Constitutive Model and Microstructure Evolution Finite Element Simulation of Multidirectional Forging for GH4169 Superalloy

This study investigates three processes of multidirectional forging (MDF), namely, closed MDF (CMDF), single-open MDF, and double-open MDF, by using a constitutive equation and a dynamic recrystallization model of hot deformation of the GH4169 superalloy. The microstructure evolution of the three processes is simulated and compared. Among the three processes, the double-open MDF obtains the highest recrystallization degree, followed by the CMDF and the single-open MDF under the same reduction. The recrystallization degree of CMDF reaches 99.5% at 1000 °C and 9 passes, and the average recrystallized grain size is small, which is approximately 8.1 μm. The double-open MDF can obtain a fine grain size of forgings at 9 passes and 1000 °C, and it is easy to obtain forgings with the single-open MDF with uniform performance. The temperature is 850 °C–1000 °C, the compression rate is 0.15–0.2, and the pass is 5–9, which are the suitable parameter selection ranges for the CMDF. The temperature is 950 °C–1000 °C, the compression rate is 0.1–0.2, and the pass is 7–9, which are the suitable parameter selection ranges for single-open MDF. The temperature is 850 °C–1000 °C, the compression rate is 0.1–0.2, and the pass is 6–9, which are the suitable parameter selection ranges for the double-open MDF.


Introduction
The GH4169 superalloy not only has good high-temperature oxidation and corrosion resistance, but also high-temperature strength and good fatigue resistance. They are used frequently in manufacturing several components with complex shapes and structures in applications with high temperature, such as working blades and turbine disks of aero-engines [1][2][3]. Special application environments such as aerospace determine that superalloy parts should have excellent mechanical properties and service properties. Therefore, special processing methods are needed to effectively refine grains and improve the mechanical properties of materials [4][5][6]. The multidirectional forging process (MDF) can refine the grain of the material through the large plastic deformation method, thereby improving the mechanical properties of the material [7,8]. However, the unreasonable design of the forging process frequently leads to defects such as uneven structure and properties, even cracking and so on. Therefore, the use of finite element simulation technology to quantitatively describe the 2 of 11 evolution of the material microstructure in the multidirectional forging process is one of the most commonly used process design methods in the field of multidirectional forging [9][10][11].
Considerable MDF studies have focused on closed MDF (CMDF) or free MDF (FMDF) [12][13][14]. Mikhail et al. [15] studied the effect of isothermal multidirectional forging on the structure evolution of conventional Al-Mg-based alloys in the strain range of 1.5-6.0 and the temperature range of 200 • C-500 • C. Xing et al. [16] studied the grain refinement of magnesium alloy AZ31 at a multidirectional forging temperature of 623-423 K, and studied the law of temperature and forging passes on the grain refinement of the material. Konovalov et al. [17] solved the problem of carbon distribution in titanium carbide and liquid titanium-carbide solution through the establishment of mathematical models. However, there are few reports about the microstructure evolution simulation of multidirectional forging. Zafari et al. [18] proposed a single-open multidirectional forging process and conducted research on the simulation of the microstructure evolution of titanium alloy in multidirectional forging. On this basis, the double-open MDF is proposed, and MDF is divided into CMDF, single-open MDF, and double-open MDF. The constitutive equation and dynamic recrystallization model of GH4169 superalloy thermal deformation are established in Deform-3D software, and the software is redeveloped to predict the recrystallization volume during the thermal deformation of materials. Recrystallization volume fraction (RVF) and recrystallized grain size (RGS) are used to simulate the grain size, recrystallization volume fraction, and strain of three MDF processes. The results are compared and analyzed to guide the design of MDF.

Multidirectional Forging Process
Multidirectional forging is a plastic processing method in which the forging is compressed from different directions to obtain a fine-grained structure by continuously changing the direction of the external load axis. In comparison with other plastic processing methods, the mechanical properties of forgings obtained by MDF exhibit isotropy, and their deformation degree is relatively uniform in all directions. In this study, MDF is divided into CMDF, single-open MDF, and double-open MDF. Figure 1 shows the principle and process of MDF.
Metals 2020, 10, x FOR PEER REVIEW 2 of 11 describe the evolution of the material microstructure in the multidirectional forging process is one of the most commonly used process design methods in the field of multidirectional forging [9][10][11].
Considerable MDF studies have focused on closed MDF (CMDF) or free MDF (FMDF) [12][13][14]. Mikhail et al. [15] studied the effect of isothermal multidirectional forging on the structure evolution of conventional Al-Mg-based alloys in the strain range of 1.5-6.0 and the temperature range of 200 °C -500 °C. Xing et al. [16] studied the grain refinement of magnesium alloy AZ31 at a multidirectional forging temperature of 623-423 K, and studied the law of temperature and forging passes on the grain refinement of the material. Konovalov et al. [17] solved the problem of carbon distribution in titanium carbide and liquid titanium-carbide solution through the establishment of mathematical models. However, there are few reports about the microstructure evolution simulation of multidirectional forging. Zafari et al. [18] proposed a single-open multidirectional forging process and conducted research on the simulation of the microstructure evolution of titanium alloy in multidirectional forging. On this basis, the double-open MDF is proposed, and MDF is divided into CMDF, single-open MDF, and double-open MDF. The constitutive equation and dynamic recrystallization model of GH4169 superalloy thermal deformation are established in Deform-3D software, and the software is redeveloped to predict the recrystallization volume during the thermal deformation of materials. Recrystallization volume fraction (RVF) and recrystallized grain size (RGS) are used to simulate the grain size, recrystallization volume fraction, and strain of three MDF processes. The results are compared and analyzed to guide the design of MDF.

Multidirectional Forging Process
Multidirectional forging is a plastic processing method in which the forging is compressed from different directions to obtain a fine-grained structure by continuously changing the direction of the external load axis. In comparison with other plastic processing methods, the mechanical properties of forgings obtained by MDF exhibit isotropy, and their deformation degree is relatively uniform in all directions. In this study, MDF is divided into CMDF, single-open MDF, and double-open MDF. Figure 1 shows the principle and process of MDF.

Constitutive Equation and Deform-3D Software Secondary Development
The material used in this study is subjected to a compression test by using the Gleeble-3800 Thermal Simulation Tester (DSI, New York City, USA). The stress-strain curves of the material during thermal compression have been reported in the previous paper [19]. The corresponding parameter values of the material are obtained on the basis of the stress-strain curve. According to the dynamic recrystallization model of the material, the relationship between the mechanical properties and the microstructure of the material is established to quantitatively describe the hot working process of the material [20][21][22][23].
The constitutive equation for the alloy is expressed as follows: .
In the research of the recrystallization model, the Avrami equation is generally used to describe the degree of recrystallization quantitatively. The dynamic recrystallization model of GH4169 alloy obtained is: where X is the dynamic RVF of the material, ε is a dependent variable, ε c is the critical strain, and ε 0.5 is a dependent variable when the material recrystallization amount reaches 50%. The Deform-3D software (version 11, SFTC, Columbus, OH, USA) secondary development programming interface is based on the Windows platform, and several user-defined subroutines are used in the secondary development. The constitutive equation of the dynamic recrystallization of materials is embedded in Deform simulation software by establishing an external file, and the Deform simulation software is redeveloped. Figure 2 presents the finite element model. In the forging process, due to the constraint of the die, the blank is compressed in the height direction, stretched in the length direction, and does not deform in the width direction. For the CMDF, when the amount of deformation reaches a certain level, the length of the left and right directions of the forging is no longer increased, due to the restraining action of the lower die, and the groove of the lower die is gradually filled. Meanwhile, the single-open MDF is full on one side, whereas the other side maintains a free surface. The sides of the double-open MDF maintain a free surface. After each forging pass, the forging is rotated 90 • around the axis, and then the last forging process is repeated. The simulation utilizes a relative net partitioning method, which uses a tetrahedral mesh with 20,000 cells. Table 1 shows the setting of relevant simulation parameters of the forging process. restraining action of the lower die, and the groove of the lower die is gradually filled. Meanwhile, the single-open MDF is full on one side, whereas the other side maintains a free surface. The sides of the double-open MDF maintain a free surface. After each forging pass, the forging is rotated 90° around the axis, and then the last forging process is repeated. The simulation utilizes a relative net partitioning method, which uses a tetrahedral mesh with 20,000 cells. Table 1 shows the setting of relevant simulation parameters of the forging process.

Variation of Internal Grain Size of Forgings Under Different Conditions
Under different multidirectional forging process conditions, the constraint of the die on forging is different, which will lead to different grain evolutions in forging. Figures 3 and 4 show the grain distribution and grain evolution of the CMDF under different passes.

Variation of Internal Grain Size of Forgings Under Different Conditions
Under different multidirectional forging process conditions, the constraint of the die on forging is different, which will lead to different grain evolutions in forging. Figures 3 and 4 show the grain distribution and grain evolution of the CMDF under different passes.
As shown in Figures 3 and 4, the recrystallization degree of forging in CMDF is gradually increased as the forging pass increases. The dynamic recrystallization zone begins to occur in the center of the forging. The dynamic recrystallization of the forging starts from the central region along the direction of 45° to all directions. Thus, the recrystallization degree of forging is gradually increased.  As shown in Figures 3 and 4, the recrystallization degree of forging in CMDF is gradually increased as the forging pass increases. The dynamic recrystallization zone begins to occur in the center of the forging. The dynamic recrystallization of the forging starts from the central region along the direction of 45 • to all directions. Thus, the recrystallization degree of forging is gradually increased.  Figures 5 and 6, in the single-open multidirectional forging process, the area where the forging first undergoes dynamic recrystallization is also the central area. As the forging progresses, the forging begins to undergo significant dynamic recrystallization from the central area along the 45° direction to only two directions with greater deformation, and then dynamic recrystallization occurs in the other two directions. Thus, the recrystallization degree of forging is gradually increased.     Figures 5 and 6, in the single-open multidirectional forging process, the area where the forging first undergoes dynamic recrystallization is also the central area.
As the forging progresses, the forging begins to undergo significant dynamic recrystallization from the central area along the 45 • direction to only two directions with greater deformation, and then dynamic recrystallization occurs in the other two directions. Thus, the recrystallization degree of forging is gradually increased.  Figures 5 and 6, in the single-open multidirectional forging process, the area where the forging first undergoes dynamic recrystallization is also the central area. As the forging progresses, the forging begins to undergo significant dynamic recrystallization from the central area along the 45° direction to only two directions with greater deformation, and then dynamic recrystallization occurs in the other two directions. Thus, the recrystallization degree of forging is gradually increased.    Figures 5 and 6, in the single-open multidirectional forging process, the area where the forging first undergoes dynamic recrystallization is also the central area. As the forging progresses, the forging begins to undergo significant dynamic recrystallization from the central area along the 45° direction to only two directions with greater deformation, and then dynamic recrystallization occurs in the other two directions. Thus, the recrystallization degree of forging is gradually increased.    Consequently, the cumulative strain of the sample is large, and the dynamic recrystallization degree of the forging is larger than those of the other two processes. Forgings also start dynamic recrystallization from the central region, and then as the forging passes increase, dynamic recrystallization occurs from the central region along the 45° direction to the surrounding region. More forging passes also indicate a higher recrystallization degree.  During hot deformation, the grain size inside the forging constantly changes with the increase in forging time and the change in forging pass and temperature. The grain size is also different under different conditions, which will have a great impact on the final performance of the forging. The minimum grain size inside the forgings obtained under different conditions is drawn in MATLAB software to obtain the grain size distribution of forgings under different forging processes, as shown in Figure 9.  Figures 7 and 8, the transverse dimension of the sample after each pass for the double-open MDF is larger than those of the closed and single-open MDFs because the two faces of the forging have no constraints. Consequently, the cumulative strain of the sample is large, and the dynamic recrystallization degree of the forging is larger than those of the other two processes. Forgings also start dynamic recrystallization from the central region, and then as the forging passes increase, dynamic recrystallization occurs from the central region along the 45° direction to the surrounding region. More forging passes also indicate a higher recrystallization degree.  The double-open MDF also has the highest dynamic recrystallization degree, followed by the closed MDF, and the single-open MDF has the lowest recrystallization degree. This is because the single-open MDF process always has one face constrained by the die.
During hot deformation, the grain size inside the forging constantly changes with the increase in forging time and the change in forging pass and temperature. The grain size is also different under different conditions, which will have a great impact on the final performance of the forging. The minimum grain size inside the forgings obtained under different conditions is drawn in MATLAB software to obtain the grain size distribution of forgings under different forging processes, as shown in Figure 9. Among the three different forging processes, the double-open MDF has the smallest grain size.
The double-open MDF also has the highest dynamic recrystallization degree, followed by the closed MDF, and the single-open MDF has the lowest recrystallization degree. This is because the single-open MDF process always has one face constrained by the die.
During hot deformation, the grain size inside the forging constantly changes with the increase in forging time and the change in forging pass and temperature. The grain size is also different under different conditions, which will have a great impact on the final performance of the forging. The minimum grain size inside the forgings obtained under different conditions is drawn in MATLAB software to obtain the grain size distribution of forgings under different forging processes, as shown in Figure 9.  Figure 9a shows the grain size distribution under different conditions for CMDF. When the temperature is constant and with the same compression rate, the recrystallized grain size of the forging decreases significantly as the forging passes increase, but it will stabilize after a certain pass amount of deformation. Under the same pass, the grain size decreases slightly with the increase in the reduction; overall, the more forging passes and the larger the compression rate, the smaller the recrystallized grain size. The RGS is relatively small when the temperature is 850 °C-1000 °C, the compression rate is 0.15-0.2, and the pass is 5-9 for the CMDF, which are the suitable parameter selection ranges for this process. Figure 9b shows the grain size distribution under different conditions for the single-open MDF. When the forging pass is constant, under the same compression rate, the recrystallized grain size of the forging shows a decreasing trend as the temperature increases. At the same temperature, the recrystallized grains of the forgings show a small decrease with the increase in the compression rate. On the whole, the higher the temperature and the greater the compression rate, the smaller the recrystallized grain size. The RGS of the forgings obtained is relatively small when the temperature is 950 °C-1000 °C, the compression rate is 0.1-0.2, and the pass is 7-9, which are the suitable parameter selection ranges for single-open MDF. Figure 9c shows the grain size distribution under different conditions for the double-open MDF. When the compression rate is constant, under the same pass, the recrystallized grain size of the forging shows a decreasing trend as the temperature rises. At the same temperature, the recrystallized grain size of the forging decreases significantly with the increase in forging passes, but it will also stabilize after a certain amount of pass deformations. On the whole, the higher the temperature and the more forging passes, the smaller the recrystallized grain size. The obtained forgings have relatively small grain size when the temperature is 850 °C-1000 °C, the compression rate is 0.1-0.2, and the pass is 6-9, which are the suitable parameter selection ranges for the double-open MDF.

Microstructure Changes of Forgings Under Different Conditions
The graphs of recrystallization volume fraction (RVF) and recrystallized grain size (RGS) of forgings at different temperatures under different processes are shown in Figures 10 and 11, respectively.  Figure 9a shows the grain size distribution under different conditions for CMDF. When the temperature is constant and with the same compression rate, the recrystallized grain size of the forging decreases significantly as the forging passes increase, but it will stabilize after a certain pass amount of deformation. Under the same pass, the grain size decreases slightly with the increase in the reduction; overall, the more forging passes and the larger the compression rate, the smaller the recrystallized grain size. The RGS is relatively small when the temperature is 850 • C-1000 • C, the compression rate is 0.15-0.2, and the pass is 5-9 for the CMDF, which are the suitable parameter selection ranges for this process. Figure 9b shows the grain size distribution under different conditions for the single-open MDF. When the forging pass is constant, under the same compression rate, the recrystallized grain size of the forging shows a decreasing trend as the temperature increases. At the same temperature, the recrystallized grains of the forgings show a small decrease with the increase in the compression rate. On the whole, the higher the temperature and the greater the compression rate, the smaller the recrystallized grain size. The RGS of the forgings obtained is relatively small when the temperature is 950 • C-1000 • C, the compression rate is 0.1-0.2, and the pass is 7-9, which are the suitable parameter selection ranges for single-open MDF. Figure 9c shows the grain size distribution under different conditions for the double-open MDF. When the compression rate is constant, under the same pass, the recrystallized grain size of the forging shows a decreasing trend as the temperature rises. At the same temperature, the recrystallized grain size of the forging decreases significantly with the increase in forging passes, but it will also stabilize after a certain amount of pass deformations. On the whole, the higher the temperature and the more forging passes, the smaller the recrystallized grain size. The obtained forgings have relatively small grain size when the temperature is 850 • C-1000 • C, the compression rate is 0.1-0.2, and the pass is 6-9, which are the suitable parameter selection ranges for the double-open MDF.

Microstructure Changes of Forgings Under Different Conditions
The graphs of recrystallization volume fraction (RVF) and recrystallized grain size (RGS) of forgings at different temperatures under different processes are shown in Figures 10 and 11, respectively.
As shown in Figures 10 and 11, the RVF of the forging gradually increases with temperature, and the RGS gradually decreases with temperature for the three forging conditions. When the temperature is 1000 • C, for the CMDF, the RVF is up to 99.5%, and the RGS is approximately 8.1 µm; for the single-open MDF, the RVF is up to 93%, and the RGS is approximately 13 µm; for the double-open MDF, the RVF and RGS are approximately 99.6% and 2.9 µm, respectively. It can be seen that when the forging temperature is the same, the double-open MDF has the largest cumulative strain, so the degree of recrystallization is the highest, and the RGS is the smallest; the CMDF is followed by the degree of temperature and the more forging passes, the smaller the recrystallized grain size. The obtained forgings have relatively small grain size when the temperature is 850 °C-1000 °C, the compression rate is 0.1-0.2, and the pass is 6-9, which are the suitable parameter selection ranges for the double-open MDF.

Microstructure Changes of Forgings Under Different Conditions
The graphs of recrystallization volume fraction (RVF) and recrystallized grain size (RGS) of forgings at different temperatures under different processes are shown in Figures 10 and 11, respectively.  As shown in Figures 10 and 11, the RVF of the forging gradually increases with temperature, and the RGS gradually decreases with temperature for the three forging conditions. When the temperature is 1000 °C, for the CMDF, the RVF is up to 99.5%, and the RGS is approximately 8.1 μm; for the single-open MDF, the RVF is up to 93%, and the RGS is approximately 13 μm; for the double-open MDF, the RVF and RGS are approximately 99.6% and 2.9 μm, respectively. It can be seen that when the forging temperature is the same, the double-open MDF has the largest cumulative strain, so the degree of recrystallization is the highest, and the RGS is the smallest; the CMDF is followed by the degree of recrystallization and the RGS; the single-open MDF has the lowest recrystallization degree and the largest RGS.

Inhomogeneity of Forgings
The internal strain and microstructure evolution of the forgings are irregular during forging. Thus, the deformation distribution of the forgings must be uniform in practical applications to obtain an ultrafine crystal material with a relatively uniform microstructure, improve the overall mechanical properties of the forgings, and make the forgings isotropic. This section calculates and compares the strain and grain size uniformity of each section of forgings under different passes and temperatures for the three multidirectional forging processes.
The nonuniformity parameters of forging deformation are: where max is the maximum equivalent plastic strain of the section, min is the minimum equivalent plastic strain of the section, and avc is the average equivalent plastic strain of the section. Nonuniformity parameter C is a required condition to obtain a uniform deformation distribution of the forging. Thus, ensuring a small nonuniformity parameter C of section deformation is the primary condition to obtain a fine-grained material. The nonuniformity parameter of recrystallization grain size can be calculated as: where max is the maximum RGS of the section, min is the minimum RGS of the section, and avc is the average equivalent plastic strain of the section. The nonuniformity parameter C of recrystallization grain size can be used to describe the grain uniformity of the forging.

Inhomogeneity of Forgings
The internal strain and microstructure evolution of the forgings are irregular during forging. Thus, the deformation distribution of the forgings must be uniform in practical applications to obtain an ultrafine crystal material with a relatively uniform microstructure, improve the overall mechanical properties of the forgings, and make the forgings isotropic. This section calculates and compares the strain and grain size uniformity of each section of forgings under different passes and temperatures for the three multidirectional forging processes.
The nonuniformity parameters of forging deformation are: where ε max is the maximum equivalent plastic strain of the section, ε min is the minimum equivalent plastic strain of the section, and ε avc is the average equivalent plastic strain of the section. Nonuniformity parameter C is a required condition to obtain a uniform deformation distribution of the forging. Thus, ensuring a small nonuniformity parameter C of section deformation is the primary condition to obtain a fine-grained material. The nonuniformity parameter of recrystallization grain size can be calculated as: where D max is the maximum RGS of the section, D min is the minimum RGS of the section, and D avc is the average equivalent plastic strain of the section. The nonuniformity parameter C of recrystallization grain size can be used to describe the grain uniformity of the forging.
The deformation nonuniformity parameter C b , the nonuniformity parameter C d of recrystallization grain size, and the recrystallization volume fraction C t under different conditions can be calculated on the basis of Equations (3) and (4), as shown in Table 2. As shown in Table 2, in the three different forging processes, although the single-open MDF has the lowest recrystallization refinement, the obtained forgings have the lowest deformation unevenness and recrystallized grain size unevenness, so the forging performance obtained by this process is more uniform. Different temperatures have an effect on the properties of the forgings under each process. In the CMDF and single-open MDF, the deformation of the forging is relatively small, so the higher the temperature, the better the uniformity of the deformation and recrystallization grain size of the forging. In the double-open MDF, due to the large deformation of the forging, the forging performance is the most uniform at 900 • C. Moreover, the RVF of the forgings exceeds 97% at 9 passes, and the RGS is at least 4 µm.
Forgings with various properties can be obtained through different forging processes under different temperature conditions. Under various temperature conditions, the single-open MDF has the best strain and recrystallization grain size nonuniformity, followed by the CMDF, and the double-open MDF obtains the worst uniformity of forgings.
Among the three forging processes, with the increase in temperature (800 • C-1000

Conclusions
(1) The recrystallization degree of CMDF reaches 99.5% at 1000 • C and 9 passes, and the average RGS is the smallest, which is approximately 8.1 µm. The RVF of the single-open MDF is approximately 93%, and its average RGS is approximately 13 µm. The double-open MDF basically undergoes dynamic recrystallization, where its maximum recrystallization degree is approximately 99.6%, and its RGS is the smallest, which is approximately 2.9 µm.
(2) Among the three processes, the double-open MDF has the highest recrystallization degree under the same reduction, and its maximum value is approximately 99.75%; the closed MDF has the second highest recrystallization degree, and the single-open MDF has the lowest recrystallization degree. The double-open MDF can obtain a fine grain size of forgings at 9 passes and 1000 • C, and the single-open MDF can easily obtain uniform forgings.
(3) The RGS is relatively small when the temperature is 850 • C-1000 • C, the compression rate is 0.15-0.2, and the pass is 5-9, which are the suitable parameter selection ranges for the CMDF. The obtained RGS is relatively small when the temperature is 950 • C-1000 • C, the compression rate is 0.1-0.2, and the pass is 7-9, which are the suitable parameter selection ranges for single-open MDF. The obtained forgings have a relatively small grain size when the temperature is 850 • C-1000 • C, the compression rate is 0.1-0.2, and the pass is 6-9, which are the suitable parameter selection ranges for the double-open MDF.