4.2.1. Recrystallization Critical Condition
The effect of dynamic recrystallization on the stressstrain curve of the material is shown in
Figure 10. If there is no dynamic recrystallization during the deformation process, the flow stress of the material increases slowly with the increase of strain, as shown in the black solid line in
Figure 10. When the strain reaches the critical strain of dynamic recrystallization, the flow stress decreases, which is manifested as the recrystallization softening effect as shown in the red line in
Figure 10. However, when the strain reaches a certain value (critical strain
ε_{r}), the material undergoes dynamic recrystallization. At this time, the flow stress shows a significant downward trend, as shown in the red solid line in
Figure 10, which is the flow softening phenomenon. Therefore, the determination of the critical condition of dynamic recrystallization, that is, the critical strain, is the key to the study of dynamic recrystallization flow softening.
The dynamic recrystallization critical strain of the PM superalloy obtained under the experimental conditions in this paper is shown in
Table 5. It can be seen from the data in
Table 5 that the critical strain of dynamic recrystallization of PM superalloy is not only related to the deformation temperature, but also related to the strain rate, which verifies the conclusion of Denguir [
21]. The critical strain decreases with the increase of temperature and decreases with the increase of strain rate.
Table 5.
Critical strain ε_{r} under different temperature and strain rate conditions.
Table 5.
Critical strain ε_{r} under different temperature and strain rate conditions.
strain rate (s^{1}) 
Temperature, T (℃) 
200 
400 
600 
700 
800 
6000 10000 12000 


0.4233 
0.3621 
0.3211 
0.4033 
0.3640 
0.3199 
0.3144 
0.2911 
0.3999 
0.3443 
0.3091 
0.3051 
0.2698 
Based on the data in
Table 3, the fitting surface of the critical strain is obtained by polynomial fitting (Equation (11)), is shown in
Figure 11.
The equation of calculating the critical strain of dynamic recrystallization of PM superalloy under experimental conditions in this paper is shown in Equation (16).
4.2.2. Identification of Constitutive Model’s Coefficients
 (1)
Linear regression method
The modified JC constitutive model (Equation (7)) represents strain hardening effect, strain rate strengthening effect, thermal softening effect and recrystallization softening effect from left to right. According to the linear regression parameter solving method, A, B, C, n, m and H_{i} are the parameters to be fitted, ${\dot{\epsilon}}_{0}$, T_{0} and T_{m} are 0.001s^{1}, 25°C and 1350°C, respectively.
The strain hardening coefficient can be obtained by processing the quasistatic compression test data at room temperature. The proposed constitutive model is simplified as shown in Equation (17). The quasistatic compression tests permitted to determine the yield stress, which is represented by the coefficient
A (
Figure 12a). The Equation (18) was obtained by taking the logarithm on both sides of the Equation (17). The solution of the coefficients
n and
B can be obtained by a slope and an intercept of fitting straight line (
Figure 12b). As shown in
Figure 12,
A =
σ_{0.2} = 773
MPa,
n = 0.667,
B = 1271MPa.
Figure 12.
Solution of coefficients A, B and n. a) Solution of coefficient A; b) Solution of coefficients B and n.
Figure 12.
Solution of coefficients A, B and n. a) Solution of coefficient A; b) Solution of coefficients B and n.
The strain rate sensitivity coefficient C, the thermal softening index m and the recrystallization softening correction coefficient H_{i} in the modified JC constitutive model were obtained from the result of SHPB experiments.
According to SHPB tests at room temperature for different strainrates, the constitutive equation simplified to Equation (19). The stressstrain curves of PM materials at different strain rates at room temperature in this paper are shown in
Figure 13.
The thermal softening coefficient
m is determined according to Equation (20). The data of the SHPB tests under high temperature conditions were used. Finally, the fitting relationship between
m and strainrate is obtained as shown in
Figure 14.
The coefficients h_{i} (i=0, 1, 2) related to the dynamic recrystallization can be obtained according to Equation (21).
In summary, the modified constitutive model obtained by the linear regression method is shown in
Table 6.
 (2)
Function iteration method
According to Function iteration method [
26,
27], the proposed model is shown in Equation (22). The iterative function method determines the coefficients of the prediction model by continuously iterating the function through the optimization algorithm.
The stress values corresponding to the room temperature
T_{0} and the reference strain rate
${\dot{\epsilon}}_{0}$ conditions are selected as the initial values. The quasistatic compression test data at room temperature are selected for polynomial fitting to obtain the results of
f (
ε), which are shown in
Figure 15a. Then, according to the relationship between measured stress in SHPB experiment at room temperature and
f (
ε), the relationship between stress and strain rate is obtained by iteration, which is
f (
$\dot{\epsilon}$), which are shown in
Figure 15b. Finally, the results of
f (
T) and
f (
H_{i}) can be obtained according to the high temperature SHPB experimental data by iteration process, as shown in
Figure 15c,d, respectively.
In the iterative process, the error R^{2} is used to judge the accuracy of the results. When the result meet Equation (23), the result is considered to be accepted.
Where, ${R}_{k}^{2}$ is the error of determination of the ith iterated function, ${R}_{k1}^{2}$ is the error of determination of the (i1)th iterated function.
Figure 15.
Function iteration method model solution results. a) Optimization results of f (ε); b) Optimization results of f ($\dot{\epsilon}$); c) Optimization results of f (T); d) Recrystallization fitting f (H_{i}).
Figure 15.
Function iteration method model solution results. a) Optimization results of f (ε); b) Optimization results of f ($\dot{\epsilon}$); c) Optimization results of f (T); d) Recrystallization fitting f (H_{i}).
Finally, the constitutive model constructed by the function iteration method is obtained, which is shown in Equation (24).
 (3)
Comparison of different methods
In order to optimize the solution method of the parameters, the stressstrain curves obtained by the constitutive equations solving by the two methods described in the previous section are compared with the experimental results. Under the condition of strain rate is 10000
s^{1} and the temperature is 25℃, 200
℃, 400
℃, 600
℃, 700
℃, 800
℃, the comparison results of stressstrain curves are shown in
Figure 16.
Figure 16a,b are the experimental comparison results of linear regression solving method and functional iteration solving method, respectively. According to
Figure 16, both of the equations obtained by linear regression solving method and functional iteration solving method can predict the trend of flow stress.
In order to quantitatively evaluate the overall error of the two methods, the scatter plot is used to calculate the correlation value. The results of the correlation between the calculated stress and the experimental stress obtained by the two methods are shown in
Figure 17.
Figure 17a,b are the results of linear regression solving method and functional iteration solving method, respectively. Compared with
Figure 17b (
R^{2} = 0.889), the data correlation index in
Figure 17a is higher (
R^{2} = 0.985), that is, the data concentration is higher.
The maximum relative error θ between the measured stress and the predicted stress is also calculated to evaluate prediction accuracy, as shown in Equation (25).
Where,
σ_{p} is the predicted stress,
σ_{m} is the measured stress. The maximum relative error of the constitutive model obtained by the linear fitting method and the function iteration method are shown in
Table 7. According to
Table 7, the accuracy of the model obtained by the linear regression method (11.21%) is much higher than that obtained by the function iteration method (4.74%) in the nondynamic recrystallization stage. Correspondingly, in the recrystallization stage, the accuracy of the model obtained by the function iteration method is improved (4.11%), which is very small compared with the accuracy of the model obtained by the linear regression method (5.11%). By calculating the average value of the maximum error
$\stackrel{}{\theta}$ before and after dynamic recrystallization, the model accuracy obtained by the linear regression method is greater than the model obtained by the functional regression method.
In summary, the JC constitutive equation of PM superalloy proposed in this paper solved by linear regression method is shown in Equation (26).