Modeling of flow behavior of Ti–6Al–4V alloy at elevated temperatures

Modeling of flow behavior of Ti–6Al–4V alloy at elevated temperatures

Materials Science & Engineering A 599 (2014) 212–222 Contents lists available at ScienceDirect Materials Science & Engineering A journal homepage: w...

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Materials Science & Engineering A 599 (2014) 212–222

Contents lists available at ScienceDirect

Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea

Modeling of flow behavior of Ti–6Al–4V alloy at elevated temperatures J. Porntadawit a, V. Uthaisangsuk a,n, P. Choungthong b a Department of Mechanical Engineering, Faculty of Engineering, King Mongkut's University of Technology Thonburi, 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand b Department of Production Engineering, Faculty of Engineering, King Mongkut's University of Technology Thonburi, 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand

art ic l e i nf o

a b s t r a c t

Article history: Received 22 November 2013 Received in revised form 21 January 2014 Accepted 22 January 2014 Available online 6 February 2014

Titanium alloy grade Ti–6Al–4V has been widely used for many applications such as aircraft structural component, machine parts and especially parts for medical equipment. To better understand the deformation behavior and microstructure evolution of the material under the hot working process is significant for achieving desired dimension and final mechanical properties of a product. In this study, stress–strain responses of the Ti–6Al–4V alloy were investigated using compression tests at different elevated temperatures and strain rates. The determined flow behaviors of the alloy were subsequently calculated according to the constitutive models based on the hyperbolic sine equation, the Cingara equation, and the Shafiei and Ebrahimi equation. Influences of both work hardening and dynamic recrystallization on the hot deformation behavior of the material were described. Then, the predicted flow curves were compared with experimental results and obtained discrepancies were characterized. Accurate modeling of flow curves can considerably assist in design of the forming process. & 2014 Elsevier B.V. All rights reserved.

Keywords: Ti–6Al–4V alloy Flow behavior Dynamic recrystallization

1. Introduction Ti–6Al–4V alloy is a titanium alloy, which has been developed to be less interstitial so that it is difficult to react with surrounded media. Therefore, this alloy is suitable to apply for many artificial organs such as bone plant, hip prosthesis, joint implants, heart valves and various kinds of implants in the dental area [1]. To manufacture components made of such alloys, machining is often used that causes much material loss. Cold forming though needs high load capacity. Therefore, the hot forming process has gained more interest, since complex shapes can be attained. Besides, mechanical properties of components can be improved at the same time. To obtain required final dimension and properties of hot formed parts, understanding of plastic deformation of material at high temperature is important [2]. The deformation of metals or alloys at temperatures above 0.5Tm involve a complex process, in which mechanical working interacts with various metallurgical phenomena such as dynamic restoration including recovery and recrystallization and phase transformation of polymorphous materials. It is necessary to take into account these in-parallel occurred mechanisms. It was reported that alloys, which have relatively low or medium values of the stacking fault energy (SFE) could

n

Corresponding author. Tel.: þ 662 470 9274; fax: þ662 470 9111. E-mail address: [email protected] (V. Uthaisangsuk).

http://dx.doi.org/10.1016/j.msea.2014.01.064 0921-5093 & 2014 Elsevier B.V. All rights reserved.

exhibit dynamic recrystallization, but for alloys with high SFE only recovery takes place during high temperature deformation [2,3]. Ding et al. [3] investigated variation of microstructure of a Ti–6Al–4V alloy under different hot working conditions using Optical Microscopy (OM) and Scanning Electron Microscopy (SEM). Influences of hot forming parameters on the microstructural evolution of the Ti–6Al–4V alloy were determined. Luo et al. [4] analyzed effects of deformation temperature, strain and strain rate on the flow stress in order to describe mechanical behavior of an isothermally compressed Ti–6Al–4V alloy. The apparent activation energy for deformation of the isothermally compressed Ti– 6Al–4V alloy at different strains was calculated. Then, deformation mechanisms in the two phase region and single phase region were clarified through this activation energy in comparison to the activation energy for self-diffusion of α-Ti and β-Ti alloy. Finally, processing map of the compressed Ti–6Al–4V alloy at a strain of 0.6 was established in order to optimize hot working parameters. Jonas et al. [5] investigated flow curves of 11 different steels with regard to dynamic recrystallization (DRX). The double differentiation method was used to define the critical strain εc for initiation of DRX. The “athermal” hardening parameter h and the dynamic recovery parameter r were evaluated from the work-hardening region prior to εc. The saturation stresses ssat according to the nonrecrystallized zones as well as the associated work-hardening curves srecov were obtained by this manner. The net softening due to the DRX was then defined as the difference between the srecov

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and experimental curves. Avrami plots were prepared from experimental data. However, it was found that strain rate and temperature dependences differed considerably from their static values. The dependency of the time of half-softening t50 on strain rate and temperature was used to predict the DRX curves under industrial conditions. Momeni et al. [6] examined behavior of the Ti–6Al–4V alloy in both single phase (β) and two phase (α þ β) regions under straining by hot compression. The results were analyzed in terms of different dependences of flow stress on temperature, strain rate and various microstructure evolutions. It was found that in the two phase region strain values at the peak point and the highest rate of flow softening were almost independent from the Zener–Hollomon parameter. Mirzadeh and Najafizadeh [7] developed a stress–strain equation to predict flow stress curves using peak stress, peak strain, and four materials constants, which were independent of deformation conditions. These parameters were calculated by a nonlinear regression of one or more experimental stress–strain curves. Although predictions of the stress–strain responses became more accurate than those using the hyperbolic sine equation, a strong physical basis was still lacked. Qin et al. [8] proposed a model for representing flow stresses of magnesium alloy under hot deformation condition. The model was capable of describing flow behavior including work hardening, dynamic recovery (DRV) and softening caused by dynamic recrystallization. By the modeling, the doubledifferentiation method was used to identify the critical strain for initiation of DRX and the DRV parameter. Cai et al. [9] established constitutive equations for predicting flow stresses of Ti–6Al–4V alloy at elevated temperatures. Isothermal hot compression tests were performed to develop a constitutive equation for different phase regions. Effects of temperature and strain rate on deformation behaviors were represented by the Zener–Hollomon parameter in an exponent-type equation. Although hot deformation behavior of Ti–6Al–4V alloy in either α þ β or β phase region has been numerously investigated, few models the flow behavior of this alloy were based on microstructure evolution. In Shafaat et al. [10] experimental flow curves of the Ti–6Al–4V alloy were obtained from isothermal hot compression tests. The results were then described using two types of constitutive equation. The first one was Sellars equation. For the second one, modeling of flow curves up to the peak point was carried out with the Cingara model and modeling beyond that was performed with a model developed on the basis of the Johnson–Mehl–Avrami–Kolmogorov (JMAK) theory. The reliability of each model was then evaluated by means of the root mean square error (RMSE). Hot working behavior of 26NiCrMoV 14-5 steel was examined by hot compression tests for the temperature range of 850–1150 1C and strain rate of 0.001– 1 s  1 in Mirzaee et al. [11]. The obtained flow curves at temperatures higher than 1000 1C exhibited typical dynamic recrystallization, whereas those at lower temperatures represented work hardening without any indication of dynamic recrystallization. The flow stresses were correlated with strain rates and temperatures using the hyperbolic sine function. It was observed that flow curves at high temperatures described using a combination of Cingara and Avrami equations were considerably different from experimental curves. Regarding low work hardening rate and increasing of flow stress at the temperatures below 1000 1C, the possibility of dynamic precipitation was supposed. In order to determine the potential of precipitation in the material stress relaxation tests were performed in different temperature ranges. Different dependences of flow stress on temperature at low and high temperatures were thus considered according to the results. Shafiei et al. [12] proposed a constitutive equation, in which extrapolation of dynamic recovery flow stress curves and kinetic equation for DRX was used. Hereby, the dynamic recovery and work hardening region before critical strain were not taken into account.

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The description of flow stress of metals and alloys at high temperatures is important for estimating required forming forces, dimensional accuracy of final products and process simulation. Therefore, it is necessary to provide a model, which can overcome limitations of previously mentioned models to some extent. In addition, Finite Element Analysis (FEA) has been an effective tool for identifying proper working conditions of the precise hot forming process. Reliability of FE simulation results is significantly influenced by given material property parameters. Therefore, prediction accuracy of stress–strain behavior at high temperature of investigated alloy must be considered. In this study, stress–strain curves of the Ti–6Al–4V alloy were investigated using hot compression tests at the temperatures of 900, 950, 1000 and 1050 1C with the strain rates of 0.1, 1 and 10 s  1. By the modeling, experimentally determined flow curves were described using three constitutive approaches. First, the hyperbolic sine equation was used to describe the dependency between flow stress and strain, strain rate and temperature. Secondly, model according to the Cingara equation was applied for representing the flow curves up to their peak points. Then, successive flow stresses were characterized by the Shafiei and Ebrahimi model, which was based on an extrapolation of DRV and kinetic equation for DRX [11,8]. Finally, numerically calculated flow curves were compared with the experimental curves and evaluated using the RMSE method.

2. Material In this work, Ti–6Al–4V alloy commercial grade was taken from the LHM Company, PR China. The as-received alloy was heattreated by the mill annealing process and was in a round bar shape with a diameter of 16 mm. The chemical composition of the examined alloy is provided in Table 1. The Ti–6Al–4V alloy is classified in the group of α/β phase based titanium alloy, which contains 6% weight of aluminum and 4% weight of vanadium. During the processing, aluminum and vanadium act as an α phase stabilizer and a β phase stabilizer, respectively. Consequently, the equilibrium microstructure of the alloy at room temperature mainly consists of the α phase matrix and some retained β phase. The primary α phase has a hexagonal close-packed structure, while the β phase exhibits a body-centered cubic structure. Fig. 1 shows a simplified phase diagram of the Ti–6Al–4V alloy. The β transus temperature defines the temperature, above which the Table 1 Chemical composition (mass content in %) of the investigated Ti–6Al–4V alloy. N

C

H

Fe

O

Al

V

Ti

0.021

0.079

0.009

0.140

0.126

6.030

3.990

Balance

Fig. 1. Phase diagram of Ti–6Al–4V alloy.

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equilibrium microstructure is only available in the form of β phase. The β transus temperature of the Ti–6Al–4V alloy is approximately 975 1C [1]. The α/β phase titanium alloys can transform to α' martensite when undergoing fast cooling from the β phase field to room temperature. The martensite start temperature of the Ti–6Al–4V was about 800 1C [1]. Microstructure of the as-received Ti–6Al–4V alloy is shown in Fig. 2, in which α phase matrix including retained β phase is observed.

3. Experimental procedure Cylindrical specimens of the investigated alloy were prepared with a diameter of 5 mm and a height of 10 mm. Compression tests were carried out using a deformation dilatometer. Details of the deformation dilatometer are shown in Fig. 3a. All specimens were initially heated up and homogenized at 1100 1C for 1 min. After that, the specimens were cooled down to different temperatures for upsetting. The compression tests were isothermally performed at the temperatures of 900, 950, 1000 and 1050 1C so that both single phase (β) and two phase (α þ β) regions could be examined. Additionally, strain rates were varied to be 0.1, 1 and 10 s  1. The compression was terminated at the final sample height of 4 mm. Finally, the specimens were quenched with argon gas immediately after the compression. The temperature–time route of the experiments is depicted in Fig. 3b. Subsequently, stress– strain responses of each testing conditions were determined and metallography analyses were carried out for each deformed sample.

4. Modeling and results 4.1. Stress–strain relationships at high temperatures The stress–strain curves obtained from the hot compression tests at various temperatures and strain rates are illustrated in Fig. 4. It was found that typically the flow stresses increased with

rising strain rate at a given temperature and decreased with increasing temperature at a given strain rate. At the beginning of the hot deformation, flow stress steeply increased because of strain hardening effect and accordingly increase of dislocation density. The rate of work hardening became lower and the flow stresses then reached a maximum value. After that, the flow stresses began to soften. This softening phenomenon was due to a combination of adiabatic heating, dynamic recovery and dynamic recrystallization. The flow stress softening, which occurred during deformation in the temperature range of α þ β phase (900 and 950 1C), was caused by the dynamic recovery and adiabatic heating. The adiabatic heat led to an increasing of actual forming temperature of the specimens. Thus, proportion of the existing β phase, which is a softer phase, enhanced during deformation in these temperature ranges. The effect of the adiabatic heating could be more noticeable when forming at high strain rate and lower temperature. When deformation took place in the temperature range of β phase (1000 1C and 1050 1C) dynamic recrystallization could be additionally observed. The flow softening was then controlled by both adiabatic heating and dynamic recrystallization. 4.2. Hyperbolic sine constitutive equation First, the constitutive equations according to Arrhenius were applied in this study to describe the relationship between flow stress, strain rate and temperature. The material constants for the constitutive equations could be directly determined from experimental results of the hot compression test. Hereby, the Zener– Hollomon parameter with an exponent-type equation was used. The Arrhenius equation provided functions between the Zener– Hollomon parameters and material flow stress, given as [4,7–10]   Q Z ¼ ε_ exp ð1Þ RT 

ε_ ¼ AFðsÞ 

Q RT

 ð2Þ

where F(s) is the stress function which can be expressed by any of the following: 8 sn αs o 0:8 > < αs 41:2 expðβsÞ FðsÞ ¼ ð3Þ > : sinh ðαsÞn for all

ε_ is the strain rate (s  1), R is the universal gas constant

Fig. 2. Microstructure of as received Ti–6Al–4V alloy.

(8.31 J mol  1K  1), T is the absolute temperature (K), Q is the activation energy for hot deformation (kJ mol  1), s is the flow stress (MPa) for a given strain and A, α, n are the material constants, in which α ¼ β/n. The values at the strain of 0.5 and temperature of 1000 1C and 1050 1C were taken as an example for demonstrating solution

Fig. 3. (a) Schematic of deformation dilatometer and (b) temperature–time route of the isothermal hot compression test.

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Fig. 4. True stress–strain curves obtained from hot compression tests at different strain rates and temperatures for the investigated Ti–6Al–4V alloy.

Fig. 5. Relationships between (a) lnðsÞ and lnð_εÞ and (b) s and lnð_εÞ.

procedure of the involved material constants. Eqs. (2) and (3) could be applied to a wide range of stresses. At low stress levels (αs o 0:8) stress function could be approximated by the power law equation, while at high stress levels (αs 4 1:2) it approached the exponential law. Taking logarithm on both sides of the equations: 1 1 lnðsÞ ¼ ln ðε_ Þ  lnðBÞ n' n' 1

1

s ¼ lnðε_ Þ  lnðcÞ β β ln ½ sin ðαsÞ ¼

ln ε_ Q ln A  þ nRT n n

ð4Þ

ð5Þ

ð6Þ

From Eqs. (4) and (5), the material constants n' and β were calculated. The mean values of n' and β were then obtained by the slope of ln(ε_ ) vs. ln(s) plots and ln(ε_ ) vs. s plots for different temperatures, as shown in Fig. 5 for the temperatures of 1000 1C and 1050 1C. For the α þ β phase region, the mean values of n'¼ 5.2167, β ¼0.0594 and α ¼0.0109 were calculated. In case of

the β phase region, the mean values of n' ¼6.0623, β ¼0.1238 and α ¼0.0201 were obtained. For any particular temperatures, differentiating Eq. (6) gives 1 df sinhðαsÞg ¼ n dðln ε_ Þ

ð7Þ

where n is the slope of lnðε_ Þ vs. ln[sinhðαsÞ] plots at different temperatures, as seen for the temperatures at 1000 1C and 1050 1C in Fig. 6. The mean value n of 3.9019 for the temperature range of the α þ β phase and 4.5770 for the temperature range of the β phase was determined. The activation energy Q is an important physical parameter of the material and could be derived by partial differentiation of Eq. (6). Then, the activation energy is expressed by the following equation: Q ¼ Rn

df ln ½ sinhðαsÞg dð1=TÞ

ð8Þ

The value of Q could be simply obtained from a mean slope of ln[sinhðαsp Þ] vs. (1000/T) plots at various strain rates, as illustrated in Fig. 7. The mean Q value of 613.4026 kJ mol  1 for the alloy in the α þ β phase region and 574.1788 kJ mol  1 for the alloy

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could be directly obtained. Additionally, the mean value of n was computed according to Eq. (10) to be 3.5357 for the α þ β phase region and 4.2688 for the β phase region. These n values for both regions were comparable to those from the literatures. The determined material parameters n', β, α, ln A and Q were plotted against increasing strain in Fig. 9 for the temperature in the β phase region. It was obvious that the applied polynomial functions could fairly represent relationships between these parameters and plastic strain. From Eq. (9), the plastic flow stress can be expressed as h z i1n 1 s ¼ sin  1 ð11Þ A α   ð12Þ arcsinhx ¼ ln x þ ðx2 þ 1Þ1=2

Fig. 6. Relationship between ln ½ sinhðαsÞ and lnð_εÞ.

Substituting x ¼ ðz=AÞ1=n in an inverse sine function as shown in Eq. (12), the flow stress in Eq. (11) could be then rewritten. Finally, constitutive equation that related flow stress and Zener–Hollomon parameters could be described using the hyperbolic sine function in Eq. (13). Influences of strain rate and temperature on the plastic flow stress of material were incorporated in this function. 8 "  #1=2 9   = 1 < Z 1=2 Z 2=n s¼ þ þ1 ð13Þ ; A α: A As shown in Fig. 10, the calculated flow curves for the temperatures of 900, 950, 1000 and 1050 1C using the hyperbolic sine equation were compared with the experimentally obtained stress– strain curves. It was found that flow stresses of the investigated alloy could not be precisely described by this approach, in particular at high temperatures and high strain rates. However, at low temperature of about 900 1C and low strain rate of 0.1 and 1 s  1, the predicted flow curves by the hyperbolic sine function acceptably agreed with those from the experiments.

Fig. 7. Relationship between ln ½ sinhðαsÞ and 1=T.

4.3. Determination of peak stress and critical stress

Fig. 8. Relationship between ln z and ln ½ sinhðαsÞ.

in the β phase region was obtained. These values were somewhat different from those found from the literatures [9]. It was likely due to different determination methods and constitutive equations used. From Eqs. (1) and (2),   Q Z ¼ ε_ exp ¼ A½ sinhðαsÞn ð9Þ RT Taking the logarithm to both sides of Eq. (9): ln z ¼ n ln ½ sinhðαsÞ þ ln A

ð10Þ

From the experimentally resulting stress–strain curves, relationships between ln[sinh(αs)] and ln Z could be determined, as depicted in Fig. 8. Then, the values of ln A and n in Eq. (10) were calculated as the intercept and slope of ln Z and ln[sinh(αs)] plot, respectively. The value of the parameter A of 2.28  1024 s  1

The critical stress for DRX of material is a significant parameter for the flow stress description. When the flow stress of a material was below the critical stress, effects of work hardening and DRV were dominant due to the generation and reorientation of dislocations, respectively. When the flow stress was raised above the critical stress and deformation continued to increase, effects of volume fraction of dynamically recrystallized grains became gradually more effective. As a result, material softening occurred with an increasing rate because of nucleation and growth of new dislocation-free grains. The critical stress for DRX at different forming conditions could be determined by either quantitative metallography analysis or evaluating the corresponding plastic stress–strain curves. In this work, change in slope of the stress– strain curve (θ ¼ ds/dε) plotted against the flow stresses was used as an indication of microstructure alteration taking place in the investigated alloy. The θ–s curves basically originated from a common intercept θ0 at s ¼0 and decreased with rising flow stress. As shown in Fig. 11, the θ–s curves exhibited three distinguished segments, in which two of them were approximately linear. The first linear section decreased with stress for an initial range of strain up to the point, where sub-grains began to form with a low increased rate of DRV. The curve gradually changed to a lower slope of the second linear segment up to the point, where the critical stress sc was attained for initiating the dynamic recrystallization. The curve then dropped off rather sharply to θ ¼0 at the peak stress sp. Extrapolation of the second linear segment of the θ–s curve intercepted the s axis at the saturation stress ss. If dynamic recrystallization was present, shape of the θ–s curve would be similar to the solid line in Fig. 11.

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Fig. 9. Variation of n', β, α, ln A and Q with strain for the temperature range of β phase.

Fig. 10. Comparison between flow stresses experimentally determined and predicted by the hyperbolic sine equation at the temperatures of (a) 900 1C, (b) 950 1C, (c) 1000 1C and (d) 1050 1C (symbol – experiments, solid line – calculations).

But if only dynamic recovery occurred as the restorative mechanism operation, the θ–s curve would be identical to the dashed line [2,7,12].

The simple method according to Najafizadeh and Jonas [2,7,12] was used to determine the critical stress sc describing initiation of DRX. The inflection point was identified by fitting a third order

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polynomial to the θ–s curves up to the peak stress, as expressed by

θ ¼ As3 þ Bs2 þ C s þ D

ð14Þ

where A, B, C, and D are constants for a given set of hot deformation conditions. The second derivative of this equation with respect to s could be written as d θ ¼ 6As þ 2B d s2 2

ð15Þ

equation, which was given as [2,6,10]  ε  ε m  p X DRX ¼ 1  exp  kd ε_

ð17Þ

where XDRX is the volume fraction of dynamic recrystallized grain, εp is the peak strain, ε_ is the strain rate. kd and m are materials constants. XDRX could be interpreted as the drop of flow stress from the peak stress with respect to the whole softening. The x value was determined by [10]

sp  s sp  ss

At the critical stress for DRX initiation, the second derivative became zero. Therefore,



sc ¼ ¼ 2B=6A

By plotting the function between ln[ln(1/(1 x)] vs. ln[(ε  εp)/ε_ ], the values of m and kd were obtained for each testing condition, as depicted for example in Fig. 13. Subsequently, the XDRX of the investigated alloy during isothermal compression could be calculated for the different temperatures and strain rates, as shown in Fig. 14. Considering the same deformation temperature, higher strain rate led to delayed recrystallization with slower rate. However, at the

ð16Þ

The corresponding third order polynomials of work hardening rate vs. flow stress at different deformation temperatures and strain rates were determined and shown in Fig. 12. The peak stress could be obtained from these third order polynomials when θ ¼0. The critical stresses of each test condition were identified from the point where the second derivative of θ was zero. Principally, the kinetics of DRX considered as a solid-state transformation could be described by the Johnson–Mehl–Avrami–Kolmogorov (JMAK)

Fig. 11. Strain hardening rate vs. flow stress [2].

ð18Þ

Fig. 13. Determination of the constants m and kd for the temperature range of β phase.

Fig. 12. Plots of work hardening rate vs. true stress for different temperatures and strain rates of (a) 0.1 s  1, (b) 1 s  1, (c) 10 s  1 and (d) for the temperature of 1000 1C and strain rate of 0.1 s  1.

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strain rate of 10 s  1, the recrystallization started at lower strain than that of 0.1 and 1 s  1 and proceeded with retarded rate due to an adiabatic effect. When samples were formed at high strain rates, heat generated by the work of deformation could not be removed and temperature of the samples thus increased during deformation. Furthermore, it could be seen that higher temperature caused accelerated recrystallization for low strain rates of 0.1 and 1 s  1. For the Ti–6Al–4V alloy, the recrystallization process should be described with regard to individual deformation temperatures. The temperatures of 900 1C and 950 1C were in the two phase α/β region, whereas the temperature of 1050 1C was in the single β phase region. The temperature of 1000 1C was approximately among the transition temperature, in which β phase transformation was likely not complete. In case of FCC metals with medium to low stacking fault energy (SFE), before reaching the critical strain, strain hardening and dynamic recovery were the principle mechanisms responsible for stress–strain response. Afterwards, DRX accompanied the process. On the other hand, high SFE metals like aluminum alloys, alpha titanium alloys, and ferritic BCC steels were expected to be dynamically softened by dynamic recovery

219

(DRV) instead of DRX [13]. The stress–strain manners were changed from DRX to DRV behavior with decrease of deformation temperature or increase of strain rate. Note that hints of recrystallization were only observed at both these temperatures by metallography. Otherwise it was reported that discontinuous DRX as a predominant nucleation via grain boundary sites was observed at higher temperature and lower strain rate, while the continuous DRX occurred at lower temperature and higher strain rate [14]. Because of these mechanisms different recrystallization characteristics could take place at higher strain rates. The predicted volume fractions of dynamic recrystallization were further used for description of flow stress. 4.4. Cingara constitutive equation By another modeling approach, flow curves up to the peak stress were firstly described using the Cingara equation [2,7,10].     ε ε c s ¼ sp ð19Þ exp 1 

εp

εp

where sp and εp are the peak stress and peak strain, respectively. In Eq. (19), c is a parameter, which controlls the shape of flow curve up to the peak point. It could be determined by taking natural logarithm to Eq. (19) leading to the following expression:      s ε ε ln ¼ c 1  þ ln ð20Þ

sp

Fig. 14. Predicted volume fractions of DRX during isothermal compression of Ti–6Al–4V alloy at different temperatures and strain rates.

εp

εp

This equation is a linear function between lnðs=sp Þ and 1  ðε=εp Þ þ lnðε=εp Þ. It could be used to determine the value of the constant c, as presented in Fig. 15a–d for both temperature ranges of α þ β and β phase, respectively. Since these slopes were approximately constant, the c values were evaluated for different deformation temperatures and strain rates by the linear fitting method. The mean value of the parameter c of 0.153 was obtained and further applied.

Fig. 15. Plot for calculation of parameter c in the Cingara equation when the temperature is (a) 900 1C, (b) 950 1C, (c) 1000 1C and (d) 1050 1C.

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Fig. 16. Predicted flow curves at the temperatures of 900, 950, 1000, and 1050 1C using the Cingara equation in comparison with experimental results (symbol – experiments, solid line – calculations).

Fig. 16 represents the entire flow curves predicted by using the Cingara equation in comparison with the experimentally obtained curves. Obviously, the resulting stress–strain responses after peak stresses were much underestimated by the calculations, especially for the temperature of 900 1C and at high strain rates. However, at low strain rate conditions and other temperatures, stress–strain curves calculated by the Cingara model were better in agreement with the experimental ones, since the flow stress softening was not significant, but rather remained constant.

4.5. Shafiei and Ebrahimi constitutive equation Based on the extrapolation of DRV flow stress curves and kinetics equation of DRX, Shafiei and Ebrahimi proposed the subsequent equation for describing flow behavior with DRX in the strain range between εc r ε Z εss [2,12].

s ¼ ss ðss  sc Þexp½c'ðεc  εÞ ðss  sss ÞX

ð21Þ

where sc, sp, sss, ss, εc and X are the critical stress, peak stress, steady state stress, DRV saturation stress, critical strain and volume fraction of DRX, respectively. c'is a constant with metallurgical aspect according to  sp εp π ∅ ¼ þarctan ð22Þ 2 ðεc  εp Þεc tan ∅ ¼

1 c'

ð23Þ

The parameters determined in Sections 4.3 and 4.4 were then applied to Eq. (21). The flow curves described by using the Cingara model in combination with the Shafiei and Ebrahimi model were illustrated in Fig. 17 in comparison with those from the experiments. It was found that by this approach the calculated stress– strain responses fairly agreed with the experimental curves for the entire strain range of all testing conditions. The Shafiei and Ebrahimi model took into account the recrystallization mechanism and could better predict the stress–strain responses of the

investigated alloy, in particular at high strain rates, by which flow softening remarkably occurred at early deformation states. To evaluate discrepancies between predictions and experiments, the root mean square error (RMSE) method with the following function was applied for the results calculated by each applied approach. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N RMSE ¼ ∑ ðY t  St Þ2 ð24Þ Nt¼1 Yt are the experimental values and St are the predicted values. The obtained RMSEs, shown in Table 2, exhibited average deviations of stress–strain results from experiments and predictions using the Cingara, Cingara combined with Shafiei and Ebrahimi and the hyperbolic sine model. It could be clearly seen that flow behaviors at high temperatures of the investigated Ti–6Al–4V alloy could be properly described by the combination of the Cingara and Shafiei and Ebrahimi equation. The Cingara model was applied for the flow stresses from the beginning up to the peak points. Subsequent flow stresses were then represented by the Shafiei and Ebrahimi model. By this manner, recrystallization was also considered using the JMAK equation. This model was preliminary intended for materials with relatively low stacking fault energy, nevertheless the predictions were acceptable. Regarding conditions of low temperature and high strain rate, all methods provided the highest discrepancies. 4.6. Microstructure observations The microstructures obtained from metallography examinations are depicted in Fig. 18 for the deformation at the strain rate of 0.1 s  1 and temperatures of 900, 950, 1000 and 1050 1C. The microstructure occurred during formation at the temperatures of 900 1C and 950 1C, which were in the range of α þ β phase, contained α' martensite and some α phases. Grain boundaries of the former β grain could be observed and a grain size of about 47 μm was measured. At the temperatures of 1000 1C and 1050 1C, α' martensite was mostly found in the entire area. Some evidences

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221

Fig. 17. Predicted flow curves at the temperatures of 900, 950, 1000 and 1050 1C using the Cingara model combined with the Shafiei and Ebrahimi model in comparison with experimental results (symbol – experiments, solid line – calculations).

Table 2 Maximum of errors calculated for flow curves obtained by the Cingara, the Shafiei and Ebrahimi, and the hyperbolic sine models in comparison. Temperature (1C)/strain rate (s  1)

RMSE (MPa) 0.1

Cingara combined Shafiei and Ebrahimi's equation 900 950 1000 1050 Cingara equation 900 950 1000 1050 Hyperbolic sine equation 900 950 1000 1050

of recrystallized Fig. 18c and d.

β grain could be seen at these temperatures in

5. Conclusion In this work, isothermal hot compression tests were performed at different temperatures and strain rates for the Ti–6Al–4V alloy within the areas of both α þ β and β phase. Then, stress–strain curves were determined and described using the hyperbolic sine equation, the Cingara equation, and the Shafiei and Ebrahimi equation. The following conclusions could be drawn. 1. The flow curves obtained under each condition exhibited a single peak point. The stress–strain responses indicated that the main restoration mechanism during hot deformation of the

1

10

9.2430 0.4076 0.0995 0.7743

11.5793 3.8363 0.9193 1.0572

30.3585 8.9119 3.4621 2.2491

180.8699 0.6655 0.2511 0.1524

106.1944 2.8501 2.5511 0.3765

232.1138 38.5183 12.1316 4.6933

40.6770 40.0155 3.9533 11.7731

30.2041 12.9079 4.2431 3.2525

75.0597 78.3234 8.3911 3.2401

Ti–6Al–4V alloy in the two phase region was the DRX and adiabatic heating. 2. The flow curves of the Ti–6Al–4V alloy at elevated temperatures were successfully represented using a combination of the Cingara and Shafiei and Ebrahimi model. 3. The mean errors of the predicted flow stresses were calculated at a wide range of strains for all experimental conditions. The results showed that the combined approach gave the most accurate description of the flow behavior, since both work hardening and dynamic recrystallization processes were considered. At low temperatures and high strain rates, all methods exhibited the highest discrepancies, because different recrystallization mechanisms and recovery should be also incorporated. 4. According to the primarily metallographic analyses, microstructures of the Ti–6Al–4V alloy showed the α lamellar structure and α' martensite, when the samples were deformed

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Fig. 18. Microstructures of the Ti–6Al–4V alloy formed at the strain rate of 0.1 s  1 and temperatures of (a) 900 1C, (b) 950 1C, (c) 1000 1C and (d) 1050 1C.

in the α þ β phase region. In case of hot forming in the β phase region, dynamically recrystallized β grains were observed. Acknowledgments The authors would like to acknowledge the National Research Council of Thailand for financial supports, the King Mongkut's University of Technology Thonburi (KMUTT) for supporting press machines and the Institute of Thailand for supporting deformation dilatometer. References [1] G. Luetjering, J.C. Williams, in: Engineering Materials and Processes, 2nd edition, 978-3-540-71397-5, 2007, Springer, New York.

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