A study on hot deformation behavior of Ti-6Al-2V-0.5Zr-1.5Mo alloys

The true stress-strain curves of the Ti-6Al-2V-0.5Zr-1.5Mo alloy under different deformation conditions are shown in figure 3. It showed that the flow stress increased when the deformation temperature decreased and the strain rate rose. During the thermal deformation, the changes in the flow stress were dependent on the interaction between work hardening and softening caused by dynamic recovery and recrystallization [18, 19]. It was changed as a result of dislocation multiplication and disappearance, as well as dislocation slip and climb.

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The flow stress decreased with temperatures when the deformation rate was unchanged. This is because when the temperature rises, the internal energy of atoms increases continuously, and the dislocation is easy to break away from the pinning effect of solid solution atoms, thus promoting dislocation slip and climb and making grains coordinated. In addition, the alloy has enough energy at grain boundaries and dislocation entanglement to provide dynamic recrystallization nucleation, so the softening effect caused by dynamic recovery and recrystallization is more significant, resulting in lower flow stress [20]. Moreover, the increase of deformation temperatures enables the temperature of the alloy to exceed that of the phase transformation, which turns the (α+β) alloy into a β-phase one. Since α-Ti is close-packed and hexagonal structured, while β-Ti is body-centered and cubic structured, there are more slip systems in the β phase than in the α phase, thus slip is easier to happen in the β phase, which is also a reason for the lower flow stress.

When the deformation temperature is unchanged, the flow stress increased with strain rates. The deformation rate mainly affects the time frame of softening mechanism during thermal deformation. At low strain rates, there is enough time for dislocation slip and climb caused by deformation and for dynamic recrystallization to be nucleated and grow. Under such a condition, the softening mechanism takes a dominant role, making the flow stress tend to decrease after crossing the peak value. At higher strain rates, there is only a short time for dynamic recovery and recrystallization, and work hardening becomes dominant, which brings greater deformation resistance and higher flow stress [21].

In the end, the true stress-change curves of the rear segment are likely to go up with the increase of the strain, work hardening, and higher flow stress as a result of the multiplication and accumulation of dislocations in the alloy [22].

3.2.1. Macrostructure

Figure 4 is the macroscopic morphology of the samples before and after hot compression. The samples without receiving hot compression are cylinders of φ8 mm × 12 mm. After hot compression, they become drum-shaped, with a maximum width of about 12 mm and a height of about 6 mm. Different drum deformation was observed in all samples receiving hot compression, indicating that the deformation inside the samples was uneven during hot compression. No cracks or damage were found on their surfaces. The results show that the temperature range of 800 °C–950 °C is good for the Ti-6Al-2V-0.5Zr-1.5Mo alloy to be produced. In the current research, microstructure analysis focuses on the central area of deformed samples.

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3.2.2. Effect of deformation temperatures on the microstructure evolution of the Ti-6Al-2V-0.5Zr-1.5Mo Alloy

At a strain rate of 0.001 s⁻¹, the microstructures of the samples receiving deformation at the true strain of 0.69 and at different temperatures are shown in figure 5. As the deformation temperature increases, more microstructure areas underwent dynamic recrystallization. As shown in figure 5(a), at the deformation temperature of 800 °C, the deformation temperature required was relatively low, and the dynamic recrystallization was not completely carried out. There are many coarse and long grains, among which there are fine grains formed by dynamic recrystallization. At the deformation temperature of 850 °C, the proportion of dynamic recrystallization rose, but coarse grains without dynamic recrystallization were found mixed with the fine grains of dynamic recrystallization, generating a mixed crystal structure. This makes the microstructure of the alloy less uniform, which is not conducive to its performance (see figure 5(b)). As shown in figure 5(c), at a higher deformation temperature of 900 °C, the original structure of the strip was not observed, which meant that complete dynamic recrystallization had occurred. Now the microstructure was relatively uniform, and the grains looked small. Despite the disappearance of the original structure, it can be seen that this high temperature has coarsened the dynamic recrystallization, and the dynamic recrystallization grains enjoy significant growth (see figure 5(d)). This is similar to the result given by Ji et al [13]. When the temperature is lower than 900 °C, the α phase is dominant. With the increase of the temperature, recrystallized grains grow significantly. When the temperature rises to 950 °C, both dynamic recrystallization and the shift from α phase to β phase happen.

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3.2.3. Effect of strain rates on the microstructure evolution of the Ti-6Al-2V-0.5Zr-1.5Mo Alloy

In figure 6, the microstructures of the Ti-6Al-2V-0.5Zr-1.5Mo alloy which was deformed at the true strain of 0.69 at 900 °C and different strain rates are shown. As shown in figures 6(a)–(d), as the strain rate rose from 0.001 s⁻¹ to 1 s⁻¹, the grains gradually became smaller. This is very similar to the results offered by Peng et al when conducting hot compression on TC4-DT alloys [23]. This can be explained by the fact that lower strain rates mean longer deformation time when the deformation temperature is constant. At high temperatures, the activity of grain boundaries is enhanced, and grains are more likely to grow. Thus grains are more likely to grow when the strain rate is lower and the deformation time at a high temperature is longer. However, a higher strain rate will shorten the deformation, which is not conducive to the complete dynamic recrystallization of the alloy and some grains are not completely refined [24]. As shown in figures 6(b), a small number of large grains in the shape of strips can be observed. In figure 6(d), an obvious zonal structure is formed. The effects of grain growth and dynamic recrystallization, therefore, need to be taken into consideration when deciding the strain rates.

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Based on the analysis above, the microstructures of the Ti-6Al-2V-0.5Zr-1.5Mo alloy after deformation under different conditions are shown in figure 7. It shows that deformation temperatures and strain rates play a role in the microstructure evolution of the alloy during hot deformation [25, 26]. As the deformation temperature rose, more dynamic recrystallization was seen, but grains enjoyed some growth. When the temperature was constant, as the strain rates went up, the dynamic recrystallization grains became smaller, less dynamic recrystallization occurred, and the microstructures looked more nonuniformed. The reason behind this is that temperatures can prevent the movement of grain boundaries, through which the dynamic recrystallization is completed. Thus more dynamic recrystallization occurs and grains are more likely to grow when the temperature rises. The strain rate can prevent dislocation accumulation. When the strain rate increases, dislocation accumulation happens more quickly, and it is easier to reach the critical strain required for dynamic recrystallization. However, the deformation time will be shortened, which may lead to incomplete dynamic recrystallization and increase the nonuniformity of the structure [27].

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In the process of hot compression, flow stress can be affected by deformation temperatures, deformation rates, and deformation amounts [28, 29]. When elements in the alloy and its initial structure are unchanged, the classical hyperbolic-sinusoidal Arrhenius equation [30] and Zener-Hollomon parameter (Z) [31] can reflect how flow stress changes with deformation temperatures and deformation rates.

$$\begin{eqnarray}&&\unicode{x003AD}={A}_{1}{\sigma }^{{n}_{1}}\exp \left(-\displaystyle \frac{Q}{RT}\right),\alpha \sigma \lt 0.8\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\unicode{x003AD}={A}_{2}\exp \left(\beta \sigma \right)\exp \left(-\displaystyle \frac{Q}{RT}\right),\alpha \sigma \gt 1.2\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&\unicode{x003AD}=A{\left[sinh\left(\alpha \sigma \right)\right]}^{n}\exp \left(-\displaystyle \frac{Q}{RT}\right),{\rm{for}}\,{\rm{all}}\,\sigma \end{eqnarray} \tag{ 3 }$$

In these equations, constants $A,$ $n,$ and $\alpha$ are unrelated to deformation temperatures; $Q$ represents the thermal deformation activation energy (KJ mol⁻¹), and $R$ the universal gas constant (8.314 J·mol⁻¹K⁻¹).

In the Zener-Hollomon parameter, the stress-strain relationship is as follows:

$$\begin{eqnarray}&&Z=\unicode{x003AD}\,\exp \left(\displaystyle \frac{Q}{RT}\right)=A{\left[sinh\left(\alpha \sigma \right)\right]}^{n}\end{eqnarray} \tag{ 4 }$$

Equations (5)–(7) were built based on equations (1)–(3). The relation of $In\sigma -In\unicode{x003AD},$ $\sigma -In\unicode{x003AD},$ and $In\left[sinh(\alpha \sigma )\right]-In\unicode{x003AD}$ was figure out after a linear regression analysis by using the least square, as shown in figures 8(a)–(c). After the calculation, the values of ${n}_{1},$ $\beta ,$ $\alpha ,$ $n$ were found to be 5.556, 0.046, 0.0102, and 3.663, respectively.

$$\begin{eqnarray}&&In\unicode{x003AD}=In{A}_{1}+{n}_{1}In\sigma -\displaystyle \frac{Q}{RT}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&In\unicode{x003AD}=In{A}_{2}+\beta \sigma -\displaystyle \frac{Q}{RT}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{n}In\unicode{x003AD}=\displaystyle \frac{1}{n}InA+In\left[sinh\left(\alpha \sigma \right)\right]-\displaystyle \frac{Q}{nRT}\end{eqnarray} \tag{ 7 }$$

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After adjustment, equation (7) can be expressed as follows:

$$\begin{eqnarray}&&Insinh\left(\alpha \sigma \right)=\displaystyle \frac{Q}{nRT}+\displaystyle \frac{In\unicode{x003AD}-InA}{n}\end{eqnarray} \tag{ 8 }$$

The $Insinh\left(\alpha \sigma \right)-1000/T$ relation is shown in figure 8(d), and the value of $Q$ is 739996.7 J mol⁻¹.

Equation (9) can be formed based on equation (4):

$$\begin{eqnarray}&&InZ=InA+nIn\left[\sin \,h\left(\alpha \sigma \right)\right]\end{eqnarray} \tag{ 9 }$$

The $InZ-In[sinh(\alpha \sigma )]$ relation is shown in figure 9, and the value of $A$ is 2.017 × 10³¹. The constitutive equation of Ti-6Al-2V-0.5Zr-1.5Mo alloy can be expressed in equation (10).

$$\begin{eqnarray}&&\unicode{x003AD}=2.017\times {10}^{31}{\left[sinh\left(0.0102\sigma \right)\right]}^{3.663}\exp \left(\displaystyle \frac{-739996.7}{RT}\right)\end{eqnarray} \tag{ 10 }$$

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It can be transformed into the Zener-Hollomoll parametric form, which is equation (11).

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{1}{0.0102}In\left\{{\left(\displaystyle \frac{z}{2.017\times {10}^{31}}\right)}^{\displaystyle \frac{1}{3.663}}+{\left[{\left(\displaystyle \frac{z}{2.017\times {10}^{31}}\right)}^{\displaystyle \frac{2}{3.663}}+1\right]}^{\displaystyle \frac{1}{2}}\right\}\end{eqnarray} \tag{ 11 }$$

Apart from deformation rates and temperatures, the strain variables also affect flow stress. It is commonly believed that the parameters $\alpha ,$ $n,$ $Q,$ and $A$ in equations (7) and (8) are affected by the strain variables during the construction of the constitutive model [32]. The flow stress in the true stress-strain curves was calculated, and the strains were between 0.05 and 0.65 with an interval of 0.05 to ensure that the prediction based on the Arrhenius model is accurate. Then much regression analysis was conducted on the flow stress value to work out the parameters at each strain. Table 1 shows the results. The sixth-order polynomial equation shown in (12) is used to demonstrate the relation between the strain and the correlation constant.

$$\begin{eqnarray}&&Y\left(\varepsilon \right)={B}_{0}+{B}_{1}\varepsilon +{B}_{2}{\varepsilon }^{2}+{B}_{3}{\varepsilon }^{3}+{B}_{4}{\varepsilon }^{4}+{B}_{5}{\varepsilon }^{5}+{B}_{6}{\varepsilon }^{6}\end{eqnarray} \tag{ 12 }$$

Figure 10 shows how variables $\alpha ,$ $n,$ $Q,$ and $InA$ changed with the strain, which can be expressed in equation (13). The square of the regression coefficient of each parameter is over 0.96, which means that the fitting effect is good.

$$\begin{eqnarray}&&\left\{\begin{array}{c}\alpha =0.0121-0.0565\varepsilon +0.7056{\varepsilon }^{2}-3.8896{\varepsilon }^{3}\\ \,+\,10.5422{\varepsilon }^{4}-13.6073{\varepsilon }^{5}+6.9667{\varepsilon }^{6}\\ n=2.90+27.26\varepsilon -332.52{\varepsilon }^{2}+1812.39{\varepsilon }^{3}\\ \,-\,4761.35{\varepsilon }^{4}+5933.48{\varepsilon }^{5}-2825.84{\varepsilon }^{6}\\ Q=579.89+6070.25\varepsilon -68443.46{\varepsilon }^{2}+34352.96{\varepsilon }^{3}\\ \,-\,87494.43{\varepsilon }^{4}+98749.52{\varepsilon }^{5}-52224.53{\varepsilon }^{6}\\ InA=56.95+593.83\varepsilon -689.31{\varepsilon }^{2}+3539.70{\varepsilon }^{3}\\ \,-\,9158.12{\varepsilon }^{4}+9503.45{\varepsilon }^{5}-5641.42{\varepsilon }^{6}\end{array}\right.\end{eqnarray} \tag{ 13 }$$

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The constitutive model of the predicted flow stress of the Ti-6Al-2V-0.5Zr-1.5Mo alloy during hot deformation can be worked out after using equation (13) in equations (4) and (14). In the model, the influence of deformation temperatures, rates, and strain variables on the flow stress is taken into consideration.

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{1}{\alpha }In\left\{{\left(\displaystyle \frac{z}{A}\right)}^{\displaystyle \frac{1}{n}}+{\left[{\left(\displaystyle \frac{z}{A}\right)}^{\displaystyle \frac{2}{n}}+1\right]}^{\displaystyle \frac{1}{2}}\right\}\end{eqnarray} \tag{ 14 }$$

The flow stress calculated based on the constitutive model was compared with that from tests to ensure that the model was accurate, as shown in figure 11. The changes in the calculated value are almost consistent with that of the one from tests, and the deviation is small. The calculated flow stress decreased with deformation temperatures, and it fell when the deformation rate went down, which is consistent with the flow stress in the tests. This means that the constitutive model is able to reflect the thermal deformation behavior of the Ti-6Al-2V-0.5Zr-1.5Mo alloy accurately.

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Correlation coefficient (R) and absolute mean error (AARE) were introduced to work out the deviation between the calculated flow stress and the experimental one in a quantitative way [33, 34]. The equations are shown in (15) and (16):

$$\begin{eqnarray}&&R=\displaystyle \frac{\displaystyle {\sum }_{i=1}^{N}\left({E}_{i}-\bar{E}\right)\left({P}_{i}-\bar{P}\right)}{\sqrt{\displaystyle {\sum }_{i=1}^{N}{\left({E}_{i}-\bar{E}\right)}^{2}\displaystyle {\sum }_{i=1}^{N}{\left({P}_{i}-\bar{P}\right)}^{2}}}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&AARE\left( \% \right)=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}\left|\displaystyle \frac{{E}_{i}-{P}_{i}}{{E}_{i}}\right|\times 100\end{eqnarray} \tag{ 16 }$$

In the equation, $E$ represents the flow stress gained in the thermal deformation experiment; $P$ represents the flow stress worked out based on the constitutive model, and $\bar{E}$ and $\bar{E}$ represent the mean values of $E$ and $P,$ respectively. $R$ can reflect the linear correlation between the measured stress value and the calculated one, and AARE represents the error degree of the built model. Thus when the value of $R$ is larger and that of AARE is smaller, the model will bring a more accurate result. As shown in figure 12(a), most of the data points are distributed near the optimal regression line, and the square of $R$ reaches 0.9811 after fitting, indicating that the flow stress obtained from the experiment matches well with the calculated one. The AARE value at each deformation rate is shown in figure 12(b), and AARE is 6.84%, which means that the error caused by the model calculation is insignificant.

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The thermal processing map, which was established based on the DMM theory by Prasad et al [35], can reflect the changes in the internal microstructures of materials when processed under different parameters. It can also directly show instability areas and safe areas during thermal deformation, which offers a theoretical foundation for deciding the optimal processing parameters of metal materials.

According to the hypothesis of the DMM theory, the external input energy (P) consists of two parts: dissipative energy (G) and dissipative covariate (J) [36]:

$$\begin{eqnarray}&&p=\sigma \unicode{x003AD}=G+J=\displaystyle {\int }_{0}^{\unicode{x003AD}}\sigma d\unicode{x003AD}+\displaystyle {\int }_{0}^{\sigma }\unicode{x003AD}d\sigma \end{eqnarray} \tag{ 16 }$$

The ratio of $G$ to $J$ depends on m, which is the sensitive factor of deformation rates. The factor $m$ can be obtained by the cubic spline function of $Log\sigma$ and $Log\unicode{x003AD}.$

$$\begin{eqnarray}&&m={\left[\displaystyle \frac{\partial \left(lg\sigma \right)}{\partial \left(lg\unicode{x003AD}\right)}\right]}_{\varepsilon ,T}={a}_{2}+2{a}_{3}\,\mathrm{lg}\,\unicode{x003AD}+3{a}_{4}{\left(\mathrm{lg}\,\unicode{x003AD}\right)}^{2}\end{eqnarray} \tag{ 18 }$$

During thermal deformation, the dissipated power dissipation coefficient as a result of changes in the microstructures of the alloy is represented by the dimensionless parameter $\eta .$

$$\begin{eqnarray}&&\eta =\displaystyle \frac{J}{{J}_{{\rm{\max }}}}=\displaystyle \frac{2m}{m+1}\end{eqnarray} \tag{ 19 }$$

In general, metal materials enjoy the best thermal deformation performance when $\eta \gt 0.3,$ [28]. During thermal processing, if $\eta$ is high, it means that the dynamic recrystallization of the material is more complete, and it is easier to produce fine equiaxed grains. When $\varepsilon =0.5,$ the power dissipation map of the Ti-6Al-2V-0.5Zr-1.5Mo alloy is shown in figure 13. The figure shows that $\eta$ is the smallest when the strain rates are 0.03 s⁻¹–0.3 s⁻¹ and the temperature is low, or when the strain rate is as high as 0.1 s⁻¹–1 s⁻¹ and the temperatures are between 800 °C–930 °C. The value of $\eta$ increased significantly, be the deformation rates rose or fell. When the deformation rate was below 0.03 s⁻¹ or above 0.3 s⁻¹, $\eta$ was basically above 0.3. Thus better-recrystallized structure can be produced if thermal deformation was conducted under such conditions.

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But even if the $\eta$ of an area is high, it does not necessarily mean it is safe for processing, as further analysis needs to be conducted on the instability parameter $\xi (\unicode{x003AD})$ [37]. From the definition of the DMM model, the instability criterion of materials in the process of thermal deformation can be expressed as follows:

$$\begin{eqnarray}&&\xi \left(\unicode{x003AD}\right)=\displaystyle \frac{\partial \,\mathrm{lg}\left(\displaystyle \frac{m}{m+1}\right)}{\partial lg\unicode{x003AD}}+m={b}_{2}+2{b}_{3}lg\unicode{x003AD}+3{b}_{4}{\left(lg\unicode{x003AD}\right)}^{2}+m\lt 0\end{eqnarray} \tag{ 20 }$$

Here, ξ $\left(\unicode{x003AD}\right)$ is an instability parameter. When it is less than 0, the external part of the materials will be unstable. Twin crystals, adiabatic shear, cracks, and other defects will be seen in instability areas, which should be avoided during hot working.

The maps of thermal working are composed of a power dissipation map and a flow instability map. As shown in figure 14, instability areas are calculated based on the Prasad criterion [38], which are mainly located in the upper left, middle, and lower right of the hot working map, namely the area with low temperatures and high strains, as well as that with high temperatures and low strains. Specifically, one area is at temperatures between 800 °C to 845 °C and deformation rates between 0.006 s⁻¹ to 1 s⁻¹, and the other at temperatures between 800 °C to 845 °C and deformation rates between 0.001 s⁻¹ to 0.18 s⁻¹. When conducting hot working, one should choose safe areas, which are outside the instability ones, and the value of η needs to be high. Combining the processing map with the microstructures and the processing map to divide them into different areas so that the microstructure maps of the alloy under different processing conditions can be worked out, as shown in figure 15.

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A and B in figure 15 are instability areas. Microcracks, adiabatic shear bands, voids, and other defects are seen[39–41] in these areas during processing. E is a flow localization area, and its main softening mechanism lies in DRV. Due to the lack of energy, the recrystallization in this area is incomplete. Area F is overheated. Long-term high temperatures and slow strain rates lead to the rapid growth of DRX grains, which increases the risk of instability. It is found that areas C and D are the most desirable areas for the processing of the Ti-6Al-2V-0.5Zr-1.5Mo alloy as fine DRX equiaxed grains can be formed to improve the plastic deformation of the alloy. The optimum hot processing parameters are as follows: deformation temperatures of 810 °C−850 °C and deformation rates of 0.001 s⁻¹–0.003 s⁻¹, and deformation temperatures of 870 °C–950 °C and deformation rates of 0.3 s⁻¹–1 s⁻¹.