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Hot Deformation Behavior and Constitutive Models of Ti600 Alloy in (α+β) Phase Region  PDF

  • Li Huiming 1,2
  • Zhang Jingli 2
  • Mao Xiaonan 2
  • Hong Quan 2
  • Zhang Yongqiang 2
  • Pan Hao 2
  • Cai Jianhua 2
1. Northeastern University, Shenyang 110819, China; 2. Northwest Institute for Nonferrous Metal Research, Xi'an 710016, China

Updated:2021-07-08

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Abstract

The high-temperature deformation behavior of Ti600 alloy with a lamellar initial microstructure was investigated in the temperature range of 800~960 ℃ and the strain rate range of 10-3~1 s-1. Subsequently, the strain hardening exponent (n) was proposed to characterize the competition of flow softening and work hardening. The softening behavior of this alloy was also studied according to flow curve analysis and microstructure observation. The results indicate that deformation parameters have significant influences on the flow behavior of Ti600 alloy. The n-value gradually decreases after the peak strain, which indicates that the dynamic softening begins to take dominant. The dynamic softening behavior of Ti600 alloy mainly attributes to the bending, fragmentation, dynamic recovery and recrystallization of α phase during the high-temperature deformation according to the microstructure characterization. Based on experimental data, original strain-compensated Arrhenius, Hensel-Spittel and modified Arrhenius constitutive models are established to describe the deformation behavior of Ti600 alloy. The flow stresses predicted by three models are compared with experimental results, and the calculated correlation coefficients are 0.965, 0.989, and 0.997. Also, the values of average absolute relative error are 12.86%, 9.74%, and 3.26%. These results suggest that three models can descript the flow behavior of Ti600 alloy, and the modified Arrhenius model exhibits the highest prediction accuracy.

The thermal-mechanical processes of metals are often complicated due to the deformation parameters such as temperature, strain rate, and strain[

1,2]. The final mechanical properties of workpieces are not only determined by the initial microstructural state, but also closely related to the hot working process. The hot deformation of metals is often accompanied by the competition of work hardening (WH) and dynamic softening which mainly includes dynamic recovery (DRV) and dynamic recrystallization (DRX)[3,4]. Therefore, modelling high-temperature deformation behavior and under-standing microstructure evolution of metals are critical to optimize thermoforming parameters and to fabricate high-quality workpieces[5].

In recent years, researchers have carried out many experi-mental studies on the hot deformation of metals, and established various models to characterize the hot deformation behavior. Cai et al[

6] researched the high-temperature defor-mation of 3Cr23Ni8Mn3N heat-resistant steel and proposed a strain-compensated Arrhenius equation based on the influence of deformation strain on flow stress to forecast the hot deformation stress. Long et al[7] used genetic algorithms to calculate the material parameters of strain-compensated Arrhenius model, and established optimized Arrhenius model. Wang et al[8] combined iterative method with regression analysis to obtain material constants in the Arrhenius model. Hence, the Arrhenius constitutive model was widely used for characterizing hot deformation features of metals. Also, similar Arrhenius equations were established to predict deformation characteristics of titanium alloys[9-11], aluminum alloys[12-14], magnesium alloys[15-17], Ni-based alloys[18-20], and steel[21-23]. Under various deformation conditions, the micro-structure evolution of metals is generally different. Resear-chers usually combine the microstructure evolution with flow behavior to establish constitutive models with physical meaning. Based on the theory of dislocation density, Bobbili et al[24] proposed a physically-based constitutive model, and brought this model and dynamic recrystallization kinetic into the ABAQUS finite element software to successfully predict the hot deformation behavior of Ti-10V-2Fe-3Al alloy. Haghdadi et al[25] improved the E-M model to characterize the WH and DRV of LDX 2101 duplex steel. In addition, various types of models such as K-H-L[26], J-C[27] and F-B[28] have also been proposed to depict the flow behavior of metals. Artificial neural network[29] is the model established through computer analysis, which is more and more popular in industrial production due to its efficiency and convenience. It is mainly used to predict the hot deformation behavior of metals, establish hot processing maps, and model process-structure-performance relationships[30].

The above models can be roughly classified into three types: phenomenological models[

31], physically-based models[32] and artificial intelligence models[33]. Among them, phenomeno-logical models only require conventional mathematical calculations rather than complex physical parameters. They are more efficient, and the prediction accuracy can also be achieved. Therefore, phenomenological models are commonly applied in practical researches.

Owing to the excellent comprehensive properties, such as high temperature tensile strength, superior creep performance and great fatigue resistance, Ti600 alloy has always been considered as the candidate material applied in aerospace engine parts. The aerospace engine has always been employed under the high-temperature or high-pressure environment, which requires excellent mechanical properties. The mecha-nical performance of the metal is largely linked to its deformation process, and the workpieces with great comprehensive performance can be obtained by reasonable processing techniques.

In this work, the hot deformation tests were conducted on Ti600 alloy, and its deformation behavior was investigated within a wide range of deformation temperatures (800~960 ℃) and strain rates (10-3~1 s-1). Based on experimentally obtained flow curves, original strain-compensated Arrhenius model (OSCA), Hensel-Spittel model (HS) and modified Arrhenius model (MA) were introduced to depict the deformation behavior of Ti600 alloy. Additionally, the estimated accuracy of the three models was also quantitatively evaluated.

1 Experiment

The alloy used in this experiment was the forged Ti600 alloy with a chemical composition of Ti-6Al-2.8Sn-4Zr-0.4Mo-0.4Si-0.1Y (wt%). The β transus temperature was determined to be 1010±5 ℃ through the metallographic method. The initial microstructure of studied alloy was a typical lamellar microstructure. The thermal compression experiment was performed on a Gleeble-3800 thermal-mechanical simulation, and the compressive temperatures were 800~960 ℃, and strain rates were 10-3~1 s-1. The specimens for hot compression experiment were cylinders of Φ8 mm×12 mm. Fig.1 shows a schematic diagram illustrating the experimental procedure of hot deformation. Before hot compression, the samples were firstly heated up to the preset temperature at the heating rate of 10 ℃/s and kept for 5 min to ensure a uniform temperature distribution. After that, the samples were compressed to a true strain of 0.916, followed by immediately quenching in order to obtain hot deformation microstructure. Subsequently, the quenched samples were sliced along the direction parallel to compression axis. After grinding, polishing, and etching (10 mL HF+30 mL HNO3+60 mL H2O), the deformed microstructures were observed through OLYMPUSPM3 optical microscope (OM).

Fig.1 Schematic depiction for specimen preparation, isothermal compression and microstructure observation

2 Hot Deformation Results and Discussion

2.1 Flow curves of hot deformation

Fig.2 shows the flow curves of Ti600 alloy after hot compression over the deformation temperatures of 800~960 ℃ and the strain rates of 10-3~1 s-1. It clearly shows that all stress curves exhibit the continuous softening behavior, in which no steady-state stress appears after the peak stress because of the insufficient softening. And it also can be seen from Fig.2 that the deformation parameters have prominent effects on flow stress. The peak stress decreases with the reduction of strain rates and the elevation of temperatures, which is consistent with other titanium alloys. During the hot deformation, a great deal of dislocation is generated within the deformed alloy at the initial work hardening stage. Subsequently, the dislocation entanglement and pileup occur, which cause the rapid increase of flow stress. It is worth mentioning that the rapid increase of dislocation content will lead to a higher stress level at the high strain rate. With further deformation, the flow curve begins to soften, which results in a decrease of work hardening rate. During this stage, dislocation rearrangement and climb are carried out as a result of the DRV, which lead to the decrease of dislocation density. When the DRV is difficult to overcome the WH and the deformation activation energy reaches a critical level, the DRX process is activated within the alloy. Subsequently, grain boundary migration consumes a large amount of dislocation, and the true stress continues to decline. Besides, at the lower strain rate, the deformed alloy has longer residence time at the high temperature, that is to say, it has sufficient time to release the distortion energy which can promote the softening process. Therefore, the DRV and DRX are responsible for the flow softening of Ti600 alloy during high-temperature deformation according to flow curves.

Fig.2 Flow curves of Ti600 alloy at different temperatures and strain rates: (a) 10-3 s-1, (b) 10-2 s-1, (c) 10-1 s-1, and (d) 1 s-1

2.2 Softening behavior

The competitive relationship between dynamic softening and work hardening can be reflected by the strain hardening exponent (n)[

34] during the hot deformation. The n-values calculated by Eq.(1) are shown in Fig.3. As the strain increases, most n-values gradually decrease under different deformation conditions. The dynamic softening and work hardening reach the balance at the peak strain, and the n-value is zero at this time. With the continuous increase of deformation strain, the dynamic softening begins to dominate, causing a decreased n-value. It can be noted that n-value shows a linear decrease when the strain rate increases to 1 s-1 by comparing the variation of n-value at different strain rates. This can be explained that the dislocation density inside deformed grain is high at the high strain rate, and a large amount of dislocation must be eliminated in subsequent softening process, which makes the n-value decrease rapidly. Moreover, the rapidly decreased n-value also reveals that the softening rate is faster at the high strain rate. For α+β titanium alloy, the close-packed hexagonal structure α phase is believed to have fewer slip systems than β phase which is the body-centered cubic structure and is hard to deform. The α phase is often termed as strengthening phase in α+β titanium alloy[35]. Therefore, the strain hardening exponent and softening behavior are significantly affected by the microstructure evolution of α phase during hot deformation. The strain hardening exponent can be expressed as:

Fig.3 Variation of strain hardening exponent (n) at diffident strain rates: (a) 10-3 s-1, (b) 10-2 s-1, (c) 10-1 s-1, and (d) 1 s-1

n=dlgσdlgεε˙,T (1)

where σ is the true stress (MPa), ε is the strain, ε˙ is the strain rate (s-1) and T is the absolute temperature (K).

According to microstructure characterization, the deforma-tion behavior of studied alloy can be understood more distinctly. Fig.4 displays the thermal compression microstruc-tures of Ti600 alloy at 880~960 ℃ and 10-3~10-1 s-1. It is evident that the original grain boundaries of equiaxed β are broken and disappear, and the lamellar α phase structure is retained. Owing to the compression deformation, the lamellar α phase is elongated along the direction perpendicular to the compression direction, as shown in Fig.4b. Fig.4c includes the deformed α phase, fragmentized α phase, and recrystallized α phase, which indicate that the non-uniform deformation occurs at the deformation temperature of 920 ℃ and strain rate of 10-3 s-1. During the compression deformation process, the grain boundaries deform firstly due to the anisotropy of grain boundaries. A large number of dislocations are formed at the grain boundaries, and these generated dislocations can promote the formation of sub-grains. The Ti600 alloy belongs to high stacking fault energy metal for a high Al content. For high stacking fault energy metal, continuous dynamic recry-stallization is prone to occur during hot deformation[

36]. This means that dislocations will be consumed by the formation of sub-grains and low-angle grain boundaries will eventually be transformed into high-angle grain boundaries. Therefore, the DRX process occurs firstly at the grain boundaries. With the further deformation, local strain occurs in lamellar structure within the grains, which may become the second nucleation site for DRX. And the lamellar structure with soft orientation is easier to undergo DRX than the hard-oriented lamellae. When the alloy is deformed at 960 ℃ and 10-3 s-1, sufficient DRX takes place and all original structures are replaced by recrystallized grains (re-grains). In addition, full DRX can lead to the refinement of initial microstructure of Ti600 alloy (Fig.4d).

Fig.4 Microstructures of Ti600 alloy after the hot deformation under different conditions: (a) initial structure, (b) 880 ℃/10-3 s-1, (c) 920 ℃/10-3 s-1, (d) 960 ℃/10-3 s-1, (e) 960 ℃/10-2 s-1, (f) 960 ℃/10-1 s-1

Under the deformation temperature of 960 ℃, the elongated α lamellae along the metal flow direction can be readily observed as the strain rate increases to 10-1 s-1 (Fig.4f). At this time, there is a higher dislocation density in the deformed microstructure. When the strain rate decreases from 10-1 s-1 to 10-2 s-1, some fine equiaxed α grains appear in the micro-structure, which indicate that insufficient DRX occurs (Fig.4e). When the strain rate reaches to 10-3 s-1, the re-grains grow slightly because the longer time contributes to the coarsening of re-grains during hot compression, as shown in Fig.4d. Hence, the softening behavior of Ti600 alloy during high-temperature deformation is mainly induced by the bending, fragmentation, DRV and DRX of α phase.

3 Constitutive Models and Discussion

The constitutive model is mainly applied to evaluate the flow behavior of alloys during the hot deformation, to calculate the deformation parameters, and to provide a theore-tical basis for the finite element numerical simulation[

37]. The thermoplastic deformation of titanium alloy is a thermally activated process, which includes the WH, DRV and DRX processes. In this study, three kinds of constitutive models including original strain-compensated Arrhenius (OSCA), Hensel-Spittel (HS) and modified Arrhenius (MA) model are conducted to predict the flow behavior of Ti600 alloy. And the prediction accuracy of three models is also evaluated.

3.1 Original strain-compensated Arrhenius (OSCA) model

Sellars and Tagart[

38] did lots of pioneering works on the hot deformation, and proposed the Arrhenius equation to describe the flow stress of metals. The equations are as follows:

ε˙=AF(σ)exp(-QRT) (2)
F(σ)=σn1                 ασ<0.8exp(βσ)        ασ>1.2[sinh(ασ)]n   for  all σ (3)

where ε˙ represents the strain rate (s-1); σ is the flow stress (MPa); A, α, β are material constants; n1, n are flow stress exponents and α=β/n1; Q represents the activation energy of high-temperature deformation (kJ/mol); R is the gas constant (8.314 J/mol/K), and T is the absolute temperature (K).

Eq.(4) shows the equation of Zener-Hollomon parameter[

39] which is considered to evaluate the dependence of the deformation behavior on the deformation parameters during high-temperature deformation. Eq.(2) and Eq.(3) are substi-tuted into Eq.(4), and the flow stress can be described by Eq.(5).

Z=ε˙exp(QRT)=A[sinh(ασ)]n (4)
σ=1αlnZA1/n+ZA2/n+11/2 (5)

However, it can be found that the deformation strain has a significant influence on the flow stress of Ti600 alloy through observing the flow curves. To correct the effect of strain and to improve the prediction accuracy, all material constants are calculated by above Eq.(2) and Eq.(4) with an interval of 0.05 within the strain range of 0.1~0.75. Fig.5 shows the variation of material constants at different strains. The relationships between material constants and strains can be determined by the polynomial fitting, and the fitted results are listed in Table 1. Finally, all material constants can be obtained by following equations.

Fig.5 Variation of material constants at different true strains: (a) α, (b) n, (c) Q, and (d) lnA

Table 1 Coefficients of the fitted polynomial
αnQlnA
B0=0.00364 C0=5.012 D0=366.766 E0=35.424
B1=0.00426 C1=-17.811 D1=4356.999 E1=424.281
B2=0.00432 C2=66.221 D2=-48611.427 E2=-4810.435
B3=-0.00865 C3=-120.662 D3=255861.498 E3=25468.258
B4=0.00682 C4=110.779 D4=-706232.32 E4=-70367.298
B5=-0.00108 C5=-41.338 D5=1.056×106 E5=105077.191
D6=-810173.70 E6=-80367.249
D7=249823.776 E7=24677.067

αε=B0+B1ε+B2ε2+B3ε3+B4ε4+B5ε5nε=C0+C1ε+C2ε2+C3ε3+C4ε4+C5ε5Qε=D0+D1ε+D2ε2+D3ε3+D4ε4+D5ε5+D6ε6+D7ε7lnAε=E0+E1ε+E2ε2+E3ε3+E4ε4+E5ε5+E6ε6+E7ε7 (6)Substituting the material constants obtained by above fitted polynomial equations into Eq.(5), the flow stresses of Ti600 alloy under all experimental conditions can be determined. So as to evaluate the accuracy of this constitutive model, the stresses calculated by the strain-compensated Arrhenius equa-tion are compared with the measured stresses. Fig.6 shows the contrast of flow stresses obtained by OSCA model with the experimental results. It can be observed from Fig.6 that the OSCA model can predict the flow stress of Ti600 alloy. And a similar trend exists between measured and predicted flow stress. However, under some deformation conditions, the predicted results have some differences compared to the experimental stresses, and the differences are positive or negative. It can be explained that the model only considers the effect of strain when calculating the material constants, whose average values are used under the same strain at various deformation conditions, whereas ignores the influences of deformation temperatures and strain rates.

Fig.6 Comparison between the predicted stresses by OSCA model and experimental stresses: (a) 10-3 s-1, (b) 10-2 s-1, (c) 10-1 s-1, and (d) 1 s-1

3.2 Hensel-Spittel (HS) model

Hensel and Spittel[

40] explored another constitutive equation to depict the flow stress of metals. HS model considers the effects of strain, deformation temperature, and strain rate, and this model can be written as:

sinh(ασ)=Aexp(m1T)εm2ε˙m3exp(m4/ε)×                       (1+ε)m5Texp(m6ε)ε˙m7TTm8 (7)

where m1, m2, m3, m4, m5, m6, m7 and m8 are material coefficients. To simplify the calculation, m7 and m8 are generally ignored. The α value can be computed by above fitted polynomial.

Taking logarithm of Eq.(7), the stress can be expressed as Eq.(8). Substituting experimental data into Eq.(8), a mathe-matical software is used to perform multiple linear regression analysis. Subsequently, the material coefficients are given in Table 2. The predicted stresses and experimental stresses are compared to verify the accuracy of HS model (Fig.7). Note that the overall stress difference is small, but under some particular conditions such as low temperatures (800 and 840 ℃), the gap of the predicted flow stress and the experi-mental stress is large.

lnsinh(ασ)=lnA+m1T+m2lnε+m3lnε˙                     +m4/ε+m5Tln(1+ε)+m6ε (8)
Table 2 Material coefficients in HS model
Am1m2m3m4m5m6
277451.105 -0.0124 -0.149 0.289 -0.0623 -0.0058 3.458

Fig.7 Comparison between the precited stresses by HS model and experimental stresses: (a) 10-3 s-1, (b) 10-2 s-1, (c) 10-1 s-1, and (d) 1 s-1

3.3 Modified Arrhenius (MA) model

The OSCA model only considers the single influence of strain, deformation temperatures and strain rates on material constants, in which the flow stress cannot be accurately predicted under some experimental conditions. A modified Arrhenius model, considering multiple effects of the thermodynamic parameters, is expressed as:

ε˙=A(ε,ε˙,T)sinh(α(ε,T)σ)n(ε,T)exp(-Q(ε,ε˙,T)RT) (9)

Taking logarithm of Eq.(9), Eq.(10) can be defined as:

lnε˙=lnA(ε,ε˙,T)+n(ε,T)sinh(α(ε,T)σ)-Q(ε,ε˙,T)RT (10)

Material constants in Eq.(10) can be determined by following equations:

α(ε,T)=lnε˙σ(ε,T)/lnε˙lnσ(ε,T) (11)
n(ε,T)=lnε˙lnsinh(α(ε,T)σ)ε,T (12)
Q(ε,ε˙,T)=Rlnε˙lnsinh(α(ε,T)σ)ε,Tlnsinh(α(ε,T)σ)(1/T)ε,ε˙=Rn(ε,T)n3(ε,ε˙) (13)

Eq.(10) also can be transformed as Eq.(14):

lnA(ε,ε˙,T)=Q(ε,ε˙,T)RT+lnε˙-n(ε,T)sinh(α(ε,T)σ)=n(ε,T)n3(ε,ε˙)/T+S(ε,T) (14)

The values of α, n, and n3 under different experimental conditions can be calculated by above equations, and the results are shown in Fig.8. Subsequently, the results are fitted by binary polynomial, and the fitted equation is Eq.(15). The coefficients of polynomial are shown in Table 3, and the values of material constants can be obtained through Eq.(15). Fig.9 shows the deviation between predicted stresses and experimental stresses of MA model. It can be seen that all precited stresses are consistent with measured stresses, and the MA model shows the higher accuracy.

Fig.8 Relationship between α (a), n (b), and n3 (c) and deformation temperatures, strain rates, and strain

α(ε,T)=F0+F1ε+F2T+F3ε2+F4T2+F5ε3+F6T3+F7εT+F8ε2T+F9εT2n(ε,T)=G0+G1lnε+G2lnT+G3(lnε)2+G4(lnT)2+G5(lnε)3+G6(lnT)3+G7(lnε)(lnT)+G8(lnε)2(lnT)+G9(lnε)(lnT)2/G10εT+G11n3(ε˙,T)=H0+H1ε˙+H2T+H3ε˙2+H4T2+H5ε˙3+H6T3+H7ε˙T+H8ε˙2T+H9ε˙T2 (15)
Table 3 Coefficients of the binary polynomial
αnn3
F0=-5013.222 G0=2.309×106 H0=10.173
F1=269.228 G1=2.069×106 H1=47.449
F2=13.501 G2=-3.663×105 H2=1.335
F3=34.944 G3=-3991.929 H3=-99.622
F4=-0.012 G4=-35693.922 H4=0.384
F5=2.437 G5=-293.073 H5=59.126
F6=3.628×106 G6=5827.749 H6=0.00984
F7=-0.537 G7=-5.886×105 H7=-5.9067
F8=-0.033 G8=279.809 H8=-1.865
F9=0.00027 G9=41759.283 H9=-1.095
G10=-0.704
G11=-3.594

Fig.9 Comparison between the predicted stresses by MA model and experimental stresses: (a) 10-3 s-1, (b) 10-2 s-1, (c) 10-1 s-1, and (d) 1 s-1

3.4 Accuracy evaluation of constitutive models

Three kinds of original strain-compensation Arrhenius constitutive model (OSCA), Hensel-Spittel constitutive model (HS) and modified Arrhenius constitutive model (MA) are established to describe the thermal deformation behavior of Ti600 alloy. The flow stresses predicted by three models and the measured stresses are compared in Fig.10. It can be confirmed that the stress distribution of the OSCA model and HS model is relatively discrete. However, the stress distribution by the MA model is convergent and the flow stresses are uniformly distributed on one line.

Fig.10 Comparison between experimental stresses and predicted stresses: (a) OSCA model, (b) HS model, and (c) MA model

The average absolute relative error (AARE) and correlation coefficient (R) of each model can be computed through Eq.(16) and Eq.(17) to quantitatively assess the prediction accuracy of each model.

AARE=1Ni=1NYi-XiYi×100% (16)
R=i=1N(Xi-X¯)(Yi-Y¯)i=1N(Xi-X¯)2i=1N(Yi-Y¯)2 (17)

where Y is the experimental stress, X is the predicted stress, and X¯ and Y¯ represent average value of predicted and measured flow stress, respectively.

Here, the R reflects linear correlation between predicted results and measured data. And the AARE reveals the deviation degree of predicted stress from experimental stress. The R values of OSCA model, HS model, and MA model are 0.965, 0.989, and 0.997, and the AARE values of the three models are 12.86%, 9.74%, and 3.26%. Among three models, the MA model has the largest R value and the smallest AARE value, which indicates that the MA model has the highest prediction accuracy for the Ti600 alloy.

The above constitutive models are proposed based on mathematical parameters. However, the softening mechanisms of metals may change under different deformation conditions during the hot deformation. Therefore, there may be an appropriate deviation between the flow stress predicted by the above constitutive models and the experimentally measured stress. Through the accuracy evaluation, the accuracy of the three models is within the controllable range, which means that these constitutive models can predict the hot deformation behavior of Ti600 alloy.

4 Conclusions

1) The thermal deformation behavior of Ti600 alloy is signi-ficantly influenced by the strain, deformation temperatures and strain rates. The increased strain rates and decreased deformation temperatures can result in an elevated flow stress.

2) After the peak strain, the n-value of all curves gradually decreases due to the dynamic softening. Moreover, it can be deduced that the dynamic softening rate increases with the increase of strain rate through observing the curve slope of strain hardening exponent. And the flow softening behavior of Ti600 alloy is mainly resulted from the bending, fragmentation, DRV and DRX of α phase according to microstructure observation.

3) The OSCA model, HS model and MA model are conducted to describe the hot deformation flow properties of Ti600 alloy. Among these models, the MA model has the highest prediction accuracy, which considers the simultaneous influences of strain, temperatures and strain rates.

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