Introduction

Compared to the conventional die forging technology, isothermal die forging is widely used in the forming of alloy components because it can effectively decrease the deformation resistance of materials and improve the homogeneity of metal flow (Ref 1). Therefore, isothermal die forging technology is suitable for manufacturing the components with complicated geometry and large deformation resistance. Over the last decades, isothermal die forging technology has been applied in the hot forming of different alloy components, such as several aluminum alloy forgings (Ref 2, 3), turbine blade (Ref 4), compressor disks (Ref 5), etc. During isothermal die forging, one most important task is to control the billet temperature within the optimal forming processing range. However, the inevitable heat loss and the generation of deformation heat greatly affect the billet temperature. So, the accurate methods to control the temperatures of billet and die are essential for isothermal die forging technology (Ref 6).

Due to its excellent mechanical, physical, and anticorrosion properties in extreme service environment, Ni-based superalloy is widely applied in gas turbine and aerospace engines (Ref 7-10). Some researches have been carried out to investigate the effects of processing parameters on hot deformation behavior and microstructural evolution of some typical Ni-based superalloys (Ref 11-17). The previous researches show that the forming temperature range of Ni-based superalloy is extremely narrow. However, there is little research on the accurate method to control billet or die temperature during isothermal die forging of Ni-based superalloy components. Superalloy casing, a typical thin-walled ring or sleeve, is one of the most important mechanical components used in aero engines. Due to the extreme service environment, the excellent mechanical properties must be guaranteed. However, due to its complicated geometry (thin-wall, multi bulges, etc.) and narrow forming temperature range (Ref 11, 12), the conventional forming processes cannot accomplish the high-quality superalloy casing. Therefore, it is necessary to propose an appropriate and optimal processing to satisfy the forming requirements of superalloy casing.

In this study, a new method is proposed to accurately control the billet temperature of complex superalloy casing by the finite element simulation and response surface methodology (RSM). The proposed method is accomplished by the following two steps. Firstly, the thermal compensation process is designed and optimized to overcome the inevitable heat loss of dies during hot forging. i.e., the layout and opening time of heaters assembled on die sleeves are optimized. Then, the effects of forging speed (the pressing velocity of hydraulic machine) and its changing time on the maximum billet temperature are discussed. Furthermore, the optimized forging speed and its changing time are obtained by RSM. Finally, the volume fraction of dynamic recrystallization and grain size in final forging under conventional and optimized forging processes are compared to verify the proposed new method.

Structures of Isothermal Die Forging Assembly and Optimization Method

Structures of Isothermal Die Forging Assembly

In Fig. 1(a), the numbers 1-6 represent the top die sleeve, heater, top die, bottom die, bottom die sleeve, and billet, respectively. Figure 1(b) illustrates the geometry of final forging. As shown in Fig. 1(c), heaters are assembled on the surfaces A, B, and C of bottom die sleeve. Meanwhile, heaters in top die sleeve are divided into two parts marked by D and E in Fig. 1(e). Figure 2 shows the two-dimension geometry of bottom and top dies. In Fig. 2(a), the marked points P1, P2, and P3 are three tracked points on the inner surface of bottom die. l represents the distance between the adjacent heaters. Figure 2(b) shows the two-dimension geometry of top die. The marked points P4, P5, and P6 are three tracked points on the inner surface of top die.

Fig. 1
figure 1

Structures and geometries of (a) the section view of die forging assembly; (b) final forging; (c) bottom die sleeve (half); (d) bottom die (half); (e) top die sleeve (half); (f) top die (half)

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Fig. 2
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Two-dimension geometry of (a) bottom die and bottom die sleeve (half); (b) top die and top die sleeve (half)

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Optimization Methods and Finite Element Model

In order to optimize the layouts of heaters and processing parameters for isothermal die forging technology, the finite element simulation and response surface methodology (RSM) are used. Moreover, in order to simplify the optimization of thermal compensation process, the heaters in surfaces A and B, parts D and E are firstly optimized by two-dimension simulation. Then, the heaters in surface C are optimized by three-dimension simulation. Based on the optimized results of thermal compensation process, the forging speed (the pressing velocity of hydraulic machine) and its changing time are optimized at last. The flow chart of optimization process is shown in Fig. 3. Here, the processes of heat transfer and die forging are simulated by DEFORM software. From Fig. 1(a), it can be found that the dies are plane symmetrical. So, half dies are used in finite element simulation. The finite element model is based on the rigid-viscoplastic variation principle. Usually, the governing equations for the solution of the mechanics of rigid-viscoplastic deformation do not consider the volume force and neglect the elastic deformation of materials. Also, the material conforms to the von Mises yield criterion. For the established finite element model, the element numbers of billet, die sleeves, top, and bottom dies are set as 200,000, 50,000, 100,000, and 250,000, respectively. The tetrahedron element is used. The dies and die sleeves are set as rigid body, while the billet is set as plastic body. The material of dies and die sleeves is K403 alloy. Other parameters can be found in Table 1. The material of the studied casing is a commercial nickel-based superalloy (Inconel 718), and its chemical composition (wt.%) is 52.82Ni-18.96Cr-5.23Nb-3.01Mo-1.00Ti-0.59Al-0.01Co-0.03C-(bal.)Fe. The initial microstructure is composed of fine equiaxed grains with a mean size of 75 μm. The kinetics models for dynamic recrystallization can be expressed as (Ref 12, 18)

Fig. 3
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The flow chart of the optimization process

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Table 1 Parameters used in finite element simulation
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$$ \left\{ \begin{aligned} \, X_{\text{drx}} & = 1 - \exp \left[ { - 0.693\left( {\frac{{\varepsilon - \varepsilon_{\text{c}} }}{{\varepsilon_{0.5} - \varepsilon_{\text{c}} }}} \right)^{{n_{\text{d}} }} } \right] \\ Z & = \dot{\varepsilon }\exp \left( {4.74 \times 10^{5} /{\text{RT}}} \right) \\ \varepsilon_{0.5} & = \left\{ \begin{aligned} = 0.00037Z^{0.17729} \left( {T = 1193 - 1233\,{\text{K}}} \right) \hfill \\ 0.00232Z^{0.13167} \left( {T = 1233 - 1313\,{\text{K}}} \right) \hfill \\ \end{aligned} \right. \\ n_{\text{d}} & = \left\{ \begin{aligned} 1.84\left( {T = 1193 - 1233\,{\text{K}}} \right) \hfill \\ 1.73\left( {T = 1233 - 1313\,{\text{K}}} \right) \hfill \\ \end{aligned} \right. \\ \varepsilon_{\text{c}} & = 0.00150Z^{0.11741} \\ \end{aligned} \right. $$
(1)

In addition, RSM is used to optimize the thermal compensation process and forging speed. RSM is a collection of statistical and mathematical techniques, which is employed to analyze the engineering problems (Ref 19, 20). The central composites design (CCD) and Box-Behnken design (BBD) are the common design methods of RSM. RSM can be used to evaluate the relationships between the design variables and response variables. The relationship between the design variables (\( x_{1} ,x_{2} , \ldots ,x_{m} \)) and the response variable Y can be written as follows:

$$ Y = f\left( {x_{1} ,x_{2} , \ldots ,x_{m} } \right) + \varepsilon, $$
(2)

where \( f \) represents the response function, and \( \varepsilon \) is the statistical error.

For the RSM, the most common model is the polynomial based on Taylor series expansion (Ref 21, 22). The quadratic polynomial model in Eq 3 is used in this study.

$$ y = \beta_{0} + \sum\limits_{i = 1}^{m} {\beta_{i} } x_{i} + \sum\limits_{i = 1}^{m} {\beta_{ii} } x_{i}^{2} + \sum\limits_{i < j}^{m} {\beta_{ij} } x_{i} x_{j} + \varepsilon, $$
(3)

where \( \beta_{0} \) is the model constant, \( \beta_{i} \) and \( \beta_{ii} \) represent the linear coefficient and the quadratic coefficient, respectively, and\( \beta_{ij} \) represents the interaction coefficient.

A series of the response variables can be recorded in the vector Y. Then, Eq 3 can be written in the matrix form as follows:

$$ Y = X\beta + \varepsilon $$
(4)

where \( X \) is a matrix of explanatory variables, \( \beta \) is a column vector containing the pending coefficient, and \( \varepsilon \) is the residual error vector.

In order to evaluate the regression coefficient vector \( \beta \), the least square method is used. It can be described as follows:

$$ \beta = \left( {X^{\text{T}} X} \right)^{ - 1} X^{\text{T}} Y, $$
(5)

where \( X^{\text{T}} \) is the transpose of the matrix \( X .\)

Results and Discussions

Optimization of Thermal Compensation Process

During isothermal die forging, the billet and dies are heated to the same temperature, and the billet is deformed under low strain rate. The forging process is simply shown in Fig. 4. Due to the inevitable heat loss of dies, the billet temperature easily decreases during isothermal die forging. So, the thermal compensation process is essential to control the billet temperature within the optimal forming processing range. For the thermal compensation process, the layout and opening time of heaters are the main variables, which greatly affect the billet temperature. So, the distance between adjacent heaters and opening time of heaters are selected as the design variables when the thermal compensation process is optimized. In Table 2, l represents the distance between adjacent heaters. \( t_{\text{a}} \) and \( t_{\text{b}} \) are the opening times of heaters in surfaces A and B, respectively. According to the design requirements of RSM, 17 groups of variables combinations are designed. Table 3 shows the tracked temperatures (named simulated temperatures in the following sections) for the designed combinations. Here, \( T_{1} \), \( T_{2} \) , and \( T_{3} \) represent the tracked temperatures at points P1, P2, and P3 during forging, respectively. Based on the simulated results shown in Table 3, the regression models (Eq 6-8) are established by response surface methodology. Figure 5 shows the comparisons between the predicted and simulated values of \( T_{1} \), \( T_{2} \) , and \( T_{3} \). It can be found that the predicted values are in good agreement with the simulated ones. The values of the correlation coefficient between the predicted and simulated values of \( T_{1} \), \( T_{2} \) , and \( T_{3} \) are 1.000, 0.9989, and 0.9991, respectively. It means that the established regression models can accurately predict the evolution of bottom die temperature during isothermal die forging.

Fig. 4
figure 4

Illustration of isothermal die forging process

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Table 2 Values of design variables
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Table 3 Simulated temperatures of bottom die for the designed combinations
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Fig. 5
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Comparisons between the predicted and simulated values of (a) \( T_{1} \); (b) \( T_{2} \); (c) \( T_{3} \)

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$$ \begin{aligned} T_{1} =\; & 1241.525 - 3.791 \times 10^{ - 3} l + 1.481\\ &\times 10^{ - 3} t_{\text{a}} - 1.619t_{\text{b}} - 1.756 \times 10^{ - 16} lt_{\text{a}} + 5.951\\ &\times 10^{ - 4} lt_{\text{b}} - 6.938 \times 10^{ - 20} t_{\text{a}} t_{\text{b}} - 5.683 \times 10^{ - 5} l^{2}\\ &- 1.234 \times 10^{ - 5} t_{\text{a}}^{2} - 1.004 \times 10^{ - 3} t_{\text{b}}^{2} \hfill \\ \end{aligned} $$
(6)
$$ \begin{aligned} T_{2} = 1168.143 + 1.506l + 0.057t_{\text{a}} + 0.139t_{\text{b}} - 1.789 \times 10^{ - 3} lt_{\text{a}} - 1.464 \times 10^{ - 3} lt_{\text{b}} - \hfill \\ \, 2.678 \times 10^{ - 4} t_{\text{a}} t_{\text{b}} - 7.709 \times 10^{ - 3} l^{2} - \, 3.148 \times 10^{ - 4} t_{\text{a}}^{2} - 2.276 \times 10^{ - 3} t_{\text{b}}^{2} \hfill \\ \end{aligned} $$
(7)
$$ \begin{aligned} T_{3} = 1246.024 + 0.072l - 0.325t_{\text{a}} - 0.044t_{\text{b}} - 1.404 \times 10^{ - 4} lt_{\text{a}} - 3.305 \times 10^{ - 6} lt_{\text{b}} + \hfill \\ \, 3.339 \times 10^{ - 5} t_{\text{a}} t_{\text{b}} - 4.991 \times 10^{ - 4} l^{2} + 2.164 \times 10^{ - 4} t_{\text{a}}^{2} + 1.033 \times 10^{ - 3} t_{\text{b}}^{2} \hfill \\ \end{aligned} $$
(8)

In order to keep the die temperature within the optimal forming processing range, the minimum temperatures of the three tracked points (P1, P2, and P3) should be higher than 1223 K. So, the optimization problem can be described as follows:

$$\left\{ \begin{array}{l} {\text{Variables: }}l,t_{\text{a}} ,t_{\text{b}}\\ {\text{Goals:}} \,\left\{ \begin{aligned} T_{1} \ge 1223\,{\text{K}} \hfill \\ T_{2} \ge 1223\,{\text{K}} \hfill \\ T_{3} \ge 1223\,{\text{K}} \hfill \\ \end{aligned} \right. \\ {\text{subject to:}} \hfill \\ \, \left\{ \begin{array}{l} 30{\text{ mm}} < l \le 90\,{\text{mm}} \hfill \\ 0 < t_{\text{a}} \le 120\,{\text{s}} \hfill \\ 0 < t_{\text{b}} \le 40\,{\text{s}} \hfill \\ \end{array} \right. \hfill \\ \end{array} \right.$$
(9)

The optimization problem expressed in Eq 9 is solved by the Expert-Design software, and the optimized results are shown in Table 4. Also, the optimized values of \( t_{\text{a}} \), \( t_{\text{b}} \) , and \( l \) are implemented into the finite element model to verify the reliability of the above-optimized results. The comparisons between the predicted (by Eq 6-8) and simulated values are shown in Table 5. Obviously, the errors between the predicted and simulated values are less than 1%, which means that the above optimization is accurate and effective. In Table 5, the error is evaluated by \( \frac{{\left| {X - Y} \right|}}{X} \times 100\% \), where X and Y are the simulated and predicted values, respectively.

Table 4 Optimized values of variables in surfaces A and B
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Table 5 Comparisons between the predicted and simulated values
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Similarly, the optimized layout and opening times of heaters in surface C, parts D and E are shown in Table 6. Here, h represents the distance between adjacent heaters in top die.\( t_{\text{d}} \) and \( t_{\text{e}} \) represent the opening times of heaters in parts D and E of top die, respectively.\( d \) and \( t_{\text{c}} \) represent the distance between adjacent heaters and opening time of heaters in surface C, respectively. Based on the above-optimized results, a new die forging process with thermal compensation is simulated, and the billet temperature is analyzed, as shown in Fig. 6 and 7. The physical meaning of y-axis, percent of nodes (%) in the finite element simulated results, indicates the percentage of nodes with the same temperature in the deformed forging. From Fig. 6 and 7, it can be found that the minimum billet temperature after the optimized die forging process is around 1223 K, which is within the optimal forming temperature range. But, the maximum billet temperature under the die forging process with thermal compensation is 1293 K, which is obviously out of the optimal forming temperature range. The reason is that a huge amount of heat generates when the billet undergoes large plastic deformation. So, the billet temperature rapidly rises, and exceeds the optimal forming processing range. During isothermal die forging, the deformation heat is mainly affected by the forging speed. From Fig. 6, it can be also found that the maximum temperature of billet exceeds 1263 K at 1876s. It means the forging speed should be decreased before 1876s to decrease the maximum billet temperature. So, in the following sections, the effects of forging speed and its changing time on the maximum billet temperature will be discussed.

Table 6 The optimized values of variables
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Fig. 6
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The maximum and minimum temperatures of billet in die forging with optimized thermal compensation

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Fig. 7
figure 7

The temperature distribution in final forging after die forging with optimized thermal compensation (Color figure online)

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Optimization of Forging Speed and Its Changing Time

Based on the above analysis, the forging speed and its changing time are selected as the design variables. In Table 7, \( v \) and \( t_{\text{ch}} \) represent the forging speed and its changing time, respectively. According to the design requirements of RSM, 13 groups of variables combinations are designed. Table 8 shows the maximum simulated temperatures (\( T_{ \hbox{max} } \)) of billet under the designed combinations. Then, the regression model (Eq 10) is established. Figure 8 shows the comparisons between the predicted and simulated \( T_{ \hbox{max} } \). It can be found that the predicted values are in good agreement with the simulated ones. Furthermore, the correlation coefficient (R) between the predicted and simulated \( T_{ \hbox{max} } \) is 0.9804, which means that the developed regression model can accurately predict the maximum temperature of billet.

Table 7 Values of design variables
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Table 8 Simulated temperatures of billet for designed combinations
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Fig. 8
figure 8

Comparisons between the predicted and simulated values of \( T_{ \hbox{max} } \)

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$$ T_{ \hbox{max} } = 5141.699 - 4.384t_{\text{ch}} + 5261.95v - 2.432vt_{\text{ch}} + 0.001t_{{_{\text{ch}} }}^{2} - 6748.275v^{2} $$
(10)

Considering the optimal forming temperature range (1223-1263 K) (Ref 10, 11), the maximum temperature of billet should be less than 1263 K. Moreover, the forging speed must be higher than 0.02 mm/s. So, the optimization problem can be described as follows:

$$ \left\{\begin{array}{l} {\text{Varibles:}} v,t \hfill \\ {\text{Goal:}} \, T_{ \hbox{max} } \le 1263{\text{ K}} \hfill \\ {\text{subject to:}} \hfill \\ \, \left\{ \begin{array}{l} 0.02{\text{ mm/s}} \le v \le 0.05{\text{ mm/s}} \hfill \\ 1736{\text{ s}} \le t_{\text{ch}} \le 1952{\text{ s}} \hfill \\ \end{array} \right. \hfill \\ \end{array} \right. $$
(11)

Similarly, the optimization problem expressed in Eq 11 is solved by the Expert-Design software and the optimized values of \( t_{\text{ch}} \) and \( v \) can be obtained as 1786s and 0.02 mm/s, respectively. Also, the optimized values of \( t_{\text{ch}} \) and \( v \) are implemented into the finite element model to verify the accuracy of the optimized results. It can be evaluated that the error between the predicted and simulated values is only 0.11%, which means that the above-optimized results are accurate and effective.

Combined with the optimized thermal compensation process and forging speed, a new isothermal die forging process is simulated, and the billet temperature is analyzed. As shown in Fig. 9 and 10, it can be found that the billet temperature is totally within the optimal forming processing range for the optimized die forging process. Also, the difference between the minimum and maximum temperatures in final forging is only 13 K, and the distribution of temperature in final forging is desired.

Fig. 9
figure 9

The maximum and minimum temperatures of billet in die forging with optimized thermal compensation and forging speed

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Fig. 10
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The temperature distribution in final forging after die forging process with optimized thermal compensation and forging speed (Color figure online)

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Comparisons Between Conventional and Optimized Die Forging Processes

Figure 11 shows the evolutions of maximum and minimum billet temperatures under the conventional and optimized die forging processes. From Fig. 11, it can be found that the minimum billet temperature decreases from 1253 to 1046 K during conventional die forging. However, in the die forging process with the optimized thermal compensation, the maximum billet temperature increases from 1253 to 1293 K. Obviously, the billet temperatures of the above two die forging processes are out of the optimal forming temperature range. But, for the die forging process with optimized thermal compensation and forging speed, the maximum and minimum billet temperatures are 1223 and 1263.62 K, respectively, which are actually within the optimal forming temperature range.

Fig. 11
figure 11

The maximum and minimum temperatures of billet under different die forging processes. (cases 1-3 represent the conventional die forging process, die forging process with optimized thermal compensation, and die forging process with optimized thermal compensation and forging speed, respectively) (Color figure online)

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Figure 12 shows the temperature ranges (from the minimum to the maximum) and the standard deviations (S.D.) of temperature distribution under three die forging processes. From Fig. 12, it can be found that the temperature ranges in final forging after three die forging processes are significantly different. For the conventional die forging process (case 1), the temperature ranges from 1046 to 1217 K, which is seriously out of the optimal forming temperature range. For the die forging process with optimized thermal compensation (case 2), the temperature range is 1257-1293 K, which exceeds the optimal forming temperature range. However, for the die forging process with optimized thermal compensation and forging speed (case 3), the minimum and maximum temperatures of the final forging are 1249 and 1262 K, respectively, which are totally within the optimal forming temperature range. Additionally, the standard deviations of temperature distribution in final forging are 34.7, 7.81, and 4.7 K, respectively, for cases 1, 2, and 3. Obviously, the temperature distribution in final forging under case 3 is the most homogeneous.

Fig. 12
figure 12

Comparisons among three die forging processes: (a) temperature range; (b) standard deviation of temperature distribution. (The meanings of cases 1-3 are the same to those in Fig. 11)

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In hot forming processes, metals or alloys generally undergo the large plastic deformation (Ref 23). The hot deformation mechanisms such as work hardening (WH), dynamic recovery (DRV), and dynamic recrystallization (DRX) often occur (Ref 24, 25), which lead to the complex microstructural evolutions (Ref 26, 27). Generally, the dynamic recrystallization is an effective mechanism to refine grains (Ref 25, 26, 28). Figure 13 shows the distributions of dynamic recrystallization volume fractions in final forging under three die forging processes. From Fig. 13, the volume fractions of dynamic recrystallization can be evaluated as 17.2, 95.1, and 97.2%, respectively, for cases 1, 2, and 3. Moreover, the standard deviations of dynamic recrystallization volume fraction distribution can be also computed as 20.1, 10.4, and 9.2%, respectively, for cases 1, 2, and 3. It is obvious that the dynamic recrystallization is complete under case 3. Figure 14 shows the distributions of grain size in final forging under three die forging processes. Similarly, the average grain sizes can be evaluated as 62.8, 19.6, and 11.5 µm, respectively. Moreover, the standard deviations of the grain size distribution are 15, 3.71, and 3.39 µm, respectively, for cases 1, 2, and 3. According to the above analysis, it can be concluded that the proposed new method can effectively control the billet temperature within the optimal forming temperature range, and guarantee the quality of final forging.

Fig. 13
figure 13

Distributions of dynamic recrystallization volume fraction under: (a) case 1; (b) case 2; (c) case 3. (The meanings of cases 1-3 are the same to those in Fig. 11) (Color figure online)

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Fig. 14
figure 14

Distributions of grain size under: (a) case 1; (b) case 2; (c) case 3. (The meanings of cases 1-3 are the same to those in Fig. 11) (Color figure online)

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Conclusion

In this study, a new method is proposed to accurately control the billet temperature based on finite element simulation and response surface methodology (RSM). The proposed control method includes the optimization of thermal compensation process, forging speed, and its changing time. For the optimized die forging process, the temperature range of final forging is 1249-1262 K, which is actually within the optimal forming temperature range. Meanwhile, the average grain size and dynamic recrystallization volume fraction are 11.5 µm and 97.2%, respectively, which satisfy the forming requirements of superalloy casing. Comparisons between the optimized and conventional die forging processes indicate that the proposed method can effectively control the billet temperature, and guarantee the quality of final forging.

Additionally, this work may be a good guide for designing the layout and opening time of heaters assembled on die sleeves, as well as the forging speed, for the typical thin-walled rings, which are the most important mechanical components used in aero engines and other important equipment. In other words, the proposed optimization methods/routines are useful for some practice engineering issues.