A probabilistic fatigue model based on nonlinear Kohout-Věchet function: Application to 42CrMo4+QT steel

This paper presents a fatigue model designed to provide probabilistic-stress-life (P-S-N) curves at any probability level. Grounded in the Weibull probabilistic model’s framework, the model describes two scenarios: P-Case A, which considers deviations in stress, and P-Case B, centered on deviations in cycles to failure from the median S-N curve. A significant aspect of the proposed model is the incorporation of the Kohout-Věchet function as the representative median S-N curve, allowing for the Kohout-Vechet P-S-N curves across any probability level. To demonstrate the model’s practical relevance, an experimental example focused on specimens made from 42CrMo4+QT steel and subjected to push-pull loading is explored. Not only do the results show the robustness of the model but they also emphasize the role of the characteristic P-S-N curves, especially at lower probability level, in assuring the safety of engineering components.


Introduction
Fatigue testing is pivotal in assessing the reliability and durability of engineering components and structures.Over the years, fatigue models have been vital to deriving stress-life (S-N) curves, which have influenced design methodologies.However, relying solely on the median S-N curve, which refers to a 50% probability of failure, might not be sufficient to guarantee engineering safety.This aspect highlights the importance of probabilistic models, designed to generate P-S-N curves at lower probabilities of failure.Some standards, including ASTM E739-10 and ISO 12107:2003, address the inherent variability in fatigue data, anchoring their methodologies in the classical Basquin function [1] [2] [3].Yet, the Basquin model has a noticeable weakness-it does not accommodate the low and high-cycle regions.To fill in this gap, the Palmgren model has been introduced to capture the S-N curve's nonlinearity across various cycle regions [4] [5].Still, like the Basquin one, the Palmgren model also represents a median S-N curve.To consider S-N curves for other failure probabilities, the Weibull probabilistic model was introduced, offering two distinct cases: P-Case A, considering deviations in stress, and P-Case B, focusing on deviations in cycles to failure from the median S-N curve [5] [6].These cases stand out for their simplicity and efficiency, enabling the creation of P-S-N curves associated with various probabilities of failure.
A significant limitation of the Palmgren function lies in its asymmetry in the log-log fit concerning the inflection point [7].This asymmetry represents a fundamental shortcoming despite the function's adaptable shape across various cycle regions and alignment with empirical tests.Addressing this shortcoming, Kohout-Věchet in [7] introduced a function that not only extends the Basquin model to encompass both low and high-cycle regions, akin to the Palmgren function, but it also demonstrates symmetry in the log-log fit relative to the inflection point.Thanks to its symmetry, the Kohout-Věchet function has an advantage over the Palmgren function, notably it is more suitable for interpolating fatigue curves when there is no fatigue data in a specific region of cycles.Additionally, it provides a better fit for experimental results transitioning to the fatigue limit region and in the quasi-static domain [7].
Despite its advantages, the Kohout-Věchet (K-V) function, likewise the Palmgren and Basquin functions, only represents a median S-N curve.This leaves a particular gap as the comprehensive range of P-S-N curves had not been formulated for the K-V function.Addressing this lack, this paper introduces a probabilistic fatigue model designed to obtain the percentile S-N curves based on the K-V function.
Drawing parallels with Weibull probabilistic model, the proposed model accounts for deviations in both stress and cycles to failure from the median S-N curve.This leads to the distinction of two cases: P-Case A and P-Case B. The key new element of this proposal lies in utilization the Weibull model with the Kohout-Věchet function as its representative median S-N curve.Consequently, this model can produce Kohout-Věchet fatigue curves for any probability of failure.
This model is applied to experimental data from fatigue tests conducted by the authors on unnotched cylindrical specimens of 42CrMo4+QT steel under push-pull loading.

Weibull probabilistic model
The Weibull probabilistic fatigue model, first introduced in [5], is rooted in the Palmgren function.This function is a popular choice for depicting S-N curves and is expressed as: This equation, characterized by parameters, , , the position of the knee of the curve,  1 , and fatigue limit,  ∞ , offers an extensive representation of S-N curves, covering from low to high-cycle fatigue regions.However, a significant limitation of the Palmgren function is its representation as a median S-N curve defined for a 50% probability of failure.This might fall short in assuring engineering safety in certain contexts.Addressing this limitation, the Weibull probabilistic model introduces two distinct cases, P-Case A and P-Case B, each discussed in the subsections below.These cases are characterized by their simplicity and effectiveness, offering P-S-N curves associated with any probabilities of failure.
For both A and B cases, the probability of failure adheres to the three-parameter Weibull distribution: where   = () is also the cumulative distribution function (CDF).The Weibull parameters  (scale),  (location) and  (shape) dictate the distribution's characteristics.For the Weibull distribution to be valid, it is essential to adhere to few constraints:  ≥ , −∞ <  < ∞,  > 0 and  > 0. These requirements are consistently considered throughout this paper.

P-case A: Deviations in stress
P-Case A centers its attention on the difference between the observed stress value and the median S-N curve provided by the Palmgren function in Equation (1).The variation from this median S-N curve is characterized as: Substituting ∆ for the random variable  in the Weibull CDF in Equation ( 2), P-Case A can be expressed in terms of the probability of failure.This approach incorporates the inherent variability of stress deviations from the mean value into the probabilistic model, offering a more comprehensive measure of probability of failure than a deterministic S-N curve alone.

P-case B:
Deviations in cycles to failure P-Case B, in contrast, emphasizes the deviations of cycles to failure from the median S-N curve.The deviation in cycles from this curve is defined by: By associating ∆ log( +  1 )  with the random variable  in Equation ( 2), P-Case B is expressed in terms of the probability of failure.
It is important to highlight that Weibull in [5][6] did not included 'run-out' fatigue data in either Pcase A or B. For this reason, run-out data will also be omitted in our analysis.

Kohout-Věchet function
The Kohout-Věchet (K-V) function, defined by parameters  ∞ ,  1 ,  2 and , is formulated as: Both exponent, , and the knee point,  1 , share a similar meaning with the Palmgren function in Equation (1), whereas cycle bend point,  2 , indicates the bend's position between the sloping and lower horizontal sections of the fatigue curve.These parameters,  ∞ ,  1 ,  2 and , are estimated by minimizing the sum of squares [7].
The K-V function has some advantages over other functions, including the Palmgren function.While the latter covers a range from tensile strength to fatigue limit, it lacks symmetry in the log-log fit or in another fit with simple transformation [7].This asymmetry impedes its ability for interpolating fatigue curves when there is no fatigue data in a specific region of cycles.Conversely, the K-V function is symmetrical in log-log fits around its inflection point.Not only does this symmetry ensure superior accuracy in data interpolation but it also provides a more accurate fit for experimental results in the transition to the fatigue limit and in the quasi-static domain.
Despite its advantages, the K-V function only represents a median S-N curve.The whole family of percentile curves, so typical for the commonly used power-law S-N functions, was not derived for the K-V function till now.To overcome this limitation, this study proposes a probabilistic fatigue model to obtain the P-S-N curves for the K-V function.

Proposed probabilistic model
Based on the Weibull probabilistic model, the proposed model adapts the fundamental assumptions made by Weibull about deviations in stress and cycles to failure from the median S-N curve.As a result, this model also divides into P-Case A and P-Case B. However, the distinguishing element of this proposal regarding the Weibull probabilistic model lies in its utilization of the Kohout-Věchet function as its representative median S-N curve.An in-depth examination of both cases follows.

P-case A: Deviations in stress
The deviation in stress is established by the difference between the observed stress value and the median S-N curve, which in this context is represented by the Kohout-Věchet function.Mathematically, this deviation can be represented as: Upon substituting  with ∆ in Equation ( 2), the probabilistic equation is derived: This equation is subject to the constraints consistent with the Weibull distribution, i.e.,  − For effective model application, it is crucial to first estimate the K-V function parameters:  ∞ ,  1 ,  2 and  by a least-square fitting procedure.Subsequent to this, it is possible to determine the random for  = 1,2, … , , where   and   are the stress and number of cycles to failure of the -th specimen tested, and  denotes the sample size.Once this is obtained, the parameters ,  and  can be estimated by maximizing their log-likelihood: The log-likelihood maximization method has been recognized for its effectiveness in probabilistic model parameter estimation, showcasing consistent results in various studies [8][9].

P-case B: Deviations in cycles to failure
For the P-case B, the attention is shifted towards the deviations in cycles to failure from the median S-N curve.This is expressed as: Following the P-case A,  is substituted for ∆ log ( in Equation ( 2) to derive the corresponding probabilistic equation: Equation ( 10) adheres to the Weibull distribution constraints, i.e., log ( + 1 + 2 ) − log  + log  ∞ ≥ , −∞ <  < ∞,  > 0 and  > 0.
To extract the model's parameters, one can adopt the approach used in P-case A. The only difference lies in the selected random variable, given by the values   = log (

Experimental application
This section demonstrates the practical application of the proposed model in estimating the P-S-N curves for different specimens through the analysis of fatigue data.The data herein is taken from the literature, providing an initial validation for the proposed model.Importantly, while the data is drawn from [10], [11] and [12], the actual fatigue tests were conducted at the Czech Technical University in Prague under the direct guidance of the authors.
The specimens tested were unnotched cylindrical specimens subjected to push-pull loading.They were made from 42CrMo4+QT high-strength steel.Four distinct geometries (series S1 to S4) were manufactured, see Figure 1.The Amsler HFP422 resonator fatigue machine was utilized for the testing.Specimens experienced fully reversed push-pull loading set at a resonant frequency corresponding to the unique specimenmachine pairing.This approach ensured consistent test frequencies within each series.Yet, between the series, frequencies differed between 100 and 160 Hz, depending on the specimen geometry.
A frequency drop of 5 Hz determined the end of each test.If the specimens remained unbroken after stopping the test or if no clear fatigue cracks were apparent, a dye penetrant check was employed to detect any latent cracks.For a more comprehensive understanding of the experimental methodology, readers are encouraged to refer to [10].
The fatigue data are depicted as markers in Figure 2  Upon applying the proposed model as per Section 4, both the K-V function parameters and the Weibull parameters for P-case A and B were determined, see Table 1 and Table 2.In Table 1, the S4 series displays the lowest values for parameters  ∞ and  1 , and highest values for  2 and , indicating that its P-S-N curves tend to sit at the bottom left compared to other series.In Table 2, for the S2 series, the Weibull parameters, ,  and , for both P-case A and B, approach zero, signifying a tighter distribution than the other series.After parameter estimation, the next step is to check the model's fitting accuracy.The Kolmogorov-Smirnov test, applied with a 5% significance level, serves this purpose [8].It measures the fit between the hypothesized Weibull CDF in Equation ( 9) for P-case A and Equation (10) for P-case B against the empirical CDF [13].The null hypothesis, suggesting no significant variance between the variables, is accepted or rejected based on the p-value's relation to the 5% significance threshold.
The test results, although not detailed here, revealed p-values over 5% for all series, often exceeding 60%.Consequently, all series, inclusive of both P-case A and B, were selected for subsequent analysis of the P-S-N curves.
Figure 2 illustrates the P-S-N curves for series S1 and S2, estimated excluding run-outs, for various probabilities of failure:   =2.3%, 50%, and 97.7%.The median P-S-N curve for   =50%, aligns perfectly for both P-Case A and B -seen as an overlapping dashed line.The fit between these median P-S-N curves and the available data is reasonable.Additionally, these median P-S-N curves capture the nonlinearity of the fatigue curves across different cycle regions, particularly the transitions from the upper horizontal to the sloping parts of the curves, as well as between the sloping sections to the lower horizontal (fatigue limit).Examining all the P-S-N scatter bands, their widths appear visually similar between P-case A and Pcase B. However, P-Case A shows a variability in width over cycles to failure.This is confirmed by the scatter index 1   ⁄ , representing the ratio between stress amplitudes for   =97.7% and 2.3% [14], that differs with cycles to failure only for P-Case A. Another observation is that the characteristic P-S-N curve, typically founded on the lower probability of failure,   =2.3%, reveals that all data points in Similarly, Figure 3 displays the P-S-N curves for series S3 and S4 for the same probabilities of failure.While the median P-S-N curve remains identical for both cases P-case A and B, the transitions between the curve segments are not as evident for the S4 series as they are for the S3.For series S3 and S4, the observations concerning P-S-N scatter band widths and changes in width over cycles to failure in P-Case A reflect those from series S1 and S2.The characteristic P-S-N curve based on the lower probability of failure,   =2.3%, continues to reflect that all data in Figure 3 in favour of a conservative side, indicating a safe estimation.

Conclusion
The paper has proposed a probabilistic fatigue model designed to generate P-S-N curves at any probability level.Based on the Weibull probabilistic model framework, the proposed model considers two cases: P-Case A addressing deviations in stress, and P-Case B, focusing on deviations in cycles to failure from the median S-N curve.A noteworthy aspect of the proposed model is the incorporation of the Kohout-Věchet function as the representative median S-N curve, allowing the user to define the Kohout-Věchet fatigue curves at any probability of failure.
The application of the proposed fatigue model was demonstrated through experimental fatigue data collected in our previous study.Data refer to fatigue tests on unnotched cylindrical specimens from 42CrMo4+QT steel subjected to push-pull loading.The outcomes of this application led to important insights: • Across all specimen series, there was a notable fit between the P-S-N curves and the fatigue data, as confirmed both visually and by the Kolmogorov-Smirnov test.• The characteristic P-S-N curve, defined at probability of failure   =2.3%, was to the left of all data points, showing that all data points fell on the conservative side considering the P-case B (deviations in cycles to failure from the median S-N curve).• The scatter index 1   ⁄ , which refers to the ratio between stress amplitudes for   =97.7% and 2.3%, is not constant thus highlighting a scatter band of a variable width over cycles to failure occurring only in P-Case A (deviations in stress from the median S-N curve).

References
[1] ASTM E739-10 2015 Standard practice for statistical analysis of linear or linearized stress-life

Figure 1 .
Figure 1.Standard cylindrical solid specimens with dimensions of the central part.Configurations range from S1 to S4.

and Figure 3 .
Specifically, Figure 2(a) presents the S1 series, marked with circles; Figure 2(b) showcases the S2 series, indicated by squares; Figure 3 (a) represents the S3 series, using upward-pointing triangles; and Figure 3(b) highlights the S4 series, denoted by diamonds.Each series contains approximately 10 samples, which aligns with the 6-12 specimens recommended by ASTM E739-10 for preliminary and exploratory fatigue research [1].

Figure 2
Figure2lie on the conservative side.This implies safe estimations, with only one exception: a data point from the S1 series in P-Case A.Similarly, Figure3displays the P-S-N curves for series S3 and S4 for the same probabilities of failure.While the median P-S-N curve remains identical for both cases P-case A and B, the transitions between the curve segments are not as evident for the S4 series as they are for the S3.

Figure 3 .
Figure 3. Fatigue data and P-S-N curves for probabilities of failure   = 2.3%, 50%, and 97.7%:(a) pertains to the S3 series, and (b) to the S4 series.

Table 1 .
The Kohout-Věchet function parameters correspond to each specimen series.

Table 2 .
The Weibull parameters correspond to P-case A and B of the proposed model.