4.2.1. Recrystallization Critical Condition
The effect of dynamic recrystallization on the stress–strain curve of the material is shown in
Figure 11. If there is no dynamic recrystallization during the deformation process, the flow stress of the material increases slowly with the increase in strain, as shown in the black solid line in
Figure 11. 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
εr), 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 11, which is the flow softening phenomenon. Therefore, the determination of the critical condition of dynamic recrystallization, namely, 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 FGH96 is not only related to the deformation temperature but also to the strain rate, which verifies the conclusion of Denguir [
21]. The critical strain decreases with the increase in temperature and decreases with the increase in strain rate.
Based on the data in
Table 3, the fitting surface of the critical strain is obtained by polynomial fitting (Equation (11)), as shown in
Figure 12.
The equation for calculating the critical strain of dynamic recrystallization of FGH96 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 J-C constitutive model (Equation (7)) represents the 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 Hi are the parameters to be fitted, ε0, T0, and Tm are 0.001 s−1, 25 °C, and 1350 °C, respectively.
The strain hardening coefficient can be obtained by processing the quasi-static compression test data at room temperature. The proposed constitutive model is simplified as shown in Equation (17). The quasi-static compression tests permitted to determine the yield stress are represented by coefficient
A (
Figure 13a). 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 a fitting straight line (
Figure 13b). As shown in
Figure 12,
A =
σ0.2 = 773 MPa,
n = 0.667,
B = 1271 MPa.
The strain rate sensitivity coefficient C, the thermal softening index m, and the recrystallization softening correction coefficient Hi in the modified J-C constitutive model were obtained from the result of SHPB tests.
According to SHPB tests at room temperature for different strain rates, the constitutive equation is simplified to Equation (19). The stress–strain curves of materials at different strain rates at room temperature in this paper are shown in
Figure 14. The mean value of multiple sets of test results is taken as the final result (
C = 0.031).
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 strain rate is obtained as shown in
Figure 15.
The coefficients
hi (
i = 0, 1, 2) related to the dynamic recrystallization can be obtained according to Equation (21). The ratio of the actual flow stress after recrystallization to the predicted stress of the J-C model can be obtained by taking the above parameters into account. The value of parameter
hi (
i = 0, 1, 2) can be obtained by fitting the experimental results.
In summary, the modified constitutive model obtained by the linear regression method is shown in
Table 6.
- (2)
Function iteration method
According to the 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
T0 and the reference strain rate
ε0 conditions are selected as the initial values. The quasi-static compression test data at room temperature are selected for polynomial fitting to obtain the results of
f(
ε), which are shown in
Figure 16a. Then, according to the relationship between measured stress in the SHPB test at room temperature and
f(
ε), the relationship between stress and strain rate is obtained by iteration, which is
f(
) as shown in
Figure 16b. Finally, the results of
f(
T) and
f(
Hi) can be obtained according to the high-temperature SHPB experimental data by the iteration process, as shown in
Figure 16c and
Figure 16d, respectively.
In the iterative process, the error
R2 is used to judge the accuracy of the results. When the result meets Equation (23), the result is considered to be accepted.
where
is the error of determination of the
ith iterated function,
is the error of determination of the (
i − 1)th iterated function.
Finally, the constitutive model constructed by the function iteration method is obtained. The constitutive model after reaching the critical strain of recrystallization is shown in Equation (24).
- (3)
Comparison of different methods
To optimize the solution method of the parameters, the stress–strain curves obtained by the constitutive equations solved by the two methods described in the previous section are compared with the experimental results. Under the condition of strain rate is 10,000 s
−1 and the temperature is 25 °C, 200 °C, 400 °C, 600 °C, 700 °C, 800 °C, the comparison results of stress–strain curves are shown in
Figure 17.
Figure 17a,b are the experimental comparison results of the linear regression solving method and functional iteration solving method, respectively. According to
Figure 16, both of the equations obtained by the linear regression solving method and functional iteration method can describe the trend of stress in the plastic deformation stage.
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 18.
Figure 18a,b are the results of the linear regression solving method and functional iteration solving method, respectively. Compared with
Figure 18b (
R2 = 0.889), the data correlation index in
Figure 18a is higher (
R2 = 0.985), namely, the data concentration is higher.
The maximum relative error
θ between the measured and predicted stress is also calculated to evaluate prediction accuracy, as shown in Equation (25).
where
σp is the predicted stress,
σm 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 plastic deformation of the non-dynamic 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
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 the whole plastic deformation stage.
In summary, the modified J-C constitutive equation considering the recrystallization softening effect proposed in this paper is solved by the linear regression method, and the results are shown in Equation (26).