Normalized Parameter Creep Model of DD6 Nickel-Based Single Crystal Superalloy

: For nickel-based single crystal superalloy DD6 (AECC Beijing Institute of Aeronautical Materials, Beijing, China) material, a method for predicting creep rupture time was proposed based on a newly defined equivalent stress method. An anisotropic creep model for describing the orien-tation-dependent creep behavior and lifetime of a nickel-based single crystal superalloy was proposed. The creep subroutine was written based on the proposed nickel-based single crystal creep model. The stability of the model was improved by adjusting the iterative algorithm. The creep calculation results in [001], [011], and [111] loading directions were compared with the experimental results. The accuracy of the calculation results by the nickel-based single crystal creep subroutine was verified. The initial time step and maximum time step of the creep subroutine were studied.


Introduction
Nickel-based single crystal superalloy turbine blades are one of the key technologies for aero engines since the 1980s. Over the past dozen years, the first, second, and third generation of nickel-based single crystal superalloys have been developed and applied successively, to enhance the temperature resistance of aero-engine turbine rotor blade materials by nearly 90 °C [1] compared with directional solidification superalloys. At present, almost all advanced aero-engine turbine rotor blades have adopted nickelbased single crystal superalloy.
In the 1970s, the United States first used PWA1422 (Pratt & Whitney Group, Connecticut, United States) directional blades on military engines, and then on civil aircrafts. In the 1980s, PWA1480 (Pratt & Whitney Group, Connecticut, United States) single crystal blades were used in the F100 engines. Since then, directional and single crystal blades have become important features of various advanced engines. The development of directional solidification technology has greatly improved the high temperature capability of cast superalloys. After the 1980s, the thrust-to-weight ratio was increased from 8 to 10. The first-generation single crystal superalloy PWA1480 was used on turbine blades. Subsequently, the second-generation single crystal superalloy PWA1484 (Pratt & Whitney Group, Connecticut, United States) and CMSX-4 (Cannon Muskegon Corporation, Muskegon, United States) were used. The 100 h rupture strength reached 140 MPa at 1100 °C. After the 1990s, the third-generation single crystal alloys RenéN6 and CMSX-10 (Cannon Muskegon Corporation, Muskegon, United States) were developed. The melting point, initial melting temperature and service temperature of the alloys were multiple temperatures at the same time. In literature [18], a constitutive model for the mechanical behavior of single-crystalline superalloys at high temperatures has been developed. The model relies on the slip system theory and is able to predict rafting and its influence on plastic flow. In literature [19], new internal variables representing the microstructural changes under those specific thermal loadings have been introduced in the framework of crystal plasticity using a macroscopic approach to account for the transient creep behavior induced by microstructure changes. In literature [20], a homogenization method including modified γ / γ ' microstructure area surrounding pores and topologically close-packed (TCP) phase particles was developed and correlated to creep life. In literature [21], a modified crystal plasticity constitutive model considering microstructure evolution is developed. In the literature [22], a physics-based model is proposed to predict the γ / γ ' microstructure evolution of single crystal (SC) superalloy at medium temperature and high stress level. The anisotropy of mechanical properties of nickel-based single crystal materials [5,23] is still a major challenge in the deformation simulation of aerospace engine turbine nickel-based single crystal blades.
Creep models considering nickel-based single crystal orientation in this paper include: Second-stage creep strain rate prediction, creep model establishment and parameter fitting, creep rupture time prediction, flow law based on newly defined equivalent stress, creep model algorithm, usermat subroutine writing and model verification. Since the creep strain-time curve is incomplete, and is highly dispersive at different crystal orientations and temperatures, this paper has not proposed a more accurate model to describe the creep curves at different temperatures in different crystal orientations. According to literature [24], the creep curves in different orientations have similar shapes. Assuming that the creep curves in different orientations have similar shapes, the creep deformation behavior of all orientations was described with the creep curve shape of [001] orientation.

Basic Characteristics of DD6 Material
DD6 material is a second-generation nickel-based single crystal superalloy with a melting point of 1370.5 °C. In view of the cubic symmetry of nickel-based single crystal materials, the elastic properties of DD6 material [24] at different temperatures are shown in Table 1.
For the yield strength data in different orientations at different temperatures, the yield strength under the calculation condition can be obtained by linear interpolation or polynomial parameter fitting. The uniaxial tensile experimental data [24]   The predicted tensile yield strength of DD6 material with different orientations at different temperatures is shown in Figure 1. The plane composed of the x and y axes corresponds to the standard projection plane. The fitting function used in Figure 1 is Equation (7). It will be described in detail in Section 2.4. It can be observed from the database that the yield strength of DD6 material tends to isotropic with increase of temperature at higher temperature. The yield strengths corresponding to different orientations are not much different. Due to the particularity of the experimental data for the 1070 °C, the yield strength in the [111] direction is the smallest, and in the [011] direction is the largest. The yield surface shape corresponding to the yield function is the shape of Figure 1b. The special data leads to the maximum yield strength in the near [001] direction. This abnormal situation requires more material data for further verification.

Second Stage Creep Strain Rate Prediction
The creep strain-time curve is processed to obtain the second stage creep strain rate of the corresponding creep under different orientations, different temperatures, and stress levels, and fit it by the following activation energy formula [25]. The second stage creep strain rate is the minimum creep rate, which is the steady-state creep rate.   Table 3. Base on the potential different deformation mecha-nisms in this large temperature window, the stress exponent n evolves for physical reasons. Due to the lack of material data, the same n is used in this article. The second stage creep strain rate corresponding to different orientations is calculated and compared with the experimental results [24]. The percentage error is defined as Only the [001] orientation results are listed here, see Table 4. The maximum percentage error of the prediction results is 25.2% at 980 °C and 200 MPa. The percentage error of most results is less than 20%. This is acceptable for engineering.

Creep Model Establishment And Parameters Fitting
The creep strain-time curves under different orientations, temperatures, and stress levels were fitted using the creep equation [26] 5 4 t is the lifetime at a given temperature and stress, and  The five parameters in the creep model are taken the natural logarithm, and they are considered to be dependent on temperature and stress, thus having the following form: In the formula,     Finally, the creep three-stage model in reference [26] is If only the [001] orientation of the nickel-based single crystal material is considered, this paper combines the creep data of the nickel-based single crystal DD6 material to modify the three-stage creep model to (1 e ) e ( e (1 e )) Further, the above formula can be simplified as (1 e ) e ( ) (  Table 7. The creep rupture elongation of [001] orientation is analyzed, as shown in Figure 5, in which the different colors correspond to different temperatures. The stress level of the same color gradually increases along the positive direction of the x axis. The creep rupture elongations at different temperatures take the mean value. It can be found that the creep rupture elongation is dispersive under different temperatures and different stress levels. The correlation between the creep rupture elongation and stress level is not obvious at the same temperature. The correlation between the creep rupture elongation and temperature is not obvious. In view of the limited creep curves data, it is impossible to get enough data of the third stage creep. In this paper, the creep rupture elongation c  in the [001] direction of the DD6 material is taken as 27.0%. A nickel-based single crystal yield criterion is proposed, which can be written into the following form without considering the asymmetry of tension and compression.  can be obtained. All the parameters in the yield function are dimensionless parameters. Of course, the parameters in the function can also be obtained from the tensile or torsional yield strengths of other different orientations. It is considered that the yield surface and the potential energy surface have similar shapes, thus the above formula is applied to the creep deformation process of the nickel-based single crystal material.

Creep Rupture Life Prediction
Using the durable stress-life curve equation and the DD6 material parameters, the durability stresses at different temperatures corresponding to different lifetimes in different crystal orientations are obtained. Combined with the yield function, the durable stress-life data is calculated to obtain the parameters , , I J k for different given creep rupture lifetimes at different temperatures, see Table 8. The parameters for the given different creep rupture lifetimes are averaged, and the parameters , I J are linearly regressed to temperature, as shown in Figure 6. As the temperature exceeds 1100 °C and approaches the melting point of the material, the parameters , I J all approach 4, that is, the directionality of the material properties will gradually become insignificant. The results obtained by the method in this paper show this trend. However, both the values will diverge at higher temperatures.
I J ＞ at low temperature and I J ＜ at very high temperature. Finally, using the regression coefficients of the obtained parameters , I J and the durable stress-life curve equation parameters in the [001] orientation, the durability life prediction of the nickel-based single crystal material can be performed. The durability life prediction results of different temperatures and different orientations are shown in Table 9. The life prediction results are better.

Flow Rule Based on Newly Defined Equivalent Stress
Based on the newly defined yield function form, the yield function without considering the asymmetry of tension and compression can be written into the following form Among them, I J 、 are yield function parameters, ij s is the partial stress component. Assuming that the plastic potential energy surface has the same shape as the yield surface, thus Since the plastic strain increment is In the above formula, nickel-based single crystal creep model during the iterative process is the equivalent creep strain rate c  . The time unit of the subroutine is unified as h. The subroutine calculation flow chart corresponds to the iterative solution part in the subroutine in Figure 8, as shown in Figure 9. Only the algorithms corresponding to normal stress and normal strain are listed in the figure. The algorithms corresponding to shear stress and shear strain are similar so they are not listed. The iterative process mainly includes four modules: The main iterative module, the creep strain rate solving module, the newly defined equivalent stress solving module and the intermediate variable solving module.

Subroutine Verification
(  Figure 11(1). The creep calculation results are the same for different maximum time steps. The conclusion can be drawn that the maximum time step has no effect on the creep calculation results, with a certain range  Figure 11. The creep calculation results for different initial time steps are the same. It can be seen that the creep calculation results are not affected by the initial time step when the initial time step is smaller than Due to the large number of nickel-based single crystal hollow turbine blade elements, it is necessary to select a larger initial time step and maximum time step to improve the calculation efficiency. However, in the actual calculation process, a larger initial time step often leads to non-convergence in parts of Gaussian integration point. To ensure the convergence of most Gaussian integration points, the initial time step shoule not be too large. At the same time, the creep convergence curve of the initial stage of creep calculation (0-0.1 h) oscillates greatly, and the subsequently creep calculation convergence curve is relatively stable. Hence, in the initial stage, the initial time step is

Conclusions
(1) Based on the proposed equivalent stress that can characterize the orientation characteristics of nickel-based single crystals, the creep rupture life in different orientations is predicted. The corresponding flow rule based on the proposed equivalent stress is derived. Finally, creep constitutive model and model algorithm for single crystal materials are proposed. (3) By writing the usermat subroutine, the high precision creep deformation simulation of the structural parts for nickel-based single crystal materials at different temperatures and different stress levels can be realized, and the subroutine calculation efficiency meets engineering application.
(4) Initial time step and maximum time step of the usermat creep subroutine are studied, and some suggestions for the selection of initial time step and maximum time step are provided.

Data Availability Statement:
The data presented in this study are available in article.

Conflicts of Interest:
The authors declare no conflict of interest.