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International Journal of Fracture

, Volume 156, Issue 1, pp 21–35 | Cite as

Modeling complex crack problems using the numerical manifold method

  • G. W. Ma
  • X. M. An
  • H. H. Zhang
  • L. X. Li
Original Paper

Abstract

In the numerical manifold method, there are two kinds of covers, namely mathematical cover and physical cover. Mathematical covers are independent of the physical domain of the problem, over which weight functions are defined. Physical covers are the intersection of the mathematical covers and the physical domain, over which cover functions with unknowns to be determined are defined. With these two kinds of covers, the method is quite suitable for modeling discontinuous problems. In this paper, complex crack problems such as multiple branched and intersecting cracks are studied to exhibit the advantageous features of the numerical manifold method. Complex displacement discontinuities across crack surfaces are modeled by different cover functions in a natural and straightforward manner. For the crack tip singularity, the asymptotic near tip field is incorporated to the cover function of the singular physical cover. By virtue of the domain form of the interaction integral, the mixed mode stress intensity factors are evaluated for three typical examples. The excellent results show that the numerical manifold method is prominent in modeling the complex crack problems.

Keywords

Numerical manifold method Mathematical cover Physical cover Weight function Cover function Complex crack problems 

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References

  1. Babuska I, Melenk JM (1997) The partition of unity method. Int J Numer Methods Eng 40: 727–758. doi: 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N zbMATHCrossRefMathSciNetGoogle Scholar
  2. Barsoum R (1977) Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements. Int J Numer Methods Eng 11: 85–98. doi: 10.1002/nme.1620110109 zbMATHCrossRefGoogle Scholar
  3. Belytschko T, Tabbara M (1996) Dynamic fracture using element-free Galerkin methods. Int J Numer Methods Eng 39: 923–938. doi: 10.1002/(SICI)1097-0207(19960330)39:6<923::AID-NME887>3.0.CO;2-W zbMATHCrossRefGoogle Scholar
  4. Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37: 229–256. doi: 10.1002/nme.1620370205 zbMATHCrossRefMathSciNetGoogle Scholar
  5. Belytschko T, Lu YY, Gu L (1995a) Crack propagation by element-free Galerkin methods. Eng Fract Mech 51(2): 295–315. doi: 10.1016/0013-7944(94)00153-9 CrossRefADSGoogle Scholar
  6. Belytschko T, Lu YY, Gu L, Tabbara M (1995b) Element-free Galerkin methods for static and dynamic fracture. Int J Solids Struct 32(17–18): 2547–2570. doi: 10.1016/0020-7683(94)00282-2 zbMATHCrossRefGoogle Scholar
  7. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng 139: 3–47. doi: 10.1016/S0045-7825(96)01078-X zbMATHCrossRefGoogle Scholar
  8. Chen Y, Hasebe N (1995) New integration scheme for the branch crack problem. Eng Fract Mech 52: 791–801. doi: 10.1016/0013-7944(95)00052-W CrossRefGoogle Scholar
  9. Chen GQ, Ohnishi Y, Ito T (1998) Development of high-order manifold method. Int J Numer Methods Eng 43: 685–712. doi: 10.1002/(SICI)1097-0207(19981030)43:4<685::AID-NME442>3.0.CO;2-7 zbMATHCrossRefGoogle Scholar
  10. Cheung Y, Wang Y, Woo C (1984) A general method for multiple crack problems in a finite plate. Comput Mech 20: 583–589Google Scholar
  11. Chiou YJ, Lee YM, Tsay RJ (2002) Mixed mode fracture propagation by manifold method. Int J Fract 114: 327–347. doi: 10.1023/A:1015713428989 CrossRefGoogle Scholar
  12. Daux C, Moes 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: 1741–1760. doi: 10.1002/1097-0207(20000830)48:12<1741::AID-NME956>3.0.CO;2-L zbMATHCrossRefGoogle Scholar
  13. Duarte CA, Reno LG, Simone A (2007) A high-order generalized FEM for through-the thickness branched cracks. Int J Numer Methods Eng 72: 325–351. doi: 10.1002/nme.2012 CrossRefMathSciNetGoogle Scholar
  14. Duflot M, Nauyen-Dang H (2004) A meshless method with enriched weight functions for fatigue crack growth. Int J Numer Methods Eng 59: 1945–1961. doi: 10.1002/nme.948 zbMATHCrossRefGoogle Scholar
  15. 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. doi: 10.1002/(SICI)1097-0207(19970430)40:8<1483::AID-NME123>3.0.CO;2-6 CrossRefMathSciNetGoogle Scholar
  16. Huang R, Sukumar N, Prevost JH (2003) Modeling quasi-static crack growth with the extended finite element method part II: numerical applications. Int J Solids Struct 40: 7539–7552. doi: 10.1016/j.ijsolstr.2003.08.001 zbMATHCrossRefGoogle Scholar
  17. Karihaloo BL, Xiao QZ (2003) Modeling of stationary and growing cracks in FE framework without remeshing: a state-of-the-art review. Comput Struct 81: 119–129. doi: 10.1016/S0045-7949(02)00431-5 CrossRefGoogle Scholar
  18. Krysl P, Belytschko T (1999) Element-free Galerkin method for dynamic propagation of arbitrary 3-D cracks. Int J Numer Methods Eng 44: 767–800. doi: 10.1002/(SICI)1097-0207(19990228)44:6<767::AID-NME524>3.0.CO;2-G zbMATHCrossRefGoogle Scholar
  19. Kwon YW, Akin JE (1989) Development of a derivative singular element for application to crack propagation problems. Comput Struct 31(3): 467–471. doi: 10.1016/0045-7949(89)90394-5 CrossRefGoogle Scholar
  20. Li SC, Li SC, Cheng YM (2005) Enriched meshless manifold method for two-dimensional crack modeling. Theor Appl Fract Mech 44: 234–248. doi: 10.1016/j.tafmec.2005.09.002 CrossRefGoogle Scholar
  21. Lin JS (2003) A mesh-based partition of unity method for discontinuity modeling. Comput Methods Appl Mech Eng 192: 1515–1532. doi: 10.1016/S0045-7825(02)00655-2 zbMATHCrossRefGoogle Scholar
  22. Melenk JM, Babuska I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139: 289–314. doi: 10.1016/S0045-7825(96)01087-0 zbMATHCrossRefMathSciNetGoogle Scholar
  23. Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150. doi: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J zbMATHCrossRefGoogle Scholar
  24. Muravin B, Turkel E (2006) Multiple crack weight for solution of multiple intersecting cracks by meshless numerical methods. Int J Numer Methods Eng 67: 1146–1159. doi: 10.1002/nme.1661 zbMATHCrossRefGoogle Scholar
  25. Rao BN, Rahman S (2000) An efficient meshless method for fracture analysis of cracks. Comput Mech 26: 398–408. doi: 10.1007/s004660000189 zbMATHCrossRefGoogle Scholar
  26. Shi GH (1991) Manifold method of material analysis. In: Transactions of the 9th army conference on applied mathematics and computing, Minneapolis, Minnesota, pp 57–76Google Scholar
  27. Shi GH (1992) Modeling rock joints and blocks by manifold method. In: Proceedings of the 33rd US rock mechanics symposium, Santa Fe, New Mexico, pp 639–648Google Scholar
  28. Simone A, Duarte CA, Van Der Giessen E (2006) A generalized finite element method for polycrystals with discontinuous grain boundaries. Int J Numer Methods Eng 67: 1122–1145. doi: 10.1002/nme.1658 zbMATHCrossRefGoogle Scholar
  29. Strouboulis T, Babuska I, Copps K (2000a) The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 181: 43–69. doi: 10.1016/S0045-7825(99)00072-9 zbMATHCrossRefMathSciNetGoogle Scholar
  30. Strouboulis T, Copps K, Babuska I (2000b) The generalized finite element method: an example of its implementation and illustration of its performance. Int J Numer Methods Eng 47: 1401–1417. doi: 10.1002/(SICI)1097-0207(20000320)47:8<1401::AID-NME835>3.0.CO;2-8 zbMATHCrossRefMathSciNetGoogle Scholar
  31. Strouboulis T, Copps K, Babuska I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190: 4081–4193. doi: 10.1016/S0045-7825(01)00188-8 zbMATHCrossRefMathSciNetGoogle Scholar
  32. Sukumar N, Prevost JH (2003) Modeling quasi-static crack growth with the extended finite element method part I: computer implementation. Int J Solids Struct 40: 7513–7537. doi: 10.1016/j.ijsolstr.2003.08.002 zbMATHCrossRefGoogle Scholar
  33. Sukumar N, Moes N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modeling. Int J Numer Methods Eng 48: 1549–1570. doi: 10.1002/1097-0207(20000820)48:11<1549::AID-NME955>3.0.CO;2-A zbMATHCrossRefGoogle Scholar
  34. Tsay RJ, Chiou YJ, Chuang WL (1999) Crack growth prediction by manifold method. J Eng Mech 125: 884–890. doi: 10.1061/(ASCE)0733-9399(1999)125:8(884) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • G. W. Ma
    • 1
  • X. M. An
    • 1
  • H. H. Zhang
    • 1
    • 2
  • L. X. Li
    • 1
    • 2
  1. 1.School of Civil and Environmental EngineeringNanyang Technological UniversitySingaporeSingapore
  2. 2.MOE Key Laboratory for Strength and VibrationXi’an Jiaotong UniversityXi’anPeople’s Republic of China

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