A Phase Field Modeling Approach of Crack Growth in Materials with Anisotropic Fracture Toughness

Authors Christoph Schreiber , Tim Ettrich, Charlotte Kuhn, Ralf Müller

Thumbnail PDF


  • Filesize: 1.64 MB
  • 17 pages

Document Identifiers

Author Details

Christoph Schreiber
  • Applied Mechanics, Technische Universität Kaiserslautern, Germany
Tim Ettrich
  • Technische Universität Kaiserslautern, Germany
Charlotte Kuhn
  • Faculty 7, Universität Stuttgart, Germany
Ralf Müller
  • Applied Mechanics, Technische Universität Kaiserslautern, Germany

Cite AsGet BibTex

Christoph Schreiber, Tim Ettrich, Charlotte Kuhn, and Ralf Müller. A Phase Field Modeling Approach of Crack Growth in Materials with Anisotropic Fracture Toughness. In 2nd International Conference of the DFG International Research Training Group 2057 – Physical Modeling for Virtual Manufacturing (iPMVM 2020). Open Access Series in Informatics (OASIcs), Volume 89, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Within this contribution, we present a diffuse interface approach for the simulation of crack nucleation and growth in materials, which incorporates an orientation dependency of the fracture toughness. After outlining the basic motivation for the model from an engineering standpoint, the phase field paradigm for fracture is introduced. Further, a specific phase field model for brittle fracture is reviewed, where we focus on the meaning of the auxiliary parameter differentiating between material phases and the coupling of such a parameter to continuum equations in order to obtain the characteristic self organizing model properties. This specific model, as will be explained, provides the phenomenological and methodical basis for the presented enhancement. The formulation of an appropriate evolution equation in terms of a Ginzburg-Landau type equation will be highlighted and several comments on sharp interface models will be made to present a brief comparison. Following up on the basics we then introduce the formulation of a modified version of the model, which additionally to the handling of cracks in linear elastic materials under quasi static loading is also capable of taking into account the effect of resistance variation with respect to the potential crack extension direction. The strong and also the weak forms of the respective governing equations corresponding to the developed anisotropic phase field model are presented. Utilizing the weak formulation as starting point for the discretization of the two fields (displacement field and the phase field), the computational framework in terms of finite elements is introduced. We finally explain several test cases investigated within simulations and discuss the corresponding numerical results. Besides examples, which are set up to illustrate the general model properties, a comparison with crack paths obtained by experimental investigations will be presented in order to show the potential of the developed phase field model.

Subject Classification

ACM Subject Classification
  • Applied computing → Physical sciences and engineering
  • Phase field modeling
  • Brittle fracture
  • Anisotropic fracture toughness
  • Finite elements


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. R. Alessi, S. Vidoli, and L. DeLorenzis. A phenomenological approach to fatigue with a variational phase-field model: The one-dimensional case. Engineering Fracture Mechanics, 190:53-73, 2017. Google Scholar
  2. B. Bourdin. Numerical implementation of the variational formulation of brittle fracture. Interfaces and Free Boundaries - INTERFACE FREE BOUND, 9:411-430, 2007. Google Scholar
  3. B. Bourdin, G. A. Francfort, and J-J. Marigio. Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids, 48:797-826, 2000. Google Scholar
  4. S. Chan. Steady-state kinetics of diffusionless first order phase transformation. The Journal of Chemical Physics, 67(12):5755-5762, 1977. Google Scholar
  5. E. De Giorgi and G. Dal Maso. Γ-convergence and calculus of variations. In Mathematical Theories of Optimization, pages 121-143, Berlin, Heidelberg, 1983. Springer. Google Scholar
  6. Robert V. Goldstein and Rafael L. Salganik. Brittle fracture of solids with arbitrary cracks. International Journal of Fracture, 10:507-523, 1974. Google Scholar
  7. A. A. Griffith. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London, 221:163-198, 1921. Google Scholar
  8. D. Gross and Th. Seelig. Bruchmechanik-Mit einer Einfürung in dei Mikromechanik. Springer, Heidelberg, 5 edition, 2011. Google Scholar
  9. V. Hakim and A. Karma. Laws of crack motion and phase-field models of fracture. Journal of the Mechanics and Physics of Solids, 57(2):342-368, 2009. Google Scholar
  10. P. O. Judt and A. Ricoeur. Crack growth simulation of multiple cracks systems applying remote contour interaction integrals. Theoretical and Applied Fracture Mechanics, 75:78-88, 2015. Google Scholar
  11. P. O. Judt, A. Ricoeur, and G. Linek. Crack path prediction in rolled aluminum plates with fracture toughness orthotropy and experimental validation. Engineering Fracture Mechanics, 138:33-48, 2015. Google Scholar
  12. A. Karma, D. Kessler, and H. Levine. Phase-field model of mode iii dynamic fracture. Physical Review Letters, 87:045501, 2001. Google Scholar
  13. A.P. Kfouri. Crack extension under mixed-mode loading in an anisotropic mode-asymmetric material in respect of resistance to fracture. Fatigue and Fracture of Engineering Materials and Structures, 19(1):27-38, 1996. Google Scholar
  14. Ryo Kobayashi. Modeling and numerical simulations of dendritic crystal growth. Physica D: Nonlinear Phenomena, 63(3):410-423, 1993. Google Scholar
  15. C. Kuhn and R. Müller. A continuum phase field model for fracture. Engineering Fracture Mechanics, 77:3625-3634, 2010. Google Scholar
  16. C. Kuhn and R. Müller. A discussion of fracture mechanisms in heterogeneous materials by means of configurational forces in a phase field fracture model. Computer Methods in Applied Mechanics and Engineering, 312:95-116, 2016. Google Scholar
  17. C. Kuhn, T. Noll, and R. Müller. On phase field modeling of ductile fracture. GAMM Mitteilungen, 39:35-54, 2016. Google Scholar
  18. C. Kuhn, A. Schlüter, and R. Müller. On degradation functions in phase field fracture models. Computational Materials Science, 108:374-384, 2015. Google Scholar
  19. B. Li, C. Peco, D. Millán, I. Arias, and M. Arroyo. Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy. International Journal for Numerical Methods in Engineering, 102(3-4):711-727, 2015. Google Scholar
  20. L. Ma and A. Korsunsky. On the use of vector j-integral in crack growth criteria for brittle solids. International Journal of Fracture, 133, 2005. Google Scholar
  21. J. J. Marigo. Modelling of brittle and fatigue damage for elastic material by growth of microvoids. Engineering Fracture Mechanics, 21:861-874, 1985. Google Scholar
  22. C. Miehe, F. Aldakheel, and T Stephan. Phase-field modeling of ductile fracture at finite strains. a robust variational-based numerical implementation of a gradient-extended theory by micromorphic regularization: Phase field modeling of ductile fracture. International Journal for Numerical Methods in Engineering, December 2016. URL: https://doi.org/10.1002/nme.5484.
  23. C. Miehe, F. Welschinger, and M. Hofacker. Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field fe implementations. International Journal for Numerical Methods in Engineering, 83(10):1273-1311, 2010. Google Scholar
  24. P. O'Hara, J. Hollkamp, C.A. Duarte, and T. Eason. A two-scale generalized finite element method for fatigue crack propagation simulations utilizing a fixed, coarse hexahedral mesh. Computational Mechanics, 57:55-74, 2016. Google Scholar
  25. N. Provatas and K. Elder. Phase Field Methodes in Material Science and Engineering. Wiley, Berlin, 1 edition, 2010. Google Scholar
  26. A.H. Rajkotwala, A. Panda, E.A.J.F. Peters, M.W. Baltussen, C.W.M. [van der Geld], J.G.M. Kuerten, and J.A.M. Kuipers. A critical comparison of smooth and sharp interface methods for phase transition. International Journal of Multiphase Flow, 120:103093, 2019. Google Scholar
  27. B. Roman, E. Hamm, and F. Melo. Forbidden directions for the fracture of thin anisotropic sheets: An analogy with the Wulff plot. Physical Review Letters, 110, April 2013. URL: https://doi.org/10.1103/PhysRevLett.110.144301.
  28. A. Schlüter, A. Willenbücher, C. Kuhn, and R. Müller. Phase field approximation of dynamic brittle fracture. Computational Mechanics, 54:1141-1161, 2014. Google Scholar
  29. C. Schreiber, C. Kuhn, and R. Müller. Phase field modeling of cyclic fatigue crack growth under mixed mode loading. Computer Methods in Materials Science, 19:50-56, 2019. Google Scholar
  30. C. Schreiber, C. Kuhn, R. Müller, and T. Zohdi. A phase field modeling approach of cyclic fatigue crack growth. International Journal of Fracture, 2020. Google Scholar
  31. M. Seiler, P. Hantschke, A. Brosius, and M. Kästner. A numerically efficient phase-field model for fatigue fracture – 1d analysis. PAMM, 18(1), 2018. Google Scholar
  32. S. Teichtmeister, D. Kienle, F. Aldakheel, and M.A. Keip. Phase field modeling of fracture in anisotropic brittle solids. International Journal of Non-Linear Mechanics, pages 100-118, July 2017. Google Scholar
  33. C. V. Verhoosel and R. de Borst. A phase-field model for cohesive fracture. International Journal for Numerical Methods in Engineering, 96(1):43-62, 2013. Google Scholar
  34. A.A. Wheeler, B.T. Murray, and R.J. Schaefer. Computation of dendrites using a phase field model. Physica D: Nonlinear Phenomena, 66(1):243-262, 1993. Google Scholar
  35. C. H. Wu. Maximum-energy-release-rate criterion applied to a tension-compression specimen with crack. Journal of Elasticity, 8:235-257, 1978. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail