**1. Introduction**

GH4169 alloy is a nickel-based superalloy that is extensively used in a steam turbine, aerospace, chemical, and nuclear industries [1–3]. However, the original blanks produced during casting are unsatisfactory due to increased performance requirements, and thus, must be improved to meet the performance standards [4–6]. Multidirectional forging is a plastic processing method for obtaining a fine grain structure by continuously changing the direction of axial external loads, thereby compressing the forgings in di fferent directions [7]. Multidirectional forging is a plastic processing method to obtain fine grain structure by continuously changing the direction of axes of external loads and compressing forgings in di fferent directions. The multi-directional forging process can improve the microstructure of the material and obtain fine-grained microstructure materials with excellent mechanical properties and uniform properties. This process is particularly suitable for the regulation of the structure and properties of GH4169 alloy materials.

In recent years, research on the multidirectional forging process in the process of thermal deformation has gradually increased at home and abroad. Mikhail et al. [8] mainly studied the influence of isothermal multi-directional forging on the microstructure evolution of conventional Al-Mg-based alloys in the strain range of 1.5–6.0 and the temperature range of 200–500 ◦C. Xia et al. [9] conducted

a multi-directional forging of Mg-Gd-Y-Nd-Zr alloy at low temperatures to study grain refinement, microstructure, and uniformity of the mechanical properties. Aoba et al. [10] systematically studied the microstructure evolution and mechanical properties of 6000 series aluminum alloys subjected to multi-directional forging and artificial aging treatment. Lin et al. [11] derived a constitutive and microstructure evolution model for GH4169 superalloy based on equivalent dislocation density. However, relevant studies on the simulation of multi-directional forging microstructures are still rare.

In this study, a double-open multidirectional forging with relatively few deformation passes and uniform deformation was generated. The constitutive equation and dynamic recrystallization model were formulated based on a thermal compression experiment on the GH4169 superalloy. The model was then imported into the DEFORM simulation software. The design process of double-open multidirectional forging was explored, and specifically, the predictions of the recrystallization volume fraction and grain size as functions of temperature during the microstructure evolution simulation of GH4169 superalloy were discussed.

#### **2. Experimental Materials and Methods**

The material used in the experiment was a GH4169 alloy cylinder (ϕ 6 mm × 9 mm). The original microstructure of the sample was approximately uniform equiaxed, and the average grain size was approximately 25 μm, as shown in Figure 1. Heat treatment is necessary for the dynamic-recrystallization hot compression test of the samples. The equipment used in the experiment was the Gleeble-3800 Thermal Simulation Tester, and the specific process parameters are shown in Table 1. Figure 2 shows a flowchart of the GH4169 superalloy hot compression test. For the Gleeble-3800 thermal compression test, the alloy samples were initially heated at a rate of 5 ◦C/s to the required temperature (900 ◦C, 950 ◦C, 1000 ◦C, 1050 ◦C, or 1120 ◦C) for the experiment. Once the desired temperature was maintained for 2 min, the sample was subjected to a hot compression test at different strain rates (0.001, 0.01, 0.1, and 1 s<sup>−</sup>1). Each sample was immediately water-cooled at the end of the experiment [12–14].

**Figure 1.** The original microstructure of the GH4169 forge piece.

**Table 1.** Deformation parameters for the GH4169 superalloy hot compression experiment.


**Figure 2.** Flowchart of the GH4169 superalloy hot compression experiment.

#### **3. Results and Discussion**

#### *3.1. True Stress-Strain Curve at Di*ff*erent Strain Rates*

Figure 3 shows the true stress-strain curve obtained for the GH4169 superalloy at different strain rates and temperatures. The stress gradually decreased with an increase in temperature at a given strain and strain rate. Both the peak stress and strain point corresponding to the peak stress gradually decreased with increasing temperature. The material had a similar response to thermal deformation at different strain rates; that is, the stress value increased with the deformation amount. Meanwhile, the stress value decreased with an increase in deformation when the stress value reached its maximum (peak stress). Furthermore, the stress of the material remained constant with the increase in strain when the stress value reached steady-state stress.

**Figure 3.** Thermal compression strain curve of GH4169 alloy under different strain rates and temperatures. (**a**) .ε = 0.001*s*<sup>−</sup>1; (**b**) .ε = 0.01*s*<sup>−</sup>1; (**c**) .ε = 0.1*s*<sup>−</sup>1; (**d**) .ε = 1*s*<sup>−</sup>1.

The sharp increase of stress in the initial stage of thermal compression deformation was due to work hardening. Although recovery and recrystallization might occur, work hardening played a leading role. Because of the slippage of grains in the process of plastic deformation and the tangling of dislocations, the grains were elongated or broken and fibrozed, which resulted in the formation of residual stress in the metal. As the deformation continued to increase, the stress was gradually reduced due to work softening. The subsequent stress reduction during deformation was mainly due to recovery and recrystallization, and work softening caused by recovery and recrystallization played a leading role in this. The stress gradually approached a stable value and remained constant with an increase in deformation. At this point, the work hardening and work softening of the material were balanced and entered a steady deformation stage [15,16].

The stress and peak stress gradually decreased with an increase in the strain rate under certain temperature conditions. The strain point corresponding to the peak stress increased with the temperature and amount of deformation. The deformation showed the same response at different temperatures; that is, the stress began to rapidly increase due to work hardening and reached a peak as the amount of deformation increased. Then, the stress gradually decreased and stabilized due to the influence of work softening [17,18].

#### *3.2. Construction of Constitutive Equations*

The flow stress of the metal during hot deformation is related to the thermodynamic parameter Z and deformation rate .ε of the material. The relation is expressed as follows [19,20]:

$$Z = \dot{\varepsilon} \exp(\frac{\mathcal{Q}}{RT}),\tag{1}$$

where *Q* is the thermal deformation activation energy of the material, *R* is the gas constant, and *T* is the absolute temperature.

Materials have different stress function forms under different conditions, as follows:

$$F(\sigma) = A\_1 \sigma^\eta (\alpha \sigma < 0.8) \tag{2}$$

$$F(\sigma) = A 2 \exp(\beta \sigma) (a \sigma \rhd 1.2) \tag{3}$$

$$F(\sigma) = A\_2 \exp(\beta \sigma) (\alpha \sigma > 1.2) \tag{4}$$

where *Ai* (I = 1, 2, 3); α, β, and *n* are the material parameters.

The sinusoidal equation proposed by Sellars et al. is generally used to describe the relationship between various material parameters during plastic deformation and is expressed as follows:

$$\dot{\varepsilon} = A \left[ \sinh(\alpha \sigma) \right]^n \exp \left[ -\frac{Q}{RT} \right] \tag{5}$$

where α, *n*, *A*, and β are the material parameters. *Q* is the thermal deformation activation energy of the material. *R* is the gas constant, which is a fixed value (*R* = 8.31). .ε is the deformation rate. Σ is the flow stress, and *T* denotes absolute temperature.

Substituting Equations (2) and (3) into Equation (6) and deriving on two sides of the equation, Equation (7) could be obtained.

$$
\ln \dot{\varepsilon} = \ln A\_1 + n \ln \sigma - \frac{\dot{Q}}{RT} \tag{6}
$$

$$
\ln \dot{\varepsilon} = \ln A\_2 + \beta \sigma - \frac{Q}{RT} \tag{7}
$$

For convenience, the logarithmic form of the equation is often employed. Then, the corresponding graphs are made with the coordinates of *ln* .ε and *ln*σ and *ln* .ε and σ, respectively. As shown in Figure 4, the peak stress corresponding to different strain rates at different temperatures was calculated. The data are also shown in Table 2.

The diagram in Figure 4 could be fitted, where the average slope of each *ln* .<sup>ε</sup>-*ln*<sup>σ</sup>*p* line could be calculated based on Equation (6) and Figure 4a, and the *ln* .<sup>ε</sup>-*ln*<sup>σ</sup>*p* lines could be calculated based on Equation (7) and Figure 4b. The resulting average slopes were *n* = 5.954252 and β = 0.031217. Given α = β/*<sup>n</sup>*, α = 0.0052428 could also be obtained.


**Table 2.** Strain rate (.ε), peak stress (<sup>σ</sup>*p*), and corresponding logarithmic value of the GH4169 superalloy at various temperatures.

**Figure 4.** Relation among *ln*<sup>σ</sup>*p*, <sup>σ</sup>*p*, and *ln* .ε curves at different temperatures. (**a**) *ln*<sup>σ</sup>*p* and *ln* .ε curves at different temperatures; (**b**) <sup>σ</sup>*p* and *ln* .ε curves at different temperatures.

Deriving on both sides of Formula (6):

$$
\ln \dot{\varepsilon} = \ln A + n \ln [\sinh(\alpha \sigma)] - \frac{Q}{RT} \tag{8}
$$

Performing partial derivation of 1/*T* at a certain strain rate for Equation (8) yields:

$$Q = R \left[ \frac{\partial \ln \dot{\varepsilon}}{\partial \ln [\sinh(a\sigma)]} \right]\_T \left[ \frac{\partial \ln [\sinh(a\sigma)]}{\partial (1/T)} \right] \tag{9}$$

The activation energy of the material is constant when the strain rate is constant. The *n* values fitted in Figure 4 were substituted into Equation (9) to obtain *ln* .ε and *ln*[*sinh*(ασ)] at different deformation temperatures and *ln*[*sinh*(ασ)] and 1000/*T* at different strain rates, where 1000 times 1/*T* was used for convenient calculation, as shown in Figures 5 and 6.

The average slope *k* = 0.231755 of each straight line could be calculated based on the straight line fitted in Figure 5, and the average slope of each straight line *t* = 13.90354 could be calculated based on the straight line fitted in Figure 6. Based on Equation (9), the thermal deformation energy of the material could be calculated as follows:

$$Q = \frac{R \ast T}{k} = 498.54 \frac{kJ}{mol} = 498540 \text{ J/mol} \tag{10}$$

**Figure 5.** The relation between *ln*[*sinh*(ασ*p*)] and *ln* .ε at various deformation temperatures.

**Figure 6.** The relation between *ln*[sinh(ασ*p*)] and 1000 K/*T* at various strain rates.

From Equations (1) and (6), Equation (11) was obtained:

$$Z = \dot{\varepsilon} \exp\left(\frac{\mathcal{Q}}{RT}\right) = \left.A[\sinh(\alpha \sigma)]\right|^n \tag{11}$$

Performing derivation on both sides of Equation (11):

$$
\ln Z = \ln \dot{\varepsilon} + \frac{Q}{RT} \tag{12}
$$

$$
\ln \mathcal{Z} = \ln A + \ln \text{ln} [\sinh(\alpha \sigma)] \tag{13}
$$

*Metals* **2019**, *9*, 1146

The corresponding value of *lnZ* could be calculated by comparing the peak stress values for different temperatures and strain rates, and the corresponding *lnZ* and *ln*[*sinh*(ασ*p*)] maps were fitted using Equation (13), as shown in Figure 7. The slope of the line was *n* = 4.56568, and the intercept was *lnA* = 42.21596 (*A* = 2.1585 × 1018). Thus, *Z* = 2.1585 × 10<sup>18</sup> × [*sinh*(0.0052428σ)]4.56568 was obtained.

Substituting all e values obtained into Equation (6) yielded the constitutive equation of the alloy, as shown in Equation (14).

$$
\dot{\varepsilon}\_{1} = 2.1585 \times 10^{18} \times \left[ \sinh(0.0052428 \sigma) \right]^{4.5658} \times \exp(-\frac{498540}{RT}) \tag{14}
$$

$$
\begin{array}{c|c}
\text{52.5} \\
\text{50.0} \\
\text{47.5} \\
\text{42.5} \\
\text{53.0} \\
\text{40.0} \\
\text{37.5} \\
\text{1.5} \quad \text{-1.0} & \text{0.0} \quad \text{1.5} \quad \text{-2.5} \\
\text{1.5} & \text{0.0} \quad \text{-1.0} & \text{0.0} \quad \text{0.5} \\
\text{1.5} & \text{-1.0} & \text{0.0} \quad \text{0.0} & \text{0.0} \quad \text{1.5} \quad \text{-2.0} & \text{2.5}
\end{array}
$$

**Figure 7.** Relation between *lnZ* and *ln*[*sinh*(ασ*p*)] at various strain rates and deformation temperatures.

#### *3.3. Dynamic Recrystallization Model*

#### 3.3.1. Proposed Model

Existing studies generally use the Avrami equation to describe the recrystallization degree quantitatively [21,22]:

$$X = \begin{bmatrix} 1 - \exp\left[-k\left(\frac{\varepsilon - \varepsilon\_{\mathcal{E}}}{\varepsilon\_{0.5}}\right)^{n}\right] \end{bmatrix} \tag{15}$$

where *X* is the dynamically recrystallized volume fraction of the material; *k* and *n* are the material parameters; ε is a dependent variable; ε*c* is the critical strain; ε0.5 is the recrystallized amount of the material when 50% strain is reached.

Peak strain model:

$$
\varepsilon\_p = A Z^{\mathfrak{m}} \tag{16}
$$

where *A* and *m* are the material parameters; *Z* is the temperature compensation factor.

Critical strain model:

$$
\kappa\_{\mathcal{L}} = k \varepsilon\_p \tag{17}
$$

where *k* ranges from 0.6 to 0.85, and a value of 0.8 was used in this study.

A quantitative description of the recrystallization quality is usually given by:

$$D\_{2-d\text{rxn}} = A\_1 Z^{A\_2} \tag{18}$$

where *A*1 and *A*2 are material-dependent constants.

#### 3.3.2. Model Establishment

The peak strain value of the material during hot deformation could be obtained through experiments, and the corresponding recrystallization volume fraction and *lnZ* value were calculated, as shown in Table 3.

Simplifying Equation (16) yielded:

$$
\ln \varepsilon\_p = \ln A + \text{mlnZ} \tag{19}
$$

As shown in Equation (18), the corresponding *ln*<sup>ε</sup>*p* and *lnZ* maps could be constructed and fitted (Figure 8). The intercept of the line was *lnA* = −4.46775, that is, *A* = 1.15 × <sup>10</sup>−2, and the slope of the line was *m* = 0.067.

Substituting the obtained *A* and *m* values into Equation (16) yielded Equation (20).

$$
\varepsilon\_p = 1.15 \times 10^{-2} Z^{0.067} \tag{20}
$$

Then, by Equation (17):

$$
\varepsilon\_{\varepsilon} = 9.2 \times 10^{-3} Z^{0.067} \tag{21}
$$

The strain value corresponding to material recrystallization of 50% could be obtained by performing Newton interpolation on the parameter values obtained from the experiment, as shown in Equation (22).

$$
\varepsilon\_{0.5} = 0.29 Z^{0.016} \tag{22}
$$

Simplifying Equation (15) yielded:

$$\ln\left[\ln\left(\frac{1}{1-X}\right)\right] = \ln k + n\ln\left(\frac{\varepsilon-\varepsilon\_c}{\varepsilon\_{0.5}}\right) \tag{23}$$

The corresponding *ln*{*ln*[1/(1 − *X*)]} and *ln*[(ε − <sup>ε</sup>*c*)/<sup>ε</sup>*0.5*] maps could be constructed from Equation (23), and a linear fit could be performed, as shown in Figure 9. By fitting the data in Figure 9, one found *k* = 0.812 and *n* = 0.92.

Substituting the values of *k* and *n* into Equation (15) yielded:

$$X = \ 1 - \exp[-0.812(\frac{\varepsilon - \varepsilon\_c}{\varepsilon\_{0.5}})^{0.92}] \tag{24}$$



**Figure 8.** The relation between *lnZ* and *ln*<sup>ε</sup>*p* at various deformation temperatures.

**Figure 9.** Relation between *ln*{*ln*[1/(1 − *X*)]} and *ln*[(ε − <sup>ε</sup>*c*)/<sup>ε</sup>*0.5*] at various deformation temperatures.

The grain size of the microstructure calculated according to the GH4169 high-temperature dynamic recrystallization metallographic structure diagram is shown in Table 4. According to the data in the table, the logarithmic fitting of *D2-drex* and *Z* could be used to find the corresponding model parameters. The fitted image is shown in Figure 10. The obtained dynamic recrystallization crystal mass equation is shown in Equation (25).

(25)

**Figure 10.** *D2-drex* and *Z* fitted image.


**Table 4.** Strain rate .ε, dynamic recrystallization grain size, and *Z* of the GH4169 superalloy at various temperatures.

#### *3.4. Double-Open Multidirectional Forging Simulation*

#### 3.4.1. Process and Finite Element Model

The finite element model used in the simulation is shown in Figure 11. During forging, the initial billet was compressed in the height direction, stretched in the longitudinal direction, and did not deform in the width direction, owing to the restraining effect of the mold. When the amount of deformation reached a certain level, the closed multidirectional forging occurred due to the restraining action of the lower die, the length of the forging was no longer increased in the longitudinal direction, and the groove of the lower die was gradually filled. Meanwhile, the single-open multidirectional forging side was full, the other side maintained a free surface, and the sides of the double-open multidirectional forging maintained a free surface. The forging was rotated by 90◦ every time it was swept, and the last forging was repeated. The simulation utilized a relative net partitioning method and a tetrahedral mesh with 20,000 cells. Table 5 shows the parameter settings used for the double-open multidirectional forging simulation.



**Figure 11.** Finite element model.

#### 3.4.2. Numerical Simulation Results and Analysis

Figure 12 shows the dynamic recrystallization volume fraction cloud diagram of forgings from three to nine passes at 800 ◦C. The dynamic recrystallization volume fraction of the forging was symmetrically distributed and could be divided into three deformation zones, namely difficult, easy, and free. The equivalent strain was minimized on the outer surface of the forging due to the restraint of the mold. Therefore, the recrystallization degree in this area was relatively low, the area with the largest recrystallization was approximately 30%, and some areas were not recrystallized. In the interior of the forging, the equivalent strain was large, and the recrystallization degree was relatively high because it was not affected by the mold friction. The recrystallization degree near the inner forging was high, and the center portion reached approximately 82%.

Figure 13 shows the recrystallized grain size graph of forging from three to nine passes at 800 ◦C. The cloud diagram shows that the recrystallized grain size of the forging was symmetrically distributed. The closer it was to the forging center, the smaller was the grain size. The equivalent strain of the forging was also small on the outer surface of the forging due to the limitation of the die on the forging, thereby affecting the recrystallization. Thus, the average grain size of the forging in this area was large, and no recrystallization occurred. The equivalent strain in the interior of the forging was large because it was not affected by the mold friction. Thus, the average grain size was small, with a minimum value of approximately 33.5 μm.

The recrystallization degree of the forgings was remarkably improved when it reached six passes. The closer it was to the internal, the higher was the recrystallization degree, which resulted in the center portion to approximately reach 95.5%. Therefore, the recrystallization degree was high, and the average grain size was small. The minimum value was approximately 12.2 μm. When forging nine passes, the recrystallization degree of the forgings reached 98.5%, and the minimum average grain size was approximately 5.1 μm.

**Figure 12.** Dynamic recrystallization volume fraction cloud diagram for three to nine passes forging at 800 ◦C.

**Figure 13.** Forging of three to nine passes for recrystallized grain size cloud map at 800 ◦C.

Figure 14 shows the dynamic recrystallization volume fraction cloud diagram of forging from three to nine passes at 900 ◦C. From the figure, the dynamic recrystallization volume fraction of the forging was symmetrically distributed and could be divided into three deformation zones: difficult, easy, and free deformation. The equivalent strain was minimized on the outer surface due to the mold constraint. Therefore, the recrystallization degree in this area was relatively low, the area with the largest recrystallization was approximately 50%, and some areas were not recrystallized. In the interior of the forging, because it was not affected by the mold friction, the equivalent strain was large; the recrystallization degree was relatively high, and the closer it was to the internal, the higher was the recrystallization degree, which resulted in the center portion achieving a value of approximately 87%.

Figure 15 shows the recrystallized grain size cloud diagram of forging from three to nine passes at 900 ◦C. From the figure, the recrystallized grain size of the forging was symmetrically distributed, and the closer it was to the inside of the forging, the smaller was the grain size, and the closer the outer grain size was, the larger was the grain size. The equivalent strain was small on the outer surface of the forging due to the limitation of the die on the forging, which a ffected its recrystallization. Thus, the average grain size of the forging in this area was large, and no recrystallization occurred. In the interior of the forging, because it was not a ffected by the mold friction, the equivalent strain was large. Thus, the recrystallization degree in this region was high, the average grain size was small, and the minimum value was approximately 26 μm.

The center portion reached approximately 97.5% under the forging of six passes. Therefore, the recrystallization degree was high in this region, the average grain size was small, and the minimum value was approximately 10 μm. When forging nine passes, the recrystallization degree of the forgings reached approximately 99.7%, and the average grain size minimum was approximately 4 μm.

Figure 16 shows the dynamic recrystallization volume fraction cloud diagram of forgings from three to nine passes at 1000 ◦C. From the figure, the dynamic recrystallization volume fraction of the forging was symmetrically distributed and could be divided into three deformation zones: di fficult, easy, and free deformation zones. The equivalent strain was minimized on the outer surface of the forging due to the constraint of the mold. Thus, the recrystallization degree in this area was relatively low, and the area with the largest recrystallization was approximately 70%. In the interior of the forging, because it was not a ffected by the mold friction, the equivalent strain was large, and the recrystallization degree was relatively high. The closer it was to the internal, the higher was the recrystallization degree, which resulted in the highest value of approximately 74% at the center; moreover, the degree of internal recrystallization did not change considerably with the outside. No obvious increasing trend was observed.

**Figure 14.** Forging from three to nine passes for dynamic recrystallization volume fraction cloud image at 900 ◦C.

**Figure 15.** Forging from three to nine passes for recrystallized grain size cloud image at 900 ◦C.

**Figure 16.** Dynamic recrystallization volume fraction cloud image of forging from three to nine passes at 1000 ◦C.

Figure 17 shows the recrystallized grain size graph of forging from three to nine passes at 1000 ◦C. From the figure, the recrystallized grain size of the forging was symmetrically distributed; moreover, the closer it was to the inside of the forging, the smaller was the grain size, and the closer it was to the outer grain size, the larger was the grain size. The equivalent strain was small on the outer surface of the forging due to the restraining effect of the die on the forging, which affected the recrystallization of the forging. The average grain size of the forging in this area was relatively large. In the interior of the forging, because it was not affected by the mold friction, the equivalent strain was large. Thus, the recrystallization degree in this area was relatively high, the average grain size was small, the minimum value was approximately 24 μm, and the internal and external grain sizes of the forging did not change significantly. This situation occurred because the deformation temperature of the forging was relatively high, and the strain had no considerable effect on recrystallization.

The center portion approximately reached 99% with the forging of six passes. Therefore, the recrystallization degree was high in this region, the average grain size was small, and the minimum value was approximately 9.1 μm. When forging nine passes, all areas of recrystallization volume fraction exceeded 95%, the grain size was 3–5.5 μm, the recrystallization degree of forgings reached 99.7%, and the average grain size minimum was approximately 3 μm.

**Figure 17.** Recrystallization grain size cloud image of forging from three to nine passes at 1000 ◦C.
