*3.2. Flow Stress Modeling*

Although a flow stress model of the GH4698 alloy was established by Zhang et al. [22], different shapes of stress-strain curves were obtained under various compression conditions, as shown in Figure 2. Therefore, a flow stress model describing the stress-strain relationships of GH4698 should be established. The Arrhenius equation was expressed as in [26]:

$$
\dot{\varepsilon} \exp\left(\frac{\mathcal{Q}}{RT}\right) = A(\sinh(a\sigma))^n. \tag{1}
$$

Here, *A*, *n*, and α were material constants, was the strain rate in s<sup>−</sup>1, *Q* was the thermal activation energy in kJ/(mol·K), *R* was the gas constant, which equaled 8.314 J/(mol·K), and *T* was the deformation temperature in K. As a constraint, the following could be applied for different stress levels [27,28]:

$$
\dot{\varepsilon} = A\_1 \sigma^\text{\prime} \exp\left(-\frac{Q}{RT}\right) \qquad (a\sigma < 0.8) \tag{2}
$$

$$
\dot{\varepsilon} = A\_2 \exp\left(\beta \sigma\right) \exp\left(-\frac{Q}{RT}\right) \qquad (a\sigma > 1.2). \tag{3}
$$

*A1*, *A2*, *n*, and β are constants, which follows:

$$
\alpha = \frac{\beta}{n} \tag{4}
$$

Taking logarithms of both sides of Equations (2) and (3) gives:

$$
\ln \dot{\varepsilon} = \ln A\_1 + n \ln \sigma - \frac{Q}{RT} \tag{5}
$$

$$
\ln \dot{\varepsilon} = \ln A\_2 + \beta \sigma - \frac{Q}{RT} \,. \tag{6}
$$

Thus, *<sup>n</sup>* and <sup>β</sup> could be calculated by the slopes of the fitted line of ln <sup>σ</sup> versus ln . ε, and σ versus ln . ε. The value of α is calculated according to Equation (4). Taking a logarithm of Equation (1) results in:

$$
\ln \dot{\varepsilon} = \ln A + \nu \ln(\sinh(a\sigma)) - \frac{Q}{RT}.\tag{7}
$$

Therefore, the value of *Q* could be obtained via the slope of the fitted line of <sup>1</sup> *<sup>T</sup>* versus ln(sinh(ασ)), and the value of ln *A* could be obtained via the intercept of the fitted line of ln(sinh(ασ)) versus ln . ε+Q/*RT*. Taking ε = 0.05 as example, the calculation process of the model parameters is shown in Figure 3. For effectiveness, the calculations were made using MATLAB software at various strains. The values of the model parameters with the increasing strain are shown in Figure 4.

**Figure 3.** The calculation process of *<sup>n</sup>*, <sup>β</sup>, *<sup>Q</sup>*, and ln *<sup>A</sup>* from the slope of the fitted line of (**a**) ln <sup>σ</sup> versus ln . ε, (**b**) <sup>σ</sup> versus ln . ε, (**c**) <sup>1</sup> *<sup>T</sup>* versus ln(sinh(ασ)), and from the intercept of the fitted line of (**d**) ln(sinh(ασ)) versus ln . ε + <sup>Q</sup> *RT*.

**Figure 4.** The Arrhenius model parameter values of GH4698 alloy at various strains. (**a**) *n*, (**b**) β, (**c**) *Q*, and (**d**) ln *A*.

Fifth-order polynomial fittings were then applied to describe the relationships between the model parameters and the strain, resulting in the following:

$$G = k\_0 + k\_1 \varepsilon + k \varepsilon^2 + k\_3 \varepsilon^3 + k\_4 \varepsilon^4 + k\_5 \varepsilon^5. \tag{8}$$

Here, *G* denotes *n*, β, *Q*, and ln *A*, respectively. *k*<sup>0</sup> − *k*<sup>5</sup> denote the coefficients, whose values are determined by polynomial fitting in the *Origin* software, shown in Table 2.


**Table 2.** Coefficients of the polynomials.

The comparisons between the calculated and experimental flow stress values are shown in Figure 5. Basically, the flow stress could be predicted with good accuracy, showing that the Arrhenius model was capable of accurately calculating the flow behaviors of GH4698 under various strain rates. The scatter plots of the calculated versus experimental flow stress are shown in Figure 6. The calculated stresses fit well with the experimental stresses. The adjusted R square value to describe the goodness of fit was 0.990. The average calculation error was 10.96 MPa (5.90%). The errors may be attributed to the following two reasons. First, there were still computational deviations in the fitting of model parameters by the fifth-order polynomial, which could be improved by trying better fitting expressions. Additionally, an abnormal softening occurred in the final part of the stress–strain curve under 1000 ◦C and 0.01 s<sup>−</sup>1, which may be caused by experimental deviations. The number of repetitions under the same hot compression conditions could be increased to reduce such deviations.

**Figure 5.** Comparisons of the calculated and experimental flow stress at the strain rates of (**a**) 0.001 s<sup>−</sup>1, (**b**) 0.01 s<sup>−</sup>1, (**c**) 0.1 s<sup>−</sup>1, (**d**)1s<sup>−</sup>1, and (**e**)3s−1.

**Figure 6.** Scatter plots of the calculated versus experimental flow stress.

#### *3.3. Processing Maps*

The processing maps were significant references for process parameter optimizations of GH4698 large forgings. It was suggested by Prasad et al. [26] that the deformation work (*P*) during forging is consumed by temperature rising (*G*) and microstructure evolution (*J*), as represented by:

*Processes* **2019**, *7*, 491

$$P = G + J = \int\_0^{\dot{\kappa}} \sigma \, \mathbf{d}\dot{\varepsilon} + \int\_0^{\sigma} \dot{\varepsilon} \, \mathbf{d}\sigma. \tag{9}$$

The power dissipation coefficient (η) quantifying the fraction of energy by microstructure evolution was calculated by [4]:

$$\eta = \frac{2\frac{\partial \ln \sigma}{\partial \ln \varepsilon}}{\frac{\partial \ln \sigma}{\partial \ln \varepsilon} + 1}. \tag{10}$$

An instability coefficient ζ for predicting deformation instability was expressed as in [4]:

$$\zeta = \frac{\partial \log(\eta/2)}{\partial \log(\dot{\varepsilon})} + \frac{\partial \ln \sigma}{\partial \ln \dot{\varepsilon}}.\tag{11}$$

The value of ζ was negative when deformation instability occurred according to previous research [4,12–15]. The value of σ could be obtained from the flow stress model. Thus, the thermal processing maps of GH4698 were established according to Equations (10) and (11), as shown in Figure 7. The deformation instability was shown as shaded areas, and the dissipation coefficients were shown as contours. At a strain of 0.2 (Figure 7a), three deformation instability domains were found. One was located at 950–1150 ◦C and 0.25–3 s−1, another at 985–1015 ◦C and 0.001–0.002 s−1, and the third at 1060–1140 ◦C and 0.001–0.004 s−1. As the strain increased to 0.4, the deformation instability domain at high strain rates split into two, one at 950–1025 ◦C and 0.5–3 s<sup>−</sup>1, and the other at 1060–1150 ◦C and 0.5–3 s−1. Another deformation instability domain was located at 960–1010 ◦C and 0.002–0.025 s−1. A comparison of data presented in Figure 7a,b showed that the majority of the deformation instability domain had dissipation coefficients lower than 35%, and the dissipation coefficients of the other parts of the map were generally greater than 35%. One reason for this could be that at higher strain rates and lower temperatures, dynamic recrystallization was insufficient, and deformation instability occurred more easily. Thus, a much larger proportion of the deformation work was converted into heat. At a strain of 0.6 (Figure 7c), two deformation instability domains were found. One was located at 950–975 ◦C and 0.3–3 s−1, and the other at 1070–1140 ◦C and 0.5–3 s−1. It is worth noting that the dissipation coefficients were high in the lower right part of the map in Figure 7c, which corresponds to high temperature and low strain rate conditions, and this may be because of the dramatic grain growth after the dynamic recrystallization was completed. At a strain of 0.8 (Figure 7d), two deformation instability domains occurred. One existed at 950–1070 ◦C and 0.05–0.63 s<sup>−</sup>1, and the other at 1080–1150 ◦C and 0.4–3 s<sup>−</sup>1. The overlapping of the processing maps in Figure 7a–d reveals that to avoid deformation instability, the hot working parameters should drop in the area of 1080–1150 ◦C and 0.004–0.05 s<sup>−</sup>1.

The processing maps at various temperatures or strain rates are extremely important for the hot working parameter optimization when the deformation temperature or the punch speed are restrained. The processing maps of the GH4698 alloy at various temperatures could be obtained by slicing and interpolating the processing maps at specific temperatures, as shown in Figure 8. A deformation instability domain at a strain of 0.2–0.8 and at a strain rate of 0.1–3 s−<sup>1</sup> was found under 950 ◦C (Figure 8a). The dissipation coefficients were below 25%. A comparison with data provided in Figure 8a indicated that much smaller deformation instability domains existed under 1050 ◦C, as shown in Figure 8b. At 1150 ◦C (Figure 8c), the deformation instability domain disappeared. Figure 8 shows that the workability of the GH4698 alloy was improved by dynamic recrystallization under higher hot working temperatures. It is worth noting that the dissipation coefficients generally increased with increasing temperature, because the dynamic recrystallization was more complete at higher temperatures, thereby consuming a much larger proportion of energy.

**Figure 7.** Processing maps of GH4698 at the strains of (**a**) 0.2, (**b**) 0.4, (**c**) 0.6, and (**d**) 0.8.

**Figure 8.** Processing maps of GH4698 at the temperatures of (**a**) 950 ◦C, (**b**) 1050 ◦C, and (**c**) 1150 ◦C.

Hot working maps of a GH4698 alloy at various strain rates are shown in Figure 9. Good hot working abilities were obtained at 0.001 s<sup>−</sup>1, as no deformation instability domain was found, as shown in Figure 9a. A dissipation coefficient of ~45% at 1050–1150 ◦C and at the strain of 0.4–0.8 showed that a full dynamic recrystallization followed by dramatic grain growth might have occurred. A deformation instability domain at 950–1050 ◦C and at the strain of 0.75–0.8 could be observed at 0.1 s<sup>−</sup>1, as shown in Figure 9b, and a relatively high dissipation coefficient of ~40% could still be obtained, meaning that the thermal processing parameters were still acceptable. However, the deformation instability domain occupied the majority of the processing map, as shown in Figure 9c, indicating that the GH4698 alloy should not be hot-formed at such a fast strain rate, as deformation instability and local material flow would occur.

Based on the results presented in Figures 7–9, the optimal hot working parameters were suggested as 1080–1150 ◦C and 0.004–0.1 s<sup>−</sup>1. The forging temperature should be no lower than 1050 ◦C to ensure that dynamic recrystallization occurs completely, and the strain rate should be neither too high to avoid deformation instability, nor too low to prevent grain coarsening. The visible differences in the shape of the stress-strain curves in this research and in the literature [22] might lead to the difference of the shape of the processing maps. Therefore, further research should be carried out to quantify the influence of chemical compositions. In the practical production of GH4698 large forgings, the forging temperature can be determined by the pressure that the forging device can provide. The results also showed that the GH4698 alloy was suitable for low-speed isothermal forgings at high temperatures (1080–1150 ◦C).

**Figure 9.** Processing maps of GH4698 at the strain rates of (**a**) 0.001 s<sup>−</sup>1, (**b**) 0.1 s<sup>−</sup>1, and (**c**)3s−1.
