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[Estimation the S-N Curve by Machine Learning Random Forest Method_Mater. Trans. 65(2024)428-433.pdf](https://mdr.nims.go.jp/filesets/0e3046f4-55f8-4101-8cb1-dd2ea2f5bac2/download)

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[Nobuo Nagashima](https://orcid.org/0000-0003-3588-980X), [Masao Hayakawa](https://orcid.org/0000-0001-5143-8350), [Hiroyuki Masuda](https://orcid.org/0000-0003-2847-1616), Kotobu Nagai

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[Estimating the S-N Curve by Machine Learning Random Forest Method](https://mdr.nims.go.jp/datasets/4f674c5c-49ad-4d2e-8afa-98cfb4d162de)

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Estimating the S-N Curve by Machine Learning Random Forest MethodEstimating the S-N Curve by Machine Learning Random Forest Method+Nobuo Nagashima, Masao Hayakawa, Hiroyuki Masuda and Kotobu NagaiNational Institute for Materials Science, Tsukuba 305-0047, JapanFatigue limit is well predicted by tensile strength or hardness, and the relationship is often analyzed by linear regression using theminimum squared approximation. However, the prediction of the number of cycles to failure at a given stress amplitude, meaning the estimate ofthe S–N curve, has not been realized. Therefore, we aim to investigate the estimability of the S–N curve using the random forest method basedon the data described in the NIMS fatigue data sheet. The random forest method is a machine learning algorithm and an ensemble learningalgorithm that integrates weak learners of multiple decision tree models to improve generalization ability. It was clarified that the machinelearning of the multiple decision tree model is excellent in fatigue limit prediction. The S–N curve can be accurately estimated by combining theprediction of fatigue limit and the number of cycles to failure at a given stress amplitude. [doi:10.2320/matertrans.MT-Z2023006](Received November 6, 2023; Accepted December 7, 2023; Published February 9, 2024)Keywords: fatigue, high-cycle fatigue, data sheet, machine learning, random forest method1. IntroductionNIMS has accumulated fatigue test data of variousstructural materials for approximately 40 years, known asNIMS fatigue data sheets (FDS).1) These FDS showempirical correlations between fatigue limits (i.e., fatiguestrength at 107 cycles) and other mechanical properties(Fig. 12)). From these FDS, it is empirically known that thereis a correlation between fatigue limit and other mechanicalproperties (Fig. 12)). In addition to the fatigue limit, theestimation of fatigue strength (S–N curve) is attempted bynormalizing the stress amplitude using the tensile strength.3)Table 1 lists the index properties of fatigue. In Table 1,fatigue is first classified into high- and low-cycle fatigueaccording to the life range. The high-cycle fatigue strengthproperty is generally expressed by the curve ·a-Nf, which isthe relationship between stress amplitude and life. In thiscase, the index is the strength property, with tensile strength·B denoting the static index and cyclic yield stress ·yccharacterizing the dynamic index. The reasons for this aredescribed in a later. Conversely, the low-cycle fatiguestrength property is represented by the relationship betweenstrain and life, ¾a-Nf. Therefore, the deformation character-istic is considered an index. In this case, the static index is therupture ductility ¾f and the dynamic index is the exponent nAof the cyclic stress–strain curve.3) It is empirically known thatan excellent correlation exists between tensile strength ·Band fatigue limit ·w. The correlation between yield stress ·y(or 0.2% proof stress ·0.2) and ·w has also been investigated,but it is not as strong as the ·B–·w relationship because ·yis affected by an instability phenomenon called yielding.However, a linear relationship is established between thecyclic yield stress ·yc and ·w because ·yc corresponds to theinternal microstructure, reaching a certain steady state afterrepeated plastic deformation. Thus, it is reasonable to adopttensile strength ·B as a static index of high-cycle fatiguestrength and cyclic yield stress ·yc as a dynamic index. Thedynamic index should essentially be adopted because fatigueis caused by repeated plastic strain, but there are somebarriers to adopting ·yc. First, ·yc must be measured by astrain control test using the companion specimen method orthe incremental step method,4) and the measurement data arenot plentiful. As shown in Fig. 2, the two index properties·B and ·yc are proportional, so we believe it is acceptableto use the static index for practical purposes. Figure 3 shows·a/·B-Nf normalized by ·B. However, the entire normalizedresults in a wide band, which is not an accurate estimation.Therefore, we attempted to estimate the S–N curve (relation-ship between stress amplitude and fatigue life) throughmachine learning.The random forest method is an algorithm in machinelearning. It is an ensemble learning algorithm that improvesgeneralization ability by integrating weak learners of multipledecision tree models and is mainly used for classification(discrimination) and regression (estimation) applications. Thekey issues are (1) whether more accurate data can be sampledfor the target data population and (2) whether decision treemodels can be created for each training component. Inconventional mathematical model regression, the regressionis based on the least-squares approximation to find thecorrelation between two data sets of interest. However,machine learning can create a regression model that relatesmultiple decision tree models of the learning elements, whichis expected to provide a more accurate estimation.In this study, we explored the improvement of theestimation accuracy of the fatigue limit using theexperimental data available from the NIMS FDS by therandom forest method. Next, the possibility of estimatingthe S–N curve was also examined by predicting the fatiguestrength below 106 cycles using the same method.2. Analysis MethodThe data population for estimating fatigue limits was basedon the experimental data of S25C (FDS No. 1) and S55C(FDS No. 4) by rotating bending fatigue tests. A randomforest method was used to examine the effect of each studyelement. Next, fatigue limit data from torsional fatigue testswere added to the data population to study the effects ofdifferent fatigue test methods. Furthermore, the effect ofstress ratio was examined by adding the fatigue test data+This Paper was Originally Published in Japanese in J. Soc. Mater. Sci.,Japan 70 (2021) 876–880.Materials Transactions, Vol. 65, No. 4 (2024) pp. 428 to 433©2024 The Society of Materials Science, Japanhttps://doi.org/10.2320/matertrans.MT-Z2023006with R = 0 and with stress ratio R = ¹1. On the basis ofthe results of previous studies, estimation accuracy wasexamined using fatigue data for different types of steels:S35C (FDS No. 2), SNCM439 (FDS No. 25), SmN438(FDS No. 16), SmN443 (FDS No. 17), SUS403 (FDSNo. 30), SUS304 (FDS No. 33), and S25C and S55C. Theestimation accuracy of the fatigue data of different types ofTable 1 Index property of fatigue.0 500 1000 1500050010001500: S25C,S35C,S45C Normalized: S25C,S35C,S45C QT: SCr440,SCM435,SNCM329 QT: Ti, Ti-6Al-4V, Ti-6Al-4V ELI: SUS304: Aluminum allysTensile strength B (MPa)Cyclic yield stressyc (MPa)Fig. 2 Relationship between tensile strength and cyclic yield stress.102 103 104 105 106 107 1080.511.52Steel S25C, S35C, S45C N S25C, S35C, S45C QT SCr440, SCM325, SCM329 QT Number of cycles to failure (Cycles)a /BAluminum alloys5083-O, 7N01-T5, 7N01-T6Fig. 3 S-N curves were normalized in tensile strength.(b)(a) 0 100 200 300 4000200400600Hardness of Vickers Fatigue limit,w (MPa) S25C, S35C, S45C N S35C, S45C QT SCr440, SCM435        , SNCM439, QT  SUS304 ST Al Ti  w = 1.63HV  107 Fatigue limitFig. 1 Relationship between mechanical properties and fatigue limit. (a) versus Vickers Hardness (b) versus tensile strength.Estimating the S-N Curve by Machine Learning Random Forest Method 429steel was examined. Next, fatigue life estimation wasattempted using fatigue data of 106 cycles or less for varioussteels. Finally, the estimation of the S–N curve was attemptedfor S45C (FDS No. 3) tempered at 550°C, Heat A, bypredicting the fatigue strength under 106 cycles for eachstress amplitude. Until now, “elongation” and “reduction ofarea” have not been focused on because they correlatewell with tensile strength and hardness for estimating fatiguelimits. However, in the finite life range of the S–N curve,especially in the low-cycle range of short life, ruptureductility is an indicator of low-cycle fatigue, so a decisiontree model was adopted to relate tensile strength, hardness,elongation, and reduction of area.A commercial personal computer was used for machinelearning, and Python 3.6.1,5) available for free download, andthe external library Anaconda6) were used.The target data were the fatigue test results described inthe FDS. For the sake of fairness of analysis, 80% of the datawere training data and 20% were test data randomly extractedeach time. Therefore, it is impossible to determine which dataare the test data. The mean absolute percentage error (MAPE)was obtained from the test data as one of the evaluationresults of the analysis.MAPE ð%Þ ¼ 100NXNi¼1by�� yiyi�������� ð1Þwhere by� is the value of the data used in the analysis and yi isthe estimate obtained from the analyzed data.The root mean square error (RMSE), mean squared error(MSE), and coefficient of determination (R2) are used asindicators to evaluate the fit accuracy of the model obtainedin the regression analysis. However, when calculated withthe RMSE and MSE error functions, the + and ¹ data aresummed, resulting in a canceled mean error. Conversely,MAPE can localize discrepancies in prediction data becauseof absolute values, and problems with MAPE include caseswhere the measured value is zero, or the prediction is toosmall. Additionally, without cross-validation and grid search,biased conclusions may be obtained. However, for allpredictions, a relationship diagram between experimentaland predicted values, as shown in Fig. 4, is developed andvisually observed, which is considered a substitute for cross-validation and grid search. For these reasons, we consideredit appropriate to use MAPE rather than RMSE and MSE asthe error function in this study.3. Analysis Results and Discussion3.1 Fatigue limit estimation by machine learning3.1.1 Fatigue limit analysis of S25C and S55C underrotating bending testsUsing the data from S25C and S55C rotating bendingfatigue tests (total = 218), four decision tree models werecreated as learning factors for Vickers hardness, tensilestrength, elongation, and reduction of area. Table 2 showsthe results. The MAPE of Vickers hardness and tensilestrength is <2%, signifying a high estimation accuracy.These results confirm the excellent correlation betweenhardness and tensile strength and fatigue limit shown in(a)(b)Fig. 4 Relationship between fatigue limit by AI prediction and fatigue limitby experiment using 107 times unbroken data of rotational bending fatiguetest and torsional fatigue test of S25C and S55C. (a) Prediction using adecision tree model for HV only. (b) Prediction using a decision treemodel for HV and test method.Table 2 Analysis results by machine learning.N. Nagashima, M. Hayakawa, H. Masuda and K. Nagai430FDS No. 5 (Fig. 1) by machine learning, and the estimationaccuracy is much improved.3.1.2 Influence of test methodTorsion test data were added to the rotating bending testdata for S25C and S55C conducted in Section 3.1.1. (total =279). A test method section was added as a learning element.The analysis results are shown in Table 3 and Fig. 4. Thefatigue limit estimated only using the Vickers hardness inFig. 4(a) was approximately 12% of MAPE. Alternatively,the MAPE of the fatigue limit estimated from the regressionmodel that links Vickers hardness and the decision tree modelof the test method in Fig. 4(b) is 2.23%, dramaticallyimproving estimation accuracy. This result indicates that theregression model by machine learning, which can relatemultiple learning factors, is effective for fatigue limitestimation.3.1.3 Effect of stress ratioA decision tree model was added to the rotating bendingand torsion test results for S25C and S55C conducted inSection 3.1.2, using the test data from the axial loading tests(R = 0 and ¹1) as stress ratios (total = 306). The analyticalresults are shown in Table 3 and Fig. 5. The MAPE of thefatigue limit estimated only by the tensile strength and testmethod in Fig. 5(a) was 3.02%. The MAPE of the fatiguelimit estimated from the regression model with three learningfactors based on the decision tree model of stress ratio, tensilestrength, and the test method in Fig. 5(b) is 2.35%, which isan enhancement in the estimation accuracy.3.1.4 Influence of various steel dataFatigue limit data (total = 892) from rotating bendingfatigue tests of S35C, SNCM439, SmN438, SmN443,SUS403, and SUS304 were added to the fatigue test resultsof S25C and S55C. The analysis results are shown in Table 3and Fig. 6. The MAPE of the fatigue limit estimated fromthe regression model linking the hardness and decision treemodel of the test method was 2.94%, which is a highestimation accuracy.3.2 Estimation of fatigue strength below 106 times bymachine learningA decision tree model of Vickers hardness, tensile strength,reduction of area, and elongation was developed byrestricting the analysis to the S25C and S55C fatigue data(total = 515) of 106 cycles or less, and the fatigue strength of106 cycles or less was estimated by relating all decision treetraining elements. The results of the analysis are shown inTable 4 and Fig. 7. The regression model with the decisiontree model for Vickers hardness, tensile strength, elongation,and reduction of area showed a high estimation accuracy of92.0% for the training data, but 65.8% for the randomlyselected test data, and 38.7% for the MAPE. This is thoughtto be because the training data distinguish between S25C andS55C fatigue data, resulting in fatigue strength estimatesTable 3 Analysis results by machine learning.(a)(b)Fig. 5 Relationship between fatigue limit by AI prediction and fatiguelimit by experiment using 107 times unbroken data (total 306) of axialload test (R = 0, ¹1) for rotating bending fatigue test and torsion fatiguetest of S25C and S55C. (a) Prediction by tensile strength and test method.(b) Prediction by tensile strength, test method and stress ratio.Estimating the S-N Curve by Machine Learning Random Forest Method 431closer to the original data. Conversely, since the test data areextracted randomly, S25C and S55C, which have differentfatigue strengths, are not distinguished, and the estimateddata vary. It is unknown which data correspond to each ofS25C and S55C (because the data are extracted at random),but it is thought that it is probably the band indicated by thecircle in the figure.Next, the Vickers hardness, tensile strength, elongation,and reduction of area were estimated by linking the decisiontree models using a total of 2478 pieces of fatigue data(106 cycles or less) for different types of steels (S25C, S55C,S35C, SNCM439, SmN438, SmN443, SUS403, andSUS304). The results are shown in Table 4 and Fig. 8,which show that MAPE was estimated 29.8% moreaccurately than for the two steel grades, S25C and S55C,as shown in Fig. 7. This result is due to the increase in thetotal number of data by a factor of five compared to Fig. 7,and a further improvement in estimation accuracy can beexpected with more experimental data in the future.3.3 Estimation of the S–N curve of S45C steelThe relationship between the fatigue strength under5 © 106 cycles was obtained by machine learning using thedecision tree models of Vickers hardness, tensile strength,elongation, and reduction of area, based on the fatigue data(total = 2834) under 5 © 106 cycles for different types ofsteels (see Table 4). The fracture strength at each stressamplitude was estimated from the mechanical properties ofS45C. Additionally, the fatigue limit at 2.12 © 107 cycles wasobtained from the Vickers hardness of S45C based on therelationship between Vickers hardness and the fatigue limit at2.12 © 107 cycles obtained by machine learning from varioussteel materials. The analysis results are shown in Fig. 9.Experimental and estimated data are indicated by and ,respectively. First, as shown in Table 2, the fatigue limitestimated from the Vickers hardness agreed very well withthe estimated accuracy of 99% and MAPE of 1.76. Theestimated fatigue strength below 5 © 106 cycles is also inFig. 6 Prediction of 107 times fatigue limit using data of S25C, S35C,S55C, SNCM439, SmN438, SmN443, SUS403, SUS304.Fig. 7 Prediction result of fracture life using data of S25C and S55C (onlydata with fracture life of 106 times or less is used).Table 4 Analysis results by machine learning.Fig. 8 Prediction result of fracture life using data of S25C, S35C, S55C,SNCM439, SmN438, SmN443, SUS403, SUS304 (only data withfracture life of 106 times or less is used).N. Nagashima, M. Hayakawa, H. Masuda and K. Nagai432good agreement, even though the MAPE is 29.8%. The testdata did not include steel grades with different strengths, asshown in Figs. 7 and 8; thus, there was no variation in theprediction accuracy. Moreover, the estimation of the S–Ncurve by machine learning was highly accurate when thefatigue strength and limit were estimated separately. Thisresult reveals that the approximation of the S–N curve ispossible by utilizing the accumulated experimental data in theFDS.4. ConclusionsUsing the experimental data provided in the NIMS FDS,we attempted to estimate the fatigue limit and the fatiguestrength below 106 cycles by the random forest method andexamined the possibility of estimating the S–N curve. Theresults obtained are as follows:(1) The regression model by machine learning, which canassociate multiple learning factors, was superior inestimating the fatigue limit.(2) The S–N curve could be estimated with high accuracyby estimating the fatigue strength and fatigue limitseparately by machine learning.REFERENCES1) National Institute for Materials Science (NIMS): Fatigue data sheet(FDS), https://smds.nims.go.jp/fatigue.2) S. Nishijima, A. Ishii, K. Kanazawa, S. Matsuoka and T. Masuda:“Standard fatigue characteristics of JIS machine structural steel”, NIMSMaterials Strength Data Sheet Technical Document, No. 5 (1989).3) S. Matsuoka, N. Nagashima and S. Nishijima: “Index property for thefatigue of engineering alloys”, NIMS Materials Strength Data SheetTechnical Document, No. 17 (1997).4) JSMS Committee on Fatigue of Materials: Syoshinnsya no tamenohirousekkeihou, (The Society of Materials Science, Japan, 2004) p. 28.5) Python 3.6.1 (https://www.python.org/).6) Anaconda (https://www.anaconda.com/).Fig. 9 Prediction of S-N curve of fracture life using data of S25C, S35C,S55C, SNCM439, SmN438, SmN43, SUS403, SUS304 (data of fracturelife of 5 © 106 times or less, fatigue limit considers only hardness).Estimating the S-N Curve by Machine Learning Random Forest Method 433https://smds.nims.go.jp/fatiguehttps://www.python.org/https://www.anaconda.com/https://www.anaconda.com/