1 Introduction

There are two basic concepts for mean stress effect evaluating in the stress-life prediction. The concept of the reduced fatigue limit amplitude is a traditional solution, which shifts the S–N curve of the fully reversed load case to a new position based on the value for the reduced fatigue limit amplitude σFL, red. This value is set from the Haigh diagram relevant to the number of cycles at the fatigue limit NFL. Determining the complete reduced S–N curve from the reduced fatigue limit can differ in various applications.

The second approach is in fact much simpler in use. Instead of focusing the Haigh diagram at some given lifetime, it uses the same core formula but with switched meaning of its parameters to compute the equivalent stress amplitude at fully reversed loading. This equivalent stress amplitude should cause the same damage as the cycle described by the stress amplitude and mean stress. The curve in the Haigh diagram goes directly through the loading point, because it is related to the final computed lifetime Nx. The equivalent stress amplitude can be thus immediately used with the S–N curve in fully reversed loading.

Both approaches use the Haigh diagram as the basic part of the solution. Its description by the Goodman formula:

$$\sigma_{a,eq} = \frac{{\sigma_{a} }}{{1 - \frac{{\sigma_{m} }}{{S_{u} }}}}$$
(30.1)

is widely used in the industry for evaluating the mean stress effect (MSE). If conclusions of the relatively recent papers (see e.g. [1,2,3,4,5,6]) are checked, it becomes apparent that the Goodman method leads to unreliable and in most cases to very conservative results. Often, a new MSE model is suggested and proved to be better than the Goodman method, but the test set used for validating remains small, limited to less than 10 materials [2,3,4,5]). If a sufficient number of experimental validation data is used, the data sets contain various issues as discussed e.g. in [7, 8].

The recent work by Dowling et al. [1] is very extensive in comparison with other papers. Its authors compared results for four methods for the MSE inclusion (Goodman, Morrow, SWT and Walker) via the equivalent stress amplitude concept. Goodman, Morrow and SWT methods are strictly defined using the existing material parameters, but the Walker method contains additional fitting exponent:

$$\sigma_{a,eq} = \sigma_{max} \left( {\frac{1 - R}{2}} \right)^{\gamma }$$
(30.2)

By using the Basquin formulation of the S–N curve

$$\sigma_{a,eq} = AN^{b}$$
(30.3)

Dowling et al. transformed the mix of Eqs. (30.2) and (30.3) to the multiple linear formulation

$$\log N = \frac{1}{b}\left[ {\log \sigma_{\hbox{max} } + \gamma \cdot \log \left( {\frac{1 - R}{2}} \right) - \log A} \right]$$
(30.4)

After completing the regression on collected experimental data in this way, the authors compared computed equivalent stress amplitudes with the stress amplitudes for a fully reversed cycle at the same number of cycles for all experimental points. Use of the linear regression leads to the best fit of the Walker parameter to all data of the given data set. Thanks to it, this method achieved best results in [1].

Papuga et al. [8] recently objected that the comparison of the optimized Walker method with other non-optimized methods could not lead to another conclusion. They decided to give a chance also to the generalized Goodman formula marked as the Linear method (see Table 30.1). M parameter was optimized in the same way, though the formulation enforced a non-linear regression analysis. Results showed that both optimized methods clearly exceeded the prediction quality of any other tested method (9 non-optimized methods were scrutinized) and the results for the Walker method exhibited a lower scatter compared with the Linear method. When the proposed estimate for γ was applied to the data, the prediction quality worsened, but it got similar with the Linear method.

Table 30.1 Formulas for evaluated generalized methods, and names of methods, to which they can degenerate with adequately set material parameters
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This paper goes farther with the evaluation, while placing the question: Could some other trend result in a better approximation of the data than the Walker method does?

2 Evaluation and Its Results

2.1 MSE Methods

Table 30.1 lists the methods evaluated within this study. The Walker and the Bergmann methods are similar—they are generalized versions of the SWT parameter. It should be noted that while the Bergmann method can also be transformed to the reduced fatigue limit amplitude concept, the Walker method cannot. Two more methods working with an additional regression parameter are present in Table 30.1—the Linear method that have been already analyzed in [8] and the Kwofie method [4].

It can be anticipated that the additional parameter included in the formulation should bring better overall results compared with any non-optimized methods. This statement was confirmed in [8]. On the other hand, the optimized methods are useful only if their parameters can be easily set. If this is not true, they are of limited practical applicability.

2.2 Used Experimental Data

The data set used here for the analysis conforms to the testing set described in [8]. It comprises 19 different data sets, which refer to 48 different S–N curves, see their conditions in Table 30.2. The only difference is removing the only experimental case from Ra1 data set with negative mean stresses. It does not have any sense to do any conclusions on validity of any method in the compression region on one set only.

Table 30.2 Numbers of data sets establishing the test set used here. See more details in [8]
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2.3 Analysis and the Results

All experimental items in each data set are first used to optimize the sought material parameters. The minimized parameter is the sum of squares of the logarithms of cycles. To compare the ability of each method to conform to the inputted experimental data, this stage is finished by the comparison presented in Table 30.3.

Table 30.3 Mean coefficients of determination R2 for each group of loadings and materials
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Numbers of S–N curves and data points in each set differ. Table 30.3 thus does not describe the prediction capability, it refers only to the ability to follow some trend.

Dowling et al. [1] checked the prediction quality of different methods by comparing the computed equivalent stress amplitude for each experimental point with the stress computed for the same fatigue life from the optimized Basquin curve (Eq. 30.3). Papuga et al. [8] used a similar solution, and final verdicts stated there correspond to it.

There is an issue anyhow. The optimization leading to final material parameters was controlled by the minimum deviation in fatigue life, but not in fatigue strength. If the relevant parameter had been based on differences in fatigue strength, material parameters obtained would differ. The results obtained in [1] and [8] are thus worth a further analysis. The parameter used for the comparison here is the fatigue life error ΔFL:

$$\Delta FL = \frac{{\log (N_{\exp } ) - \log (N_{\text{opt}} )}}{{\log (N_{\exp } )}}$$
(30.9)

Nexp corresponds to the fatigue life measured for the given combination σa−σm. The same stress combination using the formula of the MSE criterion (Table 30.1) provides the σa, eq. Lifetime Nopt is obtained from the Basquin formula with optimized material parameters while using σa, eq. ΔFL is defined in Eq. (30.9) in this way to keep the rule that positive values mean conservative results (shorter lifetime), negative values are non-conservative. All experimental data points in the set (763 items in total) are evaluated.

Table 30.4 presents the output for the comparison in the form of standard deviations of ΔFL parameter. Because the analysis is done with perfect optimized material parameters, it does not have any sense to compare the mean values for ΔFL over the groups, because they are similarly good.

Table 30.4 Standard deviations of ΔFL if evaluated for individual data groups
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3 Discussion

Results in Table 30.4 support the idea that the previous conclusions in [8] were misleading at least partly. The Walker method (and the Bergmann method) are getting close to the ideal trend compared with the Linear method only for aluminum alloys. For steels, this is on contrary. The Kwofie method works well in both situations.

Such conclusions are anyhow applicable only then, if the derived material parameters are either constant, or if they exhibit some clear dependency on widely available parameters. This paper must keep a limited scope, so only tabular values for ranges of optimized parameters could be shown in Table 30.5 (in parentheses), in addition to the derived values for each material group. Any trends are not very well visible, and thus the average values are proposed in most cases. There are more data items among steels, so simple linear trends are proposed (note anyhow that R2 < 0.5 in all cases).

Table 30.5 Proposals for estimating material parameters for individual methods. Ranges of optimized values relevant to individual sets are in parentheses. For steels, the last value in parentheses concerns the only test set loaded in plane bending, values for which differ substantially
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If these estimates are used in the analysis together with the Basquin curves relevant to S–N curves in fully reversed loading, the results start to differ substantially, see Table 30.6. The results of the Linear method worsened—the method is too conservative and exhibits large scatter. A check was done whether it is not due to the single case of plane bending in steels, but the results did not change much, if this case was removed.

Table 30.6 Statistics of ΔFL in the case when the material parameters were estimated
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Kwofie’s method is better regarding the statistics, though it is still weaker than the other two methods. Kwofie [4] expected α parameter to be close 1. Table 30.5 shows that this assumption is quite close for aluminum alloys and cast irons, but not for most steels.

The Bergmann method and the Walker method exhibit very similar behaviour. Similarly to [1] or [8], their performance is to be rated high above all in the case of aluminum alloys.

4 Conclusion

The paper focuses on evaluating the mean stress effect in the high-cycle fatigue life prediction. Four criteria were selected for evaluation through optimizing their material parameters on available experimental data of 19 data sets: the Walker and the Bergmann methods, the Linear method and the Kwofie method. Each of them contains one additional fitting parameter, which was retrieved from the optimization procedure.

To enable practical applicability of the results, the optimized parameters were analyzed and proposals for their estimates stated. Their analyses lead to these conclusions:

  1. 1.

    The Linear method achieves the worst results—too scattered and too conservative.

  2. 2.

    The Kwofie method is closer to the best performing methods, but it remains worse.

  3. 3.

    The methods by Bergmann and by Walker result in a comparable output, and they can be recommended for a general use.

  4. 4.

    More data sets would help in better understanding, whether the observed trends are really universal.