Fatigue crack propagation modeling in 2D structures using the CPCN-FEM node-based approach
Keywords:
crack propagation, Stress Intensity Factor (SIF), finite element method, CPCN-FEM, node coordinates, fracture mechanics
Abstract
Introduction/purpose: The study aims to model fatigue-related crack propagation by introducing a numerical method, CPCN-FEM (Crack Propagation by Coordinates of Nodes – Finite Element Method), which predicts crack trajectories through the systematic generation of nodal coordinates around the crack front.
Methods: The approach defines four principal nodes to control the propagation direction and computes the stress intensity factors (K I) and (KII), along with the crack inclination angle (β). The method was implemented in FORTRAN to automate node tracking, coordinate updating, and stepwise crack advance in a two-dimensional elastic isotropic model. Simulations were conducted and compared with analytical crack-path solutions, using the Richard criterion to determine crack orientation.
Results: The numerical model reproduced expected crack trajectories with high agreement with analytical predictions and demonstrated stable displacement behavior across multiple propagation cases. Mesh integrity was preserved during all propagation steps, and no remeshing was required.
Conclusion: The findings show that CPCN-FEM provides an accurate, efficient, and mesh-preserving technique for modeling crack growth under fatigue, offering reliable predictions of crack path evolution in fracture mechanics.
References
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Bentahar M. ( 2023). ALLDMD Dissipation Energy Analysis by the Method Extended Finite Elements of a 2D Cracked Structure of an Elastic Linear Isotropic Homogeneous Material. Journal of Electronics, Computer Networking and Applied Mathematics, v. 3, n. 2, pp. 1-8, 2023. Available at: https://doi.org/10.55529/jecnam.32.1.8.
Bentahar, M. (2023). Fatigue Analysis of an Inclined Crack Propagation Problem by the X-FEM Method. International Journal of Applied and Structural Mechanics (IJASM), v. 3, n. 4, pp. 23–31, Available at: https://doi.org/10.55529/ijasm.34.23.31.
Bentahar, M. Benzaama, H. and Mahmoudi, N. (2021). Numerical modeling of the evolution of the strain energy ALLSE of the crack propagation by the X-FEM method. Revue des matériaux et énergies renouvelables, v. 5, n. 2, pp. 24-31,
Bilby, B. A., and Cardew, G. E. (1975). The crack with a kinked tip. International Journal of Fracture, 11, pp. 708-712.
Borges, M. F. Lopez-Crespo, P. Antunes, F.V. Moreno, B. Prates, P. Camas, D. and Neto, D.M. (2021). Fatigue crack propagation analysis in 2024-T351 aluminium alloy using nonlinear parameters, International Journal of Fatigue, V153, 106478, Available at: https://doi.org/10.1016/j.ijfatigue.2021.106478.
Chang, J. H., & Jiang, J. Y. (2014). Modified Contour Integrals for Calculation of Stress Intensity Factors for Cracks in Anisotropic FGMs. Applied Mechanics and Materials, 627, 129-133. https://doi.org/10.4028/www.scientific.net/AMM.627.129
Crusat, L. Carol, I. and Garolera, D. (2021). Application of configurational mechanics to crack propagation in quasi-brittle materials, Engineering Fracture Mechanics, V 241, 107349, Available at: https://doi.org/10.1016/j.engfracmech.2020.107349.
Elguedj, T. (2006). Simulation numérique de la propagation de fissure en fatigue par la méthode des éléments finis étendus : thèse prise en compte de la plasticité et du contact-frottement, lamcos - umr cnrs 5514 - INSA de Lyon 20, avenue Albert Einstein, 69621 Villeurbanne Cedex (FRANCE).
Erdogan, F. and Sih G.C. (1963). On the crack extension in plates under plane loading and shear, Journal of Basic Engineering, 85, 4, pp. 519-527.
Fayed, A. S. (2017). Numerical analysis of mixed mode I/II stress intensity factors of edge slant cracked plates. Engineering Solid Mechanics, 5(1), 61-70. doi: 10.5267/j.esm.2016.8.001
Gdoutos, E. E. (2012). Problems of mixed mode crack propagation (Vol. 2). Springer Science & Business Media.
Guizhong, X. I. E., Chongmao, Z. H. A. O., Hao, L. I., Jun, L. I. U., Wenliao, D. U., Jiahe, L. V., & Chao, W. U. (2024). An adaptive extended finite element based crack propagation analysis method. Mechanics, 30(1), 74-82. DOI: https://doi.org/10.5755/j02.mech.33301
Ilie, P. and Ince, A. (2022). Three-dimensional fatigue crack growth simulation and fatigue life assessment based on finite element analysis, Fatigue & Fracture of Engineering Materials & Structures, v.45, n.11, pp. 3251-3266, Available at: https://doi.org/10.1111/ffe.13815.
Keprate, A., Chandima Ratnayake, R. M., & Sankararaman, S. (2019). Experimental validation of the adaptive Gaussian process regression model used for prediction of stress intensity factor as an alternative to finite element method. Journal of Offshore Mechanics and Arctic Engineering, 141(2), 021606. Doi https://doi.org/10.1115/1.4041457
Kocańda, D. and Torzewski, J. (2009). Deterministic approach to predicting the fatigue crack growth in the 2024-t3 aluminum alloy under variable amplitude loading , Fatigue of Aircraft Structures, Vol. 1, pp.102-115, 10.2478/v10164-010-0010-1.
Kumar, D., Roy, R., Kweon, J. H., & Choi, J. H. (2016). Numerical modeling of combined matrix cracking and delamination in composite laminates using cohesive elements. Applied Composite Materials, 23, 397-419. doi.org/10.1007/s10443-015-9465-0.
Mehmet, F. Yaren, M. F. Ayhan, A. O. (2024). A new method for prediction of fatigue crack propagation life under variable amplitude spectrum loading, Theoretical and Applied Fracture Mechanics, Volume 131, 104355, https://doi.org/10.1016/j.tafmec.2024.104355.
Montassir, S. Moustabchir, H. and Elkhalfi, A. (2023). Application of NURBS in the Fracture Mechanics Framework to Study the Stress Intensity Factor. Statistics, Optimization and Information Computing, v. 11, n. 1, pp. 106–115. Available at: https://doi.org/10.19139/soic-2310-5070-1553.
Nie, Y. Wang, X. Wu, B. Zhang, G. Gamage, R. P. Li, Shiyuan. and Zhang, L. (2025). Influences of the stress ratio and local micro mineral aggregates on small fatigue crack propagation in the shale containing bedding planes, International Journal of Rock Mechanics and Mining Sciences, V185, 105980, Available at: https://doi.org/10.1016/j.ijrmms.2024.105980.
Niu, Y. Fan, J. Shi, X. Wei, J. Jiao, C. and Hu, J. (2023). Application of the J-Integral and Digital Image Correlation (DIC) to Determination of Multiple Crack Propagation Law of UHPC under Flexural Cyclic Loading. Materials, v. 6, p. 1296, Available at: https://doi.org/10.3390/ ma16010296.
Nuismer, R.J. (1975). An energy release rate criterion for mixed mode fracture, International Journal of Fracture, 11, 2, pp. 245-250.
Pramana, N. et al. (2024).Experimental and numerical analysis of crack propagation material Al 2024 under spectrum load. AIP Conference Proceedings. Vol. 3003. No. 1. AIP Publishing LLC, Available at: https://doi.org/10.1063/5.0186423.
Rahman, M. Maeda, T. Osada, T. and Ozaki, S. (2024). Finite element analysis of crack propagation, crack-gap-filling, and recovery behavior of mechanical properties in oxidation-induced self-healing ceramics, International Journal of Solids and Structures, v. 306, n.3, 113104, Available at: https://doi.org/10.1016/j.ijsolstr.2024.113104.
Ricardo, L.C.H. (2018). Crack Propagation by Finite Element Method, Frattura edIntegrità Strutturale, v. 12, n. 43, pp. 57-78. DOI: 10.3221/IGF-ESIS.43.04.
Richard, H. A. (2004). Fulland, M and Sander, M. Theoretical crack path prediction Publishing Ltd. Fatigue Fract Engng Mater Struct 28, pp. 3-12.
Richard, H.A. (1985). Bruchvorhersagen bei überlagerter Normal- und Schubbeanspruchung von Rissen.VDI-Forschungsheft 631. VDI-Verlag, Düsseldorf.
Robles, J .M. Vasco-Olmo, J.M. Cruces, A.S. Diaz, F.A. James, M.N. and Lopez-Crespo, P. (2023). Fatigue crack characterisation in 2024-T351 aluminium alloy through SEM observation combined with the CJP model, International Journal of FatigueVolume 166, 107279, Available at: https://doi.org/10.1016/j.ijfatigue.2022.107279.
Sadek, M. Bergström, J. and Hallbäck, N. (2024). Computing the stress intensity factor range for fatigue crack growth testing at 20 kHz. Engineering Reports. 6(6):e12792. doi: 10.1002/eng2.12792.
Saverio, F. (2014). Modélisation tridimensionnelle de la fermeture induite par plasticité lors de la propagation d’une fissure de fatigue dans l’acier 304L. Thèse (Doctorat) – L’Ecole Nationale Supérieure de Mécanique et D’Aérotechnique. Soutenue le 24/11/2014. https://theses.hal.science/tel01129079.
Sedmak, A. (2023). Fatigue crack growth simulation by extended finite element method: A review of case studies. Fatigue Fract. Eng. Mater. Struct. c, 47, pp. 1819–1855. Available at: https://doi.org/10.1111/ffe.14277.
Sedmak, A. Grbović, A. Gubeljak, N. Sedmak, S. and Budimir, N. (2024) Numerical Simulation of Fatigue Crack Growth and Fracture in Welded Joints Using XFEM—A Review of Case Studies. Materials , 17, 5531. Available at: https://doi.org/10.3390/ma17225531.
Si, Z. Yu, T. Fang, W. Natarajan, S. (2024). An adaptive multi-patch isogeometric phase-field model for fatigue fracture. Int. J. Mech. Sci.2024, 271, 109146.
Tada H.P., Paris P.C. and Irwin G.R. (2000). The Stress Analysis of Cracks Handbook, American Society of Mechanical Engineering
Worthington, M., & Chew, H. B. (2023). Crack path predictions in heterogeneous media by machine learning. Journal of the Mechanics and Physics of Solids, 172, 105188.
Wysmulski, P. (2023). Numerical and Experimental Study of Crack Propagation in the Tensile Composite Plate with the Open Hole , Adv. Sci. Technol. Res. J. v. 17, n.4, pp. 249-261, Available at: https://doi.org/10.12913/22998624/169970.
Yaren, M. F. Ayhan, A. O. (2024). A new method for prediction of fatigue crack propagation life under variable amplitude spectrum loading, Theoretical and Applied Fracture Mechanics, Volume 131, 104355,Available at: https://doi.org/10.1016/j.tafmec.2024.104355.
Zhang, T. Yu, T. Ding, J. and Natarajan, S. (2024). Low-cycle fatigue crack growth in brittle materials: Adaptive phase-field modeling withvariable-node elements. Comput. Methods Appl. Mech. Eng. 2024, 425, 116917.
Zhang, T. Yu, T. Li, Y. Bui, T.Q. (2023). Crack growth in anisotropic brittle and polycrystalline materials by adaptive phase field modelusing variable-node elements. Finite Elem. Anal. Des. 217, 103909.
Zienkiewicz, O.C. Taylor, R.L. and Zhu, J.Z. (2005).The Finite Element Method: Its Basis and Fundamentals; Elsevier: Amsterdam, The Netherlands.
AlshoaibI, A. M. and Fageehi, Y. A. (2023). A Computational Framework for 2D Crack Growth Based on the Adaptive Finite. Element Method, Appl. Sci, v. 13, n. 1, 2023. 284, https:// doi.org/10.3390/app13010284.
Amestoy, M. Bui, H.D. and Dang Van K. (1980). [In:] International Advances in Fracture Research, Fran¸cois D. et al. (Eds.), Oxford, pp. 107-113.
Bentahar M. ( 2023). ALLDMD Dissipation Energy Analysis by the Method Extended Finite Elements of a 2D Cracked Structure of an Elastic Linear Isotropic Homogeneous Material. Journal of Electronics, Computer Networking and Applied Mathematics, v. 3, n. 2, pp. 1-8, 2023. Available at: https://doi.org/10.55529/jecnam.32.1.8.
Bentahar, M. (2023). Fatigue Analysis of an Inclined Crack Propagation Problem by the X-FEM Method. International Journal of Applied and Structural Mechanics (IJASM), v. 3, n. 4, pp. 23–31, Available at: https://doi.org/10.55529/ijasm.34.23.31.
Bentahar, M. Benzaama, H. and Mahmoudi, N. (2021). Numerical modeling of the evolution of the strain energy ALLSE of the crack propagation by the X-FEM method. Revue des matériaux et énergies renouvelables, v. 5, n. 2, pp. 24-31,
Bilby, B. A., and Cardew, G. E. (1975). The crack with a kinked tip. International Journal of Fracture, 11, pp. 708-712.
Borges, M. F. Lopez-Crespo, P. Antunes, F.V. Moreno, B. Prates, P. Camas, D. and Neto, D.M. (2021). Fatigue crack propagation analysis in 2024-T351 aluminium alloy using nonlinear parameters, International Journal of Fatigue, V153, 106478, Available at: https://doi.org/10.1016/j.ijfatigue.2021.106478.
Chang, J. H., & Jiang, J. Y. (2014). Modified Contour Integrals for Calculation of Stress Intensity Factors for Cracks in Anisotropic FGMs. Applied Mechanics and Materials, 627, 129-133. https://doi.org/10.4028/www.scientific.net/AMM.627.129
Crusat, L. Carol, I. and Garolera, D. (2021). Application of configurational mechanics to crack propagation in quasi-brittle materials, Engineering Fracture Mechanics, V 241, 107349, Available at: https://doi.org/10.1016/j.engfracmech.2020.107349.
Elguedj, T. (2006). Simulation numérique de la propagation de fissure en fatigue par la méthode des éléments finis étendus : thèse prise en compte de la plasticité et du contact-frottement, lamcos - umr cnrs 5514 - INSA de Lyon 20, avenue Albert Einstein, 69621 Villeurbanne Cedex (FRANCE).
Erdogan, F. and Sih G.C. (1963). On the crack extension in plates under plane loading and shear, Journal of Basic Engineering, 85, 4, pp. 519-527.
Fayed, A. S. (2017). Numerical analysis of mixed mode I/II stress intensity factors of edge slant cracked plates. Engineering Solid Mechanics, 5(1), 61-70. doi: 10.5267/j.esm.2016.8.001
Gdoutos, E. E. (2012). Problems of mixed mode crack propagation (Vol. 2). Springer Science & Business Media.
Guizhong, X. I. E., Chongmao, Z. H. A. O., Hao, L. I., Jun, L. I. U., Wenliao, D. U., Jiahe, L. V., & Chao, W. U. (2024). An adaptive extended finite element based crack propagation analysis method. Mechanics, 30(1), 74-82. DOI: https://doi.org/10.5755/j02.mech.33301
Ilie, P. and Ince, A. (2022). Three-dimensional fatigue crack growth simulation and fatigue life assessment based on finite element analysis, Fatigue & Fracture of Engineering Materials & Structures, v.45, n.11, pp. 3251-3266, Available at: https://doi.org/10.1111/ffe.13815.
Keprate, A., Chandima Ratnayake, R. M., & Sankararaman, S. (2019). Experimental validation of the adaptive Gaussian process regression model used for prediction of stress intensity factor as an alternative to finite element method. Journal of Offshore Mechanics and Arctic Engineering, 141(2), 021606. Doi https://doi.org/10.1115/1.4041457
Kocańda, D. and Torzewski, J. (2009). Deterministic approach to predicting the fatigue crack growth in the 2024-t3 aluminum alloy under variable amplitude loading , Fatigue of Aircraft Structures, Vol. 1, pp.102-115, 10.2478/v10164-010-0010-1.
Kumar, D., Roy, R., Kweon, J. H., & Choi, J. H. (2016). Numerical modeling of combined matrix cracking and delamination in composite laminates using cohesive elements. Applied Composite Materials, 23, 397-419. doi.org/10.1007/s10443-015-9465-0.
Mehmet, F. Yaren, M. F. Ayhan, A. O. (2024). A new method for prediction of fatigue crack propagation life under variable amplitude spectrum loading, Theoretical and Applied Fracture Mechanics, Volume 131, 104355, https://doi.org/10.1016/j.tafmec.2024.104355.
Montassir, S. Moustabchir, H. and Elkhalfi, A. (2023). Application of NURBS in the Fracture Mechanics Framework to Study the Stress Intensity Factor. Statistics, Optimization and Information Computing, v. 11, n. 1, pp. 106–115. Available at: https://doi.org/10.19139/soic-2310-5070-1553.
Nie, Y. Wang, X. Wu, B. Zhang, G. Gamage, R. P. Li, Shiyuan. and Zhang, L. (2025). Influences of the stress ratio and local micro mineral aggregates on small fatigue crack propagation in the shale containing bedding planes, International Journal of Rock Mechanics and Mining Sciences, V185, 105980, Available at: https://doi.org/10.1016/j.ijrmms.2024.105980.
Niu, Y. Fan, J. Shi, X. Wei, J. Jiao, C. and Hu, J. (2023). Application of the J-Integral and Digital Image Correlation (DIC) to Determination of Multiple Crack Propagation Law of UHPC under Flexural Cyclic Loading. Materials, v. 6, p. 1296, Available at: https://doi.org/10.3390/ ma16010296.
Nuismer, R.J. (1975). An energy release rate criterion for mixed mode fracture, International Journal of Fracture, 11, 2, pp. 245-250.
Pramana, N. et al. (2024).Experimental and numerical analysis of crack propagation material Al 2024 under spectrum load. AIP Conference Proceedings. Vol. 3003. No. 1. AIP Publishing LLC, Available at: https://doi.org/10.1063/5.0186423.
Rahman, M. Maeda, T. Osada, T. and Ozaki, S. (2024). Finite element analysis of crack propagation, crack-gap-filling, and recovery behavior of mechanical properties in oxidation-induced self-healing ceramics, International Journal of Solids and Structures, v. 306, n.3, 113104, Available at: https://doi.org/10.1016/j.ijsolstr.2024.113104.
Ricardo, L.C.H. (2018). Crack Propagation by Finite Element Method, Frattura edIntegrità Strutturale, v. 12, n. 43, pp. 57-78. DOI: 10.3221/IGF-ESIS.43.04.
Richard, H. A. (2004). Fulland, M and Sander, M. Theoretical crack path prediction Publishing Ltd. Fatigue Fract Engng Mater Struct 28, pp. 3-12.
Richard, H.A. (1985). Bruchvorhersagen bei überlagerter Normal- und Schubbeanspruchung von Rissen.VDI-Forschungsheft 631. VDI-Verlag, Düsseldorf.
Robles, J .M. Vasco-Olmo, J.M. Cruces, A.S. Diaz, F.A. James, M.N. and Lopez-Crespo, P. (2023). Fatigue crack characterisation in 2024-T351 aluminium alloy through SEM observation combined with the CJP model, International Journal of FatigueVolume 166, 107279, Available at: https://doi.org/10.1016/j.ijfatigue.2022.107279.
Sadek, M. Bergström, J. and Hallbäck, N. (2024). Computing the stress intensity factor range for fatigue crack growth testing at 20 kHz. Engineering Reports. 6(6):e12792. doi: 10.1002/eng2.12792.
Saverio, F. (2014). Modélisation tridimensionnelle de la fermeture induite par plasticité lors de la propagation d’une fissure de fatigue dans l’acier 304L. Thèse (Doctorat) – L’Ecole Nationale Supérieure de Mécanique et D’Aérotechnique. Soutenue le 24/11/2014. https://theses.hal.science/tel01129079.
Sedmak, A. (2023). Fatigue crack growth simulation by extended finite element method: A review of case studies. Fatigue Fract. Eng. Mater. Struct. c, 47, pp. 1819–1855. Available at: https://doi.org/10.1111/ffe.14277.
Sedmak, A. Grbović, A. Gubeljak, N. Sedmak, S. and Budimir, N. (2024) Numerical Simulation of Fatigue Crack Growth and Fracture in Welded Joints Using XFEM—A Review of Case Studies. Materials , 17, 5531. Available at: https://doi.org/10.3390/ma17225531.
Si, Z. Yu, T. Fang, W. Natarajan, S. (2024). An adaptive multi-patch isogeometric phase-field model for fatigue fracture. Int. J. Mech. Sci.2024, 271, 109146.
Tada H.P., Paris P.C. and Irwin G.R. (2000). The Stress Analysis of Cracks Handbook, American Society of Mechanical Engineering
Worthington, M., & Chew, H. B. (2023). Crack path predictions in heterogeneous media by machine learning. Journal of the Mechanics and Physics of Solids, 172, 105188.
Wysmulski, P. (2023). Numerical and Experimental Study of Crack Propagation in the Tensile Composite Plate with the Open Hole , Adv. Sci. Technol. Res. J. v. 17, n.4, pp. 249-261, Available at: https://doi.org/10.12913/22998624/169970.
Yaren, M. F. Ayhan, A. O. (2024). A new method for prediction of fatigue crack propagation life under variable amplitude spectrum loading, Theoretical and Applied Fracture Mechanics, Volume 131, 104355,Available at: https://doi.org/10.1016/j.tafmec.2024.104355.
Zhang, T. Yu, T. Ding, J. and Natarajan, S. (2024). Low-cycle fatigue crack growth in brittle materials: Adaptive phase-field modeling withvariable-node elements. Comput. Methods Appl. Mech. Eng. 2024, 425, 116917.
Zhang, T. Yu, T. Li, Y. Bui, T.Q. (2023). Crack growth in anisotropic brittle and polycrystalline materials by adaptive phase field modelusing variable-node elements. Finite Elem. Anal. Des. 217, 103909.
Zienkiewicz, O.C. Taylor, R.L. and Zhu, J.Z. (2005).The Finite Element Method: Its Basis and Fundamentals; Elsevier: Amsterdam, The Netherlands.
Published
2026/01/23
Section
Original Scientific Papers
Copyright (c) 2026 Noureddine Mahmoudi, Mohammed Bentahar, Hachemi Nedir , Youcef Moulai Arbi
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