Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2013, Article ID 978026, 12 pages
http://dx.doi.org/10.1155/2013/978026
Research Article

A Phantom-Node Method with Edge-Based Strain Smoothing for Linear Elastic Fracture Mechanics

1Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China
2Institute of Structural Mechanics, Bauhaus-University Weimar, Marienstraße 15, D-99423 Weimar, Germany
3Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City 70000, Vietnam
4School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, Brisbane, QLD 4001, Australia
5School of Engineering, Institute of Mechanics and Advanced Materials, Theoretical and Computational Mechanics, Cardiff University, Wales CF24 3AA, UK
6Department of Civil, Environmental & Architectural Engineering, Korea University, 5 Ga 1, An-Am Dong, Sung-Buk Gu, Seoul 136-701, Republic of Korea
7Aerospace Systems Ohio Eminent Scholar, University of Cincinnati, Cincinnati, OH 45221-0070, USA
8School of Civil, Environmental and Architectural Engineering, Korea University, 5 Ga 1, Anam-dong, Seongbuk-gu, Seoul 136-701, Republic of Korea

Received 15 January 2013; Accepted 1 May 2013

Academic Editor: Song Cen

Copyright © 2013 N. Vu-Bac et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. Belytschko and T. Black, “Elastic crack growth in finite elements with minimal remeshing,” International Journal for Numerical Methods in Engineering, vol. 45, no. 5, pp. 601–620, 1999. View at Google Scholar · View at Scopus
  2. A. Hansbo and P. Hansbo, “A finite element method for the simulation of strong and weak discontinuities in solid mechanics,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 33–35, pp. 3523–3540, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. J. Mergheim, E. Kuhl, and P. Steinmann, “A finite element method for the computational modelling of cohesive cracks,” International Journal for Numerical Methods in Engineering, vol. 63, no. 2, pp. 276–289, 2005. View at Publisher · View at Google Scholar · View at Scopus
  4. J.-H. Song, P. M. A. Areias, and T. Belytschko, “A method for dynamic crack and shear band propagation with phantom nodes,” International Journal for Numerical Methods in Engineering, vol. 67, no. 6, pp. 868–893, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. P. M. A. Areias and T. Belytschko, “A comment on the article “A finite element method for simulation of strong and weak discontinuities in solid mechanics”,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 9–12, pp. 1275–1276, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. J. Shi, D. Chopp, J. Lua, N. Sukumar, and T. Belytschko, “Abaqus implementation of extended finite element method using a level set representation for three-dimensional fatigue crack growth and life predictions,” Engineering Fracture Mechanics, vol. 77, no. 14, pp. 2840–2863, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. M. Duflot, “Industrial applications of XFEM for 3D crack propagation with Morfeo/Crack and Abaqus,” in ECCOMAS Thematic Conference on XFEM, Cardiff, UK, June 2011.
  8. T. Menouillard, J. Réthoré, A. Combescure, and H. Bung, “Efficient explicit time stepping for the eXtended Finite Element Method (X-FEM),” International Journal for Numerical Methods in Engineering, vol. 68, no. 9, pp. 911–939, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. T. Menouillard, J. Réthoré, N. Moës, A. Combescure, and H. Bung, “Mass lumping strategies for X-FEM explicit dynamics: application to crack propagation,” International Journal for Numerical Methods in Engineering, vol. 74, no. 3, pp. 447–474, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. T. Chau-Dinh, G. Zi, P.-S. Lee, T. Rabczuk, and J.-H. Song, “Phantom-node method for shell models with arbitrary cracks,” Computers and Structures, vol. 92-93, pp. 242–246, 2012. View at Publisher · View at Google Scholar · View at Scopus
  11. P. Laborde, J. Pommier, Y. Renard, and M. Salaün, “High-order extended finite element method for cracked domains,” International Journal for Numerical Methods in Engineering, vol. 64, no. 3, pp. 354–381, 2005. View at Publisher · View at Google Scholar · View at Scopus
  12. E. Béchet, H. Minnebo, N. Moës, and B. Burgardt, “Improved implementation and robustness study of the X-FEM for stress analysis around cracks,” International Journal for Numerical Methods in Engineering, vol. 64, no. 8, pp. 1033–1056, 2005. View at Publisher · View at Google Scholar · View at Scopus
  13. G. Ventura, R. Gracie, and T. Belytschko, “Fast integration and weight function blending in the extended finite element method,” International Journal for Numerical Methods in Engineering, vol. 77, no. 1, pp. 1–29, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. R. Gracie, H. Wang, and T. Belytschko, “Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods,” International Journal for Numerical Methods in Engineering, vol. 74, no. 11, pp. 1645–1669, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. S. P. A. Bordas, T. Rabczuk, N.-X. Hung et al., “Strain smoothing in FEM and XFEM,” Computers and Structures, vol. 88, no. 23-24, pp. 1419–1443, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. T. Rabczuk, G. Zi, A. Gerstenberger, and W. A. Wall, “A new crack tip element for the phantom-node method with arbitrary cohesive cracks,” International Journal for Numerical Methods in Engineering, vol. 75, no. 5, pp. 577–599, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. G. R. Liu, K. Y. Dai, and T. T. Nguyen, “A smoothed finite element method for mechanics problems,” Computational Mechanics, vol. 39, no. 6, pp. 859–877, 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. G. R. Liu, T. T. Nguyen, K. Y. Dai, and K. Y. Lam, “Theoretical aspects of the smoothed finite element method (SFEM),” International Journal for Numerical Methods in Engineering, vol. 71, no. 8, pp. 902–930, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. N. Nguyen-Thanh, T. Rabczuk, H. Nguyen-Xuan, and S. P. A. Bordas, “A smoothed finite element method for shell analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 2, pp. 165–177, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. S. P. A. Bordas and S. Natarajan, “On the approximation in the smoothed finite element method (SFEM),” International Journal for Numerical Methods in Engineering, vol. 81, no. 5, pp. 660–670, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. T. Rabczuk, P. M. A. Areias, and T. Belytschko, “A meshfree thin shell method for non-linear dynamic fracture,” International Journal for Numerical Methods in Engineering, vol. 72, no. 5, pp. 524–548, 2007. View at Publisher · View at Google Scholar · View at Scopus
  22. T. Rabczuk and T. Belytschko, “Application of particle methods to static fracture of reinforced concrete structures,” International Journal of Fracture, vol. 137, no. 1-4, pp. 19–49, 2006. View at Publisher · View at Google Scholar · View at Scopus
  23. T. Rabczuk and T. Belytschko, “A three-dimensional large deformation meshfree method for arbitrary evolving cracks,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 29-30, pp. 2777–2799, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. S. Bordas, P. V. Nguyen, C. Dunant, A. Guidoum, and H. Nguyen-Dang, “An extended finite element library,” International Journal for Numerical Methods in Engineering, vol. 71, no. 6, pp. 703–732, 2007. View at Publisher · View at Google Scholar · View at Scopus
  25. G. R. Liu, T. Nguyen-Thoi, and K. Y. Lam, “An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids,” Journal of Sound and Vibration, vol. 320, no. 4-5, pp. 1100–1130, 2009. View at Publisher · View at Google Scholar · View at Scopus
  26. G. R. Liu, “A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: part I theory,” International Journal for Numerical Methods in Engineering, vol. 81, no. 9, pp. 1093–1126, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. L. Chen, T. Rabczuk, S. P. A. Bordas, G. R. Liu, K. Y. Zeng, and P. Kerfriden, “Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth,” Computer Methods in Applied Mechanics and Engineering, vol. 209–212, pp. 250–265, 2012. View at Publisher · View at Google Scholar · View at Scopus
  28. N. Vu-Bac, H. Nguyen-Xuan, L. Chen et al., “A node-based smoothed extended finite element method (NS-XFEM) for fracture analysis,” CMES: Computer Modeling in Engineering and Sciences, vol. 73, no. 4, pp. 331–355, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  29. F. Z. Li, C. F. Shih, and A. Needleman, “A comparison of methods for calculating energy release rates,” Engineering Fracture Mechanics, vol. 21, no. 2, pp. 405–421, 1985. View at Google Scholar · View at Scopus
  30. B. Moran and C. F. Shih, “Crack tip and associated domain integrals from momentum and energy balance,” Engineering Fracture Mechanics, vol. 27, no. 6, pp. 615–642, 1987. View at Google Scholar · View at Scopus
  31. S. S. Wang, J. F. Yau, and H. T. Corten, “A mixed-mode crack analysis of rectilinear anisotropic solids using conservation laws of elasticity,” International Journal of Fracture, vol. 16, no. 3, pp. 247–259, 1980. View at Publisher · View at Google Scholar · View at MathSciNet
  32. F. Erdogan and G. Sih, “On the crack extension in sheets under plane loading and transverse shear,” Journal Basic Engineering, vol. 85, no. 6, pp. 519–527, 1963. View at Google Scholar
  33. A. Menk and S. Bordas, “Crack growth calculations in solder joints based on microstructural phenomena with X-FEM,” Computational Materials Science, vol. 50, no. 3, pp. 1145–1156, 2011. View at Publisher · View at Google Scholar
  34. D. F. Li, C. F. Li, S. Q. Shu, Z. X. Wang, and J. Lu, “A fast and accurate analysis of the interacting cracks in linear elastic solids,” International Journal of Fracture, vol. 151, pp. 169–185, 2008. View at Google Scholar