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Frontiers of Structural and Civil Engineering

, Volume 13, Issue 2, pp 417–428 | Cite as

Implementation aspects of a phase-field approach for brittle fracture

  • G. D. HuynhEmail author
  • X. ZhuangEmail author
  • H. Nguyen-Xuan
Research Article
  • 71 Downloads

Abstract

This paper provides a comprehensive overview of a phase-field model of fracture in solid mechanics setting. We start reviewing the potential energy governing the whole process of fracture including crack initiation, branching or merging. Then, a discretization of system of equation is derived, in which the key aspect is that for the correctness of fracture phenomena, a split into tensile and compressive terms of the strain energy is performed, which allows crack to occur in tension, not in compression. For numerical analysis, standard finite element shape functions are used for both primary fields including displacements and phase field. A staggered scheme which solves the two fields of the problem separately is utilized for solution step and illustrated with a segment of Python code.

Keywords

phase-field modeling FEM staggered scheme fracture 

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Notes

Acknowledgements

The authors would like to thank M.Sc Hoang-Giang Bui from Institute for Structural Mechanics, Ruhr University Bochum for his support on the implementation based on the open source software—Kratos and for helpful discussions on aspects of fracture mechanics.

References

  1. 1.
    Budarapu P R, Gracie R, Yang SW, Zhuang X, Rabczuk T. Efficient coarse graining in multiscale modeling of fracture. Theoretical and Applied Fracture Mechanics, 2014, 69: 126–143CrossRefGoogle Scholar
  2. 2.
    Talebi H, Silani M, Bordas S P A, Kerfriden P, Rabczuk T. A computational library for multiscale modeling of material failure. Computational Mechanics, 2014, 53(5): 1047–1071MathSciNetCrossRefGoogle Scholar
  3. 3.
    Griffith A A. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1921, 221(582–593): 163–198CrossRefGoogle Scholar
  4. 4.
    Irwin G R. Analysis of stresses and strains near the end of a crack traversing a plate. Journal of Applied Mechanics, 1957, 24: 361–364Google Scholar
  5. 5.
    Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 1999, 46(1): 131–150CrossRefzbMATHGoogle Scholar
  6. 6.
    Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343CrossRefzbMATHGoogle Scholar
  7. 7.
    Rabczuk T, Zi G. A meshfree method based on the local partition of unity for cohesive cracks. Computational Mechanics, 2007, 39(6): 743–760CrossRefzbMATHGoogle Scholar
  8. 8.
    Rabczuk T, Zi G, Gerstenberger A, Wall W A. A new crack tip element for the phantom-node method with arbitrary cohesive cracks. International Journal for Numerical Methods in Engineering, 2008, 75(5): 577–599CrossRefzbMATHGoogle Scholar
  9. 9.
    Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for non-linear dynamic fracture. International Journal for Numerical Methods in Engineering, 2007, 72(5): 524–548MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37–40): 2437–2455CrossRefzbMATHGoogle Scholar
  11. 11.
    Remmers J J C, de Borst R, Needleman A. The simulation of dynamic crack propagation using the cohesive segments method. Journal of the Mechanics and Physics of Solids, 2008, 56(1): 70–92MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Thai T Q, Rabczuk T, Bazilevs Y, Meschke G. A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2016, 304: 584–604MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ghorashi S S, Valizadeh N, Mohammadi S, Rabczuk T. T-spline based XIGA for fracture analysis of orthotropic media. Computers & Structures, 2015, 147: 138–146CrossRefGoogle Scholar
  14. 14.
    Xu X, Needleman A. Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids, 1994, 42(9): 1397–1434CrossRefzbMATHGoogle Scholar
  15. 15.
    Miehe C, Gürses E. A robust algorithm for configurational-forcedriven brittle crack propagation with R-adaptive mesh alignment. International Journal for Numerical Methods in Engineering, 2007, 72(2): 127–155MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Areias P, Rabczuk T. Finite strain fracture of plates and shells with configurational forces and edge rotation. International Journal for Numerical Methods in Engineering, 2013, 94(12): 1099–1122MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Francfort G A, Marigo J J. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 1998, 46(8): 1319–1342MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bourdin B, Francfort G A, Marigo J J. The variational approach to fracture. Journal of Elasticity, 1998, 91(1–3): 5–148MathSciNetzbMATHGoogle Scholar
  19. 19.
    Hakim V, Karma A. Laws of crack motion and phase-field models of fracture. Journal of the Mechanics and Physics of Solids, 2009, 57 (2): 342–368CrossRefzbMATHGoogle Scholar
  20. 20.
    Miehe C, Hofacker M, Welschinger F. A phase field model for rateindependent crack propagation: Robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering, 2010, 199(45–48): 2765–2778MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Amiri F, Millan D, Arroyo M, Silani M, Rabczuk T. Fourth order phase-field model for local max-ent approximants applied to crack propagation. Computer Methods in Applied Mechanics and Engineering, 2016, 312(C): 254–275MathSciNetCrossRefGoogle Scholar
  22. 22.
    Borden M J, Verhoosel V V, Scott M A, Hughes T J R, Landis C M. A phase-field description of dynamic brittle fracture. Computer Methods in Applied Mechanics and Engineering, 2012, 217–220: 77–95MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Areias P, Rabczuk T. Steiner-point free edge cutting of tetrahedral meshes with applications in fracture. Finite Elements in Analysis and Design, 2017, 132: 27–41CrossRefGoogle Scholar
  24. 24.
    Areias P, Rabczuk T, de Sá J C. A novel two-stage discrete crack method based on the screened Poisson equation and local mesh refinement. Computational Mechanics, 2016, 58(6): 1003–1018MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Areias P, Msekh M A, Rabczuk T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Engineering Fracture Mechanics, 2016, 158: 116–143CrossRefGoogle Scholar
  26. 26.
    Areias P, Reinoso J, Camanho P, Rabczuk T. A constitutive-based element-by-element crack propagation algorithm with local remeshing. Computational Mechanics, 2015, 56(2): 291–315MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Areias PMA, Rabczuk T, Camanho P P. Finite strain fracture of 2D problems with injected anisotropic softening elements. Theoretical and Applied Fracture Mechanics, 2014, 72: 50–63CrossRefGoogle Scholar
  28. 28.
    Areias P, Rabczuk T, Dias-da-Costa D. Element-wise fracture algorithm based on rotation of edges. Engineering Fracture Mechanics, 2013, 110: 113–137CrossRefGoogle Scholar
  29. 29.
    Areias P, Rabczuk T, Camanho P P. Initially rigid cohesive laws and fracture based on edge rotations. Computational Mechanics, 2013, 52(4): 931–947CrossRefzbMATHGoogle Scholar
  30. 30.
    Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M. Phase-field modeling of fracture in linear thin shells. Theoretical and Applied Fracture Mechanics, 2014, 69: 102–109CrossRefGoogle Scholar
  31. 31.
    Areias P, Rabczuk T, Msekh M. Phase-field analysis of finite-strain plates and shells including element subdivision. Computer Methods in Applied Mechanics and Engineering, 2016, 312(C): 322–350MathSciNetCrossRefGoogle Scholar
  32. 32.
    Msekh M A, Nguyen-Cuong H, Zi G, Areias P, Zhuang X, Rabczuk T. Fracture properties prediction of clay/epoxy nanocomposites with interphase zones using a phase field model. Engineering Fracture Mechanics, 2017Google Scholar
  33. 33.
    Msekh M A, Silani M, Jamshidian M, Areias P, Zhuang X, Zi G, He P, Rabczuk T. Predictions of J integral and tensile strength of clay/ epoxy nanocomposites material using phase field model. Composites. Part B, Engineering, 2016, 93: 97–114CrossRefGoogle Scholar
  34. 34.
    Hamdia K, Msekh M A, Silani M, Vu-Bac N, Zhuang X, Nguyen- Thoi T, Rabczuk T. Uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling. Composite Structures, 2015, 133: 1177–1190CrossRefGoogle Scholar
  35. 35.
    Ambati M, Gerasimov T, De Lorenzis L. Phase-field modeling of ductile fracture. Computer Methods in Applied Mechanics and Engineering, 2015, 55: 1017–1040MathSciNetzbMATHGoogle Scholar
  36. 36.
    Areias P, Dias-da-Costa D, Sargado J M, Rabczuk T. Element-wise algorithm for modeling ductile fracture with the Rousselier yield function. Computational Mechanics, 2013, 52(6): 1429–1443CrossRefzbMATHGoogle Scholar
  37. 37.
    Mauthe S, Miehe C. Hydraulic fracture in poro-hydro-elastic media. Mechanics Research Communications, 2017, 80: 69–83CrossRefGoogle Scholar
  38. 38.
    Franke M, Hesch C, Dittmann M. Phase-field approach to fracture for finite deformation contact problems. Proceedings in Applied Mathematics and Mechanics, 2016, 16(1): 123–124CrossRefGoogle Scholar
  39. 39.
    de Souza Neto E A, Petric D, Owen D R J. Computational Methods for Plasticity: Theory and Applications. Chichester: Wiley, 2008CrossRefGoogle Scholar
  40. 40.
    Ren H, Zhuang X, Rabczuk T. Dual-horizon peridynamics: A stable solution to varying horizons. Computer Methods in Applied Mechanics and Engineering, 2017, 318: 762–782MathSciNetCrossRefGoogle Scholar
  41. 41.
    Singh N, Verhoosel C, de Borst R, van Brummelen E. A fracture-controlled path-following technique for phase-field modeling of brittle fracture. Finite Elements in Analysis and Design, 2016, 113: 14–29MathSciNetCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Continuum MechanicsLeibniz-Universität HannoverHannoverGermany
  2. 2.Center for Interdisciplinary Research in Technology, Ho Chi Minh CityUniversity of Technology (HUTEH)Ho Chi Minh CityVietnam

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