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Data-driven approach to very high cycle fatigue life prediction

Article in Engineering Fracture Mechanics · September 2023

DOI: 10.1016/j.engfracmech.2023.109630

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Contents lists available at ScienceDirect

Engineering Fracture Mechanics

journal homepage: www.elsevier.com/locate/engfracmech

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Data-driven approach to very high cycle fatigue life prediction

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Yu-Ke Liu, Jia-Le Fan, Gang Zhu, Ming-Liang Zhu ⁎, Fu-Zhen Xuan
Key Laboratory of Pressure Systems and Safety, Ministry of Education, School of Mechanical and Power Engineering, East China University of Science
and Technology, Shanghai 200237, China

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ARTICLE INFO ABSTRACT

Keywords: The research on life prediction for mechanical structures in very high cycle fatigue regime is piv-
Life prediction otal to improve structure service, but it can be costly and time-consuming to collect fatigue data.
Very high cycle fatigue In response, the data-driven approach of machine learning emerged as a solution to data insuffi-
Machine learning
ciency. In this work, after extracting a small dataset of GCr15 bearing steel subjected to very
Z-parameter model
high cycle fatigue tests from open literature, the Z-parameter model was applied to obtain ex-
Small dataset
tended datasets to establish models driven by support vector machine, artificial neural network,
D
and Z-parameter based physics-informed neural network, respectively. With training on ex-
tended datasets and the original data as test set, fatigue life prediction for GCr15 steel was car-
ried out and evaluated between these models. Results showed that the physics-informed neural
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network calibrated by Z-parameter model trained on a larger dataset featured more accurate
and reliable prediction than other models did, which demonstrated effectiveness of Z-parameter
in data extension and model construction as priori physics knowledge for a data-driven ap-
proach. Looking into the future, Z-parameter model deserves more attention to its employment
in life prediction for more engineering materials and structures serving in the very high cycle fa-
tigue regime.
EC

1. Introduction

As major equipment pertaining to engineering sector is experiencing development featuring higher reliability, longer serviceable
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life and more intelligence, it is significant to prevent or delay the failure of structures from their design, manufacturing to service with
an accurate fatigue life prediction to meet higher production requirements [1,2]. The research on fatigue life in engineering has ini-
tially kicked off with investigation of structure failure and promoted the technology of structure design, leveling up the study of fa-
tigue and fracture mechanics from low and high cycle to very high cycle fatigue (VHCF) regime [3], i.e., loading cycles exceeding 107.
Correspondingly, recent decades have witnessed a considerable volume of research on VHCF behavior of different materials and
structures [4–10]. The initiation of fatigue cracks for a low-strength weld joint in the VHCF regime in terms of competition between
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defects of different size from the surface and interior was analyzed by Liu et al. [6]. Guided by theories of VHCF, Liu et al. [8] worked
out the major reason accounting for the failure of roller bearing cages. In addition, four aspects of VHCF research were highlighted by
Hong et al. [11], among which mapping the fatigue strength development with S-N curves and constructing a model to describe fa-
tigue mechanisms and properties in very high cycle regime were mentioned. Generally, traditional methods to describe VHCF lifetime

⁎ Corresponding author.
E-mail address: mlzhu@ecust.edu.cn (M.-L. Zhu).

https://doi.org/10.1016/j.engfracmech.2023.109630
Received 12 July 2023; Received in revised form 11 September 2023; Accepted 11 September 2023
0013-7944/© 20XX

Note: Low-resolution images were used to create this PDF. The original images will be used in the final composition.
Y.-K. Liu et al. Engineering Fracture Mechanics xxx (xxxx) 109630

Table 1
Chemical composition of GCr15 steel [46].
Composition C Si Mn Cr Ni Cu Mo S P
Wt. % 0.99 0.23 0.32 1.50 0.06 0.10 0.02 0.10 0.02

Table 2
Mechanical properties of GCr15 steel [46].
σ /MPa σ /MPa E/GPa

F
ψ/%
s b
1617 2310 210 0.3

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are based on either Paris law or Murakami equation [4], and a widely-recognized classification of models to predict fatigue life con-
sists of empirical, physics-driven, and data-driven approaches [1].
Being part of the first generation of bearing steels with a long history [12], the classic GCr15 steel has a fame to
claim for its high strength, resistance to wear and tear, and cost-effectiveness [13]. In light of these properties, GCr15
steel has been broadly applied in the bearing industry subject to complex and even extreme service conditions [14].
Under the circumstance of VHCF, however, our knowledge may fall short of the requirements to understand the fatigue
behavior and prevent unexpected failure of GCr15 structures. To solve this problem, a growing number of studies have

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been carried out to explore fatigue life of this structural material in the VHCF regime [15–19], but it can be time-
consuming and expensive to collect enough data of GCr15 components either in operation or from experiments for ac-
curate statistical analysis [20–24]. Zhang et al. [25] pointed out that a small group of data brought about uncertainties
for fatigue life prediction, especially for S-N curve establishment, as there have been new modes of failure, new materi-
als and emerging research methods. Lv et al. [26] proposed an integration of experimental data and priori information
to describe structures’ fatigue life when small sample size was available. Furthermore, the science of big data and ma-
chine learning (ML) has gained popularity in pushing the boundary of fatigue strength study as it works well with small
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sample size to describe the non-linear relations in fatigue life research [3].
When it comes to ML models, on one hand, support vector machine (SVM) and artificial neural network (ANN), be-
ing supervised, have been widely applied to process small-sized data for fatigue life prediction. Wang et al. [27] ex-
plained the reliability of ML approaches to predicting engineering material performance. Ma [28] constructed a SVM
model for residual life prediction with limited data of 23,144 bearings. Barbosa et al. [29] verified the feasibility of an
ANN model in fatigue resistance determination for P355NL steel under different stress ratios. Zhan et al. [30] com-
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pared the life predictive accuracy in additive manufacturing (AM) with SVM and ANN models on published experimen-
tal data, and they found that the ANN model showed better performance, and the models’ relative error in life predic-
tion decreased remarkably with increased training data. Sai et al. [31] established four ML models for fatigue life pre-
diction of multi-principal element alloys and draw a conclusion that the SVM model outperformed others.
On the other hand, unlike aforementioned algorithms just trying to build a connection between input and output, the physics-
informed neural network (PINN) is an integration of data-driven neural network and physics equation, which has what it takes for
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generalization and sound accuracy even on limited data [32,33]. Zhang et al. [34] constructed a PINN model for creep-fatigue life
prediction of 316 stainless steel with extended features at high temperatures, and enhanced accuracy was noticed of their model when
compared with that of general neural networks. Wang et al. [35] proposed a PINN model and demonstrated its better performance in
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life prediction in comparison to a SVM model of AM metals. Additional research also figured out that PINN models, as they acquire
knowledge from physics equation, exhibited higher accuracy and robustness and could better prevent overfitting in prediction than
traditional ML methods did [36–38]. However, Mortazavi et al. [39] figured out the efficiency of ANN model they proposed to ex-
plore growth of fatigue crack of various length was largely dependent on the size of input. He et al. [40] found their SVM and ANN
models for life prediction displayed underestimation and several obvious errors. What’s more, it remained a challenge to establish a
PINN model being well-integrated with physics information, as recognized by Si et al. [41]. Yucesan et al. [42] also noticed the num-
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ber of samples influenced fatigue prognosis of the hybrid PINN model they employed for turbine bearings. Overall, efforts still need to
be stepped up to improve structure fatigue prediction with ML methods from perspectives of sample size, algorithms and integration
with priori physics information.
In this work, we adopted three ML algorithms of SVM, ANN and PINN, respectively, to carry out life prediction for GCr15 bearings
in the VHCF regime based on experimental data from a published work. In addition, the Z-parameter model proposed by Zhu et al.
[43–45] was employed for data extension to explore the effect of sample size on model performance on one hand, and to support the
establishment of PINN models on the other. The outline of this paper is as follows: Section 2 provides data extension, pre-processing
and dataset division. Section 3 introduces the establishment of three ML models. Section 4 presents the results and compares model
performance. Conclusions are given in section 5.

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Table 3
Experimental data of GCr15 steel [46].
No. σ /MPa N /cycle /μm d /μm
a f inc
1 1000 3.58 × 103 35.01 230.00
2 900 3.34 × 104 29.20 26.00
3 850 2.62 × 104 27.40 56.11
4 800 1.30 × 105 28.00 350.00
5 800 8.39 × 105 34.56 160.00

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6 800 2.10 × 105 24.00 7.58
7 750 8.93 × 106 22.00 270.03
8 700 7.10 × 106 29.00 29.00

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9 700 1.17 × 107 28.00 157.28
10 650 2.45 × 107 25.13 402.36
11 650 2.80 × 108 15.05 38.77

2. Data preparation

2.1. Original data

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Considering the difficulty of data collection mentioned before, we extracted a set of 11 groups of experimental data from open lit-
erature, in which fatigue tests were carried out on GCr15 bearing steel of 3000 μm in diameter at room temperature (25 ℃) in the air
with stress ratio of R = -1. The chemical composition and mechanical properties of GCr15 steel in the experiment are shown in Table
1 and 2, respectively, and the experimental data in Table 3, which includes stress amplitude (σa), fatigue life (Nf), inclusion size
(square root of inclusion area, ) and inclusion depth (the shortest distance between inclusion center and the surface of speci-
men, dinc).
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Based on this original dataset, the S-N relation is mapped in Fig. 1, where the fatigue life of GCr15 steel generally increases with
the decrease of stress amplitude under axial loading, consistent with results of previous work [47,48]. However, as aforementioned,
such a small size of data might stumble the accuracy of ML models in fatigue life prediction. According to Wang et al. [49], the prob-
lem of data insufficiency can be solved with fusion of experimental and physical data. Therefore, we adopted an approach of data ex-
tension facilitated by a priori physics model of Z-parameter.

2.2. Introduction of Z-parameter model

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When crunching the data from fatigue tests of weld joints in the VHCF regime, Zhu et al. [43] figured out that, in addition to stress
amplitude and defect size, the location of inclusion also influenced fatigue life of structures. As a result, they adopted a parameter D to
describe the relative depth of the critical inclusion:
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(1)
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Fig. 1. S-N curve of GCr15 steel under axial loading (data from reference [46]).

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Table 4
Experimental data of GCr15 steel considering D and Z.
No. σ /MPa N /cycle /μm D Z
a f
1 1000 3.58 × 103 35.01 0.92 1772.98
2 900 3.34 × 104 29.20 0.99 1575.90
3 850 2.62 × 104 27.40 0.98 1468.91
4 800 1.30 × 105 28.00 0.88 1351.49
5 800 8.39 × 105 34.56 0.95 1424.19

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6 800 2.10 × 105 24.00 0.99 1357.85
7 750 8.93 × 106 22.00 0.91 1226.19
8 700 7.10 × 106 29.00 0.99 1223.99

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9 700 1.17 × 107 28.00 0.95 1203.49
10 650 2.45 × 107 25.13 0.87 1073.11
11 650 2.80 × 108 15.05 0.99 1018.02

where d is diameter of specimen, and dinc is the shortest distance between inclusion and the surface of specimen, both in μm. Obvi-
ously, the value of falls in the range of 0 to 0.5, thus the possible value of parameter D is 0.5 to 1. Note that a bigger D indicates an
inclusion closer to specimen surface.

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By incorporating D, a parameter Z was proposed to characterize the influence of relative location of inclusion on fatigue life in the
VHCF regime:

(2)

where σa is stress amplitude in MPa, is the inclusion size in μm, D is the relative depth of the inclusion, and is a material
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constant.
Furthermore, a modified equation for fatigue life, , was given in Eq. (3) with fitting parameters C and α as constants related to
materials:
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Fig. 2. PCC heatmaps between different fatigue life-related parameters for: (a) extended set of 70 groups of data, (b) extended set of 110 groups of data, (c) original
dataset.

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Fig. 3. The structure of SVM model.

Table 5
Optimized hyper-parameters for SVM models.
Hyper-parameter Dataset of 70 groups Dataset of 110 groups

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C 438.22 581.52
7.08x10-3 7.08x10-3

(3)
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The Z-parameter was then applied to explain fatigue behavior of a Cr-Mo-V steel in the VHCF regime [44]. After that, Eq. (2) was
improved with consideration of a shape factor of the inclusion, Y [45], as shown in:

(4)
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Where the value of Y falls in the range from 0 to 1, dependent on the shape of inclusions, and Y = 1 was employed in this work.
Empirically, the value of was 0.25 here, which could soundly describe the relationship between parameters Z and in fitting as
discussed in previous works [43–45].
Correspondingly, combining Eq. (4) through Eq. (3) gave an upgraded fatigue life function driven by Z-parameter:
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(5)
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Therefore, we enriched original data considering parameters D and Z with Eqs. (1) and (4), as shown in Table 4.

2.3. Extension of data

As the temperature and stress ratio were constants in original tests for GCR15 steel, their implications for fatigue life would not be
discussed in this work. Instead, the effects of the other three variables, namely, stress amplitude (σa), inclusion size ( ) and its
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relative depth (D), on fatigue life (Nf) were studied with ML algorithms. We extended the original 11 groups of data with an ap-
proach of variable control, i.e., changing the value of only one independent variable among σa, and D each time to get a corre-
sponding value of the dependent variable Nf by applying Eq. (5). Note that values of three independent variables in data extension
were restricted within intervals of experimental data: 650 to 1000 MPa for σa, 15.05 to 35.01 mm for , and 0.87 to 0.99 for D.
Finally, we got an extended set of 70 groups of data composed of four variables.
To work out the relation between sample size and accuracy in fatigue life prediction by ML models, we extended to form another
set of 110 groups of data based on the set of 70 groups of data within the same intervals mentioned above. As a result, there were two
sets of 70 and 110 groups of data respectively. Before ML model training, it was necessary to evaluate the consistency and reliability
of extended data compared to original ones. A widely adopted approach to ascertaining correlation between a group of variables is
the Pearson correlation coefficient (PCC):

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Fig. 4. The structure of ANN model.

Table 6

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Optimized hyper-parameters for ANN models.
Hyper-parameter Value
Number of neurons for input layer 3
Number of hidden layers 3
Number of neurons for each hidden layer 10, 10, 5
Number of neurons for output layer 1
Learning rate 0.01
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(6)

where Cov(X,Y) stands for the covariance of two variables X and Y, σX and σY represent the standard deviations of X and Y respec-
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tively, and the value of ranges from −1 to 1. Note that the closer the absolute value of gets to 1, the stronger linear correla-
tion is indicated between two variables.
Following on from above, we obtained PCC heatmaps between different fatigue life-related parameters for original and two ex-
tended datasets, respectively, as shown in Fig. 2. Obviously, in Fig. 2(c) of original dataset, the value of fatigue life is reversely related
to that of stress amplitude, inclusion size and parameter Z. On top of that, a similar pattern of correlation can be noticed in Fig. 2(a)-
(b) for both extended datasets, which demonstrated the reliability of data extension. Therefore, the application of the priori Z-
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parameter made a more comprehensive view to describe fatigue life of GCr15 steel in this work. Collectively, it was reasonable to em-
ploy these two extended datasets for following ML model establishment.
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2.4. Data pre-processing

Till now we have got datasets composed of four variables, namely, σa in MPa, in μm, Nf in cycle, as well as a dimensionless
D, all of which came from different dimensions and with various value intervals. Before adopting datasets into ML training and valida-
tion, a necessary step was data pre-processing to reduce dimensional complexity and redundancy of signals [29]. Therefore, σa,
and D were normalized considering their minimum and maximum values with an equation:
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(7)

where , , , stand for normalized, original, minimum and maximum values of variable X, respectively.
In addition, Nf was transformed with a logarithm function:

(8)

where and represent logarithmic and original values of fatigue life, respectively.
What’s more, an evaluation method was needed to quantify efficiency of models [50]. As a result, the coefficient of determination,
, would be adopted to evaluate performance of different ML models. The coefficient was given in:

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Fig. 5. The structure of Z-parameter based PINN model.

Table 7
Optimized hyper-parameters for Z-parameter based PINN models.
Hyper-parameter Value
Number of neurons for input layer 3

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Number of hidden layers 3
Number of neurons for each hidden layer 10, 10, 10
Number of neurons for output layer 1
Learning rate 0.002
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(9)

where is original fatigue life, the predicted life, and the mean value of original fatigue life. Note that the value of
falls in the range from 0 to 1, and a larger value of it indicates a better predictive performance.
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2.5. Dataset division

Following data extension and pre-processing, different approaches to optimization of hyper-parameters were adopted for ML mod-
els. On one hand, K-fold cross validation was used for SVM models with K = 10. In other words, each dataset was splintered into ten
subsets for ten times of life prediction, each time with one subset for validation and the other nine subsets for training, to get an aver-
age from ten values as the final output of predicted fatigue life. Combining this method and grid search, optimal parameters could be
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obtained for SVM models. On the other hand, for ANN and Z-parameter based PINN models, extended data took the role of training
sets with Stochastic gradient descent (SGD) as optimizer. Furthermore, original dataset would serve as the test set to evaluate accu-
racy of life prediction by different ML models. Note that stress amplitude σa, inclusion size , and its relative depth D were input,
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while fatigue life Nf was output for models in this work.

3. Model establishment

3.1. SVM model

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As one of the traditional and supervised ML algorithms, SVM showed desirable performance in handling multi-dimensional prob-
lems of classification, regression and prediction, including fatigue life investigation [30]. In this work, study of fatigue life for GCr15
steel fell into the category of SVR, and the regression function was:

(10)

where and represent the Lagrange multipliers, b the bias, and C the penalty parameter who determines accuracy of a model.
An unduly high value of C brings about overfitting and a lower one causes underfitting [51]. The kernel function we adopted was the
radial basis function (RBF):

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Fig. 6. Life prediction by SVM-70 for GCr15 steel, before optimization: (a) comparison between predicted and experimental data, (b) predictive accuracy, = 0.304;
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and after optimization: (c) comparison between predicted and experimental data, (d) predictive accuracy, = 0.700.

(11)
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where is the kernel parameter.

The structure of SVM model is shown in Fig. 3, where an input vector consists of stress amplitude σa, inclusion size , and
its relative depth D, and the output is predicted fatigue life Nf. Besides that, the optimized hyper-parameters for SVM models on
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two extended datasets are shown in Table 5.

3.2. ANN model

Inspired by the work of biological neurons, ANN was proposed to overcome occasional inaccuracy of physical models, whose pre-
vious application in fatigue life prediction has shown its generalization [52]. Generally, an ANN model can be constructed with three
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clusters of neurons, namely, input layer, hidden layer and output layer, with each neuron receiving a weighted sum of input from
prior neurons [53]. To get a desirable reflection from input to output, the number of neurons for each layer and the number of hidden
layers should be carefully selected. As aforementioned, there were three input and one output variables, so three neurons were em-
ployed for input layer and one for output layer.
When it comes to activation functions for the model, Tanh and linear functions were adopted for hidden and output layers respec-
tively, with Tanh function in Eq. (12) and Linear function in Eq. (13):

(12)
(13)

What’s more, Stochastic gradient descent (SGD) was utilized as the optimizer, and MSE the loss function as given in:

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Fig. 7. Life prediction by SVM-110 for GCr15 bearings, before optimization: (a) comparison between predicted and experimental data, (b) predictive accuracy,
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=0.545; and after optimization: (c) comparison between predicted and experimental data, (d) predictive accuracy, =0.823.

(14)
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where and are predicted and original fatigue life of GCr15 steel, respectively.
After training, the model finally got three hidden layers with ten neurons for each of the first two layers and five for the third
layer. The structure of ANN model is given in Fig. 4, and the optimized hyper-parameters for ANN models are shown in Table 6.
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3.3. Z-parameter based PINN model

Compared with traditional ML algorithms, a PINN model boasts favorable properties to integrate data-driven method with physi-
cal information [33]. By employing priori physical formula, the accuracy and reliability of a ML model can be largely improved [54,
55]. Considering the number of input and output variables, correspondingly, three input and one output neurons were adopted. In ad-
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dition, the physical model we incorporated into the neural network was Z-parameter model because of its reliable priori knowledge in
fatigue life description in the VHCF regime [43–45]. Besides that, Tanh and linear functions were applied to activate the hidden and
output layers respectively, and SGD the optimizer. According to Eq. (5) in section 2, the predicted fatigue life by Z-parameter, , can
be given as:

(15)

As a result, loss of the model came from two parts, namely, data-driven neural network loss and weighted physical model loss
by Z-parameter:

(16)

while

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Fig. 8. Life prediction by Z-parameter model for GCr15 steel: (a) comparison between predicted and experimental data, (b) predictive accuracy, =0.827; and (c)
comparison between Z-parameter model and SVM-110.
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Table 8
Life prediction for GCr15 steel by SVM and Z-parameter models.
Error band interval by absolute value & Coefficient of SVM- SVM-70 SVM- SVM- Z-
determination 70 (grid) 110 110 model
(grid)
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<2 18.2% 9.1% 27.3% 18.2% 18.2%

2∼3 54.5% 72.7% 54.5% 72.7% 54.5%
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>3 27.3% 18.2% 18.2% 9.1% 27.3%

R2 0.304 0.700 0.545 0.823 0.827

(17)
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(18)

where and represent the original and predicted life, respectively.

Finally, the Z-parameter based PINN model got three hidden layers with each one equipped with ten neurons after training. The
structure of the model is shown in Fig. 5, and the optimized hyper-parameters for Z-parameter based PINN models are shown in Table
7.

4. Results and discussion

With three ML models driven by SVM, ANN, and Z-parameter based PINN established respectively, life prediction for GCr15 steel
in the VHCF regime was carried out based on two extended datasets as mentioned in section 2.5, after which predicted results from
different models were compared with original data to explore performance of these models as follows. Note that a model would be re-

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Fig. 9. Life prediction by ANN-70 for GCr15 steel: (a) comparison between predicted and experimental data, (b) predictive accuracy, =0.598; and (c) comparison be-
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tween Z-parameter model and ANN-70.

ferred to as a combination of its algorithm and the data size it worked with, e.g., ANN-110 means an ANN model trained on extended
set of 110 groups of data.

4.1. SVM
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4.1.1. SVM-70
Life prediction for GCr15 steel by SVM-70 before and after model optimization with grid search are shown in Fig. 6, where before
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optimization, (a) compares predicted and experimental fatigue life and (b) displays the predictive accuracy by 2 and 3 error
bands, and after optimization, (c) is the line graph of two types of fatigue life and (d) shows model performance with error bands.
It is obvious that predicted datapoints overlap with experimental ones to a larger extent in Fig. 6(c) than in Fig. 6(a). Besides
that, before model improvement, there are about 54.5 percent of points falling into 2 ∼ 3 error bands and a few significant out-
liers can be seen in Fig. 6(b), while an increase of nearly 20 percentage points appears in the range of 2 ∼ 3 error bands in Fig. 6(d),
indicating a higher accuracy of life prediction after grid search. The coefficient of determination, , is 0.700 for SVM-70 after grid
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search, much higher than 0.304 for the model before optimization. Overall, the results have demonstrated the competitiveness of the
optimized SVM-70 with grid search.

4.1.2. SVM-110
The prediction by SVM-110 is given in Fig. 7, where a better fitting of prediction to original data shows in Fig. 7(c) compared to
Fig. 7(a). About 18.2 percent of datapoints are out of the 3 error bands in Fig. 7(b), which is two times as much as the figure for the
optimized SVM-110 according to Fig. 7(d). Furthermore, is 0.823 for the optimized SVM-110, with an increase by over 50 percent
of that value for a non-optimized model (0.545). Generally, SVM-110 has shown higher accuracy and stability after optimizing. Ac-
cording to the performance of SVM-70 and SVM-110, it can be concluded that a larger dataset and an introduction of optimizing algo-
rithm can bring out more desirable outcomes of fatigue life prediction by ML models.

4.1.3. Comparison between SVM-110 and Z-parameter model

To explore the reliability of SVM model we constructed, predictive accuracy of Z-parameter model and SVM-110 has been com-
pared after an evaluation of prediction from Z-parameter model. In Fig. 8(b), over 72 percent of datapoints locate in the 3 error

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Fig. 10. Life prediction by ANN-110 for GCr15 steel: (a) comparison between predicted and experimental data, (b) predictive accuracy, =0.848; and (c) comparison
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between Z-parameter model and ANN-110.

Table 9
Life prediction for GCr15 steel by ANN and Z-parameter models.
Error band interval by absolute value & Coefficient of determination ANN-70 ANN-110 Z-model
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<2 9.1% 18.2% 18.2%

2∼3 72.6% 63.6% 54.5%
>3 18.2% 18.2% 27.3%
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R2 0.598 0.848 0.827

bands, and the Z-parameter model gets a value of as high as 0.827. According to Fig. 8(c), it is noticeable that datapoints from Z-
parameter model and SVM-110 overlap with each other significantly, which indicates sound consistency between two models despite
a slightly higher of Z-parameter model. The proportion of datapoints in different error intervals and of SVM models in compari-
son to that of Z-parameter model for GCr15 steel are given in Table 8.
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4.2. ANN

4.2.1. Comparison between ANN-70 and Z-parameter model

A comparison of performance between ANN-70 and Z-parameter model in fatigue life prediction for GCr15 steel in the VHCF
regime is made here. Results from ANN-70 is presented in Fig. 9(a), and predictive accuracy in Fig. 9(b). Additionally, Fig. 9(c) com-
pares the performance of ANN-70 and Z-parameter model. There are 9.1 percent of data points in 2 error bands for ANN-70, while a
rise by 100 percent of that figure can be noticed for Z-parameter model, which demonstrates the higher accuracy of the model dri-
ven by the Z-parameter. Besides that, is 0.598 for ANN-70, obviously lower than 0.827 for Z-parameter model. Overall, ANN-70 is
less competitive than Z-parameter model in fatigue life prediction, which might be the result of the small size of dataset.

4.2.2. Comparison between ANN-110 and Z-parameter model

To figure out whether a larger dataset would make a difference in model performance, fatigue life prediction by ANN-110 is dis-
cussed in comparison with that of Z-parameter model, as shown in Fig. 10. It can be noticed that ANN-110 gets fewer datapoints out

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Fig. 11. Fatigue life prediction by Z-parameter based PINN -70 for GCr15 steel: (a) comparison between predicted and experimental data, (b) predictive accuracy,
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=0.891; and (c) comparison between Z-parameter model and Z-parameter based PINN-70.

of the 3 error bands, and it has almost 10 percentage points more of data within the 2 ∼ 3 error bands than Z-parameter model
does. What’s more, the value of jumps from 0.598 for ANN-70 to 0.848 for ANN-110, even a little bit higher than that coefficient
for Z-parameter model, which can be told from extensively overlapped datapoints from two models in Fig. 10(c). Overall, a larger
dataset could enhance accuracy and generalization of an ANN model in fatigue life prediction for GCr15 steel. The predictive error
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and of ANN models and Z-parameter model for GCr15 bearings are given in Table 9.

4.3. Z-parameter based PINN

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4.3.1. Comparison between Z-parameter based PINN-70 and Z-parameter model

The efficiency of integration of data-driven method and physical model has been explored with Z-parameter based PINN-70. As il-
lustrated in Fig. 11, there are 36.3 percent of datapoints falling into the 2 error bands for Z-parameter based PINN-70, twice as much
as the figure for Z-parameter model. In addition, PINN-70 has got a higher of 0.891 than physics model of Z-parameter and prior
ML models did. Therefore, Z-parameter based PINN-70 has shown better performance in fatigue life prediction for GCr15 steel than Z-
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parameter model did.

4.3.2. Comparison between Z-parameter based PINN-110 and Z-parameter model

With a step further on a larger dataset, the prediction by Z-parameter based PINN-110 has been conducted, where
remarkable consistency between datapoints from PINN-110 and Z-parameter model can be noticed in Fig. 12(c), where
there is a similar distribution in terms of the proportion of points from different error intervals between Z-parameter
based PINN-70 and PINN-110. Note that among the values of for all models, a peak has been reached of 0.943 for Z-
parameter based PINN-110, followed by Z-parameter based PINN-70 with = 0.891. As a result, a larger dataset has
what it takes to obtain a more favorable ML model, and a PINN model driven by Z-parameter has been feasible and reli-
able in fatigue life prediction for GCr15 steel. The predictive accuracy by error bands and of Z-parameter based
PINN models in comparison to that of Z-parameter model are given in Table 10.

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Fig. 12. Fatigue life prediction by Z-parameter based PINN-110 for GCr15 steel: (a) comparison between predicted and experimental data, (b) predictive accuracy,
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=0.943; and (c) comparison between Z-parameter model and Z-parameter based PINN-110.

Table 10
Life prediction for GCr15 steel by PINN and Z-parameter models.
Error band interval by absolute value & Coefficient of determination PINN-70 PINN-110 Z-model
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<2 36.3% 36.3% 18.2%

2∼3 36.4% 36.4% 54.5%
>3 27.3% 27.3% 27.3%
R2 0.891 0.943 0.827
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4.4. Integrated S-N curves

Here a comparison between different models in terms of fatigue life prediction for GCr15 steel is made with an integration of S-N
curves from original loading tests, Z-parameter model, as well as SVM, ANN, and Z-parameter based PINN. As shown in Fig. 13, gener-
ally, the distribution of datapoints from different models is consistent with the trendline of original data (the red line). Moreover, it is
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obvious that datapoints from Z-parameter model (hollow red rectangles), and Z-parameter based PINN-70 and PINN-110 (solid red
and blue triangles respectively) stand with close proximity to original datapoints (solid red rectangles), indicating sound performance
of Z-parameter based PINN in fatigue life prediction. In contrast, significant outliers can be noticed in results from both SVM models
(SVM-70 in black ring and SVM-110 in blue ring) when compared to original datapoints. To sum up, the Z-parameter based PINN-
110, as a combination of data-driven and prior physical approaches, outperforms other models in life prediction for GCr15 steel.

5. Conclusions

In this work, the data-driven approach featuring ML models on different algorithms, namely, SVM, ANN and Z-parameter based
PINN, for fatigue life prediction of GCr15 steel in the VHCF regime was explored. Guided by Z-parameter model, additionally, two ex-
tended datasets were obtained with sound reliability for ML model training based on a small experimental dataset from open litera-
ture. With an evaluating method constructed and then utilized, comparisons were made between different ML models in terms of pre-
dictive accuracy and reliability. Conclusions have been made as follows:

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Fig. 13. Comparison between S-N curves from original data and different models in life prediction for GCr15 steel.

(1) The size of dataset for model establishment had implications for fatigue life prediction by ML models. Specifically, a larger
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dataset made a more desirable model.
(2) The Z-parameter based PINN model enjoyed higher accuracy and reliability than SVM and ANN models did in this work,
which demonstrated effectiveness of the Z-parameter as priori knowledge when utilizing ML algorithms.
(3) Looking ahead, this work can move forward by applying Z-parameter model into advanced technologies in future fatigue life
research for more engineering materials and structures in the VHCF regime apart from GCr15 steel.
(4) Moreover, in this work, the PINN model performance might be hindered as the PINNs were established with the Z-parameter
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model as both a source of training data and the physics constraint. To obtain a more reliable and competitive PINN model,
efforts are still needed from the perspectives of experimental data sufficiency and Z-parameter model upgrade.

CRediT authorship contribution statement

Jia-Le Fan : Investigation, Data curation. Gang Zhu : Investigation. Ming-Liang Zhu : Writing – review & edit-
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ing, Methodology, Conceptualization. Fu-Zhen Xuan : Funding acquisition, Project administration.

Declaration of Competing Interest

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.

Data availability
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Data will be made available on request.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 51835003) and by Inno-
vation Program of Shanghai Municipal Education Commission (2023ZKZD42).

References

[1] Zhang X.H, Liu H.J, Zhu C.C, Wei P.T, Wu S.J. Current situation and developing trend of fatigue life prediction of components based on data-driven. J Mech
Trans 2021;45:1–14.
[2] Santecchia E, Hamouda A.M.S, Musharavati F, Zalnezhad E, Cabibbo M, El Mehtedi M, et al. A review on fatigue life prediction methods for metals. Adv Mater
Sci Engg 2016;2016:1–26.
[3] Xuan F.Z, Zhu M.L, Wang G.B. Retrospect and prospect on century-long research of structural fatigue. J Mech Eng 2021;57:26–51.
[4] Liu Y.B, Li Y.D, Li S.X, Yang Z.G, Chen S.M, Hui W.J, et al. Prediction of the S-N curves of high-strength steels in the very high cycle fatigue regime. Int J Fatigue
2010;32(8):1351–7.

15
Y.-K. Liu et al. Engineering Fracture Mechanics xxx (xxxx) 109630

[5] Li X.S, Li N, Liu G.Q, Liu Z, Zhang Y, Yan H.Y. Study on fatigue performance and life prediction of high strength bearing steel GCr15. China Metalforming Equip
Manuf Technol 2020;55:141–4.
[6] Liu L.L, Xuan F.Z, Zhu M.L. Very high cycle fatigue behavior of 25Cr2Ni2MoV steel welded joint. J Mech Eng 2014;50:25–31.
[7] Zhang W.C, Zhu M.L, Xuan F.Z. Experimental characterization of competition of surface and internal damage in very high cycle fatigue regime. Key Engng
Mater 2017;754:79–82.
[8] Liu L, Huo S, Zheng K, Wang L.Q. Analysis of roller bearing cage broken under high DN value. J Aerospace Power 2020;35:2115–22.
[9] Zhu M.-L, Zhu G, Xuan F.-Z. On micro-defect induced cracking in very high cycle fatigue regime. Fatigue Fract Engng Mater Struct 2022;45(11):3393–402.
[10] Li M.K, Zhao X.X, Kang H.M, Li Y.P, Yu H, Deng H.L. Prediction of very-high-cycle fatigue life of carburized Cr-Ni high-hardness alloy steel. Mater Sci Technol
2022;30:69–75.
[11] Hong Y.S, Sun C.Q, Liu X.L. A review on mechanisms and models for very-high-cycle fatigue of metallic materials. Adv Mech 2018;48:201801.
[12] Yu F, Chen X.P, Xu H.F, Dong H, Weng Y.Q, Cao W.Q. Current status of metallurgical quality and fatigue performance of rolling bearing steel and development

F
direction of high-end bearing steel. Acta Metall Sin 2020;56:513–22.
[13] Wang D.Y, Xia D.X, He Y, Xu R.C, Zhang M.Y, Wu S.S, et al. Failure analysis of rolling contact fatigue of GCr15 bearing steel. Spec Steel Technol 2022;28:
57–60.
[14] Gong Z.P, He T.T, Li L.F, Du S.M, Zhang Y.Z, Zhang P.J. Effect of retained austenite content on friction and wear properties of GCr15 bearing steel. Trans Mater

OO
Heat Treat 2023;44:123–34.
[15] Li Y.D, Yang Z.G, Li S.X, Yb L, Chen S.M. Correlations between very high cycle fatigue properties and inclusionsof GCr15 bearing steel. Acta Metall Sin 2008;
44:968–72.
[16] Li W, Sakai T, Li Q, Lu L.T, Wang P. Reliability evaluation on very high cycle fatigue property of GCr15 bearing steel. Int J Fatigue 2010;32(7):1096–107.
[17] Sun C, Xie J, Zhao A, Lei Z, Hong Y. A cumulative damage model for fatigue life estimation of high-strength steels in high-cycle and very-high-cycle fatigue
regimes. Fatigue Fract Engng Mater Struct 2012;35:638–47.
[18] Teng Z, Wu H, Huang Z, Starke P. Effect of mean stress in very high cycle fretting fatigue of a bearing steel. Int J Fatigue 2021;149:106262.
[19] Li G, Ke L, Peng W, Ren X, Sun C. Effects of natural aging and variable loading on very high cycle fatigue behavior of a bearing steel GCr15. Theor Appl Fract
Mech 2022;119:103360.

PR
[20] Zhang J.W, Lu L.T, Zhang Y.B, Zhang W.H. Evaluationg fatigue limits with small data samples. J Harbin Eng Univ 2010;31:336–9.
[21] Yucesan Y.A, Viana F.A.C. Hybrid physics-informed neural networks for main bearing fatigue prognosis with visual grease inspection. Comput Ind 2021;125:
103386.
[22] Xu L.H, Luo Y.S, He J.W, He X.Z. Influence of load, temperature and speed on bearing oil film thickness and life. J Mech Electr Eng 2022;39:955–60.
[23] Zhao B.F, Liao D, Zhu S.P, Xie L.Y. Probabilistic fatigue life prediction of mechanical structures: State of the art. J Mech Eng 2021;57:173–84.
[24] Xiong J.J, Shenoi R.A. A practical randomization approach of deterministic equation to determine probabilistic fatigue and fracture behaviours based on small
experimental data sets. Int J Fract 2007;145(4):273–83.
[25] Zhang J.B, Hu Z, Zhang J.L, Wang C, Zou T.G. Review of advances in fatigue S-N curve prediction models. Sci Technol Eng 2023;23:5390–411.
[26] Lv Z, Yao W.X. Progress in small sample experiments for fatigue life analysis. Mech Eng 2008;30:9–15.
[27] Wang H.K, Liu X.T, Lin L, Sun H.T, Lyu Y.H, Zhang Y.W, et al. Application of machine learning in predicting service performance of materials. Equip Environ
ED
Eng 2022;19:11–9.
[28] Ma H.L. Bearing residual life prediction based on principal component feature and SVM. Indus Mine Autom 2019;45:74–8.
[29] Barbosa J.F, Correia J.A.F.O, Júnior R.C.S.F, Jesus A.M.P.D. Fatigue life prediction of metallic materials considering mean stress effects by means of an
artificial neural network. Int J Fatigue 2020;135:105527.
[30] Zhan Z, Li H. Machine learning based fatigue life prediction with effects of additive manufacturing process parameters for printed SS316L. Int J Fatigue 2021;
142:105941.
[31] Sai N.J, Rathore P, Chauhan A. Machine learning-based predictions of fatigue life for multi-principal element alloys. Scr Mater 2023;226:115214.
[32] Cuomo S, Di Cola V.S, Giampaolo F, Rozza G, Raissi M, Piccialli F. Scientific machine learning through physics-informed neural networks: where we are and
CT

what’s next. J Sci Comput 2022;92:88.

[33] Li Y, Chen S.C. Physics-informed neural networks: recent advances and prespects. Comput Sci Comput Sci 2022;49:254–62.
[34] Zhang X.-C, Gong J.-G, Xuan F.-Z. A physics-informed neural network for creep-fatigue life prediction of components at elevated temperatures. Engng Fract
Mech 2021;258:108130.
[35] Wang H, Li B.o, Xuan F.-Z. Fatigue-life prediction of additively manufactured metals by continuous damage mechanics (CDM)-informed machine learning with
sensitive features. Int J Fatigue 2022;164:107147.
[36] He G.Y, Zhao Y.X, Yan C.L. MFLP-PINN: A physics-informed neural network for multiaxial fatigue life prediction. Eur J Mech /A Solids 2023;98:104889.
E

[37] Zhou T, Jiang S, Han T, Zhu S.-P, Cai Y. A physically consistent framework for fatigue life prediction using probabilistic physics-informed neural network. Int J
Fatigue 2023;166:107234.
[38] Hao W.Q, Tan L, Yang X.G, Shi D.Q, Wang M.L, Miao G.L, et al. A physics-informed machine learning approach for notch fatigue evaluation of alloys used in
aerospace. Int J Fatigue 2023;170:107536.
RR

[39] Mortazavi S.N.S, Ince A. An artificial neural network modeling approach for short and long fatigue crack propagation. Comput Mater Sci 2020;185:109962.
[40] He L, Wang Z, Akebono H, Sugeta A. Machine learning-based predictions of fatigue life and fatigue limit for steels. J Mater Sci Technol 2021;90:9–19.
[41] Si X.-S, Wang W, Hu C.-H, Zhou D.-H. Remaining useful life estimation – A review on the statistical data driven approaches. Eur J Oper Res 2011;213(1):1–14.
[42] Yucesan Y.A, Viana F.A.C. A hybrid physics-informed neural network for main bearing fatigue prognosis under grease quality variation. Mech Syst Sig Process
2022;171:108875.
[43] Zhu M.L, Xuan F.Z, Du Y.N, Tu S.T. Very high cycle fatigue behavior of a low strength welded joint at moderate temperature. Int J Fatigue 2012;40:74–83.
[44] Zhu M.L, Xuan F.Z, Chen J. Influence of microstructure and microdefects on long-term fatigue behavior of a Cr–Mo–V steel. Mater Sci Engng A 2012;546:90–6.
[45] Zhu M.L, Jin L, Xuan F.Z. Fatigue life and mechanistic modeling of interior micro-defect induced cracking in high cycle and very high cycle regimes. Acta Mater
CO

2018;157:259–75.
[46] Zhi B.Y. Effect of ultrasonic frequency on giga-cycle fatigue properties for GCr15 steel. Southwest Jiaotong University 2010.
[47] Sakai T, Sato Y, Oguma N. Characteristic S-N properties of high-carbon-chromium-bearing steel under axial loading in long-life fatigue. Fatigue Fract Engng
Mater Struct 2002;25(8–9):765–73.
[48] Li W, Lu L.T. Probabilistic character for S-N relations of high carbon chromium bearing steel in super-long cycle life region. J Traffic Transp Eng 2006;6:17–21.
[49] Wang H, Li B, Gong J, Xuan F.-Z. Machine learning-based fatigue life prediction of metal materials: Perspectives of physics-informed and data-driven hybrid
methods. Engng Fract Mech 2023;284:109242.
[50] Zhang G.D, Su B.L, Liao W.J, Wang X.F, Zou J.W, Yuan R.H, et al. Prediction of fatigue life of powder metallurgy superalloy disk via machine learning.
Foundary Technol 2022;43:519–24.
[51] Zhou S.W, Yang B, Wang C, Wang S.C, Xiao S.N, Yang G.W, et al. Estimation fatigue crack growth rate of 6005A–T6 aluminum alloys with different stress ratios
using machine learning methods. Chinese J Nonferrous Metals 2023;1–17.
[52] Chen D, Li Y, Liu K, Li Y. A physics-informed neural network approach to fatigue life prediction using small quantity of samples. Int J Fatigue 2023;166:
107270.
[53] Ramachandra S, Durodola J.F, Fellows N.A, Gerguri S, Thite A. Experimental validation of an ANN model for random loading fatigue analysis. Int J Fatigue
2019;126:112–21.
[54] Yang X.G, Tan L, Hao W.Q, Shi D.Q, Fan Y.S. Data-driven structural strength and life assessment of high temperature structure: progresses and challenges. J
Propulsion Technol 2023;44:2304020.
[55] Salvati E, Tognan A, Laurenti L, Pelegatti M, De Bona F. A defect-based physics-informed machine learning framework for fatigue finite life prediction in

16
Y.-K. Liu et al. Engineering Fracture Mechanics xxx (xxxx) 109630

additive manufacturing. Mater Des 2022;222:111089.

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