Modeling and Finite Element Analysis for the Dynamic Recrystallization Behavior of Ti-5Al-5Mo-5V-3Cr-1Zr Near β Titanium Alloy During Hot Deformation

Introduction

Near β type titanium alloys such as Ti-55,531 (Ti-5Al-5Mo-5V-3Cr-1Zr) show a good combination of high specific strength, ductility, and fatigue strength, which are very suitable to manufacture the large aerospace structural components with high mechanical requirements, for example, the truck beam in landing gear [1, 2]. During manufacturing of large structural components, thermomechanical processing is necessary not only for shaping, but also for obtaining the required mechanical properties by adjusting microstructure. The microstructure of near-β titanium alloys is extremely sensitive to the processing conditions. So far, a successful forging significantly depends on the cost and time-consuming empirical trial and error, which needs the profound understanding of the relationship among processing conditions, material deformation behavior, and resulting microstructures. Therefore, how to predict the microstructure evolution and deformation behavior of Ti-alloy at different processing conditions, including strain, strain rate, and temperature, has become a crucial issue.

As for the Ti-alloys, the effects of thermomechanical conditions on the microstructure and then the resulting mechanical properties have been studied by using the experimental method in the past several decades. While, researches are mainly focused on the α+β alloy Ti–6Al–4V, only the limited work has focused on the subtransus forging of near-β alloys. Furthermore, fewer researches aimed at the microstructure evolution by using numerical simulation method. During the subtransus forging of Ti–55,531, the alloy undergoes a series of microstructure evolutions such as dynamic recovery (DRV), dynamic recrystallization (DRX) and grain growth, which have the great effect on the final mechanical properties. Taking DRX as an example, it has the practical importance as it reduces the deformation resistance and refines the grains. Therefore, the study of DRX behavior during hot forging is extremely necessary.

In recent decades, DRX behavior for various materials has been investigated by a considerable amount of researchers. Sellars [3, 4] investigated the DRX of carbon-manganese steel during hot rolling and then found that the onset of DRX was before the peak strain. The critical strain corresponding to the onset of DRX depended on the chemical composition, initial grain size and deformation conditions [5]. McQueen and Poliak [6, 7] studied the DRX of austenitic stainless steel and nickel during hot compression and then found that the critical condition for the onset of DRX can be identified as a special inflection point in the work hardening rate (θ)-true stress (σ) curve. Quan [8] investigated the DRX kinetics of heat-resistant alloy by isothermal upsetting experiment. The results showed that the hot flow behavior of metals was generally reflected on the stress-strain curves resulting from the microstructure evolutions, and therefore, it is possible to model the DRX kinetics by analyzing the flow curves. Based on the above investigations, some empirical DRX kinetics equations were established, in which the DRX behavior of several kinds of steels was well predicted [9, 10, 11, 12, 13]. Few investigations also aimed at DRX of Ti-alloys with the empirical DRX kinetics model [14, 15, 16, 17]. Forexample, the FEM results about DRX of Ti-6Al-4V [18, 19] and Ti-6.5Al-3.5Mo-1.5Zr-0.3Si [20] were consistent with the experimental results, which showed a high efficiency. However, the investigation for the DRX behavior of near β Ti-alloys by FEM has been seldom reported.

In this study, DRX behavior of near β Ti-alloy Ti-55,531 was investigated by using thermal simulation tests. Sequentially, the DRX kinetic model was established based on the experimental data. Furthermore, the DRX kinetic model was implanted into DEFORM-3D finite element program by developing FORTRAN codes, and then a series of simulations were conducted to obtain the evolution of DRX volume fraction at different deformation conditions. By comparing the FEM results and experimental results of DRX volume fraction, the efficiency of numerical simulation method was verified.

Experimental details

The chemical compositions (wt. %) of the as-received Φ150 mm Ti-55,531 billet was Al-5.20, Mo-4.92, V-4.96, Cr-2.99, Zr-1.08 and Fe-0.40. The beta-transus temperature was approximately 825±5 ^oC. The billet was solution heat treated at 993 K for 2 h, followed by water quenching to room temperature. Figure 1 shows that the thermally-treated microstructure containing many nearly spherical α particles with the average diameter of 1.68 μm and the clear and smooth α/β interfaces. Sequentially, several cylindrical specimens were line cut from the thermally-treated billet and then mechanically polished to Φ8×12 mm. Isothermal compression was carried out on a Gleeble-3500 machine at 1023–1098 K and strain rate of 10^‒3–10° s^‒1 in accordance with the ASTM: E209-00. A thermocouple with a diameter of 0.08 mm was welded at the mid-height side of the specimen to measure the temperature. The graphite foil and tantalum film were placed between specimen and machine anvils for lubrication. The specimens were resistance-heated to the set temperature at the heating rate of 10 K/s, compressed to the true strain of 0.7, and then immediately cooled by an argon gas-jet to freeze the deformed microstructure. The measured load/displacement data were recorded and then transferred into the stress/strain data by testing machine automatically. The compressed specimens were cut along the cylinder axis line. The cutting faces were mechanically polished and then etched by 1.5 mL HF + 3 mL HNO₃ + 100 mL H₂O. The microstructure observation was conducted using a scanning electron microscope (SEM, NOVATM Nano SEM 230) and transmission electron microscope (TEM, JEOL JEM-2100F).

Figure 1:

Microstructure of Ti-55,531 soaked at 993 K for 2 h and then water quenching.

Results and discussion

Flow behavior

Figure 2 shows the true stress–true strain curves of Ti-55,531 obtained at the temperatures of 1023–1098 K and strain rates of 0.001–1 s^‒1 after modifying the temperature and friction by using the standard method. It is observed that all the true stress–true strain curves display a peak stress (σ_p) in the early stage of deformation, followed by a continuous flow softening till the end of thermal compression to reach the true strain of 0.7. These results are the combination of work hardening and dynamic softening. At the beginning of compression, the rapid proliferation of dislocations leads to a sharp increase in flow stress. Then the dynamic softening becomes the dominant factor, which leads to the gradual decreasing of flow.

Figure 2:

Flow stress curves at different temperatures and strain rates of (a) 0.001 s^-1, (b) 0.01 s^-1, (c) 0.1 s^-1 and (d) 1 s^-1.

In addition, it can be observed that the flow stress increases with the decreasing of temperature at a certain strain rate, which can be related to the following reasons: (1) the increasing in the content of α strengthening phase with hexagonal close-packed (hcp) structure, (2) the decreasing in the dislocation slip and the diffusion ability of grain boundaries for the annihilation of dislocations,and (3) the decreasing in the nucleation and growth of dynamically recrystallized α grains and the dynamic recovery of β gains. Meanwhile, the flow stress increases with the increasing of strain rate at a certain deformation temperature. This is mainly due to the fact that the lower strain rate provides the longer time for dislocation annihilation and then reduces the flow stress.

Calculation of DRX kinetic parameters

Material constant a₁

For the typical flow curves of DRX, the critical strain (ε_c) at the beginning of DRX and the peak strain (ε_p) are commonly determined from the work hardening rate (θ=dσ/dε) curves as shown in Figure 3. Firstly, the critical stress (σ_c) and peak stress (σ_p) need to be determined. In Figure 3, the critical stress (σ_c) is the stress at dθ/dσ=0 (the black inflection points in every θ-σ curves), while the peak stress (σ_p) is the stress at θ=0 [6, 7]. And the critical strain (ε_c) and peak strain (ε_p) are the strains at the critical stress (σ_c) and peak stress (σ_p) as listed in Table 1, respectively. Then the relationship between ε_c and ε_p are fitted linearly as shown in Figure 4, in which the slope is the value of a₁, namely 0.6053.

Figure 3:

Curves of θ (=dσ/dε) versus σ at different temperatures and strain rates of (a) 0.001 s^-1, (b) 0.01 s^-1, (c) 0.1 s^-1 and (d) 1 s^-1.

Figure 4:

Linear fitting between critical strain (ε_c) and peak strain (ε_p).

Table 1:

Values of critical strain (ε_c) and peak strain (ε_p) at different deformation conditions.

Characteristic Strain	Strain rate/s-1	Temperature
		1023 K	1048 K	1073 K	1098 K
εc	0.001	0.0310	0.0280	0.0260	0.0220
	0.01	0.0340	0.0320	0.0280	0.0270
	0.1	0.0440	0.0410	0.0369	0.0340
	1	0.0548	0.0479	0.0462	0.0442
εp	0.001	0.0489	0.0459	0.0426	0.0390
	0.01	0.0600	0.0550	0.0511	0.0468
	0.1	0.0722	0.0649	0.0621	0.0578
	1	0.0877	0.0799	0.0748	0.0687

Material constant a₂, m₂ and deformation activation energy Q₁

By taking natural logarithm and then partial derivative on both sides of eq. (3), the material constant m₂ can be expressed as:

(4)m2=∂lnεp/∂lnε˙

The relationships between lnε_p and lnε˙ at different temperatures are fitted linearly as shown in Figure 5. The slope of one fitted line is the value of m₂ at the corresponding temperature. Four fitted lines at different temperatures had the similar slope values. By taking the average of four slope values, m₂ can be obtained as 0.0821.

Figure 5:

Linear fitting between lnε_p and lnε˙.

Similarly, the deformation activation energy Q₁ can be expressed as:

(5)Q1=R∂lnεp/∂1/T

The relationships between lnε_p and 1/T at different strain rates are fitted linearly as shown in Figure 6. Q₁ is the mean value of four slope values, and here it is 28,781 J mol^-1. Then, substituting m₂ and Q₁ into eq. (3) got 16 values of material constant a₂ at different deformation condition, and then the mean value of a₂ can be obtained as 0.0031.

Figure 6:

Linear fitting between lnε_p and 1/T.

Material constant k, m₁

Since the flow behavior is determined by the microstructure evolution, DRX volume fraction can be deduced indirectly from the true stress–true strain curves as following [25]:

(6)XDRX=σ−σP/σs−σp

where σ_s is the steady-state stress. By taking natural logarithm and then partial derivative on both sides of eq. (1), the material constant m₁ can be expressed as:

(7)m1=∂ln−ln1−XDRX/∂lnε−εc/εc

Taken the deformation conditions of 1073 K and 1 s^-1 as an example, peak stress (σ_p), steady-state stress (σ_s) and critical strain (ε_c) at these conditions were 329.3671 MPa, 306.5805 MPa and 0.0462, respectively. Then by substituting these values into eqs. (6) and (7), the relationship between σ and ε (Figure 2) can be transformed into the relationship between ln[-ln(1-X_DRX)] and ln[(ε-ε_c)/ε_c]. After further linearly fitting as shown in Figure 7, the slope and intercept of the fitted line are the values of material constant m₁ and lnk at 1073 K and 1 s^-1, respectively. Through adopting the same numerical treatment, m₁ and lnk at the other deformation conditions can be obtained. Then the mean value of m₁ and k are obtained as 2.1000 and 0.0053, respectively.

Figure 7:

Linear fitting between ln[-ln(1-X_DRX)] and ln[(ε-ε_c)/ε_c] at 1073 K and strain rate of 1 s^-1.

Finally, the as-obtained DRX kinetic model is described as:

(8)XDRX=1−exp−0.0053ε−εcεc2.1εc=0.6053εpεp=0.0031ε˙0.0821exp28781RT

Based on eq. (8), the DRX volume fraction (X_DRX) at different deformation conditions were obtained as shown in Figure 8. It can be found that DRX does not occur when the true strain is less than the critical true strain (ε_c). By further increasing the true strain, X_DRX displays the S-type increasing trend, namely successively undergoing the slow increasing, fast increasing, and slow increasing till to be close to X_DRX = 100 %. The increasing of temperature and decreasing of strain rate make X_DRX close to 100 % at the lower true strain, which suggests the acceleration of DRX. This could be related to the higher grain boundary mobility at the higher temperature and lower strain rate.

Figure 8:

DRX volume fractions obtained from eq. (8) at different deformation temperatures and strain rates of (a) 0.001 s^-1, (b) 0.01 s^-1, (c) 0.1 s^-1 and (d) 1 s^-1.

Simulation of X_DRX and experimental verification

In order to investigate the X_DRX of Ti-55,531 at different deformation conditions, the true stress-true strain data obtained from the thermal simulation tests and the as-established DRX kinetic equation were implanted into DEFORM-3D finite element program by developing FORTRAN codes. During the finite element simulation, the friction between workpiece and anvil was set as the shear type with the friction coefficient of 0.3, and the heat transfer coefficient was set as 11 N s^-1 mm^{-1 o}C^-1, in which the boundary conditions were in agreement with those in experimental details about thermal simulation tests. Additionally, the workpiece was set as the plastic body, while the anvil was set as the rigid body. The total number of nodes and elements are 16,307 and 70,824, respectively. Then, a series of finite element simulations were conducted. Due to the symmetry of cylindrical specimens, only half of the simulated specimens are given in the following simulation results. Figures 9 and 10 show the effects of strain rate at 1073 K and temperature at 0.01 s^-1 on the X_DRX distribution after hot compressing to the true strain of 0.7, respectively. It can be obviously found that the X_DRX distribution is always inhomogeneous. The maximum and minimum X_DRX values locate at the center region of the cylindrical body (P1) and of the end surface (P2), respectively. With the decreasing of strain rate at 1073 K and increasing of temperature at 0.01 s^-1, P1 region (maximum X_DRX) enlarges, while P2 region (minimum X_DRX) reduces, which suggest the increasing of average X_DRX, namely the acceleration of DRX.

Figure 9:

DRX volume fraction distributions at 1023 K and strain rates of (a) 0.001 s^-1, (b) 0.01 s^-1, (c) 0.1 s^-1 and (d) 1 s^-1.

Figure 10:

DRX volume fraction distributions at the strain rate of 0.01 s^-1 and temperatures of (a) 1048 K, (b) 1073 K and (c) 1098 K.

To verify the above simulation results, Figures 11–13 show the microstructures of the thermal simulation specimens hot compressed to the true strain of 0.7 at different deformation conditions. Compared with the nearly spherical α particles and the clear and smooth α/β interfaces before hot compression (Figure 1), the hot compression at the relatively high strain rate (Figure 11d) and low temperature (Figure 12a) leads to the elongation of nearly spherical α along deforming flowing direction, which shows the DRV behavior. By decreasing the strain rate from 1 s^-1 (Figure 11d) to 0.001 s^-1 (Figure 11a) and increasing the temperature from 1048 K (Figure 12a) to 1073 K (Figure 12b), the DRX characteristics, such as the localized necking in the elongated α up to be fragmented (namely DRX refinement), and the as-resulted corrugated and serrated α/β interfaces, become more and more obvious, which suggest the acceleration of DRX. The formation of serrated α/β interfaces has been termed as geometric dynamic recrystallization by McQueen et al [26], which is different from the classical DRX. The geometrically recrystallized equiaxed α grains are evolved from the fragmentation of elongated α grains by impingement of the serrated boundaries. The mechanisms about the fragmentation of elongated α lamellae during deformation are as following. Firstly, some substructure form within the elongated α lamellae as shown in Figure 14. Then the elongated α lamellae are subsequently broken up by boundary grooving along the interfaces among substructures. The spherical α particles are finally formed with the processes of termination migration. Figures 11 and 12 also show that the low strain rate (being equivalent to the long deformation time) and high temperature cause the low α content because of the re-dissolution of α into the β matrix. Additionally, compared with the center region of end surface (1.42 μm in average diameter of α as shown in Figure 13b, corresponding to P2 region in Figure 9a), the center region of cylindrical body (Figure 13a, corresponding to P1 region in Figure 9a) has the higher DRX degree confirmed by the higher DRX refinement (1.23 μm in average diameter of α). Based on the experimental results about the influence of deformation conditions on DRX degree and the distribution of DRX degree, the FEM simulation results can be verified. Therefore, through FEM simulation, DRX evolution at the different region of specimens during hot deformation can be described numerically.

Figure 11:

Microstructures at the center region of cylindrical body at the deformation temperature of 1023 K and strain rates of (a) 0.001 s^-1, (b) 0.01 s^-1, (c) 0.1 s^-1 and (d) 1 s^-1.

Figure 12:

Microstructures at the center region of the cylindrical body at the temperatures of (a) 1048 K and (b) 1073 K and strain rate of 0.01 s^-1.

Figure 13:

Microstructures at the center region of (a) cylindrical body and of (b) end surface at the temperature of 1048 K and strain rate of 0.1 s^-1.

Figure 14:

Bright field micrographs of Ti-55,531 deformed to a true strain of 0.7 at the temperature of 1023K and strain rate of 0.001 s^-1.

Conclusions

1. All the curves of true stress σ versus true strain ε, which were obtained from thermal simulation experiments of Ti-55,531 at 1023–1098 K and strain rate of 0.001–1 s^‒1, display a peak stress (σ_p) in the early stage of deformation, followed by a continuous flow softening till the end of thermal compression to reach the true strain of 0.7. And the flow stress increases with the increasing of strain rate and decreasing of temperature.

2. By numerically treating to the essential data about σ and ε, the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation to characterize the evolution of dynamic recrystallization volume fraction (X_DRX) can be determined as X_DRX=1-exp[-0.0053((ε-ε_c)/ε_c)^2.1], where ε_c=0.6053ε_p and ε_p=0.0031ε˙^0.0081exp(28,781/RT). JMAK equation shows that dynamic recrystallization do not occur when ε is less than the critical true strain (ε_c). With the further increasing of ε, X_DRX successively experiences the slow increasing, fast increasing and slow increasing till to be close to 100 %. And the increasing of temperature and decreasing of strain rate accelerate the DRX process.

3. The essential data about σ and ε and the JMAK equation were implanted into the finite element program to simulate the hot compression of thermal simulation experiments. The finite element simulation results show the inhomogeneity of X_DRX distribution. The maximum and minimum X_DRX distribution regions locate at the center region of the cylindrical body and of the end surface, respectively. With the increasing of strain rate and decreasing of temperature, the reduced maximum X_DRX distribution region and the enlarged minimum X_DRX region suggest the decreasing of average X_DRX, which were validated by the microstructure observation