References#

  1. A.M. Alshoaibi. Fatigue crack growth analysis under constant amplitude loading using finite element method. Materials, 2022. doi:10.3390/ma15082937.

  2. M. Ambati, T. Gerasimov, and L. De Lorenzis. A review on phase-field models of brittle fracture and a new fast hybrid formulation. Computational Mechanics, 55(2):383–405, Feb 2015. URL: https://doi.org/10.1007/s00466-014-1109-y, doi:10.1007/s00466-014-1109-y.

  3. H. Amor, J.-J. Marigo, and C. Maurini. Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. Journal of the Mechanics and Physics of Solids, 57(8):1209–1229, 2009. doi:https://doi.org/10.1016/j.jmps.2009.04.011.

  4. T.L. Anderson. Fracture mechanics: fundamentals and applications. Taylor & Francis, third edition, 2005.

  5. P. Aranda and J. Segurado. A crack-length control technique for phase-field fracture in FFT homogenization. International Journal for Numerical Methods in Engineering, 126(2):e7664, 2025. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.7664, arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.7664, doi:https://doi.org/10.1002/nme.7664.

  6. I.A. Baratta, J.P. Dean, J.S. Dokken, M. Habera, J.S. Hale, C.N. Richardson, M.E. Rognes, M.W. Scroggs, N. Sime, and G.N. Wells. Dolfinx: the next generation fenics problem solving environment. 2023. doi:https://doi.org/10.5281/zenodo.10447666.

  7. R. Bharali, S. Goswami, C. Anitescu, and T. Rabczuk. A robust monolithic solver for phase-field fracture integrated with fracture energy based arc-length method and under-relaxation. Computer Methods in Applied Mechanics and Engineering, 394:114927, 2022. doi:https://doi.org/10.1016/j.cma.2022.114927.

  8. B. Bourdin, G.A. Francfort, and J.-J. Marigo. Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids, 48(4):797–826, 2000. doi:https://doi.org/10.1016/S0022-5096(99)00028-9.

  9. R. Branco, F.V. Antunes, and J.D. Costa. A review on 3d-fe adaptive remeshing techniques for crack growth modelling. Engineering Fracture Mechanics, 141:170–195, 2015. doi:https://doi.org/10.1016/j.engfracmech.2015.05.023.

  10. P. Carrara, M. Ambati, R. Alessi, and L. De Lorenzis. A framework to model the fatigue behavior of brittle materials based on a variational phase-field approach. Computer Methods in Applied Mechanics and Engineering, 361:112731, 2020. URL: https://www.sciencedirect.com/science/article/pii/S0045782519306218, doi:10.1016/j.cma.2019.112731.

  11. M. Castillón. Phasefieldx: an open-source framework for advanced phase-field simulations. Journal of Open Source Software, 10(108):7307, 2025. doi:https://doi.org/10.21105/joss.07307.

  12. M. Castillón. URL. CastillonMiguel/A-Phase-Field-Approach-to-Fracture-and-Fatigue-Analysis-Bridging-Theory-and-Simulation, 2025.

  13. M. Castillón. Zenodo DOI to be added upon acceptance. 2025.

  14. M. Castillón, I. Romero, and J. Segurado. A phase-field approach to fracture and fatigue analysis: bridging theory and simulation. 2025. URL: https://arxiv.org/abs/2509.08939, arXiv:2509.08939.

  15. G.A. Francfort and J.-J. Marigo. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 46(8):1319–1342, 1998. doi:https://doi.org/10.1016/S0022-5096(98)00034-9.

  16. A. Golahmar, C. F. Niordson, and E. Martínez-Pañeda. A phase field model for high-cycle fatigue: total-life analysis. International Journal of Fatigue, 170:107558, 2023. URL: https://www.sciencedirect.com/science/article/pii/S0142112323000592, doi:https://doi.org/10.1016/j.ijfatigue.2023.107558.

  17. L. Greco, A. Patton, M. Negri, A. Marengo, U. Perego, and A. Reali. Higher order phase-field modeling of brittle fracture via isogeometric analysis. Engineering with Computers, 40(6):3541–3560, 2024. doi:https://doi.org/10.1007/s00366-024-01949-5.

  18. J. Heinzmann, P. Carrara, M. Ambati, A.M. Mirzaei, and L. De Lorenzis. An adaptive acceleration scheme for phase-field fatigue computations. Computational Mechanics, 2024. URL: https://doi.org/10.1007/s00466-024-02551-8, doi:10.1007/s00466-024-02551-8.

  19. C. Hou, X. Jin, X. Fan, R. Xu, and Z. Wang. A generalized maximum energy release rate criterion for mixed mode fracture analysis of brittle and quasi-brittle materials. Theoretical and Applied Fracture Mechanics, 100:78–85, 2019. doi:https://doi.org/10.1016/j.tafmec.2018.12.015.

  20. H.D. Huynh, M.N. Nguyen, G. Cusatis, S. Tanaka, and T.Q. Bui. A polygonal xfem with new numerical integration for linear elastic fracture mechanics. Engineering Fracture Mechanics, 213:241–263, 2019. doi:https://doi.org/10.1016/j.engfracmech.2019.04.002.

  21. T. Ingrafea and others. Franc3d, 3d fracture analysis code. The Cornell University Fracture Group, Cornell University, Ithaca, NY, 1996.

  22. P. K. Kristensen, A. Golahmar, E. Martínez-Pañeda, and C. F. Niordson. Accelerated high-cycle phase field fatigue predictions. European Journal of Mechanics - A/Solids, 100:104991, 2023. URL: https://www.sciencedirect.com/science/article/pii/S0997753823000839, doi:https://doi.org/10.1016/j.euromechsol.2023.104991.

  23. A. Mesgarnejad, A. Imanian, and A. Karma. Phase-field models for fatigue crack growth. Theoretical and Applied Fracture Mechanics, 103:102282, 2019. URL: https://www.sciencedirect.com/science/article/pii/S0167844218306712, doi:https://doi.org/10.1016/j.tafmec.2019.102282.

  24. C. Miehe, M. Hofacker, and F. Welschinger. A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering, 199(45-48):2765–2778, 2010. doi:https://doi.org/10.1016/j.cma.2010.04.011.

  25. G. Molnár, A. Doitrand, A. Jaccon, B. Prabel, and A. Gravouil. Thermodynamically consistent linear-gradient damage model in abaqus. Engineering Fracture Mechanics, 266:108390, 2022. URL: https://www.sciencedirect.com/science/article/pii/S001379442200145X, doi:https://doi.org/10.1016/j.engfracmech.2022.108390.

  26. G.J. Pataky, H. Sehitoglu, and H.J. Maier. High temperature fatigue crack growth of haynes 230. Materials Characterization, 75:69–78, 2013. URL: https://www.sciencedirect.com/science/article/pii/S1044580312002665, doi:https://doi.org/10.1016/j.matchar.2012.09.012.

  27. C. Schreiber, C. Kuhn, R. Müller, and T. Zohdi. A phase field modeling approach of cyclic fatigue crack growth. International Journal of Fracture, 225(1):89–100, 2020. URL: https://doi.org/10.1007/s10704-020-00468-w, doi:10.1007/s10704-020-00468-w.

  28. K. Seleš, F. Aldakheel, Z. Tonković, J. Sorić, and P. Wriggers. A general phase-field model for fatigue failure in brittle and ductile solids. Computational Mechanics, 67(5):1431–1452, 2021. URL: https://doi.org/10.1007/s00466-021-01996-5, doi:10.1007/s00466-021-01996-5.

  29. H. Tada. The stress analysis of cracks handbook. ASME Press, third edition, 2000.

  30. S. van der Walt, J.L. Schönberger, J. Nunez-Iglesias, F. Boulogne, J.D. Warner, N. Yager, E. Gouillart, T. Yu, and the scikit-image contributors. Scikit-image: image processing in python. PeerJ, 2:e453, 2014. doi:https://doi.org/10.7717/peerj.453.

  31. D. Wagner. A finite element-based adaptive energy response function method for curvilinear progressive fracture. PhD thesis, The University of Texas at San Antonio, United States – Texas, 2018.

  32. D. Wagner, M.J. Garcia, A. Montoya, and H. Millwater. A finite element-based adaptive energy response function method for 2d curvilinear progressive fracture. International Journal of Fatigue, 127:229–245, 2019. URL: https://www.sciencedirect.com/science/article/pii/S0142112319302300, doi:https://doi.org/10.1016/j.ijfatigue.2019.05.036.

  33. C. Wang, K. Pereira, D. Wang, A. Zinovev, D. Terentyev, and M. Abdel Wahab. Fretting fatigue crack propagation under out-of-phase loading conditions using extended maximum tangential stress criterion. Tribology International, 187:108738, 2023. doi:https://doi.org/10.1016/j.triboint.2023.108738.

  34. H. Xin, J.A.F.O. Correia, and M. Veljkovic. Three-dimensional fatigue crack propagation simulation using extended finite element methods for steel grades s355 and s690 considering mean stress effects. Engineering Structures, 227:111414, 2021. URL: https://www.sciencedirect.com/science/article/pii/S0141029620340153, doi:https://doi.org/10.1016/j.engstruct.2020.111414.

  35. J. Zambrano, S. Toro, P.J. Sánchez, F.P. Duda, C.G. Méndez, and A.E. Huespe. An arc-length control technique for solving quasi-static fracture problems with phase field models and a staggered scheme. Computational Mechanics, 73(4):751–772, 2024. doi:https://doi.org/10.1007/s00466-023-02388-7.

  36. T.Y. Zhang and C.Y. Suen. A fast parallel algorithm for thinning digital patterns. Communications of the ACM, 27(3):236–239, 1984. doi:https://doi.org/10.1145/357994.358023.