Computational Mechanics

, Volume 63, Issue 5, pp 1047–1068 | Cite as

Identification of fracture models based on phase field for crack propagation in heterogeneous lattices in a context of non-separated scales

  • Nhu Nguyen
  • J. YvonnetEmail author
  • J. Réthoré
  • A. B. Tran
Original Paper


The construction of a homogeneous medium equivalent to a heterogeneous one under quasi-brittle fracture is investigated in the case of non-separated scales. At the microscale, the phase field method to fracture is employed. At the scale of the homogeneous medium, another phase field model either isotropic or anisotropic, depending on the microscale crack length and on the underlying microstructure, is assumed. The coefficients of the unknown phase field model for the homogeneous model are identified through the mechanical response of a sample subjected to fracture whose microstructure is fully described and estimated numerically. We show that the identified models can reproduce both the mechanical force response as well as overall crack paths with good accuracy in other geometrical configurations than the one used to identify the homogeneous model. Several numerical examples, involving cracking in regular lattices of both hard particles and pores, are presented to show the potential of the technique.


Phase field method Damage Homogenization Crack propagation Quasi-brittle materials Lattice structures 



The financial support of Institut Universitaire de France (IUF) for J.Y. is gratefully aknowledged and National Foundation for Science and Technology Development (NAFOSTED) for N.N.


  1. 1.
    Ambati M, Gerasimov T, De Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55(2):383–405MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosio L, Tortorelli VM (1990) Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun Pure Appl Math 43(8):999–1036MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Amor H, Marigo JJ, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229CrossRefzbMATHGoogle Scholar
  4. 4.
    Belytschko T, Loehnert S, Song JH (2008) Multiscale aggregating discontinuities: a method for circumventing loss of material stability. Int J Numer Methods Eng 73(6):869–894MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borden MJ, Verhoosel CV, Scott MA, Hughes TJ, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217:77–95MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borden MJ, Hughes TJ, Landis CM, Verhoosel CV (2014) A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework. Comput Methods Appl Mech Eng 273:100–118MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    de Borst R, Verhoosel CV (2016) Gradient damage vs phase-field approaches for fracture: Similarities and differences. Comput Methods Appl Mech Eng 312:78–94MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bosco E, Kouznetsova V, Geers M (2015) Multi-scale computational homogenization-localization for propagating discontinuities using x-fem. Int J Numer Methods Eng 102(3–4):496–527MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bourdin B, Francfort GA, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bourdin B, Francfort GA, Marigo JJ (2008) The variational approach to fracture. J Elast 91(1–3):5–148MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Clayton J (2015) Defects in nonlinear elastic crystals: differential geometry, finite kinematics, and second-order analytical solutions. ZAMM J Appl Math Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 95(5):476–510MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Clayton J, Knap J (2015) Nonlinear phase field theory for fracture and twinning with analysis of simple shear. Philos Mag 95(24):2661–2696CrossRefGoogle Scholar
  13. 13.
    Clayton J, Knap J (2016) Phase field modeling and simulation of coupled fracture and twinning in single crystals and polycrystals. Comput Methods Appl Mech Eng 312:447–467MathSciNetCrossRefGoogle Scholar
  14. 14.
    Coenen E, Kouznetsova V, Bosco E, Geers M (2012) A multi-scale approach to bridge microscale damage and macroscale failure: a nested computational homogenization-localization framework. Int J Fract 178(1–2):157–178CrossRefGoogle Scholar
  15. 15.
    De Lorenzis L, McBride A, Reddy B (2016) Phase-field modelling of fracture in single crystal plasticity. GAMM-Mitteilungen 39(1):7–34MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dresselhaus M, Dresselhaus G (1991) Note on sufficient symmetry conditions for isotropy of the elastic moduli tensor. J Mater Res 6(5):1114–1118CrossRefzbMATHGoogle Scholar
  17. 17.
    Feyel F, Chaboche JL (2000) Fe 2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre sic/ti composite materials. Comput Methods Appl Mech Eng 183(3):309–330CrossRefzbMATHGoogle Scholar
  18. 18.
    Fichant S, La Borderie C, Pijaudier-Cabot G (1999) Isotropic and anisotropic descriptions of damage in concrete structures. Mech Cohesive Frict Mater 4(4):339–359CrossRefGoogle Scholar
  19. 19.
    Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46(8):1319–1342MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gitman I, Askes H, Sluys L (2007) Representative volume: existence and size determination. Eng Fract Mech 74(16):2518–2534CrossRefGoogle Scholar
  21. 21.
    Guidault PA, Allix O, Champaney L, Cornuault C (2008) A multiscale extended finite element method for crack propagation. Comput Methods Appl Mech Eng 197(5):381–399CrossRefzbMATHGoogle Scholar
  22. 22.
    Hirschberger C, Ricker S, Steinmann P, Sukumar N (2009) Computational multiscale modelling of heterogeneous material layers. Eng Fract Mech 76(6):793–812CrossRefGoogle Scholar
  23. 23.
    Hossain M, Hsueh CJ, Bourdin B, Bhattacharya K (2014) Effective toughness of heterogeneous media. J Mech Phys Solids 71:15–32MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kuhn C, Müller R (2008) A phase field model for fracture. PAMM 8(1):10,223–10,224CrossRefGoogle Scholar
  25. 25.
    Kuhn C, Schlüter A, Müller R (2015) On degradation functions in phase field fracture models. Comput Mater Sci 108:374–384CrossRefGoogle Scholar
  26. 26.
    Kulkarni M, Matouš K, Geubelle P (2010) Coupled multi-scale cohesive modeling of failure in heterogeneous adhesives. Int J Numer Methods Eng 84(8):916–946CrossRefzbMATHGoogle Scholar
  27. 27.
    Lagarias JC, Reeds JA, Wright MH, Wright PE (1998) Convergence properties of the nelder-mead simplex method in low dimensions. SIAM J Optim 9(1):112–147MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Liu G, Zhou D, Bao Y, Ma J, Han Z (2017) Multiscale simulation of major crack/minor cracks interplay with the corrected XFEM. Arch Civil Mech Eng 17(2):410–418CrossRefGoogle Scholar
  29. 29.
    Loehnert S, Belytschko T (2007) A multiscale projection method for macro/microcrack simulations. Int J Numer Methods Eng 71(12):1466–1482MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Matouš K, Kulkarni MG, Geubelle PH (2008) Multiscale cohesive failure modeling of heterogeneous adhesives. J Mech Phys Solids 56(4):1511–1533MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Miehe C (1998) Comparison of two algorithms for the computation of fourth-order isotropic tensor functions. Comput Struct 66(1):37–43MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    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):2765–2778MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Miehe C, Hofacker M, Schaenzel 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.
    Nguyen T, Yvonnet J, Zhu QZ, Bornert M, Chateau C (2015) A phase field method to simulate crack nucleation and propagation in strongly heterogeneous materials from direct imaging of their microstructure. Eng Fract Mech 139:18–39CrossRefGoogle Scholar
  35. 35.
    Nguyen T, Yvonnet J, Bornert M, Chateau C (2016a) Direct comparisons of 3D crack networks propagation in cementitious materials between phase field numerical modeling and in-situ microtomography experimental images. J Mech Phys Solids 95:320–350CrossRefGoogle Scholar
  36. 36.
    Nguyen T, Yvonnet J, Bornert M, Chateau C, Sab K, Romani R, Le Roy R (2016b) 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
  37. 37.
    Nguyen T, Rethoré J, Yvonnet J, Baietto M (2017a) Multi-phase-field modeling of anisotropic crack propagation for polycrystalline materials. Comput Mech 60(2):289–314MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Nguyen TT, Réthoré J, Baietto MC (2017b) Phase field modelling of anisotropic crack propagation. Eur J Mech A/Solids 65:279–288MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Nguyen VP, Lloberas-Valls O, Stroeven M, Sluys LJ (2011) Homogenization-based multiscale crack modelling: from micro-diffusive damage to macro-cracks. Comput Methods Appl Mech Eng 200(9):1220–1236CrossRefzbMATHGoogle Scholar
  40. 40.
    Oliver J, Caicedo M, Huespe A, Hernández J, Roubin E (2015a) Continuum approach to computational multiscale modeling of propagating fracture. Comput Methods Appl Mech Eng 294:384–427MathSciNetCrossRefGoogle Scholar
  41. 41.
    Oliver J, Caicedo M, Roubin E, Huespe A, Hernández J (2015b) Continuum approach to computational multiscale modeling of propagating fracture. Comput Methods Appl Mech Eng 294:384–427MathSciNetCrossRefGoogle Scholar
  42. 42.
    Ladevéze P, Passieux DNJ-C (2010) The latin multiscale computational method and the proper generalized decomposition. Comput Methods Appl Mech Eng 199:1287–1296MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Pham K, Marigo JJ, Maurini C (2011) The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models. J Mech Phys Solids 59(6):1163–1190MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Réthoré J, Dang TBT, Kaltenbrunner C (2017) Anisotropic failure and size effects in periodic honeycomb materials: a gradient-elasticity approach. J Mech Phys Solids 99:35–49MathSciNetCrossRefGoogle Scholar
  45. 45.
    Rudoy E (2016) Domain decomposition method for crack problems with nonpenetration condition. ESAIM Math Model Numer Anal 50:995–1009MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Teichtmeister S, Kienle D, Aldakheel F, Keip MA (2017) Phase field modeling of fracture in anisotropic brittle solids. Int J Nonlinear Mech 97:1–21CrossRefGoogle Scholar
  47. 47.
    Waisman H, Berger-Vergat L (2013) An adaptive domain decomposition preconditioner for crack propagation problems modeled by xfem. J Multiscale Comput Eng 11(6):633–654CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Nhu Nguyen
    • 1
  • J. Yvonnet
    • 1
    Email author
  • J. Réthoré
    • 2
  • A. B. Tran
    • 3
  1. 1.Laboratoire Modélisation et Simulation Multi Échelle, MSME UMR 8208 CNRSUniversité Paris-EstMarne-la-ValléeFrance
  2. 2.Institut de Recherche en Génie Civil et Mécanique-GeM, UMR CNRS 6183École Centrale de NantesNantesFrance
  3. 3.National University of Civil EngineeringHa NoiVietnam

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