International Journal of Fracture

, Volume 178, Issue 1–2, pp 51–70 | Cite as

Crack propagation criteria in three dimensions using the XFEM and an explicit–implicit crack description

Original Paper


This paper studies propagation criteria in three-dimensional fracture mechanics within the extended finite element framework (XFEM). The crack in this paper is described by a hybrid explicit–implicit approach as proposed in Fries and Baydoun (Int J Numer Methods Eng, 2011). In this approach, the crack update is realized based on an explicit crack surface mesh which allows an investigation of different propagation criteria. In contrast, for the computation of the displacements, stresses and strains by means of the XFEM, an implicit description by level set functions is employed. The maximum circumferential stress criterion, the maximum strain energy release rate criterion, the minimal strain energy density criterion and the material forces criterion are realized. The propagation paths from different criteria are studied and compared for asymmetric bending, torsion, and combined bending and torsion test cases. It is found that the maximum strain energy release rate and maximum circumferential stress criterion show the most favorable results.


Three dimensional fracture Crack propagation criteria Extended finite element method XFEM Explicit crack description Implicit crack description Maximum circumferential stress Maximum strain energy release rate Minimum strain energy density Material forces 


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  1. Areias PMA, Belytschko T (2005) Analysis of three-dimensional crack initiation and propagation using the extended finite element method. Int J Numer Methods Eng 63: 760–788CrossRefGoogle Scholar
  2. Bærentzen JA, Aanæs H (2002) Computing discrete signed distance fields from triangle meshes. Technical report, informatics and mathematical modeling, Technical University of Denmark, DTU, Richard Petersens Plads, Building 321, DK-2800 Kgs. LyngbyGoogle Scholar
  3. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45: 601–620CrossRefGoogle Scholar
  4. Belytschko T, Moës N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50: 993–1013CrossRefGoogle Scholar
  5. Belytschko T, Parimi C, Moës N, Sukumar N, Usui S (2003) Structured extended finite element methods for solids defined by implicit surfaces. Int J Numer Methods Eng 56: 609–635CrossRefGoogle Scholar
  6. Bittencourt TN, Wawrzynek PA, Ingraffea AR, Sousa JL (1996) Quasi-automatic simulation of crack propagation for 2d lefm problems. Eng Fract Mech 55: 321–334CrossRefGoogle Scholar
  7. Bloomenthal J, Bajaj C, Blinn J, Cani-Gascuel MP, Rockwood A, Wyvill B, Wyvill G (1997) Introduction to implicit surfaces. Morgan Kaufmann publishers, San FranciscoGoogle Scholar
  8. Bouchard PO, Bay F, Chastel Y (2003) Numerical modeling of crack propagation: automatic remeshing and comparison of different criteria. Comput Methods Appl Mech Eng 192: 3887–3908CrossRefGoogle Scholar
  9. Bouchard PO, Bay F, Chastel Y, Tovenal I (2000) Crack propagation modeling using advanced remeshing technique. Comput Methods Appl Mech Eng 189: 723–742CrossRefGoogle Scholar
  10. Brokenshire DR (2007) Torsional fracture tests. Phd thesis, Cardiff University, UKGoogle Scholar
  11. Chang J, Xu JQ, Mutoh Y (2006) A general mixed-mode brittle fracture criterion for cracked materials. EFM 73: 1246– 1263Google Scholar
  12. Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T (2000) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Methods Eng 48: 1741–1760CrossRefGoogle Scholar
  13. De Lorenzi HG (1985) Energy release rate by the finite element method. Eng Fract Mech 21: 129–143CrossRefGoogle Scholar
  14. Destuynder P, Dajoua M, Lescure S (1983) Quelques remarques sur la mécanique de la rupture élastique. J Mech Theor Appl 1: 113–135Google Scholar
  15. Dufflot M (2007) A study of the representation of cracks with level set. Int J Numer Methods Eng 70: 1261–1302CrossRefGoogle Scholar
  16. Erdogan F, Sih GC (1963) On the crack extension in plane loading and transverse shear. J Basic Eng 85: 519–527CrossRefGoogle Scholar
  17. Eshelby JD (1951) The force on an elastic singularity. Philos Trans R Soc Lond 244: 27–112Google Scholar
  18. Fries TP, Baydoun M (2011) Crack propagation with the xfem and a hybrid explicit-implicit crack description. Int J Numer Methods Eng 89: 1527–1558CrossRefGoogle Scholar
  19. Fries TP, Belytschko T (2010) The generalized/extended finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84: 253–304Google Scholar
  20. Glaser J, Steinmann P (2007) Material force method within the framework of the x-fem-distribution of nodal material forces. Sixth international congress on industrial applied mathematics 7: 4030017–4030018Google Scholar
  21. Glávez JC, Elices M, Guinea GV, Planas J (1998) Fracture of concrete under proportional and non proportional loading. Int J Fract 94: 267–284CrossRefGoogle Scholar
  22. Gravouil A, Moës N, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets, part II: level set update. Int J Numer Methods Eng 53: 2569–2586CrossRefGoogle Scholar
  23. Gürses E (2007) Aspects of energy minimization in solid mechanics: evolution of inelastic microstructures and crack propagation. Phd thesis, University of Stuttgart, GermanyGoogle Scholar
  24. Gurtin ME (1995) The nature of configurational forces. Arch Ration Mech Anal 131: 67–100CrossRefGoogle Scholar
  25. Güsrses E., Miehe C. (2009) A computational framework of three-dimensional configurational-force-driven brittle crack propagation. Comput Methods Appl Mech Eng 198: 1413–1428CrossRefGoogle Scholar
  26. Haüsler SM, Lindhorst K, Horst P (2011) Combination of the material force concept and the extended finite element method for mixed mode crack simulations. Int J Numer Methods Eng 85: 1522–1542CrossRefGoogle Scholar
  27. Hussain MA, Pu SL, Underwood JH (1974) Strain energy release rate for a crack under combined mode i and mode ii. Fract Anal ASTM STP 560: 2–28Google Scholar
  28. Larsson R, Fagerström M (2005) A framework for fracture modeling based on the material forces concept with XFEM kinematics. Int J Numer Methods Eng 62: 1763–1788CrossRefGoogle Scholar
  29. Maiti SK, Smith RA (1984) Comparison of the criteria for mixed mode brittle fracture based on the pre-instability stress–strain field. Part ii: pure shear and uniaxial loading. Int J Fract 24: 5–22CrossRefGoogle Scholar
  30. Maugin GA (1995) Material forces: concepts and applications. Appl Mech Rev 48: 213–245CrossRefGoogle Scholar
  31. Miehe C, Gürses E, Birkle M (2007) A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization. Int J Fract 145: 245–299CrossRefGoogle Scholar
  32. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150CrossRefGoogle Scholar
  33. Moës N, Gravouil A, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets, part I: mechanical model. Int J Numer Methods Eng 53: 2549–2568CrossRefGoogle Scholar
  34. Müller R, Maugin GA (2002) On material forces and finite element discretizations. J Comput Mech 29: 52–60CrossRefGoogle Scholar
  35. Rabczuk T, Belytschko T (2007) A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Comput Methods Appl Mech Eng 196: 2777–2799CrossRefGoogle Scholar
  36. Rabczuk T, Bordas S, Zi G (2010) On three-dimensional modeling of crack growth using partition of unity methods. Comput Struct 88: 1391–1411CrossRefGoogle Scholar
  37. Rice JR (1968) A path-independent integral and the approximate analysis of strain concentrations by notches and cracks. J Appl Mech 35: 379–386CrossRefGoogle Scholar
  38. Sethian JA (1996) A fast marching level set method for monotonically advancing fronts. Proc Natl Acad Sci 4: 1591– 1595CrossRefGoogle Scholar
  39. Sih GC (1973) Mechanics of fracture 1: a special theory of crack propagation. Noordhoff International Publishing, LeydenGoogle Scholar
  40. Sih GC, Macdonald B (1974) Fracture mechanics applied to engineering applied to engineering problems—strain energy density fracture criterion. Eng Fract Mech 6: 361–386CrossRefGoogle Scholar
  41. Steinmann P (2000) Application of material forces to hyperelastostatic fracture mechanics I: continuum mechanical setting. Int J Solids Struct 37: 7371–7391CrossRefGoogle Scholar
  42. Steinmann P (2001) Application of material forces to hyperelastostatic fracture mechanics II: computational setting. Int J Solids Struct 38: 5509–5526CrossRefGoogle Scholar
  43. Stolarska M, Chopp DL, Moës N, Belytschko T (2001) Modeling crack growth by level sets in the extended finite element method. Int J Numer Methods Eng 51: 943–960CrossRefGoogle Scholar
  44. Sukumar N, Moës N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modeling. Int J Numer Methods Eng 48: 1549–1570CrossRefGoogle Scholar
  45. Ventura G, Budyn E, Belytschko T (2003) Vector level sets for description of propagating cracks in finite elements. Int J Numer Methods Eng 58: 1571–1592CrossRefGoogle Scholar
  46. Ventura G, Xu JX, Belytschko T (2002) A vector level set method and new discontinuity approximations for crack growth by EFG. Int J Numer Methods Eng 54: 923–944CrossRefGoogle Scholar
  47. Wyart E (2007) Three-dimensional crack analysis in aeronautical structures using the sub structured finite element/extended finite element method. Phd thesis, Université Catholique de Louvain, BelgiqueGoogle Scholar
  48. Yau JF, Wang SS, Corten HT (1980) A mixed mode crack analysis of isotropic solids using conservation laws of elasticity. J Appl Mech ASME 47: 335–341CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Aachen Institute of Computational Engineering SciencesRWTH AachenAachenGermany
  2. 2.Computational Analysis of Technical SystemsRWTH AachenAachenGermany

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