Probabilistic analysis of elastic–plastic stress in notched components via machine learning

Abstract

Elastic–plastic Finite-Element analysis of notched components is computationally expensive for probabilistic design. To this end, we develop Machine Learning (ML) surrogates trained on linear elastic stress to predict nonlinear elastic–plastic responses. Using Latin Hypercube Sampling, we generate datasets spanning realistic material and geometric variations. Four ML algorithms are compared, namely Deep Neural Networks, Random Forest, Gradient Boosting (GB) and Support Vector Machines, with GB providing the highest accuracy. The computationally efficient workflow eliminates iterative convergence checks, enabling large-scale probabilistic assessment. A variance-based Sobol sensitivity analysis shows that linear von Mises stress and cyclic hardening coefficient explain ~90% of response variance, enabling simplified two-parameter models for preliminary design using the surrogate models developed here.

Graphical abstract

Keywords

elastic–plastic stress / variance-based sensitivity analysis / deep neural networks / machine learning / notched components

References

[1] Stephens R I, Fatemi A, Stephens R R, Fuchs H O. Metal Fatigue in Engineering. 2nd ed. Hoboken: Wiley, 2000

[2] Suresh S. Fatigue of Materials. 2nd ed. Cambridge: Cambridge University Press, 1998

[3] Sudret B , Guédé Z . Probabilistic assessment of thermal fatigue in nuclear components. Nuclear Engineering and Design, 2005, 235(17–19): 1819–1835

[4] Neuber H . Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress–strain law. Journal of Applied Mechanics, 1961, 28(4): 544–550

[5] Glinka G . Energy density approach to calculation of inelastic strain-stress near notches and cracks. Engineering Fracture Mechanics, 1985, 22(3): 485–508

[6] Jiang Y , Xu B . Deformation analysis of notched components and assessment of approximate methods. Fatigue & Fracture of Engineering Materials & Structures, 2001, 24(11): 729–740

[7] Ye D Y , Matsuoka S , Suzuki N , Maeda Y . Further investigation of Neuber’s rule and the equivalent strain energy density (ESED) method. International Journal of Fatigue, 2004, 26(5): 447–455

[8] Seeger T , Heuler P . Generalized application of Neuber’s rule. Journal of Testing and Evaluation, 1980, 8(4): 199–204

[9] Dowling N E . Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue. 4th ed, 2013,

[10] Rovinelli A , Sangid M D , Proudhon H , Ludwig W . Using machine learning and a data-driven approach to identify the small fatigue crack driving force in polycrystalline materials. npj Computational Materials, 2018, 4(1): 35

[11] Samaniego E , Anitescu C , Goswami S , Nguyen-Thanh V M , Guo H , Hamdia K , Zhuang X , Rabczuk T . An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112790

[12] Goswami S , Anitescu C , Chakraborty S , Rabczuk T . Transfer learning enhanced physics informed neural network for phase-field modeling of fracture. Theoretical and Applied Fracture Mechanics, 2020, 106: 102447

[13] Nguyen T C H , Vu-Bac N , Budarapu P R . A kriging-based uncertainty quantification for fracture assessment. International Journal of Computational Methods, 2025, 22(1): 2450045

[14] Dsouza S M , Varghese T M , Budarapu P R , Natarajan S . A non-intrusive stochastic isogeometric analysis of functionally graded plates with material uncertainty. Axioms, 2020, 9(3): 92

[15] Hamdia K M , Msekh M A , Silani M , Thai T Q , Budarapu P R , Rabczuk T . Assessment of computational fracture models using Bayesian method. Engineering Fracture Mechanics, 2019, 205: 387–398

[16] Siruvuri S D V S S V , Budarapu P R , Paggi M . Influence of cracks on fracture strength and electric power losses in Silicon solar cells at high temperatures: Deep machine learning and molecular dynamics approach. Applied Physics A, 2023, 129(6): 408

[17] Mishra A , Anitescu C , Budarapu P R , Natarajan S , Vundavilli P R , Rabczuk T . An artificial neural network based deep collocation method for the solution of transient linear and nonlinear partial differential equations. Frontiers of Structural and Civil Engineering, 2024, 18(8): 1296–1310

[18] Nguyen T C H , Vu-Bac N . Machine learning-assisted sensitivity analysis for stochastic fatigue life modeling of metals. International Journal of Mechanical System Dynamics, 2025, 5(3): 481–494

[19] Burghardt R , Wächter M , Masendorf L , Esderts A . Estimation of elastic–plastic notch strains and stresses using artificial neural networks. Fatigue & Fracture of Engineering Materials & Structures, 2021, 44(10): 2718–2735

[20] McKay M D , Beckman R J , Conover W J . A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 1979, 21(2): 239–245

[21] Olsson A M J , Sandberg G E . Latin hypercube sampling for stochastic finite element analysis. Journal of Engineering Mechanics, 2002, 128(1): 121–125

[22] Goodfellow I, Bengio Y, Courville A. Deep Learning. Cambridge: MIT Press, 2016

[23] Breiman L . Random forests. Machine Learning, 2001, 45(1): 5–32

[24] Friedman J H . Greedy function approximation: A gradient boosting machine. The Annals of Statistics, 2001, 29(5): 1189–1232

[25] Vapnik V N. Statistical Learning Theory. New York: Wiley, 1998

[26] Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S. Global Sensitivity Analysis: The Primer. Hoboken: Wiley, 2008

[27] Budynas R G, Nisbett J K. Shigley’s Mechanical Engineering Design. 11th ed. New York: McGraw-Hill, 2020

[28] Fatemi A , Shamsaei N . Multiaxial fatigue: an overview and some approximation models for life estimation. International Journal of Fatigue, 2011, 33(8): 948–958

[29] Ramberg W , Osgood W R . Description of Stress–Strain Curves by Three Parameters. NACA-TN-902, 1943,

[30] Bai Y L , Wierzbicki T . A new model of metal plasticity and fracture with pressure and Lode dependence. International Journal of Plasticity, 2008, 24(6): 1071–1096

[31] US-ASME . Boiler and Pressure Vessel Code, Section VIII, Division 2. ASMEBPVC.VIII.2-2021, 2021,

[32] Topper T H , Wetzel R M , Morrow J . Neuber’s Rule Applied to Fatigue of Notched Specimens. Contract No. N156-46083, 1967,

[33] Molski K , Glinka G . A method of elastic–plastic stress and strain calculation at a notch root. Materials Science and Engineering, 1981, 50(1): 93–100

[34] Bäumel A Jr, Seeger T. Materials Data for Cyclic Loading. Supplement 1. Amsterdam: Elsevier, 1990

[35] Jiang Y , Sehitoglu H . Modeling of cyclic ratchetting plasticity, Part I: development of constitutive relations. Journal of Applied Mechanics, 1996, 63(3): 720–725

[36] Forrester A I J, Sóbester A, Keane A J. Engineering Design via Surrogate Modelling: A Practical Guide. Hoboken: Wiley, 2008

[37] Vu-Bac N , Rafiee R , Zhuang X , Lahmer T , Rabczuk T . Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites Part B: Engineering, 2015, 68: 446–464

[38] Vu-Bac N , Lahmer T , Zhuang X , Nguyen-Thoi T , Rabczuk T . A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[39] Vu-Bac N , Nguyen A H , Luong V H . Uncertainty analysis of static fatigue of Hi-Nicalon bundles. Engineering Analysis with Boundary Elements, 2024, 167: 105862

[40] Glinka G , Newport A . Universal features of elastic notch-tip stress fields. International Journal of Fatigue, 1987, 9(3): 143–150

[41] Kingma D P, Ba J. Adam: A method for stochastic optimization. In: Proceedings of the 3rd International Conference on Learning Representations. San Diego: OpenReview.net, 2015

[42] Akiba T, Sano S, Yanase T, Ohta T, Koyama M. Optuna: A next-generation hyperparameter optimization framework. In: Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. Anchorage: ACM, 2019, 2623–2631

[43] Bergstra J , Bengio Y . Random search for hyper-parameter optimization. Journal of Machine Learning Research, 2012, 13: 281–305

[44] Nguyen T C H , Zhuang X Y , Le A T , Luong V H , Vu-Bac N . Fatigue life prediction under multiaxial loading using machine learning and dependency-aware sensitivity analysis. International Journal of Mechanical System Dynamics, 2025, 5(4): 736–761

[45] Herman J , Usher W . SALib: an open-source Python library for sensitivity analysis. Journal of Open Source Software, 2017, 2(9): 97

[46] Chen T Q, Guestrin C. XGBoost: A scalable tree boosting system. In: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. San Francisco: ACM, 2016, 785–794

[47] Grinsztajn L, Oyallon E, Varoquaux G. Why do tree-based models still outperform deep learning on typical tabular data? In: Proceedings of the 36th International Conference on Neural Information Processing Systems. New Orleans: Curran Associates Inc., 2022, 37

RIGHTS & PERMISSIONS

Higher Education Press