A Modified Walker Model Dealing with Mean Stress Effect in Fatigue Life Prediction for Aeroengine Disks

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 andmaximum stress onmean 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.


Introduction
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 [1][2][3][4][5].It is discovered that tensile mean stresses are usually detrimental to component fatigue while compressive mean stresses are beneficial in terms of fatigue strength.Besides, it is worth noting that mean stresses have significant effect on fatigue behavior in high cycle fatigue (HCF), while they have less influence in low cycle fatigue (LCF).This is because that large amount of plastic deformation will be generated in the LCF process, which will significantly reduce any beneficial or detrimental effect of the mean stresses [6].
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 [7], Collipriest model [8], McEvily model [9], Forman model [10], and so on.These models are used in different situations and turn out to be effective [11].As to the influence of mean stress on crack initiation life, there are many models proposed, like Goodman model [1], Gerber model, Morrow model [2], Manson-Halford model, the Smith-Watson-Topper (SWT) parameter [3], the Walker equation [5], the generalized energy damage parameter, and so on [12].In this paper, we mainly focus on the effect of mean stress on crack initiation life, and the fatigue lives mentioned in rest of the paper are all crack initiation lives.
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. [12,13], Correia et al. [14], Dowling et al. [15,16], and Burger and Lee [17].Zhu et al. [12,13] have proposed the mean stress effect correction in fatigue life predictions based on energy parameters.Correia et al. [18,19] proposed the generalization of the design fatigue life curve for several fatigue damage parameters including the Walkerlike strain damage parameter and others with mean stress effects.Susmel [20] introduced a mean stress sensitivity index into the modified Wöhler curve method to account for the mean stress effect perpendicular to the critical planes under multiaxial loadings.Niesłony and Böhm [21] proposed a stress-based approach by employing two - curves under alternating stress and stress ratio  = 0. Kujawski [22] made use of analogy with Neuber's rule and proposed a deviatoric version of the SWT model to consider mean stress effect for relatively large compressive mean stress cases.Kamaya and Kawakubo [23] investigated the effects of mean stress on fatigue properties of type 316 stainless steel and indicated that mean stress correction was not necessary in component design under load control mode and for constrained ratcheting strain region.Ince [24] proposed a mean stress fatigue model based on the distortional strain energy to account for the mean stress effect on fatigue life for both positive and compressive mean stress conditions.Besides, researchers have proposed many other equations accounting for mean stress effects, some of which are discussed by Nihei et al. [25] and Kluger and Lagoda [26].

Mean Stress Models in Fatigue
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,  u , as an important parameter together with mean stress  m and stress amplitude  a to obtain the equivalent stress  ar , as is shown in (1).However, the results from the Goodman method is found to be highly inaccurate [15,27], especially for large mean stress cases.
The Morrow equation shares the same form with Goodman's, except for employing the true fracture strength  f , instead of  u .In some cases, the true fracture strength may be unavailable.Therefore, an alternate form is proposed by substituting the fitting constant,   f , the stress intercept at  f = 0.5 cycle of the - curve, to estimate the equivalent completely reversed stress amplitude at various stress ratios [28].Both of the true fracture form and the stress intercept form of Morrow equation are shown in (2a) and (2b).The assumption of   f =  f is often accurate for steels, as shown by Landgraf [29]; however, for aluminums alloys the assumption shows less accuracy, as illustrated by Dowling et al. [15].
The SWT method employs the maximum stress  max and the stress amplitude  a as parameters to calculate the equivalent completely reversed stress amplitude  ar , as is shown in (3).As can be seen, the SWT method has the advantage of simplicity and is not depending on any material constants.Besides, it is found that the SWT model provides good life prediction results in the long fatigue life region but is conservative in the low cycle fatigue life region [30,31].
Unlike the SWT method, the Walker model supposes that mean stress effect is material dependent.Thus, a fitting parameter , which varies from 0 to 1, is employed in the Walker equation to account for the variance of mean stress effect on different materials, as is shown in (4).The advantage of the Walker model is that it provides an opportunity to fit fatigue test data at various mean stresses all together, with  being obtained as part of the fitting process.The higher  is, the less sensitive of the material reacts to mean stress effect.On the contrary, a lower  means that the material is more sensitive to mean stress effect [15].
It can be observed from the comparison between the SWT and the Walker model that when  = 0.5, (4) is seen to reduce to (3), which means that the SWT equation is a special case of the Walker equation.
Basquin's equation [32] describes the relation of fatigue life and the equivalent fully reversed stress amplitude, which can be expressed as where   f is the fatigue strength coefficient,  is the fatigue strength exponent, and  f is the fatigue life.Usually, Basquin's equation is employed in conjunction with different mean stress models to estimate fatigue lives only with  ar being replaced.

The Proposed Model
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  [15][16][17]33].However, while employing the Walker model to estimate fatigue lives of some commonly used materials, such as FGH4095 powder metallurgy superalloy (600 ∘ C) and GH4169 wrought superalloy (650 ∘ C), it is noticed that the estimation results of the Walker equation distinctly changed from conservative to overestimated with the increase of stresses for large mean stress cases, which are shown in Figure 1.That is, for large mean stress cases, the Walker equation gives relatively conservative results in the low stress region; conversely, the estimated fatigue life results become nonconservative while the stresses are large, which certainly brings rather great errors.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   for each stress level  max, is determined by the fatigue data on that stress level from different stress ratios through multiple linear regression, as is shown in (6), where  1 and  2 are the fitting results of Walker model for all data sets and will not change with different stress levels.
It can be obviously seen from Figure 2 that the fitting parameter  varies with the change of logarithmic of maximum stress almost in a linear relationship for both FGH4095 and GH4169 superalloy.Since  acts as an important index of mean stress effect in the Walker model, it can be deduced that maximum stress has nonnegligible influence on mean stress effect as well and should be involved while predicting fatigue life with mean stress effect for better accuracy.Thereupon the modified Walker model is proposed.
The equivalent completely reversed stress amplitude based on the proposed modified Walker model is shown in (7), in which the material dependent fitting parameter  is replaced by + log  max compared with Walker model, with one additional fitting parameter to account for the influence of maximum stress on mean stress effect.Therefore, the modified Walker model not only contains the influence of material on mean stress effect by employing parameter , just as  do in the Walker model, but also reflects the influence of maximum stress through parameter .Taking the logarithm with base of 10 on the combination of ( 5) and (7), the modified Walker model can be expressed as (8)  data sets.Obviously, the parameters in (7) and ( 8) have the following relations:  =  3 / 2 , and  =  4 / 2 . log 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 (7) shares the same form as (4), there is no difference in the multiple linear regressions of fatigue test data at various mean stresses for the Walker model and modified Walker model.The fitting process of the Walker model has been specifically demonstrated by Dowling et al. in [15]; therefore it will not be repeated here. of the Walker model and modified Walker model result from fitting the data sets at all mean stresses.For FGH4095 superalloy (600 ∘ C), it can be recognized in Figure 3(a) that the data of stress ratio  = 0.5 distribute far above the fitted line of data  = −1 in the Goodman relation plot, and the data set of stress ratio  = 0.1 also distribute above the fitted line, but they lie comparatively closer than the  = 0.5 data set, which indicates that the Goodman relationship is applicable to estimate fatigue life for small mean stress cases; however when the mean stress gets larger, Goodman relation becomes highly inaccurate, and the larger the mean stress is, the less the accuracy may become.The results of Morrow equation for  = 0.5 and  = 0.1 data sets distribute below the zero mean stress line.Similarly, the results of  = 0.1 data set lie much closer than that of  = 0.5 data set.In Figure 3(c), reasonable results are obtained by the SWT model, which show better accuracy than the Goodman and Morrow relations.Furthermore, it is discovered that the Walker model provides slightly better results than the SWT model through comparison of Figures 1(a) and 3(c), which can be attributed to the adjustable fitting parameter  by varying the resulting curve to fit data in cases where the SWT method is not very accurate [15].Meanwhile, it is observed that either SWT or Walker model provides conservative results in the low stress region for large mean stress cases, while in the high stress region the opposite happens.Compared with other plots in Figures 1 and 3, it can be clearly observed that the modified Walker model with the consideration of mean stress effect on fatigue life influenced by both materials and maximum stresses gives superior results, which is illustrated by the fatigue test data of all stress ratios in Figure 3(d) which basically equally distribute on both sides of the fitted line.Similar conclusions can be drawn for GH4169 superalloy (650 ∘ C) in Figure 4; only the results of Goodman model distributed much far from the fitted line than Morrow model for data sets of  = 0.5 and  = 0.1.

Validation
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 where  e f is the experimental fatigue life and  p f is the predicted fatigue life of a given data point.Obviously, a positive error means that the prediction is conservative, while a negative error indicates the prediction is overestimated.
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 5 and 6 for FGH4095 (600 ∘ C) and GH4169 (650 ∘ C) superalloy, separately.As clearly seen in Figure 5, the Goodman model tends to provide conservative predictions with positive mean values of prediction error for FGH4095 superalloy (600 ∘ C).On the contrary, the Morrow model tends to overestimate fatigue lives with negative errors.Besides, the standard deviations of Morrow model are comparatively large, which indicates that the prediction accuracy of Morrow model shows less stability.The results of SWT model and Walker model share the same tendency, while the Walker model provides slightly better results with larger probability to locate near the zeroerror region.It can be seen that the modified Walker model shows best performance with mean values around zero and small standard deviations compared with other mean stress models.Particularly for the large mean stress cases like  = 0.5 data set, the modified Walker model provides much better results than other models.
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 [34], TC11 titanium alloy and 35Cr2Ni4MoA high-strength steel [35], and 2014-T6 aluminum [36] are also employed to demonstrate the accuracy of the mean stress models.The material properties and the fitting constants of Basquin's, Walker, and modified Walker equations are shown in Table 1.It should be noted that the parameters in Table 1 are determined by the analysis of fatigue data in corresponding references.
The plots of  ar versus  f by the newly proposed modified Walker model for other materials examined in this study are shown in Figure 7. Besides, the mean and standard deviation values of prediction errors for all the materials examined are shown in Table 2, in which  stands for mean value while  is standard deviation value of the prediction errors.It should be  noted that all the  and  in Table 2 are obtained by all stress ratio data sets.By analyzing plots in Figure 7 as well as the mean and standard deviation values in Table 2 for all the materials examined, it can be seen that the results of candidate mean stress models for the materials examined share the same tendency with FGH4095 and GH4169 superalloy.That is, the Goodman relation gives the most conservative results with positive mean values of error, while Morrow model overestimates the fatigue life for most cases.Besides, the standard deviation values of both Goodman and Morrow are comparatively large, which indicates that the prediction accuracy of these two models varies strongly.SWT model, as a special case of Walker equation, fails to reflect the influence of different materials on mean stress effect and thus provides less prediction accuracy than the Walker equation.The modified Walker model brings the least standard deviation values as well as mean values of error being zero, showing favorable capability to estimate fatigue lives for different materials and different mean stress cases.As to the zero values for  in the Walker and modified Walker models, because both the Walker and modified Walker model determine parameters through multiple linear regression of all stress ratio data sets, which is essentially based on the least squares principle, there is no wander that the mean values of prediction errors for the Walker and modified Walker model become zero.

Comparison of Mean Stress Models with Experimental
Data from Simulated Specimens.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 3 and the finite element (FE) models of the two specimens are shown in Figure 8.
The finite element analysis (FEA) results of the two specimens are shown in Figure 9.It should be noted that Figure 9 shows the contour plot of modified Walker equivalent stress for the two simulated specimens.Since both of the critical regions in the two specimens are obviously notched, the theory of critical distances (TCD) should be employed to properly accounting for the detrimental effect of stress concentration, and the line method (LM) of TCD is found to be an effective method [37]: where  and  are the polar coordinates of the stress point from the critical point, with the stress gradient being the  axis, and  eff is the distance from the notch tip to the minimum of relative stress gradient , as is shown in The predicted fatigue life of the two simulated specimens can be obtained by modified Walker model in conjunction with LM, as is shown in Table 4. Besides, the predicted lives by other models and experimental fatigue lives of the two specimens are also provided for comparison.Obviously, the modified Walker model provides superior results than any other models for both of the two types of simulated specimens.Although the Walker model also gives reasonable fatigue life estimation, it still shows less accuracy for failing to reflect the effect of maximum stress on mean stress effect.Morrow model overestimates the fatigue lives while SWT model generates conservative results for both of the two specimens.

Conclusions
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

Figure 5 :
Figure 5: Probability density function of prediction errors of FGH4095 superalloy (600 ∘ C) for the data set of (a) all data set and (b)  = 0.5 data set.

Figure 6 :
Figure 6: Probability density function of prediction errors of GH4169 superalloy (650 ∘ C) for the data set of (a) all data set and (b)  = 0.5 data set.

Figure 8 :
Figure 8: The finite element model for (a) bolt-hole simulated specimen and (b) run-way hole simulated specimen.

Figure 9 :
Figure 9: 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.

Table 1 :
Material properties and parameters for mean stress models.Numbers refer to the list of references at the end of the paper. a

Table 2 :
Statistical analysis for the prediction errors of different mean stress methods for all materials examined.

Table 3 :
Main geometric parameters for the two simulated specimens of turbine disk.

Table 4 :
The comparison of estimated results of each mean stress models versus experimental data for simulated specimens.