Analysis of high temperature fatigue lifetime of GH4133B superalloy used in turbine disk of aero-engine

Based on the S-H cavity model theory and the thermodynamic diffusion equation, the high temperature fatigue lifetime equation is deduced, and the influence of stress amplitude and mean stress on fatigue lifetime is quantitatively analyzed. At high temperature of 650°C, according to the test data of fatigue lifetime of GH4133B superalloy under different stress ratios or alternatively at various maximum stress levels, the nonlinear regression analysis method is applied to identify the material parameters in the fatigue lifetime equation, and a 3D Nf-σm-σa curve surface is plotted. The comparison between theoretic fatigue lifetime Nfp and test one Nft indicates that the fatigue lifetime equation derived from the microstructure evolution of metallic materials can accurately predict the fatigue lifetime of GH4133B superalloy under different cyclic loading parameters. Finally, a parameter γ is introduced to characterize the effect of mean stress σm and stress amplitude σa on fatigue lifetime Nf of GH4133B superalloy. It is suggested that the effect of mean stress σm on Nf is larger than that of stress amplitude σa on Nf under the condition of tensile-tensile fatigue loading.


Introduction
With regard to the study of high temperature fatigue lifetime of nickel-base superalloy, a lot of researches have been done. Wang [1] carried out the fatigue and creep-fatigue tests for a forged and precipitation hardened nickel-based superalloy GH4169 at 650°C, and investigated the effects of inhomogeneous microstructure and loading waveform on creep-fatigue behaviour. It was found that the S3 CSL boundaries play an important role in the inhomogeneous effect, the tensile dwells show an intergranular damage caused by precipitate-assist micro-voids, and the compressive dwells show an oxidation-assisted damage and slip-band-induced cracks. Chen [2] studied the low cycle fatigue and creep-fatigue interaction behavior of nickel-base superalloy GH4169 at 650°C. Yu [3] studied the creep and low cycle fatigue behaviour of a nickel-base superalloy. It is found that the creep curves show an obvious primary creep stage followed by a short steady-state creep stage and then an accelerating creep stage until leading to failure at 700°C. While at 900°C, the creep curves demonstrate a shorter primary stage, and a longer accelerating creep stage without steady-state creep stage. Simultaneously, the creep and low cycle fatigue properties degenerate with increasing temperature. Holländer [4] investigated the isothermal and thermo-mechanical fatigue behavior of the nickel-base superalloy IN738LC by using the standardized and advanced test methods. Based on the IOP Conf. Series: Materials Science and Engineering 531 (2019) 012029 IOP Publishing doi:10.1088/1757-899X/531/1/012029 2 continuum damage mechanics, Shi [5] carried out a creep and fatigue lifetime analysis of directionally solidified superalloy, and its brazed joints at elevated temperature.
In this work, a high temperature fatigue lifetime prediction model is constructed via the S-H cavity model theory and the thermodynamic diffusion equation, a 3D curve surface of fatigue lifetime is plotted, and the theoretical predicted value and test one are compared. A parameter is introduced to characterize the effect of mean stress and stress amplitude on fatigue lifetime.

Fatigue nucleation theory
During the process of fatigue cycle, with the accumulation of plastic deformation, the vacancies are generated and aggregated at the grain boundaries to form the original voids. The void number produced during a cycle is proportional to the stress amplitude sa, which is a half of the sum of maximum stress smax and minimum one smi n, i.e., sa=( smax +smi n)/2. Therefore, the nucleation number na of the voids in the unit volume at grain boundaries can be expressed as (1) where, P is the void nucleation factor, N is the fatigue cycles, and m is a material parameter.

Creep nucleation theory
The theoretic analysis and the experimental results indicate that the cavities are continuously nucleated with the creep time, which is sometimes called as a continuous nucleation. The previous theoretical research shows that the void number nm on grain boundaries per unit area is proportional to the creep strain e, that is (2) where, a1 is a material parameter. Under the condition of one-order approximation, the parameter a1 is independent of the stress. The nucleation rate, which represents the number of voids formed on grain boundaries per unit area in per unit time, can be written as (3) where, is the nucleation rate, is the creep rate. Supposing that there exists a relation expression between the creep rate and the mean stress, that is (4) where, sm is the mean stress, k and n are the material parameters, the Equation (3) is rewritten as (5) where, a'=a1k, which represents the product of the material parameter a1 and k.
Under the condition of stress control mode, the variables in Equation (5) are separated, and then the integral calculation on both sides of equation is operated, the following equation is (6) Due to the relation expression between the fatigue cycle N and the test frequency f, i.e., N=ft, Equation (6) can be addressed as where, a''=a'/f=a1k/f, which denotes the ratio of the parameter a' to the test frequency f.

Creep-fatigue nucleation theory
Under the condition of stress control mode, considering the influence of mean stress sm and stress amplitude sa on nucleation rate and as well the influence of creep-fatigue interaction on nucleation rate in the metallic material, the total nucleation number nt of the voids in the unit volume at grain boundaries can be expressed as (8) where, a''' is a parameter containing an operated parameter as frequency f, and n is the material parameter. In Equation (8), the terms of a'''sm n N, a'''sa n N and a'''(sm+sa) n N-a'''sm n N-a'''sa n N denote the creep nucleation term, the fatigue nucleation term and the creep-fatigue nucleation term.

Volume Diffusion
According to Fick's first law of diffusion, the mass flux on the unit section area perpendicular to the diffusion direction in the unit time is proportional to the concentration gradient at the cross section. Its mathematical expression is (9) where, J denotes the diffusion flux, D is the diffusion coefficient, C is the volume concentration of diffusion component, and x represents a position coordinate in the linear coordinates.
According to the S-H cavity growth model, the voids are homogeneously generated in the whole grain boundary volume, and the void concentration satisfies Fick's second law of diffusion, that is (10) where, DV is the void diffusion coefficient, b is the void formation rate, t is the time, and r is a polar diameter in the polar coordinates. In the steady state, i.e., , Equation (10) is simplified as The solution can be obtained from Equation (11), that is (12)

Cavity growth rate
In the case of uniaxial tension, the elastic strain energy is neglected, and the free enthalpy change dG of crystal in this process is expressed as (13) where, dN * is the void number, sm is the tensile stress, WA is the atomic volume, DV is the volume contraction caused by the void formation, and P is the hydrostatic pressure whose value is of -sm/3.
Under the condition of thermodynamic equilibrium state, i.e., dG=0, Equation (13) is reduced as Moreover, for an arbitrary surface, the normal component force P of surface tension is written as (15) where, gs is the specific surface energy, R1 and R2 are the two curvature radii of the surface. So the general expression considering a stress action is deduced as (16) For the spherical surface whose curvature radius maintains a constant R, Equation (16)  where, a denotes the diameter of grain boundary in the S-H cavity growth model.
For solid solution alloys, the chemical potential energy µi of the component i can be expressed as (19) where, µi 0 is the initiation chemical potential energy, T is the thermodynamic temperature, R is a universal gas constant, and ai =ri xi , ri represents the matter concentration. According to the boundary conditions, the void formation rate b can be determined as , The cavity growth rate is equal to the voids flow into the cavity per unit time, therefore Equation (21) can be rewritten as , Contacting Equation (8) and Equation (22), and considering dN=fdt, the total void volume is derived as The integral calculation is carried out for Equation (23), and supposing that the cavity volume reaches a critical volume Vc, the specimen is failure. So the critical volume Vc can be written as Let c= (2fVc/(a'''K)), a=-n/2, and z=2gs/R, then Equation (25) can be simplified as

High temperature fatigue tests
The fatigue tests for specimens of GH4133B superalloy are carried out at high temperature of 650ºC at atmospheric environment on an MTS809 materials testing machine. The fatigue test waveform is sine wave with a frequency of 2Hz.
The tests are divided into two groups. For the first group of high temperature fatigue tests, the maximum stress is controlled as 700MPa, and the stress ratios are set as 0.01, 0.1, 0.2 and 0.4, respectively. For the second group of high temperature fatigue tests, the stress ratio is set as 0.1, and the maximum stress ranges from 900MPa to 550MPa. The fatigue tests are conducted at 650ºC, the experimental data of displacement of the specimen are collected, and the fatigue cycle numbers are recorded. The fatigue lifetimes under different operating parameters are shown in Table 1.

High temperature fatigue lifetime prediction
According to the experimental data of fatigue lifetimes of GH4133B superalloy at different operating parameters listed in Table 4, the theoretical formula derived in this paper, i.e., Equation (26), is applied to predict the fatigue lifetime. Utilizing the nonlinear regression analysis method, the material parameters in Equation (26) are identified, and the theoretical formula of fatigue lifetime is written as (27) The curve surface of fatigue lifetime versus mean stress and stress amplitude are shown in Figure 1. The theoretical fatigue lifetimes are calculated by Equation (27), and the comparison between theoretical lifetime Nfp and experimental one Nft is shown in Figure 2. It can be found from Figure 2 that all of the experimental data points are within the range of ±3 error factor, which suggests that the theoretic formula of Equation (26) derived in this paper, can be used to predict the fatigue lifetime of GH4133B superalloy at different stress operating parameter at 650ºC.   In order to quantitatively describe the effect of mean stress and stress amplitude on fatigue lifetime, the partial derivatives of fatigue lifetime Nf in Equation (26) to the mean stress sm and the stress amplitude sa are individually calculated, and the following equations can be obtained, i.e., , A parameter g, whose value is the ratio of the partial derivative of Nf to sm and that of Nf to sa, is defined, i.e., It can be found from Equation (27)  According to Equation (31), the test data are fitted using the nonlinear regression method, the material parameters in Equation (31) are indentified, and the theoretical formula of fatigue lifetime is The curve surface of fatigue lifetime versus mean stress and stress amplitude are shown in Figure 3. The theoretical fatigue lifetimes are calculated by Equation (32), and the comparison between theoretical lifetime Nfp and experimental one Nft is shown in Figure 4. It can be found from Figure 4 that all the test data points are in the range of ±2 error factor, and the theoretical results are in good agreement with the experimental data, which indicates that comparing with Equation(26), Equation (31) can more accurately predict the fatigue lifetime.

Conclusions
The conclusions are given as following: (a) The fatigue lifetime equation is deduced, the nonlinear regression analysis method is used to identify the material parameters in the equation, and a 3D Nf-sm-sa curve surface is plotted. The comparison between theoretical fatigue lifetime Nf p and test one Nft indicates that the fatigue lifetime equation proposed in this paper can more accurately predict the fatigue lifetime of GH4133B superalloy under different operating parameters.
(b) A parameter g is introduced to characterize the influence of mean stress and stress amplitude on fatigue lifetime Nf of GH4133B superalloy. It is suggested that the effect of mean stress on Nf is larger than that of stress amplitude on Nf under the condition of tensile-tensile fatigue loading.