Constitutive Relationship for Hot Deformation of TB18 Titanium Alloy

In the present work, the hot deformation behavior of TB18 titanium alloy was investigated by isothermal hot compression tests with temperatures from 650 to 880°C and strain rates from 0.001 to 10 s . .e flow curves after friction and temperature correction show that the peak stress decreased with the temperature increase and the strain rate decrease. .ree typical characteristics of flow behavior indicate the dynamic softening behavior during hot deformation. At a strain rate of 0.001∼0.01 s , the flow stress continues to decrease as the strain rate increases after the flow stress reaches the peak stress; the flow softening mechanism is dynamic recovery and dynamic recrystallization at a lower temperature and dynamic recrystallization at a higher temperature. .e discontinuous yielding phenomenon could be seen at a strain rate of 1 s , dynamic recrystallization took place in the β single-phase zone, and flow localization bands were observed in the α+ β two-phase zone. At a higher strain rate of 10 s , the flow instabilities were referred to as the occurrence of flow localization by adiabatic heat. Constitutive equation considering the compensation of strain was also established, and the results show high accuracy to predict the flow stress with the correlation coefficient of 99.2% and the AARE of 6.1%, respectively.


Introduction
Owing to the high specific strength, remarkable fracture toughness, excellent creep and corrosion resistance, β titanium alloys have been used widely in many fields such as aerospace, aircraft, ship, and nuclear energy [1][2][3][4]. For instance, Ti-10V-2Fe-3Al was used in landing gear forgings in Boeing 777 [5], and landing gear forgings were manufactured by Ti-5Al-5Mo-5V-3Cr-0.5Fe alloys in Boeing-787 and Airbus-350 [6]. ermomechanical processing to fabricate these large products is very complicated and the key processes are related to two steps as follows: (1) to produce usable and intricate shapes through hot die forging; (2) to meet the high quality mechanical properties through the subsequent heat treatment [7]. Due to the hexagonal closepacked structure (HCP) with less slip systems, titanium alloys are more difficult to deform than other metallic materials, such as Fe alloys [8] and aluminium alloys [9]. Previous studies proved that β titanium alloys had a relatively narrow process window and were sensitive to the processing parameters, such as temperature, strain, and strain rate [7,[10][11][12]. Finite element simulation has been widely applied in the production of large forgings, and accurate material models are essential to improve the precision of the simulation results.
Considerable research studies have been carried out on the constitutive relationship and flow behavior for the hightemperature deformation by using simulative laboratory tests. Lin et al. [13] studied the flow behavior of TC18 titanium alloy in the α + β phase zone by hot compression tests with the strain rates of 0.001-0.1 s − 1 and temperature of 1033-1123K. ree constitutive models, including the strain-compensated Arrhenius type, Hensel-Spittel (HS), and artificial neural network (ANN) models were used to describe the flow behavior. Long et al. [14] investigated the flow behavior of Ti-6Cr-5Mo-5V-4Al and compared the prediction accuracy between the genetic algorithm optimized Arrhenius model (GAOAM) and the strain-compensated Arrhenius model (SCAM). Also, dynamic globularization behavior was conducted to explain the flow softening at different strain rates. Zhao et al. [15] considered that the main dynamic restoration mechanism of Ti-10V-2Fe-3Al deformed above β transus temperature is dynamic recovery. Chuan and Liang [16] studied the hot deformation and dynamic recrystallization of Ti-55531 in the β singlephase region, discontinuous yield and flow softening phenomena in the true stress-true strain curves have shown the sensitivity to temperature and strain. Fan et al. [17] found that Ti-7333 exhibits flow instabilities in the form of flow localization when strain rates reach more than 1 s − 1 and moderate-low strain rate at a lower temperature. Lei et al. [18] indicated that the hot workability of Ti-5Al4Zr8Mo7V is better at higher deformation temperature, higher strain, and lower strain rate.
It is well known that the Arrhenius equation is most widely used to describe the relationship between temperature, stress rate, and flow stress [19][20][21]. However, it is found that the effect of strain was not considered on the flow stress. In order to accurately predict the flow behavior, the straincompensated Arrhenius-type constitutive model was proposed [22]. e material constants values of Q, A, n, and α were expressed as different order polynomial functions of strain. e modified models, verified in steels [23], titanium alloys [24], aluminium alloys [25], and magnesium alloys [26], are in good agreement with the experimented flow stress.
TB18 titanium alloy is a new near beta titanium alloy that has potential applications in aerospace structural parts and whose hot workability deserves to be figured out. In the present work, isothermal hot compression tests were conducted under the temperature from 650 to 880°C with a strain rate from 0.001 to 10 s − 1 to study the hot deformation behavior of TB18, and the strain-compensated Arrheniustype constitutive model was built to describe the relationship of the flow stress, strain rate, and temperature after the friction and temperature correction. Finally, the flow of softening behavior was analysed. e research results are expected to provide guidance for the application of the alloy.
Cylindrical specimens with a size of φ10 × 15 mm were machined from the as-received billet. Hot compression experiments were carried out on a Gleeble 3800 machine with the deformation temperatures of 650, 700, 750, 800, 820, 850, and 880°C, strain rates of 0.001, 0.01, 0.1, 1, and 10 s − 1 , and the height reductions of 60%. A K-type thermocouple with a precision of ±2°C was welded in the middle of the samples to record temperature changes during deformation. All the specimens were heated up to the deformation temperature with 5°C/s and kept for 3 min to homogenize the temperature. Graphite pieces as a lubricant were covered on the two ends of specimens to reduce the friction between specimens and the anvils.

Friction Correction.
e friction between the specimen and anvils will restrict the metal flow during the deformation, although lubricants were used [27]. e velocity of metal flow gradually increases from the surface to the core with the increase of the deformation, resulting in a barrelled shape of the specimen, as shown in Figure 2. erefore, the original flow stress should be corrected with friction. A barrelling coefficient was proposed to revise the effect of friction by Roebuck et al. [28], which can be expressed as follows: where B is the barrelling coefficient, h 0 and h are the height of specimen before and after deformation, R 0 is the initial radius, and R M is the maximum radius of the specimen after deformation, respectively. When 1 < B ≤ 1.1, there is no need to revise the differences between the original flow stress and the true flow stress; when B ≥ 1.1, the original flow stress curves must be corrected. e calculated values of B at different deformation conditions are listed in Table 1. It can be seen that almost all the B values are above 1.1, which means that the original flow stress should be corrected under all the deformation conditions.  Advances in Materials Science and Engineering According to the upper-bound theory [29], the frictioncorrected flow stress can be written as follows: where P and σ represent the true stress before and after correction, b is the barrel parameter, m is the coefficient of friction, R � R 0 exp(− ε/2), and H � h 0 exp(− ε). m and b can be calculated by the following equations: R f is the average radius of the specimen after deformation, R T is the top radius of deformed samples, and R f and R T can be calculated by the following: Based on equation (2) to equation (6), the flow stress curves after friction correction are shown in Figure 3. It is obvious that the influence of friction between the specimen and the anvils is noteworthy with the increasing strain rate and the decreasing deformation temperature.

Temperature Correction.
Adiabatic heating is a common characteristic of stress-strain curves at high temperatures caused by the deformation heat or (and) phase transition. In the compression tests, the welded thermocouple is often not sensitive enough to measure the temperature and record the value by computer [30]. Figure 4 shows the instantaneous temperature recorded by thermocouple during deformation. It can be seen that at strain rates <1 s − 1 , the instantaneous temperature is almost the same as the setting value, in this situation, deformation heat is usually very small and is transmitted away with plenty of time, so the temperature corrections are not necessary to perform at lower strain rate levels; at strain rates ≥1 s − 1 , the instantaneous temperature increases fast as the deformation proceeds, and the value is much bigger than the setting one, the deformation heat could not be transferred from the core to the surface because of short deformation time and low thermal conductivity of titanium alloys. us, the flow stress data must be corrected for adiabatic heating. e temperature increase due to deformation heating may be calculated using the following equation [31]: Specimen before deformation Specimen a er deformation    Advances in Materials Science and Engineering where ΔT is the variation in temperature, η is the adiabatic correction factor, ε 0 σdε equals to the area under the true stress-strain curve, and ρ and C p are the density and the specific heat, respectively. e value of 0.95 is the fraction of mechanical work transformed to heat. e adiabatic correction factor η varies with strain rate: at strain rates ≤10 − 3 s − 1 , η � 0; at strain rates of 1 s − 1 , η � 0.75; ad at strain rates ≥10 s − 1 , η � 1 [32,33]. Figure 5 shows the calculated temperature changes during the compression test from 650°C to 880°C at strain rates of 1 and 10 s − 1 . As can be seen, a linear relationship is shown between true strain and temperature rise, where adiabatic heating is getting higher with the strain increases. When deformed at 650°C with the strain rate of 10 s − 1 and the true strain of 0.9, the temperature increases by about 90°C.  e measured flow stress curves corrected at strain rates of 1 and 10 s − 1 takes one of the following forms [34][35][36]: _ ε � A[sinh(ασ)] n exp − Q RT (for all stress levels), (10) where Q is the activation energy (kJ/mol), R is the universal gas constant (8.314 J/mol K), _ ε is the strain rate, T is the deformation temperature (K), and A 1 , A 2 , A, β, n 1 , n, and α are material constants, α � β/n 1 . e flow stress can be corrected using equations (8) and (9) for low stresses and high stresses, respectively. e correction of flow stress for deformation heating was accomplished by plotting ln(σ) against 1000/T at a lower strain rate of 1 s − 1 , and σ against 1000/T at a higher strain rate of 10 s − 1 , as shown in Figures 6 and 7, respectively; then, the corrected flow stress was calculated by extrapolating back to the preset testing temperatures. Because there is no correlation of flow stress with strain in equations (8) and (9), the correction has to be made for each selected strain value [30,37].

Flow Behavior Analysis.
e true stress-strain curves of TB18 alloy after friction and temperature correction at different experimental conditions are shown in Figures 3 and  8, respectively. It is evident that the flow behaviors are significantly sensitive to deformation temperature and strain rate. At the beginning of deformation, the flow stress increased rapidly to a peak value with increased strain rate, then dynamic softening started, and the flow stress decreases with the decrease of strain rate and the increase of deformation temperature. Figure 9 shows the relationship between peak stress and deformation temperature at different strain rates. e peak stress decreases as the deformation temperature increases while the strain rate declines. e flow softening is significant when deformed at a lower temperature in the α + β phase zone and relatively gentle in the β single-phase zone. e flow curves, reflecting the competition of work hardening and dynamic softening effect, exhibit three typical characteristics at different deformation temperatures and strain rates:    6 Advances in Materials Science and Engineering stress. is type of hot deformation behavior was also found in Ti-6554 [14] and Ti-5Al4Zr8Mo7V [18]. e deformed microstructures in Figures 10(a)∼ 10(f ) showed that recrystallization occurred on this occasion. When deformed in the α + β two-phase zone, little recrystallized grains were found at the junction of two or three grain boundaries, which indicated that dynamic recrystallization was hard to take place at lower temperature and lower strain rate, and the flow softening mechanism is dynamic recovery and dynamic recrystallization. When deformed in the β single-phase zone, small recrystallized grains distributed in the vicinity of the deformed grain boundaries, this type of recrystallization referred to as necklace recrystallization [7] was also observed in AZ31 [38], Ni-30 [39], and Ti55531 [16]. It can be concluded that at a higher temperature and lower strain rate, the dominant flow softening mechanism is dynamic recrystallization.
(2) Sharp initial peak stress followed by a softening effect to a steady-state is called discontinuous yielding phenomenon (DYP), which is presented with a strain rate of 0.1∼1 s − 1 at all deformation temperatures. A similar phenomenon occurs in the other near β titanium alloys such as Ti-5553 [40] and Ti-1300 [41]. e deformed microstructures in Figures 10(g)∼ 10(j) showed that dynamic recrystallization took place in the β single-phase zone, and flow localization bands were observed when deformed in the α + β two-phase zone. e occurrence of dynamic recrystallization reduced the dislocation density and eliminated the effect of strain hardening [42].  Advances in Materials Science and Engineering adiabatic heat, which is consistent with Ti-55531 [43].

Constitutive Relationship.
e constitutive relationship can be established by the Arrhenius type, which is listed from equations (8) to (10). Taking the nature logarithm of equations (8) and (9), gives the following: ln _ ε � βσ + ln A 2 − Q RT (for high stress level).
At the low stress level, the n 1 value can be obtained by plotting ln σ ∼ ln _ ε at fixed temperatures and the slope of the  8 Advances in Materials Science and Engineering ln σ ∼ ln _ ε gives the value n 1 . At the high stress level, the β value can be calculated by plotting σ∼ln _ ε at given temperatures and the slope of the σ-ln _ ε gives the β value. e corrected flow stress and the corresponding strain rate (0.4) were then substituted into equations (11) and (12). e relationship between flow stress and strain rates is shown in Figure 11. e average slopes of the lines are used to compute the values of n 1 and β, which are 5.57391 and 0.03144 MPa − 1 , respectively. en, α � β/n 1 � 0.00564 MPa − 1 .
For all stress levels, equation (10) can be represented as the following: Differentiating equation (12) gives the following: From equation (14), the Q value can be derived from the slopes ofln[sinh(ασ)]-ln _ ε and ln[sinh(ασ)]-1/T. Substituting the values of the corrected flow stress, deformation temperature and corresponding strain rate at the strain of 0.4 into equation (14). en, the relationships of ln _ ε-ln[sinh(ασ)] and ln[sinh(ασ)]-1000/T can be acquired, as shown in Figure 12. Because the slopes of the lines are approximately the same, the average slopes are used for acquiring the value of activated energy and it is 238.465 kJ/ mol.
Taking the natural logarithm of both sides of equation (10) gives e values of ((Q/nRT) − (ln A/n)) and 1/n are the intercept and slope of ln[sinh(ασ)]∼ln _ ε plot, respectively. Substituting the value of Q, n, R, and T into equation (15), the values of ln A at different deformation temperatures can be obtained, the average of which is 23.89918.
According to this method, all the material constants with the strain range from 0.1 to 0.9 can be obtained as listed in Table 2.

Establishment and Verification of Constitutive
Relationship. In equation (10), the influence of strain on high temperature deformation behavior is not considered. In order to consider the effect of strain, the material constants in constitutive equations (i.e., n, α, β, Q, and A) are expressed as polynomial functions of strain. First, the values of material constants (i.e., n, α, β, Q, and A) are calculated at various strain conditions; second, these values are fitted by polynomial function by the order from 2 to 9. Finally, a six-order polynomial is confirmed with a good correlation for TB18 alloy, as shown in the following equation and Figure 13: After the materials constants are obtained, the flow stress at a constant strain can be predicted. e flow stress can be expressed as the function of the Zener-Hollomon parameters. And the proposed constitutive model can be summarized as follows: e comparison between the corrected flow stress and the predicted data from the strain-compensated constitutive equation at different deformation temperatures and strain rates are shown in Figure 14. It can be seen that the predicted flow stress matches well with the corrected results at different deformation conditions.
In order to quantify the accuracy of the constitutive equation, standard statistical parameters, namely, correlation coefficient (R) and average absolute relative error (AARE) are used [44], which are expressed as follows:   where E is the original value and P is the predicted one obtained from the constitutive equation, E and P represent the mean values of E and P, respectively. N is the total number of data points. e correlation coefficient is a commonly used statistic, which reflects the strength of the linear relationship between observed data and predicted data by comparing relative errors term by term. A good correlation (R � 99.2%) is obtained between the original and the predicted data, as seen in Figure 15. e calculated AARE value equals to 6.1% at the tested deformation conditions. e statistical results show high accuracy of the established constitutive equation for the TB18 alloy ranging from the strain rate of 0.001∼10 s − 1 and the temperature of 650∼880°C.

Conclusions
In this work, the isothermal hot compression test with a temperature of 650-880°C and a strain rate of 0.001∼10 s − 1 was performed to study the hot deformation behavior of TB18 alloy. e following conclusions could be drawn from the current research: (1) e measured flow stress was revised by friction and temperature correction, and the peak stress decreased with the deformation temperature increase and the strain rate decline. (2) e flow stress curves revealed three typical characteristics at various deformation conditions: (1) At lower strain rate of 0.001∼0.01 s − 1 , the flow stress continues to decrease as the strain rate increases after the flow stress reaches the peak stress. e flow softening mechanism is dynamic recovery and dynamic recrystallization at lower temperatures and dynamic recrystallization at higher temperatures. (2) e discontinuous yielding phenomenon could be seen at a moderate strain rate of 1 s − 1 , dynamic recrystallization took place in the β single-phase zone, and flow localization bands were observed in the α + β two-phase zone. (3) At a higher strain rate of 10 s − 1 , the flow instabilities were referred to as the occurrence of flow localization by adiabatic heat.
(3) e constitutive equation considering the compensation of strain was established to predict the flow stress, whose results accurately showed the correlation coefficient and the AARE are 99.2% and 6.1%, respectively.
Data Availability e flow stress data used to support the findings of this study are available from the corresponding author upon request.