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Computational Mechanics

, Volume 63, Issue 5, pp 1019–1046 | Cite as

A phase-field crack model based on directional stress decomposition

  • Christian Steinke
  • Michael KaliskeEmail author
Original Paper

Abstract

Phase-field crack approximation relies on the proper definition of the crack driving strain energy density to govern the crack evolution and a realistic model for the modified stresses on the crack surface. A novel approach, the directional split, is introduced, analyzed and compared to the two commonly used formulations, which are the spectral split and the volumetric–deviatoric split. The directional split is based on the decomposition of the stress tensor with respect to the crack orientation, specified by a local crack coordinate system, into crack driving and persistent components. Accordingly, a modified stress strain relation is proposed to model fundamental crack characteristics properly, and a thermodynamically consistent crack driving strain energy density is postulated. The split is applied to numerical examples of initially cracked specimens and compared to results obtained by the two standard approaches.

Keywords

Numerical crack approximation Brittle fracture mechanics Phase-field method Dynamic fracture 

Notes

Acknowledgements

The authors would like to acknowledge the financial support of “German Research Foundation“ under Grant KA 1163/19 and as well the technical support of the centre for information services and high performance computing of TU Dresden for providing access to the Bull HPC-Cluster.

References

  1. 1.
    Aldakheel F (2016) Mechanics of nonlocal dissipative solids: gradient plasticity and phase field modeling of ductile fracture. Ph.D. thesis, Universität StuttgartGoogle Scholar
  2. 2.
    Aldakheel F, Wriggers P, Miehe C (2017) A modified gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling. Comput Mech.  https://doi.org/10.1007/s00466-017-1530-0
  3. 3.
    Ambati M, Gerasimov T, Lorenzis LD (2015) Phase-field modeling of ductile fracture. Comput. Mech 55:1017–1040MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ambati M, Gerasimov T, Lorenzis LD (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55:383–405MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Amor H, Marigo JJ, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57:1209–1229CrossRefzbMATHGoogle Scholar
  6. 6.
    Bleyer J, Roux-Langlois C, Molinari JF (2017) Dynamic crack propagation with a variational phase-field model: limiting speed, crack branching and velocity-toughening mechanisms. Int J Fract 204:79–100CrossRefGoogle Scholar
  7. 7.
    Borden M (2012) Isogeometric analysis of phase-field mmodel for dynamic brittle and ductile fracture. Ph.D. thesis, The University of Texas at AustinGoogle Scholar
  8. 8.
    Borden MJ, Hughes TJ, Landis CM, Anvari A, Lee IJ (2016) A phase-field formulation for fracture in ductile materials: finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Comput Methods Appl Mech Eng 312:130–166MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bourdin B, Francfort G, Marigo JJ (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48:797–826MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Braides A (2002) Gamma-convergence for beginners. Oxford University Press, OxfordCrossRefzbMATHGoogle Scholar
  11. 11.
    Clayton J, Knap J (2011) A phase field model of deformation twinning: nonlinear theory and numerical simulations. Physica D 240:841–858MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Contia S, Focardic M, Iurlano F (2016) Phase field approximation of cohesive fracture models. Ann Inst H Poincare (C) Nonlinear Anal 33:1033–1067MathSciNetCrossRefGoogle Scholar
  13. 13.
    Francfort GA, Marigo JJ (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 46:1319–1342MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gerasimov T, Lorenzis LD (2015) A line search assisted monolithic approach for phase-field computing of brittle fracture. Comput Methods Appl Mech Eng 312:276–303MathSciNetCrossRefGoogle Scholar
  15. 15.
    Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond Ser A 221:163–198CrossRefGoogle Scholar
  16. 16.
    Hilber H, Hughes T, Taylor R (1977) Improved numerical dissipation for the time intergration algorithms in structural dynamics. Earthq Eng Struct Dyn 5:283–292CrossRefGoogle Scholar
  17. 17.
    Hofacker M (2013) A thermodynamically consistent phase field approach to fracture. Ph.D. thesis, Universität StuttgartGoogle Scholar
  18. 18.
    Kuhn C, Müller R (2011) A new finite element technique for a phase field model of brittle fracture. J Theor Appl Mech 49:1115–1133Google Scholar
  19. 19.
    Kuhn C, Schlüter A, Müller R (2015) On degradation functions in phase field fracture models. Comput Mater Sci 108:374–384CrossRefGoogle Scholar
  20. 20.
    Li B, Peco C, Millán D, Arias I, Arroyo M (2015) Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy. Int J Numer Methods Eng 102:711–727MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Linse T, Hennig P, Kästner M, de Borst R (2017) A convergence study of phase-field models for brittle fracture. Eng Fract Mech 184:307–318CrossRefGoogle Scholar
  22. 22.
    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 52:72–84MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Miehe C (1993) Computation of isotropic tensor functions. Commun Numer Methods Eng 9:889–896CrossRefzbMATHGoogle Scholar
  24. 24.
    Miehe C, Aldakheel F, Raina A (2016) Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory. Int J Plast 84:1–32CrossRefGoogle Scholar
  25. 25.
    Miehe C, Apel N, Lambrecht M (2002) Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation based on incremental minimization principles for standard materials. Comput Methods Appl Mech Eng 191:5383–5425MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    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 199:2765–2778MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83:1273–1311MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Negri M (2007) Convergence analysis for a smeared crack approach in brittle fracture. Interfaces and Free Boundaries 9:307–330MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nguyen TT, Baietto JRMC (2017) Phase field modelling of anisotropic crack propagation. Eur J Mech A/Solids 65:279–288MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Schlüter A (2013) FE-Implementierung eines dynamischen Phasenfeldmodells für Bruchvorgänge. Master’s thesis, Technische Universität KaiserslauternGoogle Scholar
  31. 31.
    Steinke C, Özenç K, Chinaryan G, Kaliske M (2016) A comparative study of the r-adaptive material force approach and the phase-field method in dynamic fracture. Int J Fract 201:97–118CrossRefGoogle Scholar
  32. 32.
    Strobl M, Seelig T (2015) A novel treatment of crack boundary conditions in phase field models of fracture. Proc Appl Math Mech 15:155–156CrossRefGoogle Scholar
  33. 33.
    Teichtmeister S, Kienle D, Aldakheel F, Keip MA (2017) Phase field modeling of fracture in anisotropic brittle solids. Int J Non-linear Mech 97:1–21CrossRefGoogle Scholar
  34. 34.
    Teichtmeister S, Miehe C (2015) Phase-field modeling of fracture in anisotropic media. Proc Appl Math Mech 15:159–160CrossRefGoogle Scholar
  35. 35.
    Verhoosel CV, de Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Methods Eng 96:43–62MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Vignollet J, May S, de Borst R, Verhoosel CV (2014) Phase-field models for brittle and cohesive fracture. Meccanica 49:2587–2601MathSciNetCrossRefGoogle Scholar
  37. 37.
    van der Vorst HA (1992) Bi-CGSTAB: a fast and smoothly converging variant of bi-cg for the solution of nonsymmetric linear systems. SIAM J Sci Stat Comput 13:631–644MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    von Mises R (1928) Mechanik der plastischen Formänderung von Kristallen. Z Angew Math Mec 8:161–185CrossRefzbMATHGoogle Scholar
  39. 39.
    Zhang X, Sloan SW, Vignes C, Sheng D (2017) A modification of the phase-field model for mixed mode crack propagation in rock-like materials. Comput Methods Appl Mech Eng 322:123–136MathSciNetCrossRefGoogle Scholar
  40. 40.
    Zienkiewicz OC (1977) The finite element method. Methode der finiten Elemente, 2nd edn. Carl Hanser, MünchenGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute for Structural AnalysisTechnische Universität DresdenDresdenGermany

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