During hot deformation process, it is very widely known that the constitutive relationships between the steady-state flow stress, strain rate and deformation temperature are represented by Zener–Hollomon parameter, in an exponent-type equation, and the hyperbolic law in the Arrhenius-type equation gives better approximations between $Z$ parameter and stress, as expressed in eq. (1) [15–18]:
$Z=\dot{\mathrm{\epsilon}}exp\left(\frac{Q}{RT}\right)=AF(\mathrm{\sigma})$(1)where
$F(\mathrm{\sigma})=\{\begin{array}{cc}{\mathrm{\sigma}}^{m}& \mathrm{\alpha}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\sigma}<0.8\\ exp\phantom{\rule{thinmathspace}{0ex}}(\mathrm{\beta}\mathrm{\sigma})& \mathrm{\alpha}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\sigma}<1.2\\ [\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\phantom{\rule{thinmathspace}{0ex}}(\mathrm{\alpha}\mathrm{\sigma}){]}^{n}& \mathrm{f}\mathrm{o}\mathrm{r}\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{l}\mathrm{l}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\sigma}\end{array}$in which $\dot{\mathrm{\epsilon}}$ is the strain rate (s^{−1}), $R$ is the universal gas constant (8.31 J mol^{−1} K^{−1}), $T$ is the absolute temperature (K), $Q$ is the apparent activation energy for hot deformation (kJ mol^{−1}), $\mathrm{\sigma}$ is the flow stress (MPa) for a given stain, $A$, $\mathrm{\alpha}$, $m$ and $n$ are the material constants, $\alpha =\beta /n$.

In the current study, the experimental datasets involving deformation temperature, strain rate and peak stress (${\mathrm{\sigma}}_{P}$) at peak strain were fitted into eq. (1). The relationships between $ln\dot{\mathrm{\epsilon}}-ln{\mathrm{\sigma}}_{P}$, $ln\dot{\mathrm{\epsilon}}-{\mathrm{\sigma}}_{P}$, $ln\dot{\mathrm{\epsilon}}-\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha}{\mathrm{\sigma}}_{P})]$ and $\text{ln}[\text{sinh}(\alpha {\sigma}_{\text{P}})]-10,000/\text{T}$ in $\mathrm{\alpha}+\mathrm{\beta}$-phase temperature range are shown in Figure 3 The value of $m$ and $\mathrm{\beta}$ was obtained through taking average slope of the lines in Figure 3(a) and 3(b), respectively. Thus the $\mathrm{\alpha}$-value in $\mathrm{\alpha}+\mathrm{\beta}$-phase temperature range was calculated to be 0.005195. Subsequently, the calculation of $n$ and $Q$ has been performed according to the following relationships:
$Q=Rns$(2)
$n=\frac{dln\dot{\mathrm{\epsilon}}}{dln[sinh(\mathrm{\alpha}\mathrm{\sigma})]}$(3)
$s=\frac{dln[sinh(\mathrm{\alpha}\mathrm{\sigma})]}{d(1/T)}$(4)Hence the value of $n$ and $Q$ in $\mathrm{\alpha}+\mathrm{\beta}$-phase temperature range was calculated to be 4.1972 and 463.09 kJ/mol, respectively, from the slope of the lines in Figure 3(c) and 3(d). In the same way, the value of $\mathrm{\alpha}$, $n$ and $Q$ in $\mathrm{\beta}$-phase temperature range was calculated to be 0.010199, 3.2812 and 210.45 kJ/mol, respectively, from the plots of $ln\dot{\mathrm{\epsilon}}-ln{\mathrm{\sigma}}_{P}$, $ln\dot{\mathrm{\epsilon}}-{\mathrm{\sigma}}_{P}$, $ln\dot{\mathrm{\epsilon}}-\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha}{\mathrm{\sigma}}_{P})]$ and $\mathrm{l}\mathrm{n}[\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha}{\mathrm{\sigma}}_{P})]-$ $10,000/T$ in $\mathrm{\beta}$-phase temperature range, as shown in Figure 4.

Figure 3 Relationships between (a) $ln\dot{\mathrm{\epsilon}}-ln{\mathrm{\sigma}}_{P}$, (b) $ln\dot{\mathrm{\epsilon}}-{\mathrm{\sigma}}_{P}$, (c) $ln\dot{\mathrm{\epsilon}}-\mathrm{l}\mathrm{n}(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha}{\mathrm{\sigma}}_{P}))$, (d) $\text{ln}(\text{sinh}(\alpha {\sigma}_{P}))-10,000/T$ in $\mathrm{\alpha}+\mathrm{\beta}$-phase temperature range

Figure 4 Relationship between (a) $ln\dot{\mathrm{\epsilon}}-ln{\mathrm{\sigma}}_{P}$, (b) $ln\dot{\mathrm{\epsilon}}-{\mathrm{\sigma}}_{P}$, (c) $ln\dot{\mathrm{\epsilon}}-\mathrm{l}\mathrm{n}(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(\mathrm{\alpha}{\mathrm{\sigma}}_{P}))$, (d) $\text{ln}(\text{sinh}(\alpha {\sigma}_{P}))-10,000/T$ in $\mathrm{\beta}$-phase temperature range

The hot deformation behaviors are influenced by the process of thermal activation, and the softening mechanism is related to the thermal activation energy ($Q$), whose value reflects the difficulty of the dislocation moving, DRV and DRX. Many researchers have revealed that the self-diffusion activation energy of $\mathrm{\alpha}\u2010\mathrm{T}\mathrm{i}$ in pure titanium is in the range of 169–242 kJ/mol, while the self-diffusion activation energy of $\mathrm{\beta}\mathrm{-T}\mathrm{i}$ is 153–166 kJ/mol. As mentioned earlier, the activation energy for hot deformation in $\mathrm{\alpha}+\mathrm{\beta}$-phase temperature range is determined to be 463.09 kJ/mol, which is much larger than the activation energy of self-diffusion. However, the calculated $Q$-value (210.45 kJ/mol) in $\mathrm{\beta}$-phase temperature range is almost equal to the self-diffusion activation energy of $\mathrm{\beta}\mathrm{-T}\mathrm{i}$. In general, the DRV predominates in the work softening process of hot deformation when the activation energy for hot deformation did not appear to be much different from the self-diffusion activation energy, while the DRX is the main softening mechanism when the activation energy for hot deformation is much larger than the self-diffusion activation energy.

The variation of activation energies for as-forged Ti-10V-2Fe-3Al alloy in hot deformation along with the increase of strain from 0.02 to 0.8 is shown in Figure 5. It is obviously seen from Figure 5 that the activation energy for hot deformation decreases with strain increases in $\mathrm{\beta}$-phase temperature range, reflecting that the energy for deformation decreases with the increase of the plastic deformation. The average activation energy for hot deformation in $\mathrm{\beta}$-phase temperature range was calculated from Figure 4 to be 188.71 kJ/mol. It is almost equal to the self-diffusion activation energy of 166 kJ/mol for $\mathrm{\beta}\mathrm{-T}\mathrm{i}$, indicating that the DRV predominates in the work softening process of as-forged Ti-10V-2Fe-3Al alloy in $\mathrm{\beta}$-phase temperature range. This can be proved in Figure 2, as the variation of flow stress with strain in $\mathrm{\beta}$-phase temperature range appears to be a steady-state characteristic with significant DRV softening.

Figure 5 The relationship between $Q$ and strain

Figure 6 The microstructure of as-forged Ti-10V-2Fe-3Al alloy at different deformation conditions

However, in $\mathrm{\alpha}+\mathrm{\beta}$-phase temperature range, when the strain increases, the activation energy for hot deformation increases rapidly to a peak value and then decreases gradually to a relatively low level, as shown in Figure 5. Meanwhile, it can also be seen in Figure 5 that the activation energy for hot deformation in $\mathrm{\alpha}+\mathrm{\beta}$-phase temperature range is much larger than the self-diffusion activation energies of $\mathrm{\alpha}\mathrm{-T}\mathrm{i}$ (242 kJ/mol) and $\mathrm{\beta}\mathrm{-T}\mathrm{i}$ (166 kJ/mol), which means that the DRX predominates in softening mechanism during hot deformation of as-forged Ti-10V-2Fe-3Al alloy. Figure 6 shows the typical microstructure of as-forged Ti-10V-2Fe-3Al alloy at different deformation conditions in $\mathrm{\alpha}+\mathrm{\beta}$-phase temperature range. It can be seen from Figure 6 that the $\mathrm{\alpha}$-phase grains are fined by globularization which can be regarded as a type of DRX, and the matrix of $\mathrm{\beta}$-phase is surrounded by equiaxed $\mathrm{\alpha}$-phase grains. In fact, when the specimen is compressed, it is significantly difficult to identify the grain boundaries of $\mathrm{\beta}$-phase due to the crush, globularization and confusion of aciculate $\mathrm{\alpha}$-phase grains in the grain boundaries of $\mathrm{\beta}$-phase.

It indicates that for as-forged Ti-10V-2Fe-3Al the softening mechanism transforms from DRX to DRV with increasing temperature, which is due to an allotropic phase transformation from $\mathrm{\alpha}$-phase to $\mathrm{\beta}$-phase. As a result, the extremely complex flow behaviors of as-forged Ti-10V-2Fe-3Al in a wide temperature range make it difficult to predict the flow stresses during hot deformation with constitutive models by regression method, whereas the ANN is able to deal with the highly nonlinear relationships well.

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