Computational Mechanics

, Volume 53, Issue 2, pp 325–342 | Cite as

Three-dimensional crack propagation with distance-based discontinuous kernels in meshfree methods

Original Paper

Abstract

Distance fields are scalar functions defining the minimum distance of a given point in the space from the boundary of an object. Crack surfaces are geometric entities whose shapes can be arbitrary, often described with ruled surfaces or polygonal subdivisions. The derivatives of distance functions for such surfaces are discontinuous across the surface, and continuous all around the edges. These properties of the distance function were employed to build intrinsic enrichments of the underlying mesh-free discretisation for efficient simulation of three-dimensional crack propagation, removing the limitations of existing criteria (reviewed in this paper). Examples show that the proposed approach is able to introduce quite convoluted crack paths. The incremental nature of the developed approach does not require re-computation of the enrichment for the entire crack surface as advancing crack front extends incrementally as a set of connected surface facets. The concept is based on purely geometric representation of discontinuities thus addressing only the kinematic aspects of the problem, such to allow for any constitutive and cohesive interface models to be used.

Keywords

Fracture Crack Algorithms Meshless Discontinuities 

Notes

Acknowledgments

This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC, Grant EP/G042586/1) and Defense Science and Technology Laboratory (DSTL), both of which are gratefully acknowledged.

References

  1. 1.
    Arrea M, Ingraffea A (1982) Mixed-mode crack propagation in mortar and concrete. Technical report. Cornell University, Ithaca, pp 81–13Google Scholar
  2. 2.
    Atluri S, Zhu T (1998) A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput Mech 22(2):117–127CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Baerentzen JA, Aanaes H (2005) Signed distance computation using the angle weighted pseudonormal. IEEE Trans Vis Comput Graph 11(3):243–253CrossRefGoogle Scholar
  4. 4.
    Barbieri E, Meo M (2012) A fast object-oriented matlab implementation of the reproducing kernel particle method. Comput Mech 49(5):581–602CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Barbieri E, Petrinic N, Meo M, Tagarielli V (2012) A new weight-function enrichment in meshless methods for multiple cracks in linear elasticity. Int J Numer Methods Eng 90(2):177–195CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Belytschko T, Fleming M (1999) Smoothing, enrichment and contact in the element-free Galerkin method. Comput Struct 71(2):173–195CrossRefMathSciNetGoogle Scholar
  7. 7.
    Belytschko T, Gu L, Lu Y (1994) Fracture and crack growth by element-free Galerkin methods. Model Simul Mater Sci Eng 2:519–534CrossRefGoogle Scholar
  8. 8.
    Belytschko T, Lu Y, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37(2):229–256CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Belytschko T, Lu Y, Gu L (1995) Crack propagation by element-free Galerkin methods. Eng Fract Mech 51(2):295–315CrossRefGoogle Scholar
  10. 10.
    Brokenshire DR (1996) A study of torsion fracture tests. Ph.D. thesis, Cardiff University, CardiffGoogle Scholar
  11. 11.
    Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139(1):195–227CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    De Berg M, Cheong O, Van Kreveld M (2008) Computational geometry: algorithms and applications. Springer, BerlinGoogle Scholar
  13. 13.
    Dolbow J, Belytschko T (1998) An introduction to programming the meshless element free Galerkin method. Arch Comput Methods Eng 5(3):207–241CrossRefMathSciNetGoogle Scholar
  14. 14.
    Duarte CA, Oden JT (1996) Hp clouds: an Hp meshless method. Numer Methods Partial Differ Equ 12(6):673–706CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Duflot M (2006) A meshless method with enriched weight functions for three-dimensional crack propagation. Int J Numer Methods Eng 65(12):1970–2006CrossRefMATHGoogle Scholar
  16. 16.
    Duflot M, Nguyen-Dang H (2004) A meshless method with enriched weight functions for fatigue crack growth. Int J Numer Methods Eng 59:1945–1961CrossRefMATHGoogle Scholar
  17. 17.
    Duflot M, Nguyen-Dang H (2004) Fatigue crack growth analysis by an enriched meshless method. J Comput Appl Math 168(1–2):155–164CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Fries T, Matthies H (2003) Classification and overview of meshfree methods. Informatikbericht Nr 3. Institute of Scientific Computing, Technical University Braunschweig, Brunswick, GermanyGoogle Scholar
  19. 19.
    Fries TP, Baydoun M (2012) Crack propagation with the extended finite element method and a hybrid explicitimplicit crack description. Int J Numer Methods Eng 89(12):1527–1558CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Gasser TC, Holzapfel GA (2006) 3D crack propagation in unreinforced concrete: a two-step algorithm for tracking 3D crack paths. Comput Methods Appl Mech Eng 195(37):5198–5219CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Goodman JE, O’Rourke J (2010) Handbook of discrete and computational geometry. Chapman and Hall/CRC, Boca RatonGoogle Scholar
  22. 22.
    Gosz M, Moran B (2002) An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions. Eng Fract Mech 69(3):299–319CrossRefGoogle Scholar
  23. 23.
    Idelsohn SR, Onate E, Calvo N, Del Pin F (2003) The meshless finite element method. Int J Numer Methods Eng 58(6):893–912CrossRefMATHGoogle Scholar
  24. 24.
    Jones MW, Baerentzen JA, Sramek M (2006) 3D distance fields: a survey of techniques and applications. IEEE Trans Vis Comput Graph 12(4):581–599CrossRefGoogle Scholar
  25. 25.
    Kaczmarczyk L, Nezhad MM, Pearce C (2013) Three-dimensional brittle fracture: configurational-force-driven crack propagation. arXiv, arXiv:1304.6136 (preprint)Google Scholar
  26. 26.
    Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37(155):141–158CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Libersky LD, Petschek A (1991) Smooth particle hydrodynamics with strength of materials. In: Advances in the free-Lagrange method including contributions on adaptive gridding and the smooth particle hydrodynamics method. Springer, Berlin, pp 248–257Google Scholar
  28. 28.
    Liu W, Jun S, Zhang Y (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8–9):1081–1106CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Liu WK, Jun S, Li S, Adee J, Belytschko T (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Methods Eng 38(10):1655–1679CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Liu WK, Li S, Belytschko T (1997) Moving least-square reproducing kernel methods (i) methodology and convergence. Comput Methods Appl Mech Eng 143(1):113–154CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Nguyen V, Rabczuk T, Bordas S, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79:763–813Google Scholar
  32. 32.
    Onate E, Idelsohn S, Zienkiewicz O, Taylor R (1996) A finite point method in computational mechanics: applications to convective transport and fluid flow. Int J Numer Methods Eng 39(22):3839–3866 Google Scholar
  33. 33.
    Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Methods Eng 61(13):2316–2343CrossRefMATHGoogle Scholar
  34. 34.
    Ren B, Qian J, Zeng X, Jha A, Xiao S, Li S (2011) Recent developments on thermo-mechanical simulations of ductile failure by meshfree method. Comput Model Eng Sci 71(3):253Google Scholar
  35. 35.
    Sethian JA (1999) Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, vol 3. Cambridge University Press, CambridgeMATHGoogle Scholar
  36. 36.
    Sukumar N (1998) The natural element method in solid mechanics. Ph.D. thesis, Northwestern University, ChicagoGoogle Scholar
  37. 37.
    Walters MC, Paulino GH, Dodds RH (2005) Interaction integral procedures for 3-D curved cracks including surface tractions. Eng Fract Mech 72(11):1635–1663CrossRefGoogle Scholar
  38. 38.
    Yagawa G, Yamada T (1996) Free mesh method: a new meshless finite element method. Comput Mech 18(5):383–386CrossRefMATHGoogle Scholar
  39. 39.
    Yates J, Zanganeh M, Tomlinson R, Brown M, Garrido F (2008) Crack paths under mixed mode loading. Eng Fract Mech 75(3):319–330CrossRefGoogle Scholar
  40. 40.
    Zhu T, Zhang JD, Atluri S (1998) A local boundary integral equation (IBIE) method in computational mechanics, and a meshless discretization approach. Comput Mech 21(3):223–235CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Engineering and Materials ScienceQueen Mary University of LondonLondonUK
  2. 2.Department of Engineering ScienceUniversity of OxfordOxfordUK

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