Development of a Probabilistic Model for the Prediction of Fatigue Life in the Very High Cycle Fatigue (VHCF) Range Based on Inclusion Population

The aim of the present work is to develop a statistical approach for the correlation between the quality of metallic materials with respect to the size and arrangement of inclusions and fatigue life in the VHCF regime by using the example of an austenitic stainless steel AISI 304. For this purpose, the size and location of about 60000 inclusions on cross sections of AISI 304 sheet in both longitudinal and transversal directions were measured and subsequently modeled using conventional statistical functions. In this way a statistical model of inclusion population in AISI 304 was created. The model forms a database for the subsequent statistical prediction of inclusion distribution in fatigue specimens and the corresponding fatigue lives. By applying the extreme value theory the biggest measured inclusions were used in order to predict the maximum inclusion size in the highest stressed volume of fatigue specimens and the results were compared with the failure-relevant inclusions. The location of the crack initiating inclusions was defined based on the modeled inclusion population and the stress distribution in the fatigue specimen, using the probabilistic Monte Carlo framework. Reasonable agreement was obtained between modeling and experimental results.


Introduction
In the VHCF range the cyclic strength of metallic materials containing nonmetallic inclusions is predominantly determined by the size and location of the inclusions. They arise during manufacturing processes and result in a localized distribution of plastic strain in isolated microstructural regions at VHCF relevant load amplitudes, and their presence and influence are hardly predictable analytically. Owing to the small size of the nonmetallic inclusions in modern clean steels amounting to tens of microns [1,2] the current non-destructive analysis techniques are not able to reliably detect even their presence in material structure [2]. Thus, the prediction of fatigue behavior of a material containing randomly distributed microstructural defects can only be achieved using statistical methods together with efficient metallographic sampling strategies.
Numerous investigations showed that fatigue properties of a given material volume containing randomly distributed small defects while the material is subjected to uniformly distributed cyclic loads are related not to the average defect size but rather to the size of the maximum inclusion in the material volume [3,4]. On the basis of extreme value statistics Murakami and co-workers [3] developed a rating method for clean steels based on the observation of the largest inclusions in a defined area. By means of this method the size of maximum defect, which is assumed to be relevant for failure of a tested material volume, can be predicted. However, the present study reveals that the largest defect, which is relevant for the failure (initiation defect), measured in a batch of specimens can be considerably larger than the one predicted by means of Murakami's method. Moreover, current investigations of fatigue behavior on metallic defect-afflicted materials in the area of VHCF [5,6] show that fatigue life also relates to the location of initiation defects. This assertion can be confirmed by results for the fatigue behavior of the austenitic stainless steel AISI 304. The steel was subjected to solution annealing treatment and subsequently prestrained in order to induce 60% deformation-induced martensite volume fraction. Hence, the texture effect on fatigue life was assumed to be negligible. However, due to the difference in size and form of inclusions, the fatigue life of the specimens stressed parallel and transversal to the rolling direction is different (Fig. 1). Fig  2 shows the location of initiation inclusions in the cross section of fatigue specimens. About 50% of the inclusions are situated at the specimen surface and are related to a shorter fatigue life compared to crack initiation at interior inclusions.

Creating a data base for statistical predictions
The data base for statistical inferences about the fatigue behavior of AISI 304 was created on basis of metallographic observations on 80 plane 2 ᵡ 14 mm sample cross sections. Hence, the standard inspection area S 0 [3] was chosen to be 28 mm 2 for this study. The samples were cut from a sheet of AISI 304 in the fully austenitic condition parallel and perpendicular to the rolling direction ( Fig. 3), than mechanically ground, polished and subsequently scanned using a confocal 3D measuring laser microscope OLS4000 with a 100 magnification lens. Size and distribution of all inclusions in the obtained micrographs were defined by means of the image processing and analyzing software ImageJ. In order to express the 2-dimensional inclusion size, the parameter area [3], which is the square root of the projected area of inclusion, was used. Examples of the maximum inclusion max area measured within each standard inspection area are provided in Fig. 4. Presumably, the typical elongated and disintegrated shapes of the largest inclusions measured parallel to the RD result from the hot rolling process and favors fatigue crack initiation.

11th International Fatigue Congress
All defined max area data were modelled using Gumbel extreme value (EV) distribution. The Gumbel cumulative distribution function (cdf) is defined as with λ = 14.48, δ = 2.02 and λ = 10.37, δ = 1.52 being the location and scale parameters of the Gumbel distribution function (df) fitted to the defined max area data parallel and perpendicular to RD, respectively. The distribution of the largest inclusions within all analyzed inspection areas max area in the Gumbel probability paper is shown in Fig. 4. The presented Gumbel plot confirms the efficacy of the Gumbel df for the max area data for both sampling directions. Distributions of all measured inclusions along both RD and ND that are larger than fixed threshold sizes chosen to be 0, 6 and 12 µm are presented in Fig. 5. While the inclusion distributions in RD and TD (not presented) do not show any explicit dependence on the threshold value and can be assumed to be uniform, the plots of relative frequencies relating to larger inclusions measured in ND illustrate the tendency of larger inclusions to concentrate in the centre of examined specimens. Presumably, this phenomenon results from segregation processes during the ingot solidification progress [7] and is supported by the hot rolling process that is also responsible for the inclusion shape shown in  Because fatigue failures predominantly initiate at larger inclusions [3], only the size and location of larger inclusions exceeding some fixed size were modeled. The generalized Pareto df was proven by different authors [2,5,8] to be a reliable model for describing excesses of a random variable, such as size of the measured inclusions, above a given threshold value. The optimal threshold value for the presented data was chosen according to [8] and assumed to be 12 µm. The generalized Pareto cdf is defined as with x lim being the threshold value, λ = 1.71 being the scale parameter and k = 0.0571 being the shape parameter that were fitted to the measured inclusion size data exceeding the x lim value. In order to model the location of large inclusions exceeding the optimal threshold value (last column of Fig. 5) the uniform df for RD and TD as well as the Cauchy df for ND were selected. The Cauchy df enables to adequately describe the abrupt increase of relative frequency of large inclusions from corners to the middle in ND and has a minimum mean square deviation from the measured data. The Cauchy cdf is defined as Advanced Materials Research Vols. 891-892 with the location parameter t and the scale parameter s being equal 978 and 153 for the measured inclusions exceeding the chosen 12 µm threshold size value. Fig 6 and 7 indicate the efficacy of the chosen Pareto and Cauchy dfs to describe the size and location distributions of the larger inclusions, respectively.

Prediction of maximum inclusions in fatigue specimens
One of the important features of the inclusion rating method based on extreme value statistics is the possibility to use a distribution of maximum inclusions obtained within a relatively small inspection area in order to estimate the maximum inclusion size likely to occur in a large area or volume. According to the approach by Murakami [3], two-dimensional observations of inclusions in a plane section are equivalent to a three-dimensional observation in a lamella of a certain thickness situated below the observed plane surface. Murakami suggested to calculate this thickness as the average size t of the maximum inclusions measured in a batch of inspection areas. The resulting inspection volume is V 0 = S 0 ᵡ t . The dependence between the maximum inclusions determined in inspection volumes and the maximum inclusion likely to occur in a large volume, called prediction volume V, is defined by the return period T. On the one hand the return period is a ratio of the prediction volume to the inspection volume T =V / V 0 , on the other hand it relates to Eq. 1 as F(x) = 1-1/T and can be plotted against max area data as shown in Fig. 4. A detailed description of the maximum inclusion prediction method can be found in [3]. When the distribution of the maximum inclusion in inspection areas is of the Gumbel form (Eq. 1), then the maximum size of inclusion corresponding to T can be defined as In the present study the average size of the maximum inclusion t is 157 µm and 110 µm for specimens cut parallel and perpendicular to RD, respectively, resulting in inspection volumes V 0 of 0.44 mm 3 and 0.3 mm 3 for both cutting directions. In order to define the size of the prediction volume, i.e. the highest stressed volume in the fatigue sample (as used to generate fatigue data presented in Fig. 1), a finite element analysis was carried out. The prediction volume was defined as the volume in the middle section of the fatigue specimen, in which the stress is at least 95% of the nominal stress in the specimen (Fig. 8).
Using the calculated prediction volume V ≈ 9 mm 3 and both inspection volumes V 0 , return periods T for specimens cut in both directions were calculated. The value of maximum inclusion expected in the prediction volume may be estimated by the intersection of the fitted Gumbel cdf (Eq. 1) and the corresponding return period. Figure 9 depicts the estimation of the maximum inclusion expected in the calculated prediction volume, as well as the comparison between the predicted maximum inclusion size ("1 sample") and the size of inclusions, which were relevant for failure of fatigued specimens. The predicted value is systematically smaller than the size of the measured inclusions. Moreover, the difference should increase with increasing number of tested samples. The reason of this is that the prediction volume is calculated only for one specimen, while the inclusion size data from 6 tested specimens are presented. This problem can be solved by multiplication of the calculated return period with the number of tested samples (see "6 samples"; Fig. 9). Size of the maximum inclusions predicted by means of the described method plotted against the corresponding size of the measured inclusions for both testing directions is presented in Fig. 10.

Prediction of the location of failure-relevant inclusions
Eq. 1 and 2 in combination with uniform dfs, which describe the distribution of inclusions exceeding the chosen threshold size, form an inclusion population model, which together with the calculated stress distribution (Fig. 8) was used in order to predict the location of initiation inclusions. It was assumed that interaction among inclusions is negligible (the inclusion fraction is lower than 0.1%) and failures originate at the inclusion with a maximum stress intensity factor (SIF) defined as where σ is the stress and Y is a shape factor, which is equal to 0.65 and 0.5 for surface and internal inclusions, respectively. The prediction was performed using probabilistic Monte Carlo simulations. Each simulation consisted of the following steps: (a) generation of inclusion population within the defined prediction volume (Fig. 8) corresponding to the measured inclusion density; (b) assigning a size and location to each inclusion according to the created inclusion population model; (c) calculation of SIF (Eq. 5) for each inclusion using the assigned size and location as well as the local stress in the inclusion surrounding volume according to the calculated stress distribution (Fig. 8); (d) the inclusion with maximum SIF among other modeled inclusions is assumed to be relevant to failure. After 100 simulations the size and location of 100 inclusions with the maximum SIF were compared with experimental observations. Fig. 11 and 12 represent a reasonable agreement between simulation and experimental results.

Summary
The fusing of limited experimental data with microstructure-based probability modeling is extremely useful for the investigation and prediction of the failure behavior in the VHCF range [4]. The experimental data obtained in the VHCF range contain significant uncertainty and are often limited owing to the long duration of fatigue tests. This study describes the application of conventional statistical functions and methods in order to model an inclusion population in steels. The size of the maximum inclusions in the examined inspection areas that were cut perpendicular and parallel to RD was modeled using the Gumbel df. The modeled dfs depict the difference of the maximum inclusions measured perpendicular and parallel to RD, which apparently results in the different SN curves for fatigue specimens tested along rolling and transversal directions. In order to describe the size and location of inclusions exceeding the chosen threshold size the Pareto and Cauchy dfs were used. On the basis of the modeled inclusion population the size and location of failure-relevant inclusions were estimated. This data can be subsequently fused with obtained fatigue data [6] in order to form a reliable model for the characterization and prediction of the failure behavior in the VHCF range.