Test Data Evaluation of Very High Cycle Fatigue Based on Maximum Likelihood Method

The very high cycle fatigue tests of TC4 alloy were accomplished through ultrasonic fatigue experiment which were conducted on the ultrasonic fatigue testing system at frequency range of 20000 ±1000 Hz. In contrast to the commonly used Basquin model, a Maximum Likelihood Method to fit the three-parameter nonlinear model was used to verify the adaptability in very high cycle fatigue regime. These two methods were used to estimate S-N curves of three types of materials which were TC4 alloy, GH4169 alloy and TC17 alloy in the very high cycle fatigue regime and the results were compared with test data. The evaluation result of the MLM model based on the three-parameter nonlinear model shows a good agreement with test data. The mean and variance of relative error of the MLM model are less than those of the Basquin model, providing a suitable evaluation method for predicting very high cycle fatigue limit which often has higher scatter.


Introduction
With the increasing market needs and technical development of aero engines, the high cycle fatigue performance data based on 10 7 cycles can no longer meet the design requirements.At present, the stateof-the-art design criteria of aero engine have clearly put forward the requirement of very high cycle fatigue performance data based on 10 9 cycles.The fatigue life above 10 7 cycles is named as the very high cycle fatigue (VHCF).The Engine Structural Integrity Program (ENSIP) clearly require that the number of fatigue cycles of all engine parts should reach at least 10 9 [1].The "General specification for aviation turbojet and turbofan engine " (GJB241A•2010) issued by China also clearly stipulates that the fatigue life of titanium alloy parts needs to reach 10 9 cycles [2].It is urgent to accumulate very high cycle performance data to meet the anti-fatigue design requirements of aero engines.
The conventional fatigue test frequency does not exceed 300 Hz, and it takes nearly 40 days to complete the data of more than 10 9 cycles.Therefore, accelerated fatigue test methods and equipment based on resonance principle have been developed.For example, high-frequency resonant fatigue testing machine [3] can carry out very high cycle fatigue test with a frequency of more than 1000 Hz, and ultrasonic fatigue testing machine [4][5][6] can carry out very high cycle fatigue test with a frequency of 20 kHz.The test efficiency can be increased by hundreds of times.But it still costs much test time to describe the very high cycle fatigue behavior due to the large needs of specimens.
In order to correctly evaluate the very high cycle fatigue limit of materials, it is necessary to establish the relationship between the stress level and fatigue life, that is, the S-N curve.The S-N curve reflects the trend of fatigue life changing with cyclic load, and is the basic diagram to characterize the fatigue properties of materials.There are many models for processing fatigue data, and the commonly used models are power function and exponential function models [7], often used to describe S-N curves for high cycle fatigue regime.For example, the most commonly used is the Basquin model [8][9] (equation (1), where S is stress amplitude, N f is number of fatigue cycles, σ f ' and b are material constants), which assumes that the fatigue data meets the normal distribution.The linear relation between stress level and fatigue life in the log-log coordinate system is obtained by least square method.However, the cycle number of very high cycle fatigue is as high as 10 9 .With the increase of cycle times, the fatigue data in the very high cycle regime inevitably scatters more in comparison with the high cycle fatigue data.A lot of test data shows that the population mean and standard deviation of logarithmic fatigue life of metal materials are not linear functions of stress levels in the medium and long fatigue life regime.The Basquin model may not be suitable for describing very high cycle fatigue data, while the three-parameter nonlinear model (equation (2), where S is stress amplitude, N f is number of fatigue cycles, S0 is theoretical stress fatigue limit.S0, α and β are material constants) can describe the S-N curve in the middle and long lifetime regime.These two models are both empirical models based on deterministic macroscopic phenomena.
Ling and co-worker [10] proposed a maximum likelihood method (MLM) for estimating P-S-N curves based on the characteristic of the logarithmic fatigue life following a normal distribution.In that work, the high cycle fatigue test data of steel material was used to verifying the accuracy and convenience of this method.This kind of method can estimate S-N curves in high cycle regime using less specimen, which can be used in very high cycle fatigue test reducing cost.Through establishing the maximum likelihood function, each parameter in the formula can be obtained.Based on the threeparameter nonlinear model, this paper employed the maximum likelihood method to estimate S-N curves of three types of materials in the very high cycle fatigue regime, so as to verify the adaptability of this data processing method for very high cycle fatigue test data.

Ultrasonic Fatigue Experiment
Fatigue tests were conducted on the ultrasonic fatigue testing system (product ID: HC-DF2020GD) which was customized to work at frequency range of 20000 ±1000 Hz by AECC CAE Co. Ltd.By designing the appropriate specimen size and using the high-frequency vibration generated by the ultrasonic signal generator, the test system sends out the high-frequency displacement vibration signal and completes the resonance with the specimen.The experimental principle is shown in figure 1.The displacement in the middle of the specimen is zero and the stress is the maximum when resonance occurs.Due to the internal damping of the material, the temperature in the gauge section of the specimen will rise and has a profound influence on test results.Depending on the damping of material, compressed air or low temperature compressed air can be used for cooling.In order to ensure the accuracy of the test, the ultrasonic fatigue test system was calibrated before the test.The linear relationship between the power of the control system and the output displacement of the testing machine was obtained, so the input ultrasonic voltage value of the ultrasonic fatigue test system can match the actual stress value of the specimen.The very high cycle fatigue specimen was designed by numerical analysis and confirmed by finite element simulation.The shape of the specimen is hourglass.In the process of specimen design, the boundary condition for solving the vibration equation is that the stress at the end of the specimen is zero and the displacement amplitude is the largest [11].The elastic modulus and density of the material should be measured before the specimen size is determined.The finite element software is used to analyze the resonant frequency of the specimen.The result shows that the resonant frequency meets the requirement of 20 kHz ultrasonic vibration fatigue.The gauge section of the very high cycle specimen is longitudinally polished to ensure that the surface roughness is within Ra 0.4.The test stress ratio R is -1 and the test frequency is 20 kHz.After the calibration of the testing system is completed, the specimen is mounted to the bottom end of the horn through the thread and the resonant frequency of the specimen can be obtained by the testing Set the loading condition and operate the testing system, and the specimen will resonate at the corresponding frequency until the fracture occur or the number of vibration cycles exceeds 10 9 .This paper accomplished very high cycle fatigue tests of TC4 alloy which material property and specimen size are consistent with that in reference [12].38 test data points were obtained through ultrasonic fatigue experiment.

Maximum Likelihood Method for Model Fitting
In order to obtain the parameters of the three-parameter nonlinear model, a maximum likelihood method was used.Each test point from the fatigue testing includes two pieces of data: (1) Si, the stress level, (2) lgN i , the logarithm (base 10) of the number of cycles tested.
A suitable stress level S 0 is selected from the fatigue test results, and the logarithmic fatigue life lgN j (j= 1, 2, ..., η), the estimates of the population mean and standard deviation of these n data are respectively as follows.
The general form of three-parameter P-S-N curve equation is expressed by equation (5).Assuming that the logarithmic fatigue life follows a normal distribution, taking logarithms of both sides of equation ( 5), the expression equation ( 6) with survival probability P=50% can be obtained.Since the logarithmic fatigue life follows a normal distribution, the standard deviation σ(S) of the stress level S can be obtained as equation (7).The formula of μ ̂0 and σ ̂0 can be obtained as equation (7) and equation (8).
(S-S 0P ) β P N P =α P (5) μ(S)=lgN 50 =lgα-βlg(S-S 0 ) (6) σ ̂0=lg ( By combining the above formula, the following formula is obtained.When survival probability P equals to 84.1%, μ P equals to 1, then the standard deviation σ(S) can be expressed by equation (12).(12) According to the premise, logarithmic fatigue at any stress level S i follows a normal distribution, and its probability density function is

μ(S)=μ ̂0+βlg
Then the corresponding likelihood function is exp{- Take the natural logarithm of both sides of the equation ( 14).Set F as the function of β,β 1 ,S 0 and S 01 , equation ( 16) is obtained.As long as the minimum value of F is obtained, the maximum value of the likelihood function L can be obtained, and the maximum likelihood estimators of four parameters β,β 1 ,S 0 and S 01 can be obtained.When the parameters are obtained, the expressions of μ(S) and σ(S) can be obtained by substituting the values into equation ( 13) and equation ( 15), and the P-S-N curve determined by the maximum likelihood method can be obtained.When P=50%, the median S-N fatigue curve of the material can be obtained.

Results
There are 38 specimens of TC4 alloy totally and tests are carried on HC-DF2020GD under 8 stress levels as shown in table 1.The S-N curves shown in figure 2 (a) are obtained by fitting the test data of TC4 alloy using MLM and Basquin model.It can be seen that compared with Basquin model, the MLM has better fitting accuracy and results give good agreement with test data.To verifying the accuracy of MLM in very high cycle fatigue regime, other experimental data of GH4169 alloy in reference [13] and TC17 alloy in reference [14] are also used to validate this evaluation method.There are 18 test data points of GH4169 totally under 6 stress levels, and There are 14 test data points of TC17 alloy totally under 5 stress levels.The S-N curves obtained are shown in figure 2 (b) and figure 2 (c).The results of MLM based on maximum likelihood method for two types of materials gives a good agreement with test data.
Table 2 compare the evaluation values and the test values of the two models under the same stress level of TC4 alloy.The maximum relative error of the MLM results for TC4 is 7.8033% and the minimum is 0.05492%, while the maximum relative error of Basquin model is -17.9765% and the minimum error is 0.2699% which are larger than those of the MLM.The evaluation values and the test values of the two models under the same stress level of TC17 alloy are compared as shown in table 4. The maximum relative error of the MLM for TC17 alloy is 5.5146% and the minimum is -0.00127%, while the maximum relative error of Basquin model is 8.3814% and the minimum error is -1.7832% which are larger than those of the MLM.The mean value and variance of relative error are shown in table 5.It can be concluded that for these three materials, the evaluation accuracy of the MLM is better than that of the Basquin model.From the above results, the MLM is more suitable for fitting very high cycle fatigue test data.

Conclusions
Based on the three-parameter nonlinear model, the maximum likelihood method is used to fit the very high cycle fatigue test data of two materials.The evaluation results give a good agreement with test data.The fitting accuracy is better than that of the common Basquin model, which provides a new method for very high cycle fatigue data fitting.

Figure 1 .
Figure 1.The experimental principle of the ultrasonic fatigue testing system.

Table 1 .
Fatigue test results of TC4 alloy.

Table 2 .
Comparison results between evaluation values and test values-TC4 alloy.

Table 3
shows the comparison results of the evaluation values and the test values of the two models under the same stress level of GH4169 alloy.The maximum relative error of the MLM for GH4169 alloy is 8.04663% and the minimum is 0%, while the maximum relative error of Basquin model is -10.8410% and the minimum error is 4.8819% which are larger than those of the MLM.

Table 3 .
Comparison results between evaluation values and test values-GH4169 alloy.

Table 4 .
Comparison results between evaluation values and test values-TC17 alloy.

Table 5 .
Comparison results between evaluation values and test values-TC17 alloy.