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Generalized finite element method enrichment functions for curved singularities in 3D fracture mechanics problems

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Abstract

This paper presents a study of generalized enrichment functions for 3D curved crack fronts. Two coordinate systems used in the definition of singular curved crack front enrichment functions are analyzed. In the first one, a set of Cartesian coordinate systems defined along the crack front is used. In the second case, the geometry of the crack front is approximated by a set of curvilinear coordinate systems. A description of the computation of derivatives of enrichment functions and curvilinear base vectors is presented. The coordinate systems are automatically defined using geometrical information provided by an explicit representation of the crack surface. A detailed procedure to accurately evaluate the surface normal, conormal and tangent vectors along curvilinear crack fronts in explicit crack surface representations is also presented. An accurate and robust definition of orthonormal vectors along crack fronts is crucial for the proper definition of enrichment functions. Numerical experiments illustrate the accuracy and robustness of the proposed approaches.

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References

  1. Babuška I, Melenk JM (1997) The partition of unity finite element method. Int J Numer Methods Eng 40: 727–758

    Article  MATH  Google Scholar 

  2. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45: 601–620

    Article  MATH  MathSciNet  Google Scholar 

  3. Chopp DL, Sukumar N (2003) Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method. Int J Eng Sci 41: 845–869

    Article  MathSciNet  Google Scholar 

  4. Duarte CA, Babuška I, Oden JT (2000) Generalized finite element methods for three dimensional structural mechanics problems. Comp Struct 77: 215–232

    Article  Google Scholar 

  5. Duarte CA, Hamzeh ON, Liszka TJ, Tworzydlo WW (2001) A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Comp Methods Appl Mech Eng 190(15–17):2227–2262. http://dx.doi.org/10.1016/S0045-7825(00)00233-4

  6. Duarte CA, Reno LG, Simone A (2007) A high-order generalized FEM for through-the-thickness branched cracks. Int J Numer Methods Eng 72(3):325–351. http://dx.doi.org/10.1002/nme.2012

    Google Scholar 

  7. Duarte CAM, Oden JT (1996) An hp adaptive method using clouds. Comp Methods Appl Mech Eng 139: 237–262

    Article  MATH  MathSciNet  Google Scholar 

  8. Duflot M (2007) A study of the representation of cracks with level sets. Int J Numer Methods Eng 70: 1261–1302

    Article  MathSciNet  Google Scholar 

  9. Fleming M, Chu YA, Moran B, Belytschko T (1997) Enriched element-free Galerkin methods for crack tip fields. Int J Numer Methods Eng 40: 1483–1504

    Article  MathSciNet  Google Scholar 

  10. Jiao X (2007) Face offsetting: a unified framework for explicit moving interfaces. J Comput Phys 220(2): 612–625

    Article  MATH  MathSciNet  Google Scholar 

  11. Jiao X, Bayyana NR, Zha H (2007) Optimizing surface triangulation via near isometry with reference meshes. In: Geert YS, van Albada D, Dongarra J, Sloot PMA (eds) Computational science—ICCS 2007, pp 334–341, Beijing, China, May. Proceedings, Part I, Springer, Heidelberg

  12. Keast P (1986) Moderate-degree tetrahedral quadrature formulas. Comp Methods Appl Mech Eng 55: 339–348

    Article  MATH  MathSciNet  Google Scholar 

  13. Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37(155): 141–158

    Article  MATH  MathSciNet  Google Scholar 

  14. Lancaster P, Salkauskas K (1986) Curve and surface fitting, an introduction. Academic Press, San Diego

    MATH  Google Scholar 

  15. Lebedev LP, Cloud MJ (2003) Tensor analysis. World Scientific, New Jersey

    MATH  Google Scholar 

  16. Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comp Methods Appl Mech Eng 139: 289–314

    Article  MATH  Google Scholar 

  17. Mergheim J, Kuhl E, Steinmann P (2005) A finite element method for the computational modeling of cohesive cracks. Int J Numer Methods Eng 63: 276–289

    Article  MATH  Google Scholar 

  18. Moës N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fracture Mech 69: 813–833

    Article  Google Scholar 

  19. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150

    Article  MATH  Google Scholar 

  20. 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): 2549–2568

    Article  MATH  Google Scholar 

  21. Murakami Y (1992) Stress intensity factors handbook, vol 3, 1st edn. Pergamon, Oxford

    Google Scholar 

  22. Oden JT, Duarte CA (1997) Chapter: clouds, cracks and FEM’s. In: Reddy BD (ed) Recent developments in computational and applied mechanics, pp 302–321, Barcelona, Spain. International Center for Numerical Methods in Engineering, CIMNE (1997)

  23. Oden JT, Duarte CA, Zienkiewicz OC (1998) A new cloud-based hp finite element method. Comp Methods Appl Mech Eng 153: 117–126

    Article  MATH  MathSciNet  Google Scholar 

  24. Oden JT, Duarte CAM (1997) Chapter: solution of singular problems using hp clouds. In: Whiteman JR (eds) The mathematics of finite elements and applications—highlights 1996. Wiley, New York, pp 35–54

    Google Scholar 

  25. Park K, Pereira JP, Duarte CA, Paulino GH (2008) Integration of singular enrichment functions in the generalized/extended finite element method for three-dimensional problems. Int J Numer Methods Eng. http://dx.doi.org/10.1002/nme.2530

  26. Pereira JP, Duarte CA (2004) Computation of stress intensity factors for pressurized cracks using the generalized finite element method and superconvergent extraction techniques. In: Lyra PRM, da Silva SMBA, Magnani FS, do N. Guimaraes LJ, da Costa LM, Parente E Jr (eds) XXV Iberian Latin-American congress on computational methods in engineering. Recife, PE, Brazil, November. 15 pp. ISBN Proceedings CD: 857 409 869-8 (2004)

  27. Pereira JP, Duarte CA (2005) Extraction of stress intensity factors from generalized finite element solutions. Eng Anal Bound Elem 29: 397–413

    Article  Google Scholar 

  28. Pereira JP, Duarte CA, Guoy D, Jiao X (2008) Hp-generalized FEM and crack surface representation for non-planar 3D cracks. Int J Numer Meth Eng. http://dx.doi.org/10.1002/nme.2419

  29. Raju JC, Newman IS Jr (1979) Stress-intensity factors for a wide range of semi-elliptical surface cracks in finite-thickness plates. Eng Fracture Mech 11: 817–829

    Article  Google Scholar 

  30. Reddy JN, Rasmussen ML (1982) Advanced engineering analysis. Wiley, New York

    Google Scholar 

  31. Simone A (2004) Partition of unity-based discontinuous elements for interface phenomena: computational issues. Commun Numer Methods Eng 20: 465–478

    Article  MATH  Google Scholar 

  32. Simone A, Wells GN, Sluys LJ (2003) From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Comp Methods Appl Mech Eng 192(41–42): 4581–4607

    Article  MATH  Google Scholar 

  33. Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comp Methods Appl Mech Eng 190: 4081–4193

    Article  MATH  Google Scholar 

  34. Sukumar N, Chopp DL, Moran B (2003) Extended finite element method and fast marching method for three-dimensional fatigue crack propagation. Eng Fracture Mech 70: 29–48

    Article  Google Scholar 

  35. Sukumar N, Moës N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modelling. Int J Numer Methods Eng 48(11): 1549–1570

    Article  MATH  Google Scholar 

  36. Szabo B, Babuška I (1991) Finite element analysis. Wiley, New York

    MATH  Google Scholar 

  37. Szabo BA, Babuška I (1988) Computation of the amplitude of stress singular terms for cracks and reentrant corners. In: Cruse TA (ed) Fracture mechanics: nineteenth symposium. ASTM STP 969, pp 101–124, Southwest Research Institute, San Antonio (1988)

  38. Tada H, Paris P, Irwin G (2000) The stress analysis of cracks handbook, 3rd edn. ASME Press, New York

    Google Scholar 

  39. Walters MC, Paulino GH, Dodds RH Jr (2004) Stress-intensity factors for surface cracks in functionally graded materials under mode-I thermomechanical loading. Int J Solids Struct 41: 1081–1118

    Article  MATH  Google Scholar 

  40. Wells GN, de Borst R, Sluys LJ (2002) A consistent geometrically non-linear approach for delamination. Int J Numer Methods Eng 54: 1333–1355

    Article  MATH  Google Scholar 

  41. Wells GN, Sluys LJ (2001) A new method for modeling cohesive cracks using finite elements. Int J Numer Methods Eng 50: 2667–2682

    Article  MATH  Google Scholar 

  42. Zi G, Belytschko T (2003) New crack-tip elements for XFEM and applications to cohesive cracks. Int J Numer Methods Eng 57: 2221–2240

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 North Mathews Ave., Urbana, IL, 61801, USA

    J. P. Pereira & C. A. Duarte

  2. Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY, 11794-3600, USA

    X. Jiao

  3. Center for Simulation of Advanced Rockets, University of Illinois at Urbana-Champaign, 1304 West Springfield Avenue, Urbana, IL, 61801, USA

    D. Guoy

Authors
  1. J. P. Pereira
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  2. C. A. Duarte
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  3. X. Jiao
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  4. D. Guoy
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Correspondence to C. A. Duarte.

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Pereira, J.P., Duarte, C.A., Jiao, X. et al. Generalized finite element method enrichment functions for curved singularities in 3D fracture mechanics problems. Comput Mech 44, 73–92 (2009). https://doi.org/10.1007/s00466-008-0356-1

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  • DOI: https://doi.org/10.1007/s00466-008-0356-1

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