Arrhenius-type equation serves as a precise approach to illustrate the relationships between the deformation temperature, strain rate and flow stress. The common impact of the high temperature and strain rate over the hot deformation behavior can be shown through the exponent-type equation. It can be properly expressed through the Zener–Hollomon parameter (*Z*). These can be described using the following equations [25].

$\mathrm{Z}=\dot{\epsilon}exp\left(\frac{Q}{RT}\right)$(24)$\dot{\epsilon}=AF\left(\sigma \right)exp\left(-\frac{Q}{RT}\right)$(25)$\mathrm{F}\left(\sigma \right)=\left\{\begin{array}{c}{\sigma}^{{n}^{{\prime}^{}}}\\ exp\left(\beta \sigma \right)\\ {\left[sinh\left(\alpha \sigma \right)\right]}^{n}\end{array}\begin{array}{c}\alpha \sigma <0.8\\ \alpha \sigma >1.2\\ for\text{}all\text{}\sigma \end{array}\right)$(26)where *R* refers to the universal gas constant [8.314 J/(mol·K)], *Q* is the activation energy of hot deformation (J/mol), *A*, ${\mathrm{n}}^{{\prime}^{}}$, *β, α*, and *n* are the materials constants, and $\alpha =\beta /{\mathrm{n}}^{{\prime}^{}}$.

The true stress–strain data from isothermal compression tests under different processing conditions is applied to decide the material constants of the Arrhenius-type constitutive model. The effect of the strain over the material constants is quite clear. It impacts the predictability of the constitutive model dramatically. However, the strain is not considered in eqs. (24) and (25). Therefore, the following research over the Arrhenius-type model is based on the compensation of the strain effect. The strain of 0.4 is taken as an example to describe the solution procedures for the material constants.

Under specific deformation temperature, the $F(\sigma )$ of eq. (26) under low stress level ($\alpha \sigma <0.8$) and high stress level (*ɑσ* > 1.2) is substituted into eq. (25) in trun, the following relationships can be obtained, respectively.

$\dot{\epsilon}=B{\sigma}^{{n}^{{\prime}^{}}}$(27)$\dot{\epsilon}=Cexp\left(\beta \sigma \right)$(28)where *B* and *C* are the material constants.

Both sides of eqs. (27) and (28) are taken in the natural logarithms, and then they are transformed into a correlation function of σ, while the following equations can be obtained.

$\mathrm{l}\mathrm{n}\left(\sigma \right)=\frac{1}{{n}^{{\prime}^{}}}ln\left(\dot{\epsilon}\right)-\frac{1}{{n}^{{\prime}^{}}}ln\left(B\right)$(29)$\sigma =\frac{1}{\beta}ln\left(\dot{\epsilon}\right)-\frac{1}{\beta}ln\left(C\right)$(30)The values of the flow stress and corresponding strain rates under the strain of 0.4 are substituted into eq. (29) as well as eq. (30), respectively. The values of ${\mathrm{n}}^{{\prime}^{}}$ and *β* can be obtained from the slopes of the lines in the ln(σ) − $ln\left(\dot{\epsilon}\right)$ and σ − $ln\left(\dot{\epsilon}\right)$ plots, respectively, as presented in Figure 16(a) and (b).

Figure 16: Relationship between (a) ln (*σ*) and $ln\left(\dot{\epsilon}\right)$; (b) *σ* and $ln\left(\dot{\epsilon}\right)$.

Then, the value of $\alpha =\beta /{\mathrm{n}}^{{\prime}^{}}$ can be obtained.

For all stress levels (including low as well as high stress levels), eq. (25) can be written as follows:

$\dot{\epsilon}=A{\left[sinh\left(\alpha \sigma \right)\right]}^{n}exp\left(-\frac{Q}{RT}\right)$(31)Taking the natural logarithm of both sides of eq. (31):

$\mathrm{l}\mathrm{n}\left[sinh\left(\alpha \sigma \right)\right]=\frac{ln\dot{\epsilon}}{n}+\frac{Q}{nRT}-\frac{lnA}{n}$(32)For a particular temperature, eq. (32) can be written as:

$\frac{d\left\{ln\left[sinh\left(\alpha \sigma \right)\right]\right\}}{d\left(ln\dot{\epsilon}\right)}=\frac{1}{n}$(33)The slopes of the lines of $ln\left[sinh\left(\alpha \sigma \right)\right]$ − $ln\dot{\epsilon}$ can be applied to determine the value of material constant *n*, which is presented in Figure 17. The value of *n* is determined by averaging the values of *n* under different temperatures.

Figure 17: Relationship between $ln\left[sinh\left(\alpha \sigma \right)\right]$ and $ln\dot{\epsilon}$.

For a particular strain, eq. (32) can be written as:

$\mathrm{Q}=\mathrm{R}\mathrm{n}\frac{d\left\{ln\left[sinh\left(\alpha \sigma \right)\right]\right\}}{d\left(\frac{1}{T}\right)}$(34)The slopes of the plot of $ln\left[sinh\left(\alpha \sigma \right)\right]$ and $\frac{1000}{T}$ can determine the value of *Q*, which can be found in Figure 18 (*Q* is in KJ/mol). The value of *Q* is determined by averaging the values of *Q* under different strain rates.

Figure 18: Relationship between $ln\left[sinh\left(\alpha \sigma \right)\right]$ and $\frac{1000}{T}$.

The values of A can be determined from the intercept of $ln\left[sinh\left(\alpha \sigma \right)\right]$ − $\stackrel{\u02c9}{P}$ at a particular strain.

According to previous studies, it is usually assumed that the impact of strain over the flow stress at elevated temperatures was dramatic. Thus, it was overlooked in eqs. (24) and (25). However, the effect of the strain over the material constants (i.e. *α, n, Q* and ln*A*) is significant in the entire strain range.

As a result, the compensation of the strain should consider the constitutive model and hence to ensure that model can be more accurate. The influence of the strain of the constitutive equation is combined by the assumption that the material constants (*α, n, Q* and ln*A*) are polynomial functions of the strain. As illustrated in eq. (35), it has been found that there is a third-order polynomial to represent the influence of strain on material constants with a good correlation and generalization, as shown in Figure 19. In the meanwhile, the polynomial results of *α, n, Q* and ln*A* of TA2 are provided in .

$\alpha ={C}_{0}+{C}_{1}\epsilon +{C}_{2}{\epsilon}^{2}+{C}_{3}{\epsilon}^{3}+{C}_{4}{\epsilon}^{4}+{C}_{5}{\epsilon}^{5}$$\mathrm{n}={D}_{0}+{D}_{1}\epsilon +{D}_{2}{\epsilon}^{2}+{D}_{3}{\epsilon}^{3}+{D}_{4}{\epsilon}^{4}+{D}_{5}{\epsilon}^{5}+{D}_{6}{\epsilon}^{6}$$\mathrm{Q}={E}_{0}+{E}_{1}\epsilon +{E}_{2}{\epsilon}^{2}+{E}_{3}{\epsilon}^{3}+{E}_{4}{\epsilon}^{4}+{E}_{5}{\epsilon}^{5}+{E}_{6}{\epsilon}^{6}$$\mathrm{l}\mathrm{n}\mathrm{A}={F}_{0}+{F}_{1}\epsilon +{F}_{2}{\epsilon}^{2}+{F}_{3}{\epsilon}^{3}+{F}_{4}{\epsilon}^{4}+{F}_{5}{\epsilon}^{5}+{F}_{6}{\epsilon}^{6}$(35)Figure 19: Variation of (a) *α* and ln*A*; (b) *n* and *Q* with true strain.

Table 4: Parameters for the modified ZA model.

Table 5: Coefficients of the polynomial for *α, n, Q* and ln*A*.

Then the flow stress at a specific strain can be estimated. The hyperbolic sine function is used while the Zener–Holloman parameter is used to express the constitutive equation in the following form:

$\sigma =\frac{1}{\alpha}ln\left\{{\left(\frac{Z}{A}\right)}^{1/n}+{\left[{\left(\frac{Z}{A}\right)}^{2/n}+1\right]}^{1/2}\right\}$(36)Figure 20 presents the comparison between the predicted and experimental values by strain-compensated Arrhenius-type constitutive equation at different processing conditions.

Figure 20: Comparison between experimental and predicted flow stress using strain-compensated Arrhenius-type equation at the temperatures of (a) 933 K, (b) 983 K, (c) 1,033 K, (d) 1,083 K, (e) 1,133 K.

Figure 20 shows that the flow stress estimated by the constitutive equation is in line with the experimental results in the three lower temperatures of 933, 983 and 1,033 K. It has a deviation under the two higher temperatures 1,083 K and 1,133 K with the lower predicted flow stress compared with the experimental flow stress at high strain rates of 10 and 20s^{−1}.

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