Computational Mechanics

, Volume 49, Issue 1, pp 99–121 | Cite as

A two-scale approach for the analysis of propagating three-dimensional fractures

Original Paper

Abstract

This paper presents a generalized finite element method (GFEM) for crack growth simulations based on a two-scale decomposition of the solution—a smooth coarse-scale component and a singular fine-scale component. The smooth component is approximated by discretizations defined on coarse finite element meshes. The fine-scale component is approximated by the solution of local problems defined in neighborhoods of cracks. Boundary conditions for the local problems are provided by the available solution at a crack growth step. The methodology enables accurate modeling of 3-D propagating cracks on meshes with elements that are orders of magnitude larger than those required by the FEM. The coarse-scale mesh remains unchanged during the simulation. This, combined with the hierarchical nature of GFEM shape functions, allows the recycling of the factorization of the global stiffness matrix during a crack growth simulation. Numerical examples demonstrating the approximating properties of the proposed enrichment functions and the computational performance of the methodology are presented.

Keywords

Generalized FEM Extended FEM Fracture Crack growth Fatigue Multi-scale Global-local analysis 

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References

  1. 1.
    Areias P, Belytschko T (2005) Analysis of three-dimensional crack initiation and propagation using the extended finite element method. Int J Numer Methods Eng 63: 760–788CrossRefMATHGoogle Scholar
  2. 2.
    Babuška I, Melenk J (1995) The partition of unity finite element method. Technical Report BN-1185, Institute for Physical Science and Technology, University of MarylandGoogle Scholar
  3. 3.
    Babuška I, Melenk J (1997) The partition of unity method. Int J Numer Methods Eng 40: 727–758CrossRefMATHGoogle Scholar
  4. 4.
    Babuška I, Caloz G, Osborn J (1994) Special finite element methods for a class of second order elliptic problems with rough coefficients. SIAM J Numer Anal 31(4): 945–981CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45: 601–620CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Belytschko T, Gracie R, Ventura G (2009) A review of extended/generalized finite element methods for material modeling. Model Simul Mater Sci Eng 17: 24. doi:10.1088/0965-0393/17/4/043001 CrossRefGoogle Scholar
  7. 7.
    Ben Dhia H, Jamond O (2010) On the use of XFEM within the Arlequin framework for the simulation of crack propagation. Comput Methods Appl Mech Eng 199: 1403–1414CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bordas S, Moran B (2006) Enriched finite elements and level sets for damage tolerance assessment of complex structures. Eng Fract Mech 73: 1176–1201CrossRefGoogle Scholar
  9. 9.
    Chahine E, Laborde P, Renard Y (2008) Spider-xfem, an extended finite element variant for partially unknown crack-tip displacement. Eur J Comput Mech 15(5–7): 625–636Google Scholar
  10. 10.
    Chahine E, Laborde P, Renard Y (2009) A reduced basis enrichment for the extended finite element method. Math Model Nat Phenom 4(1): 88–105CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    dell’Erba D, Aliabadi M (2000) Three-dimensional thermo-mechanical fatigue crack growth using BEM. Int J Fatigue 22: 261–273CrossRefGoogle Scholar
  12. 12.
    Duarte C (1996) The hp Cloud Method, PhD dissertation. The University of Texas at Austin, AustinGoogle Scholar
  13. 13.
    Duarte C, Kim DJ (2008) Analysis and applications of a generalized finite element method with global-local enrichment functions. Comput Methods Appl Mech Eng 197(6–8): 487–504. doi:10.1016/j.cma.2007.08.017 CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Duarte C, Oden J (1995) Hp clouds–A meshless method to solve boundary-value problems. Technical Report 95-05, TICAM. The University of Texas at AustinGoogle Scholar
  15. 15.
    Duarte C, Oden J (1996) An it hp adaptive method using clouds. Comput Methods Appl Mech Eng 139: 237–262CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Duarte C, Oden J (1996) Hp clouds—an hp meshless method. Numer Methods Partial Differen Equ 12: 673–705CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Duarte C, Babuška I, Oden J (2000) Generalized finite element methods for three dimensional structural mechanics problems. Comput Struct 77: 215–232CrossRefGoogle Scholar
  18. 18.
    Duarte C, Hamzeh O, Liszka T, Tworzydlo W (2001) A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Comput Methods Appl Mech Eng 190(15–17): 2227–2262. doi:10.1016/S0045-7825(00)00233-4 CrossRefMATHGoogle Scholar
  19. 19.
    Duarte C, Kim DJ, Babuška I (2007) Chapter: a global-local approach for the construction of enrichment functions for the generalized fem and its application to three-dimensional cracks. In: Leitão V, Alves C, Duarte C (eds) Advances in meshfree techniques, Computational Methods in Applied Sciences, vol 5. Springer, The Netherlands. iSBN 978-1-4020-6094-6Google Scholar
  20. 20.
    Erdogan F, Sih G (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng 85: 519–525CrossRefGoogle Scholar
  21. 21.
    Fan R, Fish J (2008) The rs-method for material failure simulations. Int J Numer Methods Eng 73(11): 1607–1623. doi:10.1002/nme.2134 CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Fish J, Nath A (1993) Adaptive and hierarchical modelling of fatigue crack propagation. Int J Numer Methods Eng 36: 2825–2836CrossRefMATHGoogle Scholar
  23. 23.
    Fries TP, Belytschko T (2010) The generalized/extended finite element method: an overview of the method and its applications. Int J Numer Methods Eng 253–304Google Scholar
  24. 24.
    Galland F, Gravouil A, Malvesin E, Rochette M (2011) A global model reduction approach for 3D fatigue crack growth with confined plasticity. Comput Methods Appl Mech Eng. doi:10.1016/j.cma.2010.08.018
  25. 25.
    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(11): 2569–2586CrossRefGoogle Scholar
  26. 26.
    Guidault PA, Allix O, Champaney L, Cornuault C (2008) A multiscale extended finite element method for crack propagation. Comput Methods Appl Mech Eng 197: 381–399CrossRefMATHGoogle Scholar
  27. 27.
    Hou T, Wu XH (1997) A multiscale finite element method for elliptic problems in composite materials and porous media. J Comput Phys 134: 169–189CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Jiao X (2007) Face offsetting: A unified framework for explicit moving interfaces. J Comput Phys 220(2): 612–625CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Kim DJ, Duarte C, Proenca S (2009) Generalized finite element method with global-local enrichments for nonlinear fracture analysis. In: Mattos HDC, Alves M (eds) International Symposium on Mechanics of Solids—MECSOL 2009, ABCM—Brazilian Society of Mechanical Sciences and Engineering, Rio de Janeiro, pp 317–330. iSBN 978-85-85769-43-7Google Scholar
  30. 30.
    Kim DJ, Pereira J, Duarte C (2010) Analysis of three-dimensional fracture mechanics problems: a two-scale approach using coarse generalized FEM meshes. Int J Numer Methods Eng 81(3): 335–365. doi:10.1002/nme.2690 MATHGoogle Scholar
  31. 31.
    Kim DJ, Duarte C, Sobh N (2011) Parallel simulations of three-dimensional cracks using the generalized finite element method. Comput Mech 47(3): 265–282. doi:10.1007/s00466-010-0546-5 CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Lee SH, Song JH, Yoon YC, Zi G, Belytschko T (2004) Combined extended and superimposed finite element method for cracks. Int J Numer Methods Eng 59(1119–1136). doi:10.1002/nme.908
  33. 33.
    Loehnert S, Belytschko T (2007) A multiscale projection method for macro/microcrack simulations. Int J Numer Methods Eng 71(12): 1466–1482CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Melenk J, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139: 289–314CrossRefMATHGoogle Scholar
  35. 35.
    Menk A, Bordas P (2010) Numerically determined enrichment functions for the extended finite element method and applications to bi-material anisotropic fracture and polycrystals. Int J Numer Methods Eng 83(7): 805–828MATHMathSciNetGoogle Scholar
  36. 36.
    Mi Y, Aliabadi M (1994) Three-dimensional crack growth simulation using BEM. Comput Struct 52: 871–878CrossRefMATHGoogle Scholar
  37. 37.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150CrossRefMATHGoogle Scholar
  38. 38.
    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(11): 2549–2568CrossRefMATHGoogle Scholar
  39. 39.
    Mousavi S, Grinspun E, Sukumar N (2011) Harmonic enrichment functions: a unified treatment of multiple, intersecting and branched cracks in the extended finite element method. Int J Numer Methods Eng 85: 1306–1322. doi:10.1002/nme.3020 MATHMathSciNetGoogle Scholar
  40. 40.
    Oden J, Duarte C, Zienkiewicz O (1998) A new cloud-based hp finite element method. Comput Methods Appl Mech Eng 153: 117–126CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    O’Hara P, Duarte C, Eason T (2009) Generalized finite element analysis of three-dimensional heat transfer problems exhibiting sharp thermal gradients. Comput Methods Appl Mech Eng 198(21-26): 1857–1871. doi:10.1016/j.cma.2008.12.024 CrossRefMATHGoogle Scholar
  42. 42.
    O’Hara P, Duarte C, Eason T (2011) Transient analysis of sharp thermal gradients using coarse finite element meshes. Comput Methods Appl Mech Eng 200(5–8): 812–829. doi:10.1016/j.cma.2010.10.005 CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Oskay C, Fish J (2008) On calibration and validation of eigendeformation-based multiscale models for failure analysis of heterogeneous systems. Comput Mech 42(2): 181–195CrossRefMATHGoogle Scholar
  44. 44.
    Paris A, Erdogan F (1963) A critical analysis of crack propagation laws. J Basic Eng 85: 528–534CrossRefGoogle Scholar
  45. 45.
    Passieux J, Gravouil A, Rethore J, Baietto M (2010) Direct estimation of generalized stress intensity factors using a three-scale concurrent multigrid X-FEM. Int J Numer Methods Eng. doi:10.1002/nme.3037
  46. 46.
    Pereira J, Duarte C, Guoy D, Jiao X (2009) Hp-Generalized FEM and crack surface representation for non-planar 3-D cracks. Int J Numer Methods Eng 77(5): 601–633. doi:10.1002/nme.2419 CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Pereira J, Duarte C, Jiao X, Guoy D (2009) Generalized finite element method enrichment functions for curved singularities in 3D fracture mechanics problems. Comput Mech 44(1): 73–92. doi:10.1007/s00466-008-0356-1 CrossRefMATHGoogle Scholar
  48. 48.
    Pereira J, Duarte C, Jiao X (2010) Three-dimensional crack growth with hp-generalized finite element and face offsetting methods. Comput Mech 46(3): 431–453. doi:10.1007/s00466-010-0491-3 CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Pierres E, Baietto M, Gravouil A (2010) A two-scale extended finite element method for modelling 3D crack growth with interfacial contact. Comput Methods Appl Mech Eng 199: 1165–1177. doi:10.1016/j.cma.2009.12.006 CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Rannou J, Gravouil A, Baietto-Dubourg M (2009) A local multigrid X-FEM strategy for 3-D crack propagation. Int J Numer Methods Eng 77: 581–600. doi:10.1002/nme.2427 CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    Rashid M (1998) The arbitrary local mesh replacement method: An alternative to remeshing for crack propagation analysis. Comput Methods Appl Mech Eng 154: 133–150CrossRefMATHGoogle Scholar
  52. 52.
    Richard H, Fulland M, Sander M (2005) Theoretical crack path prediction. Fatigue Fract Eng Mater Struct 28: 3–12. doi:10.1111/j.1460-2695.2004.00855.x CrossRefGoogle Scholar
  53. 53.
    Schöllmann M, Richard H, Kullmer G, Fulland M (2002) A new criterion for the prediction of crack development in multiaxially loaded structures. Int J Fract 117: 129–141CrossRefGoogle Scholar
  54. 54.
    Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190: 4081–4193CrossRefMATHGoogle Scholar
  55. 55.
    Strouboulis T, Zhang L, Babuška I (2003) Generalized finite element method using mesh-based handbooks: Application to problems in domains with many voids. Comput Methods Appl Mech Eng 192: 3109–3161CrossRefMATHGoogle Scholar
  56. 56.
    Strouboulis T, Zhang L, Babuška I (2004) p-version of the generalized FEM using mesh-based handbooks with applications to multiscale problems. Int J Numer Methods Eng 60: 1639–1672CrossRefMATHGoogle Scholar
  57. 57.
    Sukumar N, Chopp D, Moran B (2003) Extended finite element method and fast marching method for three-dimensional fatigue crack propagation. Eng Fract Mech 70: 29–48CrossRefGoogle Scholar
  58. 58.
    Sukumar N, Chopp D, Béchet E, Moës N (2008) Three-dimensional non-planar crack growth by a coupled extended finite element and fast marching method. Int J Numer Methods Eng 76: 727–748CrossRefMATHGoogle Scholar
  59. 59.
    Tada H, Paris P, Irwin G (2000) The stress analysis of cracks handbook, 3rd edn. ASME Press, New YorkCrossRefGoogle Scholar
  60. 60.
    Ural A, Heber G, Wawrzynek P, Ingraffea A, Lewicki D, Neto J (2005) Three-dimensional, parallel, finite element simulation of fatigue crack growth in a spiral bevel pinion gear. Eng Fract Mech 72: 1148–1170CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Architectural EngineeringKyung Hee UniversityYongin, Kyunggi-DoKorea

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