Modeling metallic fatigue data using the Birnbaum–Saunders distribution

This work employs the Birnbaum–Saunders distribution to model the fatigue life of metallic materials under cyclic loading and compares it with the normal distribution. Fatigue-limit models are ﬁtted to three datasets of unnotched specimens of 75S-T6 aluminum alloys and carbon laminate with diﬀerent loading types. A new equivalent stress deﬁnition that accounts for the eﬀect of the experiment type is proposed. The results show that the Birnbaum–Saunders distribution consistently outperforms the normal distribution in ﬁtting the fatigue data and provides more accurate predictions of fatigue life and survival probability.


Introduction
Fatigue-life prediction is vital to preventing the failure of mechanical parts under cyclic loading.Stresslife models, or S-N curves, are usually used to model fatigue life [1,2,3].Although many models relate stress to the fatigue life, with probabilistic models, the fatigue life is often assumed to be a log-normal random variable [4,5,6,7] or to follow the Weibull distribution [4,8].A comprehensive review of plausible models for fatigue life is available in [9].For a specified stress-life model, a corresponding fatigue strength model can be induced and then used to make predictions [10].
This work considers Birnbaum-Saunders distributions [11] that were introduced to model fatigue failure time under cyclic loading.We aim to study the use of Birnbaum-Saunders distributions and analyze their fit results compared with the dominant choice of normal/log-normal distributions.Thus, we use two datasets of fatigue experiments applied to specimens of 75S-T6 aluminum alloys [12,13] and a dataset of bending tests of carbon laminate [14].Moreover, we study the effect of equivalent stress model when fatigue data includes different types of experiments.
Birnbaum-Saunders distributions were introduced as a two-parameter family of life distributions [11].Several studies have used these distributions to fit fatigue datasets using the maximum likelihood (ML) [15,16] and Bayesian methods [17,18,19].In addition, many variations have been proposed, such as the log-linear model for the Birnbaum-Saunders distribution [20] and bivariate log Birnbaum-Saunders distribution [21].An extensive review of the Birnbaum-Saunders distribution and its generalizations is provided in [22,23].
For the S-N models, many possible regression models could be considered.We focus only on the fatiguelimit models with constant and nonconstant variance (or the shape parameter).The fatigue-life variable is modeled by the log-normal distribution or Birnbaum-Saunders distribution.Equivalently, the logarithm of N is modeled by the normal and sinh-normal distributions, respectively.However, we demonstrate that modeling log(N ) as a Birnbaum-Saunders distribution improves the fit results.This proposed model is unprecedented in the literature, to the best of our knowledge.
The first dataset is the same data considered in [4], where fatigue-limit models and random fatiguelimit models were calibrated using the normal and Weibull distributions.We recalibrate fatigue-limit models using the Birnbaum-Saunders distribution.In addition, we propose a new equivalent stress model that accommodates different experiment types in Dataset 1.For Dataset 2, fatigue data corresponds to rotating-bending experiments applied to round bar specimens with different minimum-section diameters [13].Again, we calibrate fatigue-limit models and compare the fit of the normal and Birnbaum-Saunders distributions.As Dataset 3, we use the laminate panel data [14,5], and calibrate and compare our proposed models.
The results show that modeling log(N ) by mean of Birnbaum-Saunders distribution improves fitting systematically in all three datasets and using different variations of fatigue-limit models.It is also expected that such a choice would improve fitting with different S-N models.However, it is not our goal to find the best model for each dataset.For Dataset 1, our proposed equivalent stress also improves data fitting using different models and distributions.
This paper is organized as follows.Section 2 considers the fatigue data for unnotched sheet specimens of 75S-T6 aluminum alloys.Six variations of fatigue-limit models are introduced in Section 2, comparing stress-life models fitted to the data using the normal and Birnbaum-Saunders distributions.Next, Section 3 proposes a new equivalent stress definition to eliminate the effect of the experiment type.Then, Section 4 presents fatigue data corresponding to the unnotched round bar specimens of different sizes, followed by calibration and model comparison.Then, laminate panel data are fitted and analyzed in Section 5 using the predefined fatigue-limit models.Finally, the conclusions are presented in Section 6.

Description of Dataset 1
Dataset 1 consists of 85 fatigue experiments that applied constant amplitude cyclic loading to unnotched sheet specimens of 75S-T6 aluminum alloys [12,Table 3,.The following data are recorded for each specimen: • maximum stress, S max , measured in ksi units; • cycle ratio, R, defined as the minimum to maximum stress ratio.The ratio R is positive when the experiment corresponds to tension-tension loading and negative when the experiment corresponds to tension-compression loading; • fatigue life, N , defined as the number of load cycles at which fatigue failure occurred; and • a binary variable (0/1) to denote whether the test stopped before failure (run-out).
In 12 of the 85 experiments, the specimens remained unbroken when the tests were stopped.

Fatigue-limit models
The fatigue life should be modeled for a stress quantity defined for any cycle ratio.Therefore, we use Walker's model to define the equivalent stress: where q is a fitting parameter.In the upcoming sections, we consider fatigue-limit models where the location parameter is given by A 1 + A 2 log 10 (S eq − A 3 ) and the fatigue-limit parameter A 3 is a threshold parameter where fatigue life becomes infinite when the equivalent stress is lower than A 3 .Multiple fatiguelimit models could be created based on the choice of the distribution of the fatigue life, N .We consider three choices as follows.

Model Ia
In Model Ia, we assume fatigue life is modeled using a log-normal distribution, or equivalently, that log 10 (N ) is modeled with a normal distribution with a mean of µ(S eq ) = A 1 + A 2 log 10 (S eq − A 3 ) , if S eq > A 3 (2) and a constant standard deviation of σ(S eq ) = τ .Moreover, fatigue experiments are assumed independent, and run-outs are modeled using the survival probability.Thus, the likelihood function for Model Ia is given by (10) g(log 10 (n i ) ; µ(S eq ) , τ ) where g(t; µ, σ) , Φ is the cumulative distribution function of the standard normal distribution, and Remark.Model Ia and the likelihood function (3) have been used in [Babuška, Ivo, et

Model IIa
For Model IIa, we assume fatigue life is modeled using the Birnbaum-Saunders distribution, or equivalently, that log(N ) is modeled with a sinh-normal distribution [23] with a constant shape parameter α, a scale parameter of 2, and a location parameter µ(S eq ) given by (2).Under this assumption, the likelihood function for Model IIa is given by , y > 0, and α, µ > 0 .

Model IIIa
For Model IIIa, we assume log 10 (N ) is modeled using a Birnbaum-Saunders distribution with a constant shape parameter α and a location parameter µ(S eq ), given by (2).The resulting distribution for N is not the so-called log Birnbaum-Saunders distribution reported in [23].However, the distribution of N is obtained similarly to deriving the log-normal distribution.(10) k(log 10 (n i ) ; α, µ(S eq )) , y > 0, and α, µ > 0 .The three models are now fitted to Dataset 1 by maximizing the mentioned likelihood functions.Numerically, only the log-likelihood can be evaluated, and we maximize the log-likelihood instead.The ML estimates (MLEs) and the maximum log-likelihood value are reported in Table 1.The estimated parameters for Model IIa have a different scale compared with those in Models Ia and IIIa.The change in scale is only because Model IIa models log(N ) instead of log 10 (N ).However, all applied likelihood functions are normalized; therefore, the performance of the fit is not affected by the logarithm base selection.The results in Table 1 indicate that Models Ia and IIIa provide the best fit for Dataset 1.We visualize the fit using the 0.05 and 0.95 quantile functions and the median function.The data can also be plotted given the MLE of q.We distinguish data based on the experiment type or stress ratio.Figures 1 and 2 illustrate the quantile functions of Models Ia and IIIa obtained using the MLE parameters.The quantile functions of Model IIIa produce a better fit than those of Model Ia, which coincides with the fact that Model IIIIa has the highest log-likelihood value among the three models in Table 1.We also observe that data seem segregated by the median according to the experiment type: tension-tension (R > 0) and tension-compression (R < 0).We analyze this behavior further in Section 3.
We allow the standard deviation or shape parameter to be nonconstant to improve the fit of the three previous models.In particular, we assume this parameter is a function of the equivalent stress.With the same probability distributions previously considered, we introduce three new fatigue-limit models with nonconstant standard deviation/shape parameters.

Model Ib
Analogous to Model Ia, for Model Ib, we assume log 10 (N ) has a normal distribution with the mean function µ(S eq ) defined in 2. However, the standard deviation is assumed nonconstant and given by σ(S eq ) = 10 (B 1 +B 2 log 10 (Seq )) .The resulting likelihood function is equivalent to that derived for Model Ia.

Model IIb
For Model IIb, the fatigue life N is modeled by the Birnbaum-Saunders distribution with location parameter µ(S eq ) (defined in 2) and nonconstant shape parameter α(S eq ) = 10 (B 1 +B 2 log 10 (Seq)) .

Model IIIb
For Model IIIb, log 10 (N ) is modeled using the Birnbaum-Saunders distribution with the location parameter µ(S eq ) and nonconstant shape parameter α(S eq ) = 10 (B 1 +B 2 log 10 (Seq)) .The new Models Ib, IIb, and IIIb are calibrated to fit Dataset 1, and the MLEs of the parameters of these models are presented in Table 2. Comparing the maximum log-likelihood values in Tables 1 and  2 reveals that the fit improved considerably for all models.In contrast, the difference between the new models decreased, with Model IIIb still providing the best fit.
We compare the fit of Models Ib and IIIb using the quantile functions in Figures 3 and 4. Both models produced significantly improved fit compared with Figures 1 and 2. The fatigue-limit parameter slightly increased, and the 0.05 quantile converges rapidly to its asymptote as the equivalent stress approaches the fatigue limit.In both cases, the data remain mostly partitioned by the median into the two experiment types.

Model comparison
Using a classical approach, we compute some popular information criteria, such as the Akaike information criterion (AIC) [24], Bayesian information criterion (BIC) [25,26], and AIC with correction [27], which are based on the maximized log-likelihood values.Such measures consider the goodness of fit and complexity of the models regarding the number of parameters.Table 3 contains the maximum log-likelihood values corresponding to the models introduced in Section 2.2 and the classical information criterion computations.

Analysis of the stress ratio effect and equivalent stress for Dataset 1
In all previous models, the equivalent stress is based on Walker's model [28], which is S max (1 − R) q .The following analysis in Table 4 reveals that the parameter q is related to the sign of the cycle ratio R.    In Table 4, the fatigue-limit parameter, A 3 , has a different scale based on the stress ratio.We divided 1 − R by 2 in the equivalent stress formula to solve this.However, the estimated values of q do not change.Therefore, we defined the equivalent stress as S max ( 1−R 2 ) 1+q or S a ( 1−R 2 ) q , where S a denotes the stress amplitude.Then, we recalibrated the proposed models in Table 5.The fatigue-limit parameter has the same scale for R > 0 and R < 0. In contrast, the estimated value of q changes signs with R. Thus, it seems reasonable to propose the following equivalent stress: Next, we calibrate the parameters using the full data (R > 0 and R < 0).Table 6 presents the MLEs of Models Ia, IIIa, Ib, and IIIb, along with the maximum log-likelihood and AIC values.The fit is considerably improved in all cases using the equivalent stress 4.
Figures 5 and 6 illustrate the new quantile functions of Models Ia and IIIa with improved equivalent stress 4. The variance is reduced compared to quantiles in Figures 1 and 2. In addition, the two data types are well distributed around the median.Furthermore, the fit can be slightly improved by adapting Huang's model [29] for R > 0.5.

Profile likelihood
We compare the profile likelihood of the fatigue limit obtained using the previous models.Figure 9 depicts the profile likelihood of the fatigue limit, A 3 , using Models Ia and IIIa.For constant variance and shape parameters, the estimated profile likelihood using the Birnbaum-Saunders distribution (Model IIIa) has a noticeably higher mode and a lower variance than Model Ia, which uses the normal distribution.When adopting nonconstant variance and shape parameters, the difference between the two profile likelihoods is negligible, as displayed in Figure 10.

Survival functions
We closely examined the survival functions obtained by calibrated Models Ia, Ib, IIIa, and IIIb at different values of S max and R in Figure 11.The Birnbaum-Saunders model (Model IIIa) outperformed the counterpart Gaussian model (Model Ia) because it offers a higher survival probability before the observed failure and a lower survival probability after the observed failure.For Models IIIa and IIIb, the resulting survival probabilities are almost identical for both distributions.

Description of Dataset 2
This section introduces and studies new datasets for unnotched specimens of 75S-T6 aluminum alloys [13].These datasets correspond to 101 round bar specimens with five minimum-section diameters.In Dataset 2, the fatigue experiments are rotating-bending, and the stress ratio is −1.Out of the 101 specimens, 13 experiments are run-outs.Again, we consider fatigue-limit Models Ia, Ib, IIIa, and IIIb with the new equivalent stress (4) to fit the data introduced in 4.1.As mentioned, the stress ratio for rotating-bending experiments is −1; therefore, the equivalent stress equals S max .Table 7 provides the MLEs for Models Ia, Ib, IIIa, and IIIb when separately fitting Specimen 1 and 2. The joint fit for all specimens is also provided.The goodness of fit and estimated fatigue limit decreased when the data were combined.Figures 12 and 13 reveal the quantiles of calibrated Models Ia and Ib, respectively.

Profile likelihood and confidence intervals
We again compare the profile likelihood of the fatigue limit obtained from the four previous models using Dataset 2. Figure 14 displays the profile likelihood of the fatigue limit, A 3 , using Models Ia and IIIa.As concluded, the profile likelihood when using the Birnbaum-Saunders distribution (Model IIIa) has a higher mode and much lower variance than Model Ia.With nonconstant variance and shape parameters, the two profile likelihoods are almost identical 10.
We confirmed the mentioned conclusions by estimating the confidence intervals of the pooled MLEs, given that Dataset 2 is complete.The confidence intervals presented in Table 8 are obtained by stratified bootstrapping where the sampled dataset maintains the same proportions in the original data related to the five specimens.The results indicate that the Birnbaum-Saunders distribution provides tighter confidence intervals than the normal distribution, especially when using a constant variance.This property is essential to generate accurate survival and failure predictions.

Survival functions
Figure 16 depicts the survival probabilities of specimens from Dataset 2 under different settings using calibrated models by the pooled or specific specimen data.
Comparing the results for Datasets 1 and 2, we notice higher variability in the latter, especially when using pooled Dataset 2 with multiple specimens of different geometries and sizes.The fit results could be improved using Poisson models that consider the geometry and size of the specimen [30].However, implementing and analyzing such models is beyond the scope of the current work.

Description of Dataset 3
To generalize the previous results, we consider a well-known dataset: the laminate panel S-N dataset [14,5].This dataset contains fatigue data for 125 carbon eight-harness-satin/epoxy laminate specimens subjected to four-point out-of-plane bending tests, where 10 out of 125 experiments are run-outs.In this case, the equivalent stress needed in the fatigue-limit models is given directly in the data, and we do not have the stress ratio.
As a first illustration, we used probability plots, as suggested in [10].Figures 17 and 18 present the probability plot of the normal distribution and Birnbaum-Saunders distribution as models for the fatigue life, N .Modeling N using the normal or Birnbaum-Saunders distribution is not a good choice.Instead,  modeling log(N ) using these distributions provides better probability plots, as illustrated in Figures 19  and 20.We calibrated six fatigue-limit models: Ia, Ib, IIa, IIb, IIIa, and IIIb, slightly modified using natural instead of base 10 logarithms.This approach was conducted to make the MLE parameters comparable to the results in the literature and does not affect the goodness of fit.We also fit the data using the Weibull distribution but did not include these results, as this distribution consistently provides the worst fit.

Conclusions
Multiple variants of the fatigue-limit models were calibrated and ranked employing ML and classical information criteria.The proposed approach of modeling the logarithm of the fatigue life using the

Figure 10 :
Figure 10: Profile likelihoods of the fatigue-limit parameters using Models Ib and IIIb.

Figure 11 :
Figure 11: Survival functions of Dataset 1 specimens using calibrated Models Ia, Ib, IIIa, and IIIb for different values of Smax and R.

Figure 14 :
Figure 14: Profile likelihoods of the fatigue-limit parameters using Models Ia and IIIa.

Figure 15 :
Figure 15: Profile likelihoods of the fatigue-limit parameters using Models Ib and IIIb.

Figure 17 :
Figure 17: Probability plot of the normal distribution as a model for the number of cycles N .

Figure 18 :
Figure 18: Probability plot of the Birnbaum-Saunders distribution as a model for the number of cycles N .

Figure 19 :
Figure 19: Probability plot of the normal distribution as a model for log(N ).

Figure 20 :
Figure 20: Probability plot of the Birnbaum-Saunders distribution as a model for log(N ).

Table 1 :
Maximum likelihood estimates for Models Ia, IIa, and IIIa

Table 2 :
Maximum likelihood estimates for Models Ib, IIb, and IIIb

Table 3 :
Classical information criteria.

Table 4 :
Maximum likelihood estimates for Models I and II with Seq

Table 5 :
Maximum likelihood estimates for Models I and II with Seq

Table 6 :
Maximum likelihood estimates for Models I and II with Seq

Table 7 :
Maximum likelihood estimates for Models I and III with Seq = Smax.

Table 8 :
Confidence intervals of 90% for the pooled maximum likelihood estimates for Models I and III.
Survival functions of Dataset 2 specimens using calibrated Models Ia, Ib, IIIa, and IIIb for values of Smax and R.
Table 10 compares all six models employing classical information criteria.Figures 21 and 22 present the quantiles of calibrated Models Ia and IIIa, respectively.Figures 23 and 24 display the quantiles of calibrated Models Ib and IIIb, respectively.

Table 9 :
Maximum likelihood estimates for Models I, II, and III.