The atom which precipitated from temperature rises by tracing the as shown in 7b); Figure The atom iswhich grain boundaries firstly tends to atomic move totrajectory, the atom cluster (Figure this7a. phenomenon similar precipitated from the grain boundaries firstly tends to move to the atom cluster (Figure 7b); to [37]. It was found that small vacancy clusters have the ability to capture vacancies. Thethis atom phenomenon is similarduring to [37].cooling It was and found small vacancy clusters have the ability to represents capture clusters decompose thethat atoms enter the grain boundaries. Figure 7c–f vacancies. The atom clusters decompose cooling the4 atoms enter the respectively. grain boundaries. the distribution of point defects of Case during 1, Case 2, Case 3,and Case after annealing, It can be Figure 7c–f represents the distribution of point defects of Case 1, Case 2, Case 3, Case 4 after seen that the higher the annealing temperature, the fewer the point defects in the grain after annealing. annealing, respectively. It can time, be seen thatannealing the higheratthe annealing temperature, the exists fewer in thethe point Due to insufficient cooling after 1100 K a vacancy cluster still grain defects in the grain after annealing. Due to insufficient cooling time, after annealing at 1100 K a which cannot decompose completely, and the difference in the distribution of point defects between vacancy cluster still exists in the grain which cannot decompose completely, and the difference in the distribution of point defects between Case 1 and Case 2 after annealing is very small. According to [36,37], the enthalpy of formation and the formation energy of the anti-site defect is less than that of the vacancy defect, suggesting the anti-site defect can be seen in the grain after annealing.
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Case 1 and Case 2 after annealing is very small. According to [36,37], the enthalpy of formation and the formation energy of the anti-site defect is less than that of the vacancy defect, suggesting the anti-site defect can be seen in the grain after annealing. Materials 2018, 11, x FOR PEER REVIEW
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Figure 7. The distribution of the point defect after annealing. precipitate atoms from which Figure 7. The distribution of the point defect after annealing. (a) (a) thethe precipitate atoms from which from from grain boundaries comes when temperature rises, and the gray lines are the atomic trajectory; (b)the grain boundaries comes when temperature rises, and the gray lines are the atomic trajectory; (b) the atoms white atoms the atom clusters are precipitated the grain boundaries; white adsorbadsorb on theon atom clusters whichwhich are precipitated fromfrom the grain boundaries; (c–f)(c–f) is the is the distribution of the point defects in the grain after annealing in Case 1, Case 2, Case 3, Case distribution of the point defects in the grain after annealing in Case 1, Case 2, Case 3, Case 4. 4.
3.3. Residual Stress Distribution after Annealing with Different Parameters
3.3. Residual Stress Distribution after Annealing with Different Parameters The distribution of σxrs, σyrs, σzrs after annealing with different parameters is shown in Figures 8– The distribution of σxrs , σyrs , σzrs after annealing with different parameters is shown in 10. To observe the distribution of residual stress in three directions more intuitively, non-linear fitting Figures 8–10. To observe the distribution residualInstress directions more intuitively, of the data is carried out by using rational of functions. Figurein8,three the fitting average value of the non-linear the data is carried out by annealing using rational functions. In Figure 8, the residual fitting stress of along the X-direction after is 184 to 212 MPa, and the σxrsfitting curveaverage after value of the residual along the X-direction is 184 212 MPa, and the σxrs curve prepressing almoststress overlapped with Case 3. after The annealing distribution of σto xrs with different annealing after prepressing almost overlapped Case and 3. The of fluctuation σxrs with different parameters in the specific position iswith different, thedistribution residual stress is small annealing in other parameters thedifference specific position is different, and the is residual stress fluctuation in other positions. in The in residual stress distribution larger at 50–72 Å in Caseis1,small and the σxrs positions. The difference in residual stress distribution is larger at 50–72 Å in Case 1, and the σ shows shows a sharp drop in Case 2 at 110 Å to 116 Å . In Case 3 and Case 4, the residual stress decreases xrs a sharp drop in increases Case 2 at at 110 Å to 116 Å. In Case 3Å and Case 4,9,the stress decreases firstthe and first and then 112–116 Å and 150–160 . In Figure the residual average residual stress along Y direction annealing in all four cases less than theaverage averageresidual residualstress stressalong after the prepressing. then increases after at 112–116 Å and 150–160 Å. InisFigure 9, the Y direction The higher the annealing temperature, smaller residual the average residual stress and The at the samethe after annealing in all four cases is less than the average stress after prepressing. higher annealing temperature it can obtain smaller average residual stress after annealing at a slower cooling annealing temperature, the smaller the average residual stress and at the same annealing temperature rate.obtain The σsmaller yrs increases sharply at 82stress Å to 90 Å ,annealing 108–112 Å atina Case 1 and Caserate. 4, respectively. The it can average residual after slower cooling The σyrs increases residual stress of Case 3 decreases in the range of 174 Å to 178 Å ; the difference of σ yrs distribution in sharply at 82 Å to 90 Å, 108–112 Å in Case 1 and Case 4, respectively. The residual stress of Case 3 Case 2 before Å isofsmall, it increases first and then from the range of 184 Å to 200 decreases in the184 range 174 Åand to 178 Å; the difference of σdecreases yrs distribution in Case 2 before 184 Å is Å . In the Z direction, the average residual stress after annealing is almost lower theZ average small, and it increases first and then decreases from the range of 184 Å to 200 Å.than In the direction, residual stress after prepressing. In Case 1 and Case 4, the residual stress decreases first and then the average residual stress after annealing is almost lower than the average residual stress after increases at 52 Å to 60 Å and 72 Å to 80 Å . The σzrs in Case 3 increases first and then decreases at 12– prepressing. In Case 1 and Case 4, the residual stress decreases first and then increases at 52 Å to 60 Å 26 Å . In Case 2, there are large difference in σzrs before 14 Å and these then tend to 300 MPa which is and 72 Å to 80 Å. The σzrs in Case 3 increases first and then decreases at 12–26 Å. In Case 2, there are lower than the average residual stress after prepressing. The smaller the cooling rate at the same large difference in σzrs before 14 Å and these then tend to 300 MPa which is lower than the average annealing temperature, the lower the average residual stress after annealing.
residual stress after prepressing. The smaller the cooling rate at the same annealing temperature, the lower the average residual stress after annealing.
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88ofof1313 8 of 13 prepressing prepressing Case 1 prepressing Case 1 Case 21 Case Case 2 Case 32 Case Case 3 Case 43 Case Case 4 prepressing Case 4 curve prepressing curve Case 1 curve curve prepressing Case 1 curve Case 2 curve Case 1 curve Case 2 curve Case 3 curve Case curve Case 32 curve Case 4 curve Case curve Case 43 curve Case 4 curve
500 500 500
Residual Stress(MPa) Stress(MPa) ResidualStress(MPa) Residual
400 400 400 300 300 300 200 200 200 100 100 100 0
0 0
-100 -100 -100 -200 -200 -200
0
50 50 50
0 0
100 100
100Å) Depth( Depth( Å) Depth(Å)
150 150 150
200 200 200
Figure distribution after annealing with different annealing parameters. distribution after annealing with different annealing parameters. xrs Figure8.8.σσσxrs xrs distribution after annealing with different annealing parameters. Figure 8. σxrs distribution after annealing with different annealing parameters. prepressing prepressing Case 1 prepressing Case 1 Case 21 Case Case 2 Case 32 Case Case 3 Case 43 Case Case 4 prepressing Case 4 curve prepressing curve Case 1 curve curve prepressing Case 1 curve Case 2 curve Case 21 curve curve Case Case 3 curve Case curve Case 32 curve Case 4 curve Case curve Case 43 curve Case 4 curve
500 500 500
Residual Stress(MPa) Stress(MPa) ResidualStress(MPa) Residual
400 400 400 300 300 300 200 200 200 100 100 100 0
0 0
-100 -100 -100 -200 -200 -200
0
0 0
50 50 50
100 100
100Å) Depth( Depth( Å) Depth(Å)
150 150 150
200 200 200
Figure after annealing with different annealing parameters. Figure9. 9.σσyrsyrsdistribution distribution after annealing with different annealing parameters. Figure distribution after annealing with different annealing parameters. Figure9.9.σσyrs yrs distribution after annealing with different annealing parameters. prepressing prepressing Case 1 prepressing Case 1 Case 21 Case Case 2 Case 32 Case Case 3 Case 43 Case Case 4 prepressing Case 4 curve prepressing curve Case 1 curve curve prepressing Case 1 curve Case 2 curve Case 1 curve Case 2 curve Case 3 curve Case curve Case 32 curve Case 4 curve Case curve Case 43 curve Case 4 curve
Residual Stress(MPa) Stress(MPa) ResidualStress(MPa) Residual
650 650 650 600 600 600 550 550 550 500 500 500 450 450 450 400 400 400 350 350 350 300 300 300 250 250 250 200 200 200 150 150 150 0
0 0
20 20 20
40 60 40 60 40Depth(Å )60
Depth(Å) Depth(Å)
80 80 80
100 100 100
Figure Figure10. 10.σσzrszrsdistribution distributionafter afterannealing annealingwith withdifferent differentannealing annealingparameters. parameters. Figure 10. σzrs distribution after annealing with different annealing parameters. Figure 10. σzrs distribution after annealing with different annealing parameters.
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From Figure 7c–f, it can be seen that after annealing, the point defect concentration in thermal equilibrium is small, but the residual stress distribution is very different. Therefore, the point defect concentration is not the main factor affecting the residual stress distribution. Hammer et al. [38] investigated the residual stress gradient in W/TiN-stack on Si(100). They found that stress profiles can be related to grain size distribution by a Hall–Petch mechanism: in order to avoid extreme residual stress in metallic films, an increase the grain size can hence be suggested, as this decreases the yield strength and effectively limits the maximum possible stress level. The yield stress is related to the grain size by the Hall–Petch law equation which can be given as follows [39]: k σy = σ0 + √ , d
(3)
where σy is the yield stress, σ0 is approximately the yield stress of a very coarsegrained, untextured polycrystal, k is the strength coefficient, d is the average grain size, and σ0 and k are constants. For γ structure, σy = 175 + 0.615/d MPa [40]. However, an “inverse Hall–Petch” phenomenon occurs when the grain size is less than 10 nm because the plastic deformation of the grain boundaries replaces the dislocation plasticity mechanism within the grain when the grain boundary is excessive [41,42]. The average grain size in four cases after annealing is 4.643 nm, 4.655 nm, 4.658 nm, 4.657 nm, respectively, and more than 4.639 nm after prepressing; the average grain size increases slightly after annealing. In general, according to the “inverse Hall–Petch” phenomenon the yield stress has been increased after annealing and the residual stress may be increased after annealing. However, the fitting average residual stress in Y and Z directions show that the average residual stress after annealing in all four cases is less than the average residual stress after prepressing. The reason for this phenomenon is that the plastic deformation of the grain boundaries influences the distribution of residual stress. The simulation results show that during the annealing process, the grain boundary dislocation density decreases when heating; similar results are found in [17] when joule heating a Ti-Al-4V U-shaped screw. The dislocation increases during cooling with few dislocations in the grain. The decrease of dislocation density after annealing is closely related to the release of residual stress [18]. Table 2 shows the dislocation density after annealing with different parameters and, taking Case 1 as an example, the tend of dislocation density is shown in Figure 11. The grain boundary dislocation density after annealing is slightly larger than that after prepressing. The volume of grains increases and the grain boundary volume shrinks during cooling, the dislocation density at grain boundaries increases gradually which causes the grain boundary plastic deformation to increase, and stress is released. This finally causes the fitting average residual stress in the Y/Z direction after annealing to decrease. Table 2. Dislocation density at grain boundaries after annealing. Case
Annealing Temperature (K)
Dislocation Density (Å−2 )
1 2 3 4 prepressing
700 700 900 1100 /
0.0031 0.00278 0.0031 0.00281 0.00248
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0.0032
-2
Dislocation density(Å )
0.0030 0.0028 0.0026 0.0024 0.0022 0.0020 0.0018 300
400
500
600
700
Temperature(K)
Figure Thetrend trendof ofdislocation dislocation density cooling in Case 1. 1. Figure 11.11. The densityduring during cooling in Case
4. Conclusions
4. Conclusions
In this paper, annealing processes of γ-TiAl alloy after introducing residual stress into In this paper, annealing processes of γ-TiAl alloy after introducing residual stress into prepressing prepressing are simulated, and the dynamic evolution process of microdefects and the distribution are simulated, and the dynamic evolution of microdefects and theare distribution of residual stress before and after annealingprocess are investigated. The conclusions as follows: of residual
stress before and after annealing are investigated. The conclusions are as follows: (1)
(2)
(3)
(1) The grain boundary volume expands when the temperature rises, and the grains are The volume the grainwhen boundary shrinks whenrises, the temperature is dropped, and The compressed. grain boundary volumeofexpands the temperature and the grains are compressed. grain of size slightly after annealing. is no phase transition during The the volume the grain increases boundary shrinks whenThere the temperature is dropped, andannealing, the grain size but the atoms’ distortion occurs with the is change of temperature. slightly increases after annealing. There no phase transition during annealing, but the atoms’ (2) There are some atom clusters in the grains, with distortion occurs with the change of temperature.a few point defect and dislocations, and the main defects at the grain boundaries are different types of dislocation after prepressing. The There are some atom clusters in the grains, with a few point defect and dislocations, and the main point defect concentration increases with the rise of temperature and vice versa. The atom defects at the grain boundaries are different dislocation after prepressing. The point clusters have a certain adsorption effect ontypes the of atoms that precipitated from the grain defect concentration increases with the rise of temperature and vice versa. The atom clusters boundaries. The higher the annealing temperature, the less the point defects in the grain after have a certain adsorption effect on the atoms that precipitated from the grain boundaries. The higher annealing. (3) The distribution of residual X direction slightly. the Y direction, the the annealing temperature, thestress less in thethe point defects fluctuates in the grain after In annealing. higher the annealing temperature, the smaller the average residual stress, andthe theY same The distribution of residual stress in the X direction fluctuates slightly. In direction, annealing temperature can obtain smaller average residual stress after annealing at a slower the higher the annealing temperature, the smaller the average residual stress, and the same cooling rate. In the Y and Z directions, the average residual stress after annealing in all four cases annealing temperature can obtain smaller average residual stress after annealing at a slower is less than the average residual stress after prepressing; the reason for this phenomenon is the cooling rate. In the Y and Z directions, the average residual stress after annealing in all four cases grain boundary volume shrinkage and plastic deformation of the grain boundaries increases is less thancooling, the average residual stress after prepressing; the reason for this phenomenon is the during and stress is released.
grain boundary volume shrinkage and plastic deformation of the grain boundaries increases
Author H.L.is conceived and designed the molecular dynamics simulations; R.F., W.S., H.L. duringContributions: cooling, andR.F., stress released. and Y.Q. performed the simulation work; W.S., Y.Q. and H.Q. analyzed the data; W.S. and L.L. wrote the paper. Funding: This research was funded by the National Naturalthe Science Fundation of Chinasimulations; (No. 51665030) andW.S., the H.L. Author Contributions: R.F., H.L. conceived and designed molecular dynamics R.F., Program for Changjiang Scholarswork; and Innovative Research Team in Universities of the Ministry of Education and Y.Q. performed the simulation W.S., Y.Q. and H.Q. analyzed the data; W.S. and L.L. wrote theofpaper. China (No. IRT_15R30) and the Doctoral Research Foundation of Lanzhou University of Technology.
Funding: This research was funded by the National Natural Science Fundation of China (No. 51665030) and the Conflicts of Interest: The authors declare no conflict of interest. Program for Changjiang Scholars and Innovative Research Team in Universities of the Ministry of Education of China (No. IRT_15R30) and the Doctoral Research Foundation of Lanzhou University of Technology. Conflicts of Interest: The authors declare no conflict of interest.
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