Computational Mechanics

, Volume 59, Issue 5, pp 737–752 | Cite as

Numerical evaluation of the phase-field model for brittle fracture with emphasis on the length scale

  • Xue Zhang
  • Chet Vignes
  • Scott W. Sloan
  • Daichao Sheng
Original Paper


The phase-field model has been attracting considerable attention due to its capability of capturing complex crack propagations without mesh dependence. However, its validation studies have primarily focused on the ability to predict reasonable, sharply defined crack paths. Very limited works have so far been contributed to estimate its accuracy in predicting force responses, which is majorly attributed to the difficulty in the determination of the length scale. Indeed, accurate crack path simulation can be achieved by setting the length scale to be sufficiently small, whereas a very small length scale may lead to unrealistic force-displacement responses and overestimate critical structural loads. This paper aims to provide a critical numerical investigation of the accuracy of phase-field modelling of brittle fracture with special emphasis on a possible formula for the length scale estimation. Phase-field simulations of a number of classical fracture experiments for brittle fracture in concretes are performed with simulated results compared with experimental data qualitatively and quantitatively to achieve this goal. Furthermore, discussions are conducted with the aim to provide guidelines for the application of the phase-field model.


Phase-field model Validation Crack propagation Length scale Brittle fracture 



The authors wish to acknowledge the support of the Australian Research Council Centre of Excellence for Geotechnical Science and Engineering and the Australian Research Council Discovery Project funding scheme (Project Number DP150104257).


  1. 1.
    Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond A Math Phys Eng Sci 221(582–593):163–198CrossRefGoogle Scholar
  2. 2.
    Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chambolle A, Francfort GA, Marigo JJ (2009) When and how do cracks propagate? J Mech Phys Solids 57(9):1614–1622MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Proudhon H, Li J, Wang F, Roos A, Chiaruttini V, Forest S (2016) 3D simulation of short fatigue crack propagation by finite element crystal plasticity and remeshing. Int J Fatigue 82(Part 2):238–246CrossRefGoogle Scholar
  6. 6.
    Kuutti J, Kolari K (2012) A local remeshing procedure to simulate crack propagation in quasi-brittle materials. Eng Comput 29(2):125–143CrossRefGoogle Scholar
  7. 7.
    Bouchard PO, Bay F, Chastel Y (2003) Numerical modelling of crack propagation: automatic remeshing and comparison of different criteria. Comput Methods Appl Mech Eng 192(35–36):3887–3908CrossRefzbMATHGoogle Scholar
  8. 8.
    Réthoré J, Gravouil A, Combescure A (2004) A stable numerical scheme for the finite element simulation of dynamic crack propagation with remeshing. Comput Methods Appl Mech Eng 193(42–44):4493–4510CrossRefzbMATHGoogle Scholar
  9. 9.
    Bouchard PO, Bay F, Chastel Y, Tovena I (2000) Crack propagation modelling using an advanced remeshing technique. Comput Methods Appl Mech Eng 189(3):723–742CrossRefzbMATHGoogle Scholar
  10. 10.
    Branco R, Antunes FV, Costa JD (2015) A review on 3D-FE adaptive remeshing techniques for crack growth modelling. Eng Fract Mech 141:170–195CrossRefGoogle Scholar
  11. 11.
    Moës N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fract Mech 69(7):813–833CrossRefGoogle Scholar
  12. 12.
    Belytschko T, Gracie R, Ventura G (2009) A review of extended/generalized finite element methods for material modeling. Model Simul Mater Sci Eng 17(4):043001CrossRefGoogle Scholar
  13. 13.
    Sukumar N, Moës N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modelling. Int J Numer Meth Eng 48(11):1549–1570CrossRefzbMATHGoogle Scholar
  14. 14.
    Fries T-P, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Meth Eng 84(3):253–304MathSciNetzbMATHGoogle Scholar
  15. 15.
    Abdelaziz Y, Hamouine A (2008) A survey of the extended finite element. Comput Struct 86(11–12):1141–1151CrossRefGoogle Scholar
  16. 16.
    Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Meth Eng 83(10):1273–1311MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199(45–48):2765–2778MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schlüter A, Willenbücher A, Kuhn C, Müller R (2014) Phase field approximation of dynamic brittle fracture. Comput Mech 54(5):1141–1161MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bourdin B, Marigo J-J, Maurini C, Sicsic P (2014) Morphogenesis and propagation of complex cracks induced by thermal shocks. Phys Rev Lett 112(1):014301CrossRefGoogle Scholar
  20. 20.
    Kuhn C, Müller R (2010) A continuum phase field model for fracture. Eng Fract Mech 77(18):3625–3634CrossRefGoogle Scholar
  21. 21.
    Msekh MA, Sargado JM, Jamshidian M, Areias PM, Rabczuk T (2015) Abaqus implementation of phase-field model for brittle fracture. Comput Mater Sci 96(Part B):472–484CrossRefGoogle Scholar
  22. 22.
    Bourdin B (2007) The variational formulation of brittle fracture: numerical implementation and extensions. In: Combescure A, Borst R, Belytschko T (eds) IUTAM symposium on discretization methods for evolving discontinuities. Springer Netherlands, DordrechtGoogle Scholar
  23. 23.
    Liu G, Li Q, Msekh MA, Zuo Z (2016) Abaqus implementation of monolithic and staggered schemes for quasi-static and dynamic fracture phase-field model. Comput Mater Sci 121:35–47CrossRefGoogle Scholar
  24. 24.
    Amor H, Marigo J-J, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229CrossRefzbMATHGoogle Scholar
  25. 25.
    Ambati M, Gerasimov T, Lorenzis L (2014) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55(2):383–405MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Schillinger D, Borden MJ, Stolarski HK (2015) Isogeometric collocation for phase-field fracture models. Comput Methods Appl Mech Eng 284:583–610MathSciNetCrossRefGoogle Scholar
  27. 27.
    Kuhn C, Müller R (2011) A new finite element technique for a phase field model of brittle fracture. J Theor Appl Mech 49(4):1115–1133Google Scholar
  28. 28.
    Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hofacker M, Miehe C (2013) A phase field model of dynamic fracture: robust field updates for the analysis of complex crack patterns. Int J Numer Meth Eng 93(3):276–301MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ambati M, Gerasimov T, De Lorenzis L (2015) Phase-field modeling of ductile fracture. Comput Mech 55(5):1017–1040MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Aldakheel F, Mauthe S, Miehe C (2014) Towards phase field modeling of ductile fracture in gradient-extended elastic-plastic solids. PAMM 14(1):411–412CrossRefGoogle Scholar
  32. 32.
    Ulmer H, Hofacker M, Miehe C (2013) Phase field modeling of brittle and ductile fracture. PAMM 13(1):533–536CrossRefGoogle Scholar
  33. 33.
    Miehe C, Hofacker M, Schänzel LM, Aldakheel F (2015) Phase field modeling of fracture in multi-physics problems. Part II. Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic-plastic solids. Comput Methods Appl Mech Eng 294:486–522MathSciNetCrossRefGoogle Scholar
  34. 34.
    May S, Vignollet J, de Borst R (2015) A numerical assessment of phase-field models for brittle and cohesive fracture: \(\Gamma \)-Convergence and stress oscillations. Eur J Mech A Solids 52:72–84MathSciNetCrossRefGoogle Scholar
  35. 35.
    Vignollet J, May S, Borst R, Verhoosel CV (2014) Phase-field models for brittle and cohesive fracture. Meccanica 49(11):2587–2601MathSciNetCrossRefGoogle Scholar
  36. 36.
    Verhoosel CV, de Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Meth Eng 96(1):43–62MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Miehe C, Schänzel L-M (2014) Phase field modeling of fracture in rubbery polymers. Part I: finite elasticity coupled with brittle failure. J Mech Phys Solids 65:93–113MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Amiri F, Millán D, Shen Y, Rabczuk T, Arroyo M (2014) Phase-field modeling of fracture in linear thin shells. Theoret Appl Fract Mech 69:102–109CrossRefGoogle Scholar
  39. 39.
    Mesgarnejad A, Bourdin B, Khonsari MM (2013) A variational approach to the fracture of brittle thin films subject to out-of-plane loading. J Mech Phys Solids 61(11):2360–2379MathSciNetCrossRefGoogle Scholar
  40. 40.
    León Baldelli AA, Bourdin B, Marigo J-J, Maurini C (2011) Fracture and debonding of a thin film on a stiff substrate: analytical and numerical solutions of a 1d variational model. Springer, BerlinGoogle Scholar
  41. 41.
    León Baldelli AA, Babadjian JF, Bourdin B, Henao D, Maurini C (2014) A variational model for fracture and debonding of thin films under in-plane loadings. J Mech Phys Solids 70:320–348MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Mikelić A, Wheeler MF, Wick T (2015) A phase-field method for propagating fluid-filled fractures coupled to a surrounding porous medium. Multiscale Model Simul 13(1):367–398MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Miehe C, Mauthe S (2016) Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media. Comput Methods Appl Mech Eng 304:619–655MathSciNetCrossRefGoogle Scholar
  44. 44.
    Heider Y, Markert B (2016) A phase-field modeling approach of hydraulic fracture in saturated porous media. Mechanics Res Commun (in press)Google Scholar
  45. 45.
    Mikelić A, Wheeler MF, Wick T (2015) Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput Geosci 19(6):1171–1195MathSciNetCrossRefGoogle Scholar
  46. 46.
    Nguyen TT, Yvonnet J, Bornert M, Chateau C, Sab K, Romani R, Le Roy R (2016) On the choice of parameters in the phase field method for simulating crack initiation with experimental validation. Int J Fract 197(2):213–226CrossRefGoogle Scholar
  47. 47.
    Mesgarnejad A, Bourdin B, Khonsari MM (2015) Validation simulations for the variational approach to fracture. Comput Methods Appl Mech Eng 290:420–437MathSciNetCrossRefGoogle Scholar
  48. 48.
    Kuhn C, Schlüter A, Müller R (2015) On degradation functions in phase field fracture models. Comput Mater Sci 108(Part B):374–384CrossRefGoogle Scholar
  49. 49.
    Bourdin B, Francfort GA, Marigo J-J (2008) The variational approach to fracture. J Elast 91(1):5–148MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Philip CP, Alberto R (1995) Size effect on fracture energy of concrete and stability issues in three-point bending fracture toughness testing. Mater J 92(5):483–496Google Scholar
  51. 51.
    Nooru-Mohamed MB (1992) Mixed-mode fracture of concrete: an experimental approach. In: Civil Engineering and Geosciences (Doctoral dissertation). Delft University of TechnologyGoogle Scholar
  52. 52.
    Hamdia KM, Msekh MA, Silani M, Vu-Bac N, Zhuang X, Nguyen-Thoi T, Rabczuk T (2015) Uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling. Compos Struct 133:1177–1190CrossRefGoogle Scholar
  53. 53.
    Winkler BJ (2001) Traglastuntersuchungen von unbewehrten und bewehrten Betonstrukturen auf der Grundlage eines objektiven Werkstoffgesetzes fur Beton (Doctoral dissertation). Innsbruck University, InnsbruckGoogle Scholar
  54. 54.
    Trunk B (1999) Einfluss der Bauteilgrösse auf die Bruchenergie von Beton (Doctoral dissertation). Techn. Wiss. ETH ZürichGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Xue Zhang
    • 1
  • Chet Vignes
    • 1
  • Scott W. Sloan
    • 1
  • Daichao Sheng
    • 1
  1. 1.ARC Centre of Excellence for Geotechnical Science and EngineeringThe University of NewcastleCallaghanAustralia

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