PSI - Issue 68

Tea Marohnić et al. / Procedia Structural Integrity 68 (2025) 84 – 90 T. Marohnić and R. Basan / Structural Integrity Procedia 00 (2025) 000–000

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error function), several algorithms were used: Levenberg–Marquadt algorithm with and without early stopping, and Bayesian regularization. Since training dataset contained 70 materials, which is relatively modest in terms of ANN modeling, k -fold cross validation was utilized, with k set to ten folds. For each ANN architecture (number of neurons in hidden layer), 10 such ensembles were trained with different initial values of weights to reduce the possibility of the error function of the selected network converging to local instead of global minimum. The best ensembles were chosen from all architectures based on the value of the error function (in this case the mean square error, MSE ) and evaluated on an independent set of data that had not been utilized for ANN training. Details on development and selection of the most successful ANNs are provided in Marohnić (2017) and Marohnić and Basan (2018). 3. Comparison to selected empirical methods for estimation of strain-life parameters of steels Since comparison to other machine learning-based methods is not possible due to unavailability of existing algorithm or enough information to replicate the outcome, in this paper ANNs were compared to well-known and widely used empirical methods for estimation of strain-life parameters of steels. Selected methods are Modified Universal Slopes Method by Muralidharan and Manson (1988), Uniform Material Law by Bäumel and Seeger (1990), and Hardness Method by Roessle and Fatemi (2000). Expressions for calculation of fatigue lives 2 N f, are given in (1), (2) and (3), respectively. ! # " =0,623' $ ! % ( &,()# )2 * + +&,&, + 0,0196 * &,-.. ' $ ! % ( +&,.) )2 * + +&,./ (1) ! # " =1,5 $ ! % )2 * + +&,&(0 + 0,59 )2 * + +&,.( , (2a) = 2 1 $ ! % ≤0,003 1,375−125 $ ! % for $ ! % >0,003 (2b) ! # " = 1,#.2%34##. )2 * + +&,&, + &,)#23 " +1(%0234-,-&&& )2 * + +&,./ (3) 4. Evaluation methodology and criteria For the purpose of this investigation, Park and Song’s (1995) quantitative criteria were applied to evaluate the performance of developed ANNs: error criterion i.e. fraction of data points within a scatter band with factor of 3, E f (3), goodness of fit between the predicted and experimental values for individual datasets, ( E a ) Dset , goodness of fit between the predicted and experimental values for all data points, ( E a ) total and the average value of the above three parameters, Ē . The performance evaluation also makes use of the expanded methodology proposed by Basan and Marohnić (2024) who proposed the separate evaluations for materials divided into low strength and high strength subgroups in low-cycle fatigue (2 N f,exp ≤ 2 × 10 4 ) and high-cycle fatigue regime (2 N f,exp > 2 × 10 4 ). For low-alloy steels, division criterion regarding ultimate tensile strength R m is 1000 MPa. Such division criteria provide more detailed information. Following the mentioned procedure also ensures comparability with existing empirical methods. 5. Material data and analysis Evaluations were performed on a detailed material dataset which included monotonic properties and strain-life fatigue parameters. The dataset consisted of 23 low-alloy steels, all of which were not used for artificial neural network development. In such a way, ANNs ability to generalize i.e. their performance on “unseen” data could be evaluated. The dataset collected follows the same testing conditions and data distribution as the training dataset. Materials were tested in air at room temperature. Tests were strain-controlled, performed at minimum of 4 different strain amplitudes and with at least 0,4% range of total strain amplitude.