Mean stress effect plays an important role in fatigue life prediction, and it is discovered that maximum stress has nonnegligible influence on mean stress effect. Therefore, a modified Walker model is proposed to account for mean stress effect on fatigue life of aeroengine disks, which contains the influence of stress ratio and maximum stress on mean stress effect. Eight sets of fatigue data for standard smooth bars from six kinds of materials commonly used in aeroengine disks as well as two sets of experimental data from simulated specimens of turbine disks were employed to investigate the prediction capability of the proposed model against other candidate mean stress relationships. It is found that Goodman model generates most conservative results, while Morrow model overestimates fatigue life for most cases. SWT model yields similar results to Walker model but with less accuracy. The results of the modified Walker model turn out to be superior to those of any other candidate models for all cases examined, especially for large mean stress ones. Thus, the modified Walker model can be an effective method to predict fatigue lives of aeroengine disks influenced by mean stresses.
In aeroengine, most critical regions of disks are always subject to time varying loads with the presence of mean stresses. Many researchers have found that mean stresses have significant influence on fatigue life [
In scientific researches, most fatigue tests are conducted particularly under completely reversed conditions with mean stresses being zero. Therefore, fatigue life predicting methods on the base of these fully reversed fatigue test data should be modified to account for mean stress effect for better accuracy. Since the fatigue process of a component consists of the crack initiation phase and the crack propagation phase, thus many researchers have proposed plenty of models to consider the mean stress effect on the crack initiation life and the crack propagation life separately. The commonly used FCG models considering mean stress effect on fatigue crack growth rate are Priddle model [
The prediction accuracy of the methods to assess mean stress effect on crack initiation behavior of various materials under different loading conditions has been investigated by Zhu et al. [
Even though various models have been proposed to consider the mean stress effect, the Goodman model, Morrow model, SWT model, and Walker model are still considered to be most popular methods while predicting the crack initiation life in practical engineering. Therefore, these four mean stress models will be briefly illustrated as follows.
Goodman relationship employs the ultimate tensile strength,
The Morrow equation shares the same form with Goodman’s, except for employing the true fracture strength
The SWT method employs the maximum stress
Unlike the SWT method, the Walker model supposes that mean stress effect is material dependent. Thus, a fitting parameter
It can be observed from the comparison between the SWT and the Walker model that when
Basquin’s equation [
From previous investigations, it is known that the Walker method gives better predictions than other models when fatigue data are available to fit the adjustable parameter
Equivalent completely reversed stress amplitude versus fatigue life correlations by Walker model for the (a) FGH4095 superalloy (600°C) and (b) GH4169 superalloy (650°C).
Walker, FGH4095 600°C
Walker, GH4169 650°C
Therefore, further researches were carried out to investigate the influence of maximum stress on mean stress effect. Based on the Walker model and Basquin’s equation, the fitting parameter
It can be obviously seen from Figure
The influence of maximum stress on
The equivalent completely reversed stress amplitude based on the proposed modified Walker model is shown in (
Though the modified Walker equation contains two fitting parameters while the Walker equation merely includes one, the determination of the two parameters in modified Walker equation is not complicated compared to that of the one parameter in Walker equation. Since (
Fatigue test data of standard smooth bars for various stress ratios are employed to verify the prediction capability of the candidate mean stress models in (
The plots of
Equivalent completely reversed stress amplitude versus fatigue life correlations for FGH4095 superalloy (600°C) for the (a) Goodman, (b) Morrow, (c) SWT, and (d) modified Walker methods.
Goodman, FGH4095 600°C
Morrow, FGH4095 600°C
SWT, FGH4095 600°C
Modified Walker, FGH4095 600°C
Equivalent completely reversed stress amplitude versus fatigue life correlations for GH4169 superalloy (650°C) for the (a) Goodman, (b) Morrow, (c) SWT, and (d) modified Walker methods.
Goodman, GH4169 650°C
Morrow, GH4169 650°C
SWT, GH4169 650°C
Modified Walker, GH4169 650°C
For FGH4095 superalloy (600°C), it can be recognized in Figure
Similar conclusions can be drawn for GH4169 superalloy (650°C) in Figure
To quantitatively summarize the accuracy of the mean stress equations for different materials, the difference between the experimental logarithmic life and predicted logarithmic life for fatigue data sets is taken to calculate the prediction error of each model, which can be expressed as
Besides, a probabilistic method is applied to analyze the prediction errors in the form of fitted probability density functions (PDF). The mean and standard deviation values are effective parameters to illustrate the prediction accuracy of different models. The fitted PDFs for prediction errors of each mean stress relationship are shown in Figures
Probability density function of prediction errors of FGH4095 superalloy (600°C) for the data set of (a) all data set and (b)
Probability density function of prediction errors of GH4169 superalloy (650°C) for the data set of (a) all data set and (b)
The prediction accuracy of each mean stress model for GH4169 superalloy is similar to that of FGH4095 superalloy. Only the Goodman model shows less accuracy while Morrow model brings better results compared with FGH95 superalloy. Similarly, the modified Walker model provides better results than SWT and Walker models, even though these two models show much better results than the Goodman and Morrow models.
Besides the FGH4095 (600°C) and GH4169 (650°C) superalloy, some other materials which are commonly used in aeroengines such as FGH4095 (500°C), GH4169 (500°C), and FGH4096 superalloy [
Material properties and parameters for mean stress models.
Material | | Temperature | Ultimate | Basquin | Walker | Modified Walker | ||
---|---|---|---|---|---|---|---|---|
| | | | | ||||
FGH4095 | [ | 500 | 1490 | 7634 | −0.2077 | 0.5784 | 4.9349 | −1.4204 |
600 | 1480 | 7422 | −0.2065 | 0.4907 | 6.5782 | −1.9737 | ||
GH4169 | [ | 500 | 1230 | 5187 | −0.1729 | 0.6772 | 9.5177 | −2.8557 |
650 | 1170 | 2654 | −0.1228 | 0.5442 | 19.0449 | −6.0689 | ||
FGH4096 | [ | 550 | 1450 | 4519 | −0.1546 | 0.6465 | 2.6067 | −0.6894 |
TC11 | [ | 300 | 830 | 1734 | −0.1051 | 0.4403 | 5.5861 | −1.7912 |
35Cr2Ni4MoA | [ | 160 | 1070 | 1342 | −0.0584 | 0.3784 | 9.7109 | −3.0370 |
2014-T6 Al | [ | / | 494 | 1120 | −0.1221 | 0.4815 | 2.3949 | −0.7241 |
The plots of
Statistical analysis for the prediction errors of different mean stress methods for all materials examined.
Material | | Goodman | Morrow | SWT | Walker | Modified Walker | |||||
---|---|---|---|---|---|---|---|---|---|---|---|
| | | | | | | | | | ||
FGH4095 | 500 | 0.3230 | 0.3817 | −0.4136 | 0.5109 | 0.1462 | 0.4195 | 0 | 0.3491 | 0 | 0.2552 |
600 | 0.3185 | 0.3557 | −0.5253 | 0.6334 | 0.0603 | 0.4235 | 0 | 0.4008 | 0 | 0.2769 | |
GH4169 | 500 | 1.0847 | 1.2153 | −0.1752 | 0.7362 | 0.3505 | 0.7548 | 0 | 0.7093 | 0 | 0.6264 |
650 | 1.3863 | 1.0668 | −0.1981 | 0.8512 | 0.3472 | 0.8241 | 0 | 0.8177 | 0 | 0.5570 | |
FGH4096 | 550 | 0.1990 | 0.2909 | −0.2030 | 0.3114 | 0.1567 | 0.2915 | 0 | 0.2496 | 0 | 0.2275 |
TC11 | 300 | 0.6449 | 0.7774 | −0.5057 | 0.7527 | −0.0278 | 0.5372 | 0 | 0.5207 | 0 | 0.4510 |
35Cr2Ni4MoA | 160 | 2.6252 | 2.7821 | 0.9583 | 0.9327 | −0.0390 | 0.7505 | 0 | 0.6306 | 0 | 0.3567 |
2014-T6 Al | / | 0.4311 | 0.5573 | −0.5291 | 0.4601 | 0.0233 | 0.2884 | 0 | 0.2575 | 0 | 0.2212 |
Equivalent completely reversed stress amplitude calculated by modified Walker model versus fatigue life for (a) FGH4095 superalloy (500°C), (b) GH4169 superalloy (500°C), (c) FGH4096 (550°C) superalloy, (d) TC11 alloy (300°C), (e) 35Cr4Ni2MoA steel (160°C), and (f) 2014-T6 aluminum.
FGH4095 500°C
GH4169 500°C
FGH4096 550°C
TC11 300°C
35Cr2Ni4MoA 160°C
2014-T6 Al
By analyzing plots in Figure
The previous work has proved that the modified Walker model provides favorable results for standard smooth bars of all materials examined. However, it remains to be testified whether the modified Walker model is applicable to estimate fatigue lives of real components in aeroengine. Therefore, two sets of experimental data from a type of simulated bolt-hole specimen and a type of simulated run-way hole specimen of turbine disk at 550°C are analyzed to investigate the prediction capability of candidate models. The main geometric parameters of the two specimens are shown in Table
Main geometric parameters for the two simulated specimens of turbine disk.
Parameters | | | | | | | | |
---|---|---|---|---|---|---|---|---|
Bolt-hole specimen/mm | 3.0 | / | 8.25 | 12.0 | 34.5 | 113.0 | 4.0 | 6.0 |
Run-way hole specimen/mm | 3.2 | 4.2 | 8.25 | 11.2 | 34.2 | 126.4 | 4.0 | 6.0 |
The finite element model for (a) bolt-hole simulated specimen and (b) run-way hole simulated specimen.
Bolt-hole specimen
Run-way hole specimen
The finite element analysis (FEA) results of the two specimens are shown in Figure
The comparison of estimated results of each mean stress models versus experimental data for simulated specimens.
Specimen | Experimental fatigue lives | Median fatigue life | Predicted fatigue lives | ||||
---|---|---|---|---|---|---|---|
Goodman | Morrow | SWT | Walker | M. Walker | |||
Bolt-hole specimen | 20546, 24287, 15746, 8689, 24644, 9601, 25162 | 16977 | 28047 | 35276 | 13234 | 20276 | 14976 |
| |||||||
Error | / | 65.2% | 107.8% | −22.0% | 19.4% | −11.8% | |
| |||||||
Run-way hole specimen | 14572, 26378, 60050, 53267, 151668, 51627, 46216, 28838, 8079, 12174, 13368 | 29867 | 10823 | 48642 | 19569 | 21015 | 21188 |
| |||||||
Error | / | −63.8% | 62.9% | −34.5% | −29.6% | −29.1% |
The contour plots of modified Walker equivalent stress and relative stress gradients along the stress gradient path of the critical regions for (a) bolt-hole specimen and (b) runway-hole specimen.
To comprehensively account for the influence of stress ratio, material, and maximum stress on mean stress effect, the modified Walker model is proposed. Then, eight sets of experimental data from standard smooth bars of six different materials and two sets of fatigue test data from simulated bolt-hole and runway-hole specimens of a turbine disk have been employed to investigate the prediction capability of the proposed model against other mean stress relationships commonly used. The results of the modified Walker model turn out to be superior to other candidate models for all materials examined. Goodman model is found to be most inaccurate, and Morrow equation shows better results than Goodman model. However, the results of the two models are both far from satisfactory. The SWT model brings reasonable results; what is more, it has the advantage of brevity; therefore, the SWT method is a good choice for general use. The Walker model provides satisfactory performance when mean stresses are relatively small by employing a material dependent parameter
The proposed modified Walker model is based on the Walker model with an extra item of stress and stress ratio. Thus the modified Walker model inherits the advantage of Walker model and contains the influence of maximum stress on mean stress effect as well, showing favorable capability to estimate fatigue lives for different materials and different mean stress cases, sharing the same data pool and procedure to determine the fitting parameters, without increasing the calculation cost. Even though being a little complicated than the Walker model, the modified Walker model gives more precise results than the Walker model for all data sets examined, especially in large mean stress cases, showing better accuracy and larger scope of application.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors gratefully acknowledge the support from the China Gas Turbine Establishment.