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An improved hybrid boundary node method for 2D crack problems

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Abstract

The meshless hybrid boundary node method (HBNM) is a promising method for solving boundary value problems and is further developed and numerically implemented for 2D crack problems in this paper, in which the displacement discontinuity method (DDM) is introduced, and an iterative hybrid technique is employed. In this approach, the original problem is decomposed into two subsidiary problems, and the HBNM is used to model the finite domain of the body without crack, while DDM is utilized to represent the cracks. The results will be added and compared with the boundary conditions of the original problem. Iteration will be performed between the external boundaries and crack faces until all of the boundary conditions are satisfied. Thus, the advantages of the component methods are effectively combined. Numerical examples are given to illustrate the implementation and performance of the present method. It is shown that the high accuracy can be achieved with a small number of nodes, and the present iterative hybrid approach is very suitable for modeling complex multi-cracks and branched cracks problems and is also very easy to be extended to solve the crack propagation.

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References

  1. Barsoum R.S.: On the use of isoparametric finite elements in linear fracture mechanics. Int. J. Numer. Methods Eng. 10, 25–37 (1976)

    Article  MATH  Google Scholar 

  2. Qian G.L., Gu S.N., Jiang J.S.: A finite element model of cracked plates and application to vibration problems. Comput. Struct. 39(5), 483–487 (1991)

    Article  Google Scholar 

  3. Ayhan A.O.: Mixed mode stress intensity factors for deflected and inclined surface cracks in finite-thickness plates. Eng. Fract. Mech. 71(7), 1059–1079 (2004)

    Article  Google Scholar 

  4. Hung N.D., Ngoc T.T.: Analysis of cracked plates and shells using “metis” finite element model. Finite Elem. Anal. Des. 40(8), 855–878 (2004)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  6. Daux C., Moës N., Dolbow J. et al.: Arbitrary branched and intersecting cracks with the extended finite element method. Int. J. Numer. Meth. Eng. 48, 1741–1760 (2000)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Areias P., Belytschko T.: Analysis of three-dimensional crack initiation and propagation using the extended finite element method. Int. J. Numer. Methods Eng. 63, 760–788 (2005)

    Article  MATH  Google Scholar 

  9. Stolarska M., Chopp D.L., Moës N. et al.: Modelling crack growth by level sets in the extended finite element method. Int. J. Numer. Methods Eng. 51, 943–960 (2001)

    Article  MATH  Google Scholar 

  10. Bachene M., Tiberkak R., Rechak S.: Vibration analysis of cracked plates using the extended finite element method. Arch. Appl. Mech. 79(3), 249–262 (2009)

    Article  MATH  Google Scholar 

  11. Gonzalez-Albuixech V.F., Giner E., Tarancon J.E. et al.: Convergence of domain integrals for stress intensity factor extraction in 2-D curved cracks problems with the extended finite element method. Int. J. Numer. Methods Eng. 94, 740–757 (2013)

    Article  MathSciNet  Google Scholar 

  12. Venturini W.S., Brebbia C.A.: Some applications of the boundary element method in geomechanics. Int. J. Numer. Anal. Met. 7(4), 419–433 (1983)

    Article  MATH  Google Scholar 

  13. Telles J.C.F., Guimaraes S.: Green’s function: a numerical generation for fracture mechanics problems via boundary elements. Comput. Methods Appl. Mech. 188(4), 847–858 (2000)

    Article  MATH  Google Scholar 

  14. Silveira N.P.P., Guimarães S., Telles J.C.F.: A numerical Green’s function BEM formulation for crack growth simulation. Eng. Anal. Bound. Elem. 29(11), 978–985 (2005)

    Article  MATH  Google Scholar 

  15. Leonel E.D., Venturini W.S.: Multiple random crack propagation using a boundary element formulation. Eng. Fract. Mech. 78(6), 1077–1090 (2011)

    Article  Google Scholar 

  16. Wen P.H., Aliabadi M.H.: Dual Boundary element method for modelling curved crack paths. Int. J. Fract. 176(1), 127–133 (2012)

    Article  Google Scholar 

  17. Frangi A., Novati G., Springhetti R. et al.: 3D fracture analysis by the symmetric Galerkin BEM. Comput. Mech. 28(3-4), 220–232 (2002)

    Article  MATH  Google Scholar 

  18. Pham A.D., Mouhoubi S., Bonnet M. et al.: Fast multipole method applied to symmetric Galerkin boundary element method for 3D elasticity and fracture problems. Eng. Anal. Bound. Elem. 36(12), 1838–1847 (2012)

    Article  MathSciNet  Google Scholar 

  19. Crouch S.L.: Solution of plane elasticity problems by the displacement discontinuity method. Int. J. Numer. Methods Eng. 10, 301–343 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yan X.: A special crack tip displacement discontinuity element. Mech. Res. Commun. 31(6), 651–659 (2004)

    Article  MATH  Google Scholar 

  21. Exadaktylos G., Xiroudakis G.: A G2 constant displacement discontinuity element for analysis of crack problems. Comput. Mech. 45(4), 245–261 (2010)

    Article  MATH  Google Scholar 

  22. Ameen M., Raghu P.B.K., Gopalakrishnan A.R.: Modeling of concrete cracking-A hybrid technique of using displacement discontinuity element method and direct boundary element method. Eng. Anal. Bound. Elem. 35(9), 1054–1059 (2011)

    Article  MATH  Google Scholar 

  23. Xiao H.T., Yue Z.Q.: A three-dimensional displacement discontinuity method for crack problems in layered rocks. Int. J. Rock Mech. Min. 48(3), 412–420 (2011)

    Article  Google Scholar 

  24. Ritz E., Mutlu O., Pollard D.D.: Integrating complementarity into the 2D displacement discontinuity boundary element method to model faults and fractures with frictional contact properties. Comput. Geosci. 45, 304–312 (2012)

    Article  Google Scholar 

  25. Wang S., Zhang H.: Partition of unity-based thermomechanical meshfree method for two-dimensional crack problems. Arch. Appl. Mech. 81(10), 1351–1363 (2011)

    Article  MATH  Google Scholar 

  26. Zhuang X., Augarde C., Bordas S.: Accurate fracture modelling using meshless methods, the visibility criterion and level sets: formulation and 2D modelling. Int. J. Numer. Methods Eng. 86(2), 249–268 (2011)

    Article  MATH  Google Scholar 

  27. Rabczuk T., Belytschko T.: Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Methods Eng. 61(13), 2316–2343 (2004)

    Article  MATH  Google Scholar 

  28. Rabczuk T., Belytschko T.: A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Comput. Method Appl. Mech. Eng. 196(29), 2777–2799 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rabczuk T., AreiasP. M. A.: A new approach for modelling slip lines in geological materials with cohesive models. Int. J Numer. Anal. Methods 30(11), 1159–1172 (2006)

    Article  MATH  Google Scholar 

  30. Rabczuk T., Bordas S., Zi G.: A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. Comput. Mech. 40(3), 473–495 (2007)

    Article  MATH  Google Scholar 

  31. Rabczuk T., Zi G.: A meshfree method based on the local partition of unity for cohesive cracks. Comput. Mech. 39(6), 743–760 (2007)

    Article  MATH  Google Scholar 

  32. Zi G., Rabczuk T., Wall W.: Extended meshfree methods without branch enrichment for cohesive cracks. Comput. Mech. 40(2), 367–382 (2007)

    Article  MATH  Google Scholar 

  33. Ching H.K., Batra R.C.: Determination of crack tip fields in linear elastostatics by the meshless local Petrov–Galerkin (MLPG) method. Comput. Model. Eng. Sci. 2(2), 273–289 (2001)

    Google Scholar 

  34. Gu Y.T., Wang W., Zhang L.C. et al.: An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields. Eng. Fract. Mech. 78(1), 175–190 (2011)

    Article  Google Scholar 

  35. Chau-Dinh T., Zi G., Lee P.S. et al.: Phantom-node method for shell models with arbitrary cracks. Comput. Struct. 92, 242–256 (2012)

    Article  Google Scholar 

  36. Vu-Bac, N., Nguyen-Xuan, H., Chen, L., et al.: A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics. J. Appl. Math. 2013. Article ID 978026 (2013)

  37. Moosavi M.R., Delfanian F., Khelil A.: Orthogonal meshless finite volume method applied to crack problems. Thin Wall. Struct. 52, 61–65 (2012)

    Article  Google Scholar 

  38. Moosavi M.R.: Orthogonal meshless finite volume method applied to elastodynamic crack problems. Int. J. Fract. 179(1-2), 1–7 (2013)

    Article  Google Scholar 

  39. Karageorghis A., Poullikkas A., Berger J.R.: Stress intensity factor computation using the method of fundamental solutions. Comput. Mech. 37, 445–454 (2006)

    Article  MATH  Google Scholar 

  40. Berger J.R., Karageorghis A., Martin P.A.: Stress intensity factor computation using the method of fundamental solutions: mixed-mode problems. Int. J. Numer. Methods Eng. 69, 469–483 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhang J.M., Yao Z.H., Li H.: A hybrid boundary node method. Int. J. Numer. Methods Eng. 53, 751–763 (2002)

    Article  MathSciNet  Google Scholar 

  42. Zhang J.M.: Numerical simulation of 3-D potential problems by Regular hybrid boundary node method. Int. J. Comput. Methods Eng. Sci. Mech. 9(2), 111–120 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  43. Miao Y., Wang Y.H.: Development of hybrid boundary node method in two dimensional elasticity. Eng. Anal. Bound. Elem. 29, 703–712 (2005)

    Article  MATH  Google Scholar 

  44. Miao Y., Wang Y.H.: Meshless analysis for three-dimensional elasticity with singular hybrid boundary node method. Appl. Mathods Mech. 27(5), 673–681 (2006)

    Article  MATH  Google Scholar 

  45. Yan F., Wang Y.H., Miao Y. et al.: Dual reciprocity hybrid boundary node method for free vibration analysis. J. Sound Vib. 32, 1036–1057 (2009)

    Article  Google Scholar 

  46. Miao Y., He T.G., Yang Q. et al.: Multi-domain hybrid boundary node method for evaluating top-down crack in asphalt pavements. Eng. Anal. Bound. Elem. 34(9), 755–760 (2010)

    Article  MATH  Google Scholar 

  47. Miao Y., He T.G., Luo H. et al.: Dual hybrid boundary node method for solving transient dynamic fracture problems. Comput. Model. Eng. Sci. 85(6), 481–498 (2012)

    MathSciNet  Google Scholar 

  48. Olson, J.E.: Fracture mechanics analysis of joints and veins. Dissertation, Stanford University (1991)

  49. Tada H., Paris P.C., Irwin G.R. et al.: The Stress Analysis of Cracks Handbook. ASME Press, New York (2000)

    Book  Google Scholar 

  50. Ma G.W., An X.M., Zhang H.H. et al.: Modeling complex crack problems using the numerical manifold method. Int. J. Fract. 156(1), 21–35 (2009)

    Article  MATH  Google Scholar 

  51. Fett T.: Stress intensity factors and T-stress for single and double-edge-cracked circular disks under mixed boundary conditions. Eng. Fract. Mech. 69(1), 69–83 (2002)

    Article  Google Scholar 

  52. Liu J., Lin G., Li X. et al.: Evaluation of stress intensity factors for multiple cracked circular disks under crack surface tractions with SBFEM. China Ocean Eng. 27, 417–426 (2013)

    Article  Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071, Hubei, People’s Republic of China

    Fei Tan, Youliang Zhang & Yinping Li

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  1. Fei Tan
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  2. Youliang Zhang
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  3. Yinping Li
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Correspondence to Fei Tan.

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Tan, F., Zhang, Y. & Li, Y. An improved hybrid boundary node method for 2D crack problems. Arch Appl Mech 85, 101–116 (2015). https://doi.org/10.1007/s00419-014-0902-6

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  • DOI: https://doi.org/10.1007/s00419-014-0902-6

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