Mathematical Problems in Engineering

Volume 2018, Article ID 5148278, 12 pages

https://doi.org/10.1155/2018/5148278

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

^{1}School of Power and Energy, Northwestern Polytechnical University, Xi’an 710072, China^{2}AVIC Aircraft Strength Research Institute, Xi’an 710065, China

Correspondence should be addressed to Shan Lu; nc.ude.upwn@ulnahs

Received 3 November 2017; Revised 27 January 2018; Accepted 31 January 2018; Published 22 March 2018

Academic Editor: Gregory Chagnon

Copyright © 2018 Shan Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. 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 [1–5]. 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 Walker-like 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 . 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].

#### 2. 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, , as an important parameter together with mean stress and stress amplitude to obtain the equivalent stress , 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 , instead of . In some cases, the true fracture strength may be unavailable. Therefore, an alternate form is proposed by substituting the fitting constant, , the stress intercept at 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 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 and the stress amplitude as parameters to calculate the equivalent completely reversed stress amplitude , 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 , (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 aswhere is the fatigue strength coefficient, is the fatigue strength exponent, and is the fatigue life. Usually, Basquin’s equation is employed in conjunction with different mean stress models to estimate fatigue lives only with being replaced.

#### 3. 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 [15–17, 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.