Kinetic analysis and strain-compensated constitutive models of Ti-42.9Al-4.6Nb-2Cr during isothermal compression

Runrun Xu,Miaoquan Li,Hong Li

School of Materials Science and Engineering,Northwestern Polytechnical University,Xi'an,710072,China

ABSTRACT The hot deformation behavior of Ti-42.9Al-4.6Nb-2Cr (at.%)was investigated by isothermal compression tests at the deformation temperature range of 1373-1573 K,strain rate range of 0.001-1.0 s−1,up to the strain of 0.69.The flow stress test results of Ti-42.9Al-4.6Nb-2Cr showed negative temperature and positive strain rate sensitivity.Besides,strain had a great effect on the hot deformation behavior of Ti-42.9Al-4.6Nb-2Cr.Kinetic analysis was adopted to assess the hot workability of Ti-42.9Al-4.6Nb-2Cr via apparent activation energy(Q)of hot deformation,strain-rate sensitivity index (m) and strain hardening index (n).The Q value varied from 607.1 ± 0.7 kJ∙mol−1 to 512.6 ± 10.8 kJ mol−1 with the increasing of strain from 0.1 to 0.6.The effect of strain on the Q value at the deformation temperatures below 1473 K was mainly related to dynamic recrystallization of γ phase and kinking of γ lamellae,while the Q value at the deformation temperature above 1473 K might be linked to γ→α phase transformation and DRV of α phase.Based on the kinetic analysis,straincompensated Arrhenius model and Hensel-Spittel model were successfully established to predict the hot workability (flow stress).Average absolute relative errors of established strain-compensated Arrhenius model and Hensel-Spittel model were 7.52%and 11.95%,respectively.Moreover,both established constitutive models can be extrapolated for predicting the flow stress of Ti-42.9Al-4.6Nb-2Cr to larger strain levels.

Keywords:γ-TiAl alloys Hot deformation Kinetic analysis Arrhenius model Hensel-Spittel model

1.Introduction

Due to high melting temperature and specific stiffness,lightweight γ-TiAl alloys show great potentiality for replacing conventional aeronautical materials at service temperatures of 873-1073 K [1-3].Many processing routes,like hot-die forging and isothermal forging,are employed to produce γ-TiAl components in aero engines.However,narrow processing window and high sensitivity to temperature restrict the application of γ-TiAl alloys [4].

In order to improve the hot workability,β/β0phase is introduced in γ-TiAl alloys by the additions of Cr,Nb,Mo,W and/or other elements[5,6].However,β/β0phase in γ-TiAl alloys must be easily removed,for the fact that β/β0phase would decompose into hard and brittle ω phases at service conditions [7].Considering that β/β0phase in γ-TiAl alloys can be easily removed via hot deformation and heat treatment,it is necessary to evaluate the hot workability and optimize processing parameters of γ-TiAl alloys before industrial processing.

Kinetic analysis is widely used to evaluate the hot workability and analysis activated mechanisms of γ-TiAl alloys [8,9].Bohn et al.[10]and Zhang et al.[11]evaluated the hot workability of γ-TiAl alloys by apparent activation energy Q for hot deformation and strain rate sensitivity index m.The low Q value and high m value indicate that grain boundary sliding is activated.However,various mechanisms are activated during hot compression of γ-TiAl alloys.Thus,kinetic analysis is quite essential to investigate hot compression behavior of γ-TiAl alloy.

Constitutive model is important to predict the hot workability and optimize processing parameters via finite element simulations [12,13].Schwaighofer et al.[14]successfully established Arrhenius model for Ti-43Al-4Nb-1Mo-0.1B at the deformation temperature range of 1423-1573 K,strain rates ranging from 0.005 s−1to 0.5 s−1up to the strain of 0.9.Wang et al.[15]established Arrhenius model for Ti-43Al-4Nb-1.4W at the deformation temperatures of 1323-1473 K and strain rate range of 0.001-1.0 s−1.Godor et al.[16]established Arrhenius model of Ti-(41,45)Al-3Mo at the deformation temperatures ranging from 1423 K to 1573 K,the strain rates ranging from 0.005 s−1to 0.5 s−1up to the strain of 0.9.However,they claimed that the predicted flow stress would be illogical when the strain beyond experimental scope.Hence it is necessary to establish proper constitutive model for predicting the hot workability of γ-TiAl alloys.

The objective of the present work is to elucidate the hot deformation behavior of γ-TiAl alloys.For this purpose,Ti-42.9Al-4.6Nb-2Cr with duplex structure was isothermally compressed.Kinetic analysis was applied to assess the hot workability of Ti-42.9Al-4.6Nb-2Cr.Based on the hot deformation behavior and kinetic analysis,constitutive models that take strain into account were established.

2.Material and methods

The alloy used in the present paper was Ti-42.9Al-4.6Nb-2Cr,of which the chemical composition (at.%) was 42.89Al,4.59Nb,2.01Cr,0.25Cu,0.1Fe,0.59Si,0.07W,0.004Se and the balance Ti.Ti-42.9Al-4.6Nb-2Cr was supplied as a billet with dimensions of 170 × 60 × 30 mm3,which was obtained by hot-canned extrusion at～1500 K.Fig.1 is the initial microstructure of the as-received alloy.As seen from Fig.1,the initial microstructure is a duplex structure,which consists of α2/γ lamellar colonies,γ grains and some β0phase that distribute along grain boundaries.The γ-solvus temperature of Ti-42.9Al-4.6Nb-2Cr is～1554 K,which was measured by Φ6 mm × 25 mm specimens in thermal expansion instrument DIL402C equipped with the horizontal push rod.

Cylindrical specimens of Ti-42.9Al-4.6Nb-2Cr with a diameter of 8.0 mm and height of 12 mm were manufactured.Isothermal compression tests were conducted on a Thermecmastor-Z simulator (Fuji Electronics Industry,Japan) in vacuum (～20 Pa) at the deformation temperatures of 1373 K,1423 K,1473 K,1523 K and 1573 K,the strain rates ranging from 0.001 s−1to 1.0 s−1up to the strain of 0.69.Before compression,specimens were inductively heated with a heating rate of 10 K s−1and then soaked for 5 min at the targeted temperature to ensure homogeneous temperature distribution.The temperatures were controlled by thermocouples spot-welded in the middle surface of specimens.The mica plate was put between each specimen and SiC anvil for lubrication during the test.

After isothermal compression,the specimens were air-cooled to room temperature.Each specimen was sectioned diametrically.Before microstructure observation,the sectioned specimen was manually ground and then electro-polished in a solution of 64%methanol+30%butanol+6% perchloric at 288 K and 20 V.The microstructure of isothermally compressed Ti-42.9Al-4.6Nb-2Cr was examined via FEI Helios G4 CX scanning electron microscope (SEM) in back-scattered electron (BSE) mode at the operating voltage of 15 kV.

3.Results and discussion

3.1.Flow stress

Fig.2 exhibits some typical flow stress-strain curves of Ti-42.9Al-4.6Nb-2Cr during isothermal compression up to a strain of 0.69.As seen from Fig.2,all flow stress-strain curves firstly increased and then decreased as the strain increased to 0.69.Besides,the flow stress and peak strain (when flow stress peaked) increased as the deformation temperature decreased or strain rate increased.Additionally,the declining trend of flow stress gradually decreased with the increasing of deformation temperature or decreasing of strain rate after flow stress peaked.

3.2.Kinetic analysis

3.2.1.Apparent activation energy for hot deformation

Apparent activation energy for hot deformation reflects the energy needed for atomic transition in plastic deformation,which can evaluate the hot workability of alloys.The apparent activation energy for hot deformation is expressed as follows [8,13]:

where Q is the apparent activation energy for hot deformation(J∙mol−1),R is the gas constant(8.3145 J mol−1K−1),α is the material constant (MPa−1),σ is the flow stress (MPa),ε is the strain,is the strain rate (s−1),T is the absolute temperature (K).

According to Eq.(1),the α value should be determined before calculating the Q value.According to previous studies[11,13,18,19],the α value could be calculated as follows:

Fig.3 exhibit plots ofσ−lnand lnσ−lnat a strain of 0.6 in isothermal compression of Ti-42.9Al-4.6Nb-2Cr via linear fitting method.Combining Eq.(2) and Fig.3(a)-(b),the α value at a strain of 0.6 is 0.01795 MPa−1.In addition,the α values under other conditions can be obtained in the same way.

The Q value can be calculated by the slopes of ln [sinh (ασ)]−lnand ln [sinh(ασ)]−(1/T).Fig.4 displays the slopes of ln [sinh (ασ)]−lnand ln [sinh(ασ)]−(1/T) at a strain of 0.6.Combining with the calculated α value,Eq.(1),Fig.4(a) and (b),the Q value at a strain of 0.6 was 512.6 ± 10.8 kJ mol−1.Fig.5 presents the effect of strain on the Q value of Ti-42.9Al-4.6Nb-2Cr in isothermal compression.As seen from Fig.5,the Q value of Ti-42.9Al-4.6Nb-2Cr varied from 607.1 ± 0.7 kJ∙mol−1to 512.6 ± 10.8 kJ mol−1with the increasing of strain from 0.1 to 0.6.The mean Q value was 557.1 ± 12.7 kJ mol−1.The Q value of Ti-42.9Al-4.6Nb-2Cr was larger than the self-diffusion energy of Ti atom (～250 kJ mol−1),and that of Al atom (～358 kJ mol−1) in γ(TiAl) phase at the temperature lower than 1470 K [20].Therefore,dynamic recovery (DRV) is not an exclusive flow softening mechanism in isothermal compression of Ti-42.9Al-4.6Nb-2Cr.

Fig.6 presents SEM-BSE images of Ti-42.9Al-4.6Nb-2Cr isothermally compressed at different deformation conditions.When isothermal compression was conducted at a deformation temperature of 1373 K and a strain rate of 0.01 s−1,lamellae were kinked and elongated β/β0phase was spheroidized,as the strain increased from 0.22 to 0.69,as shown in Fig.6(a)and(b).The above-mentioned phenomenon was also observed as the strain rate decreased from 1.0 s−1to 0.01 s−1(Fig.6(b) and (c)).Additionally,some fine and equiaxed γ grains formed at the grain boundaries and phase interface,as seen from Fig.6(b).Cui et al.[21]suggested that fine γ grains formed by the rearrangement of sub-grain boundaries,and the volume fraction of DRX increased as the strain rate decreased during isothermal compression of Ti-43Al-2Cr-2Mn-0.2Y.As deformation temperature increased from 1373 K to 1523 K (Fig.6(c) and (d)),the volume fraction of β0phase significantly decreased and that of α grains (newly-formed α2/γ lamellar colonies) increased,which implying that γ→α phase transformation might contributes to the variation of Q value.

Microstructure image in Fig.6 (d) proves that γ→α phase transformation might be accounts for the variation of Q value at the deformation temperatures of>1473 K.Hence it is quite essential to elucidate the γ→α phase transformation via HRTEM as displayed in Fig.7.There were many entangled dislocations and stacking faults pairs in γ lamellae,which indicates that the γ lamella isn't the production of phase transformation during subsequent air-cooling (Fig.7a and b).Additionally,twin γ/γ orientation formed on the sides of faults pairs,and a misorientation angle of 2° (Fig.7 c) was produced in some γ variants,which means that low angle grain boundaries are formed.Besides,α2grain with few dislocations was crystallographically aligned to adjacent γ lamellae with the Blackburn orientation relationships(Fig.7 a and c).Owing to inward curves of α2/γ phase interface,it was the α2grain instead of α2lamella.Fig.7(d) presents HRTEM image of area Ⅰin Fig.7 (a),the corresponding FFT patterns are shown in Fig.7(e)-(i).As exhibited in Fig.7 (d),{111}γand (0001)α2atomic planes were deflected at the α2/γ phase interface with the angle of 12°and 15°,respectively.And deflections were accomplished by edge dislocations.Moreover,{111}γatomic planes were not parallel to (0001)α2atomic planes,which indicates that the α2/γ phase interface is semi-coherent and accommodated by interfacial edge dislocations.Furthermore,all the γ variants exhibit Blackburn orientation relationship{111}γ‖(0001)α2andto α2grain.Since there were many defects in γ variants with blurred crystal fringes,and few defects in α2grains with clear and neat fringes,it can be inferred that α2grains are formed at the expense of deformed γ variants by dislocations motion.Previous studies[22,23]suggested that α2→γ phase transformation is a diffusion-controlled step mechanism,and the phase interfaces like lamellar interface were flat.In turn,the γ→α2phase transformation is the same.However,in this study,the phase interface of α2/γ was uneven,and the γ→α phase transformation might not be a step-mechanism.

Fig.8 present microstructure images of isothermally compressed Ti-42.9Al-4.6Nb-2Cr at the deformation temperature of 1573 K which is higher than Tγ,solv.As seen from Fig.8 (a)-(d),α grains uncontrollably coarsen up to～400 μm,implying that γ→α phase transformation and DRV of α phase are the main contribution for the reduction of Q value at 1573 K.

When isothermal compression was conducted at the temperatures below 1473 K,some lamellar colonies were kinked,and the β/β0phase was elongated to release the stress concentrated at phase boundaries at the initial deformation stage,as shown in Fig.6 (a).The dislocations were intensively accumulated at the phase/grain boundaries.Thus,large deformation difficulty was large for lack of multiphase deformation coordination.With the increasing of strain (height reduction),the DRX in Ti-42.9Al-4.6Nb-2Cr was promoted,which leads to extensive consumption of dislocations and improvement of deformation homogeneity.As a result,the Q value of Ti-42.9Al-4.6Nb-2Cr decreased.Cheng et al.[24]report that the Q value of Ti-42Al-8Nb-(W,B,Y) is 427 kJ mol−1,and such a low value is ascribed to introducing of β phase by Nb addition.Li et al.[25]report that the Q value of Ti-43Al-6Nb-1Mo-1Cr is 571 kJ mol−1,which is high than that of Crfree γ-TiAl alloys.For the investigated Ti-42.9Al-4.6Nb-2Cr,the decrease of Q value with the increasing of strain implies the decrease of deformation resistance,which may be attribute to DRX of γ phase,the spheroidization of β/β0phase,and kinking of γ lamellae.When isothermal compression was conducted at temperatures above 1473 K,the volume fraction of α phase increased.It is believed that γ→α phase transformation and DRV of α phase reduce the dislocation density,which improves the deformability.

3.2.2.Strain-rate sensitivity index

Strain-rate sensitivity index reflects the homogeneous deformation ability of materials.Strain-rate sensitivity index m is defined as follows[10,26]:

where σ is the flow stress(MPa),ε is the strain,is the strain rate(s−1),T is the absolute temperature (K).

Fig.9 exhibits the effect of strain on the m value of Ti-42.9Al-4.6Nb-2Cr in isothermal compression.As seen from Fig.9,the m value of Ti-42.9Al-4.6Nb-2Cr in isothermal compression decreased with the increasing of strain rate.The m value of Ti-42.9Al-4.6Nb-2Cr increased from 0.14 to 0.30 as the strain rate decreased from 1.0 s−1to 0.01 s−1in isothermal compression at a deformation temperature of 1373 K,and the strain of 0.6,as seen from Fig.9(a).As seen from Fig.6(b)and(c),many fine and equiaxed γ grains formed as the strain rate decreased from 1.0 s−1to 0.01 s−1in isothermal compression at a deformation temperature of 1373 K,and the strain of 0.6,and therefore the coordinate deformability in Ti-42.9Al-4.6Nb-2Cr increased.Namely,the m value of Ti-42.9Al-4.6Nb-2Cr increases as the strain rate decreases,which is ascribed to improved homogeneous deformability caused by DRX.

In addition,the m value of Ti-42.9Al-4.6Nb-2Cr in isothermal compression increased as the deformation temperature increased from 1373 K to 1523 K.This phenomenon is more significant at a low strain rate.When isothermal compression was conducted at a deformation temperature of 1373 K and strain rate of 0.001 s−1,the m value ranges from 0.37 to 0.43,as seen from Fig.9(a).The volume fractions of DRX grains and activated deformation systems increased as the deformation temperature increased,which improve the homogeneous deformability.Also,homogenous deformability increased as the strain rate decreases or strain increased for enhanced DRX behavior during isothermal compression of Ti-42.9Al-4.6Nb-2Cr.However,the change of m value at constant strain rates was small as the deformation temperature increased from 1523 K to 1573 K,as seen from Fig.9(b) and (c).Combining with Figs.6(d)and Fig.8,the effect of strain rate on m value at 1523 K and 1573 K might be ascribed to γ→α phase transformation and DRV of α phase.

3.2.3.Strain hardening index

Strain hardening index (n) reflects the contribution of deformation resistance via work hardening effect,which can be calculated as follows[27-29]:

where σ is the flow stress(MPa),ε is the strain,is the strain rate(s−1),T is the absolute temperature (K).

Fig.10 displays the effect of strain rate on the n value of Ti-42.9Al-4.6Nb-2Cr in isothermal compression.As seen from Fig.10 (a),the n value of isothermally compressed Ti-42.9Al-4.6Nb-2Cr is almost negative at the deformation temperatures ranging from 1373 K to 1573 K,strains from 0.1 to 0.6 and a strain rate of 0.001 s−1.The negative n value indicates that the flow softening effect was dominant under these deformation conditions.As seen from Fig.10 (b),the n value of Ti-42.9Al-4.6Nb-2Cr in isothermal compression at a strain rate of 0.01 s−1firstly decreased and then increased at the deformation temperatures with a range of 1473-1573 K and strain range of 0.1-0.6.This demonstrates that the flow softening effect of isothermally compressed Ti-42.9Al-4.6Nb-2Cr firstly decreases and then increases with the increasing of strain.Besides,it is notable that the n value increased to zero at a deformation temperature of 1573 K,strain rate of 0.01 s−1and strain of 0.6.Namely,the flow softening effect can fully offset the work hardening effect again when the steady-state flow is achieved.Similarly,above-mentioned phenomenon occurred under other deformation conditions,for example at a deformation temperature of 1523 K and strain rate of 1.0 s−1(shown in Fig.10 (d)).

As seen from Fig.10(c),the effect of deformation temperature on n value of Ti-42.9Al-4.6Nb-2Cr in isothermal compression is significant.The n value ranged from −0.22 to −0.08 at a deformation temperature to 1523 K,strain rate of 0.1 s−1and strain range of 0.3-0.6.The n value fell into the range from −0.33 to −0.16 as the deformation temperature decreased to 1423 K.The decrease of n value demonstrates that the flow softening of Ti-42.9Al-4.6Nb-2Cr in isothermal compression decreases with the decreasing of deformation temperature.Thus,it can be reasonably explained by dislocation density.Dislocation sliding/climbing and DRX were sufficiently promoted with the increasing of deformation temperature.Therefore,the dislocation density decreased at the initial deformation stage.Consequently,dislocation-stimulated DRX was weakened.However,the lower dislocation density,the faster achievement of steady-state flow.

4.Constitutive model

As discussed in Part.3,there is the great influence of strain on hot deformation behavior of Ti-42.9Al-4.6Nb-2Cr.Hence,the strain should be considered in constitutive model for predicting plastic deformation behavior of Ti-42.9Al-4.6Nb-2Cr.

4.1.Strain-compensated Arrhenius model

Arrhenius model is widely used to describe the relationship of flow stress and processing parameters,which is given as follows [18]:

where A,α and t are material constants,Q is the apparent activation energy (kJ∙mol-1∙K-1) for hot deformation,R is the gas constant(8.3145 J mol−1K−1),T is the absolute temperature (K).

Taking logarithm of both sides,Eq.(5) is converted as follows:

The values of lnA and t are the intercept and slope of,respectively.The calculated α and Q were calculated in Part 3.2.1.

In the present work,material constants in the strain-compensated Arrhenius model for plastic deformation of Ti-42.9Al-4.6Nb-2Cr were optimized by using multiple linear regression at the deformation temperatures of 1373 K,1473 K,1523 K and 1573 K,strain rates ranging from 0.001 s−1to 1.0 s−1,and strains ranging from 0.1 to 0.6.A thirdorder polynomial function (Eq.(7)) was found to represent the strain effect of Q and material constants (A,α and t),which is shown in Fig.11.As seen from Fig.11 (a)-(d),there are good correlations between established polynomial functions and the calculated α,t,lnA and Q,with the correlation coefficients (R2) of 0.999,0.885,0.996 and 0.996,respectively.

In order to verify the modified Arrhenius model,flow stress data obtained at the deformation temperatures of 1423 K and 1473 K,strain rates ranging from 0.001 s−1to 1.0 s−1,and strains ranging from 0.1 to 0.6 were selected.Fig.12 exhibits the relationship between the predicted flow stress data and the experimental of Ti-42.9Al-4.6Nb-2Cr in isothermal compression.As seen from Fig.12 (a),the average absolute relative error(AARE)of strain-compensated Arrhenius model is 7.52%,which reveals a good predictability for the flow stress of Ti-42.9Al-4.6Nb-2Cr in isothermal compression.

4.2.Hensel-Spittel model

Many efforts have been made since Arrhenius model cannot reproduce the flow stress-strain curves before flow stress peaked.Cheng et al.[30]establish another constitutive model to reproduce flow stressstrain curves of Ti-42Al-8Nb,which displays a high computational complexity.Hensel-Spittel model with low computational complexity is also used to describe total flow stress-strain curves,which is given as follows [16,31]:

where B0,p1,p2,p3,p4,p5,p7and p8are materials constants,0is 1.0 s−1,C is the Celsius deformation temperature (°C),σ is the flow stress(MPa),ε is the strain,is the strain rate (s−1).

Taking the logarithm of Eq.(8),Hensel-Spittel model is written as follows:

Materials constants in the Hensel-Spittel model for plastic deformation of Ti-42.9Al-4.6Nb-2Cr were optimized by using multiple linear regression at the deformation temperatures of 1373 K,1473 K,1523 K and 1573 K,strain rates ranging from 0.001 s−1to 1.0 s−1,and strains ranging from 0.1 to 0.6.The established Hensel-Spittel model for Ti-42.9Al-4.6Nb-2Cr in isothermal compression is shown as follows:

Fig.12(b)shows the relationship between the predicted flow stress data via Hensel-Spittel equation and the experimental of Ti-42.9Al-4.6Nb-2Cr in isothermal compression,in which the difference between the predicted flow stress data and the experimental is 11.95%.

Fig.13 shows comparison between the predicted flow stress data and the experimental of Ti-42.9Al-4.6Nb-2Cr in isothermal compression.As seen from Fig.13 (a) and (b),the prediction accuracy via strain-compensated Arrhenius model is higher than that via Hensel-Spittel equation,especially at high strain rates (example 1.0 s−1) and/or large strains.Moreover,the predicted flow stress data via straincompensated Arrhenius model is reasonable beyond the experimental range of strain.

5.Conclusions

Ti-42.9Al-4.6Nb-2Cr with duplex structure has been isothermally compressed at the deformation temperatures ranging from 1373 K to 1573 K,strain rates ranging from 0.001 s−1to 1.0 s−1up to the strain of 0.6.Meanwhile,the kinetic analysis on the hot workability of Ti-42.9Al-4.6Nb-2Cr with duplex structure has been carried out,and constitutive models were established.

(1) Strain has the great influence on apparent activation energy(Q)for hot deformation,strain-rate sensitivity index (m) and strain hardening index (n) of Ti-42.9Al-4.6Nb-2Cr during isothermal compression.The effect of strain on the Q value at the deformation temperatures below 1473 K is mainly related to dynamic recrystallization of γ phase and kinking of γ lamellae,while the Q value at the deformation temperatures above 1473 K might be linked to γ→α phase transformation and DRV of α phase.

(2) The strain-compensated Arrhenius and Hensel-Spittel models for Ti-42.9Al-4.6Nb-2Cr with duplex structure are successfully established to predict the flow stress.Average absolute relative errors of established strain-compensated Arrhenius model and Hensel-Spittel model are 7.52% and 11.95%,respectively.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grant Nos.51505386 and 51272416) and the Fundamental Research Funds for the Central Universities(3102017gx06003).

Appendix A.Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.pnsc.2020.02.004.