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ISRN Applied Mathematics
Volume 2013 (2013), Article ID 849231, 38 pages
http://dx.doi.org/10.1155/2013/849231
Review Article

Computational Methods for Fracture in Brittle and Quasi-Brittle Solids: State-of-the-Art Review and Future Perspectives

Institute of Structural Mechanics, Bauhaus-Universitat Weimar, Marienstraße 15, 99423 Weimar, Germany

Received 1 August 2012; Accepted 3 September 2012

Academic Editors: S. Li and R. Samtaney

Copyright © 2013 Timon Rabczuk. 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. S. Li, W. K. Liu, A. J. Rosakis, T. Belytschko, and W. Hao, “Mesh-free Galerkin simulations of dynamic shear band propagation and failure mode transition,” International Journal of Solids and Structures, vol. 39, no. 5, pp. 1213–1240, 2002. View at Publisher · View at Google Scholar · View at Scopus
  2. S. Li and B. C. Simonson, “Meshfree simulation of ductile crack propagation,” International Journal of Computational Engineering Science, vol. 6, pp. 1–25, 2005.
  3. B. C. Simonsen and S. Li, “Mesh-free simulation of ductile fracture,” International Journal for Numerical Methods in Engineering, vol. 60, no. 8, pp. 1425–1450, 2004. View at Publisher · View at Google Scholar · View at Scopus
  4. D. C. Simkins Jr. and S. Li, “Meshfree simulations of thermo-mechanical ductile fracture,” Computational Mechanics, vol. 38, no. 3, pp. 235–249, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. B. Ren and S. Li, “Meshfree simulations of plugging failures in high-speed impacts,” Computers and Structures, vol. 88, no. 15-16, pp. 909–923, 2010. View at Publisher · View at Google Scholar · View at Scopus
  6. B. Ren, J. Qian, X. Zeng, A. K. Jha, S. Xiao, and S. Li, “Recent developments on thermo-mechanical simulations of ductile failure by meshfree method,” Computer Modeling in Engineering and Sciences, vol. 71, no. 3, pp. 253–277, 2011. View at Scopus
  7. B. Ren and S. Li, “Modeling and simulation of large-scale ductile fracture in plates and shells,” International Journal of Solids and Structures, vol. 49, no. 18, pp. 2373–2393, 2012.
  8. E. E. Gdoutos, Fracture Mechanics: An Introduction, vol. 123, Kluwer Academic, 2005. View at Zentralblatt MATH
  9. J. Planas and M. Elices, “Nonlinear fracture of cohesive materials,” International Journal of Fracture, vol. 51, no. 2, pp. 139–157, 1991. View at Publisher · View at Google Scholar · View at Scopus
  10. A. Carpinteri, “Post-peak and post-bifurcation analysis of cohesive crack propagation,” Engineering Fracture Mechanics, vol. 32, no. 2, pp. 265–278, 1989. View at Scopus
  11. N. Moës and T. Belytschko, “Extended finite element method for cohesive crack growth,” Engineering Fracture Mechanics, vol. 69, no. 7, pp. 813–833, 2002. View at Publisher · View at Google Scholar · View at Scopus
  12. A. Carpinteri, “Notch sensitivity in fracture testing of aggregative materials,” Engineering Fracture Mechanics, vol. 16, no. 4, pp. 467–481, 1982. View at Scopus
  13. Z. P. Bažant and G. Pijaudier-Cabot, “Nonlocal continuum damage, localization instabilities and convergence,” Journal of Applied Mechanics, Transactions ASME, vol. 55, no. 2, pp. 287–293, 1988. View at Scopus
  14. Z. P. Bažant and M. Jirásek, “Nonlocal integral formulations of plasticity and damage: survey of progress,” Journal of Engineering Mechanics, vol. 128, no. 11, pp. 1119–1149, 2002. View at Publisher · View at Google Scholar · View at Scopus
  15. R. De Borst, J. Pamin, and M. G. D. Geers, “On coupled gradient-dependent plasticity and damage theories with a view to localization analysis,” European Journal of Mechanics A, vol. 18, no. 6, pp. 939–962, 1999. View at Publisher · View at Google Scholar · View at Scopus
  16. R. de Borst, M. A. Gutiérrez, G. N. Wells, J. J. C. Remmers, and H. Askes, “Cohesive-zone models, higher-order continuum theories and reliability methods for computational failure analysis,” International Journal for Numerical Methods in Engineering, vol. 60, no. 1, pp. 289–315, 2004. View at Publisher · View at Google Scholar · View at Scopus
  17. R. De Borst, “Fracture in quasi-brittle materials: a review of continuum damage-based approaches,” Engineering Fracture Mechanics, vol. 69, no. 2, pp. 95–112, 2001. View at Publisher · View at Google Scholar · View at Scopus
  18. J. Fish, Q. Yu, and K. Shek, “Computational damage mechanics for composite materials based on mathematical homogenization,” International Journal for Numerical Methods in Engineering, vol. 45, no. 11, pp. 1657–1679, 1999. View at Scopus
  19. D. Krajcinovic and S. Mastilovic, “Some fundamental issues of damage mechanics,” Mechanics of Materials, vol. 21, no. 3, pp. 217–230, 1995. View at Scopus
  20. D. Krajcinovic, Damage Mechanics, North Holland Series in Applied Mathematics and Mechanics, Elsevier, 1996.
  21. J. Lemaitre and H. Lippmann, A Course on Damage Mechanics, vol. 2, Springer, Berlin, Germany, 1996.
  22. R. H. J. Peerlings, R. De Borst, W. A. M. Brekelmans, and J. H. P. De Vree, “Gradient enhanced damage for quasi-brittle materials,” International Journal for Numerical Methods in Engineering, vol. 39, no. 19, pp. 3391–3403, 1996. View at Scopus
  23. J. F. Shao and J. W. Rudnicki, “Microcrack-based continuous damage model for brittle geomaterials,” Mechanics of Materials, vol. 32, no. 10, pp. 607–619, 2000. View at Publisher · View at Google Scholar · View at Scopus
  24. J. C. Simo and J. W. Ju, “Strain- and stress-based continuum damage models-I. Formulation,” International Journal of Solids and Structures, vol. 23, no. 7, pp. 821–840, 1987. View at Scopus
  25. M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, Germany, 1997. View at Zentralblatt MATH
  26. L. M. Kachanov, “Time of the rupture process under creep conditions,” Izvestiya Akademii Nauk SSSR Otdelenie Tekniches, vol. 8, pp. 26–31, 1958.
  27. R. Hill, “Acceleration waves in solids,” Journal of the Mechanics and Physics of Solids, vol. 10, pp. 1–16, 1962. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. B. Loret and J. H. Prevost, “Dynamic strain localization in elasto-(visco-)plastic solids, Part 1. General formulation and one-dimensional examples,” Computer Methods in Applied Mechanics and Engineering, vol. 83, no. 3, pp. 247–273, 1990. View at Scopus
  29. J. H. Prevost and B. Loret, “Dynamic strain localization in elasto-(visco-)plastic solids, part 2. plane strain examples,” Computer Methods in Applied Mechanics and Engineering, vol. 83, no. 3, pp. 275–294, 1990. View at Scopus
  30. J. W. Rudnicki and J. R. Rice, “Conditions for the localization of deformation in pressure-sensitive dilatant materials,” Journal of the Mechanics and Physics of Solids, vol. 23, no. 6, pp. 371–394, 1975. View at Scopus
  31. T. Belytschko, W. K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons, Chichester, UK, 2000.
  32. Z. P. Bažant and T. B. Belytschko, “Wave propagation in a strain-softening bar: exact solution,” Journal of Engineering Mechanics, vol. 111, no. 3, pp. 381–389, 1985. View at Scopus
  33. R. De Borst and H. B. Muehlhaus, “Gradient-dependent plasticity: formulation and algorithmic aspects,” International Journal for Numerical Methods in Engineering, vol. 35, no. 3, pp. 521–539, 1992. View at Scopus
  34. N. A. Fleck and J. W. Hutchinson, “A phenomenological theory for strain gradient effects in plasticity,” Journal of the Mechanics and Physics of Solids, vol. 41, no. 12, pp. 1825–1857, 1993. View at Scopus
  35. R. H. J. Peerlings, R. de Borst, W. A. M. Brekelmans, and M. G. D. Geers, “Localisation issues in local and nonlocal continuum approaches to fracture,” European Journal of Mechanics A, vol. 21, no. 2, pp. 175–189, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. Z. P. Bažant, “Why continuum damage is nonlocal: micromechanics arguments,” Journal of Engineering Mechanics, vol. 117, no. 5, pp. 1070–1087, 1991. View at Publisher · View at Google Scholar
  37. G. Etse and K. Willam, “Failure analysis of elastoviscoplastic material models,” Journal of Engineering Mechanics, vol. 125, no. 1, pp. 60–68, 1999. View at Scopus
  38. T. Rabczuk and J. Eibl, “Simulation of high velocity concrete fragmentation using SPH/MLSPH,” International Journal for Numerical Methods in Engineering, vol. 56, no. 10, pp. 1421–1444, 2003. View at Publisher · View at Google Scholar · View at Scopus
  39. D. S. Dugdale, “Yielding of steel sheets containing slits,” Journal of the Mechanics and Physics of Solids, vol. 8, no. 2, pp. 100–104, 1960. View at Scopus
  40. G. I. Barenblatt, “The mathematical theory of equilibrium cracks in brittle fracture,” Advances in Applied Mechanics, vol. 7, pp. 55–129, 1962. View at Publisher · View at Google Scholar · View at Scopus
  41. A. Hillerborg, M. Modéer, and P. E. Petersson, “Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements,” Cement and Concrete Research, vol. 6, no. 6, pp. 773–781, 1976. View at Scopus
  42. K. Keller, S. Weihe, T. Siegmund, and B. Kröplin, “Generalized Cohesive Zone Model: incorporating triaxiality dependent failure mechanisms,” Computational Materials Science, vol. 16, no. 1–4, pp. 267–274, 1999. View at Scopus
  43. F. Zhou, J. F. Molinari, and T. Shioya, “A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials,” Engineering Fracture Mechanics, vol. 72, no. 9, pp. 1383–1410, 2005. View at Publisher · View at Google Scholar · View at Scopus
  44. I. Carol and P. C. Prat, “A statically constrained microplane model for the smeared analysis of concrete cracking,” in Computer Aided Analysis and Design of Concrete Structures, N. Bicanic and H. Mang, Eds., pp. 919–930, Pinedidge Press, Swansea, UK, 1990.
  45. J. Cervenka, Discrete crack modeling in concrete structures [Ph.D. thesis], University of Colorado, 1994.
  46. G. T. Camacho and M. Ortiz, “Computational modelling of impact damage in brittle materials,” International Journal of Solids and Structures, vol. 33, no. 20–22, pp. 2899–2938, 1996. View at Scopus
  47. A. Pandolfi, P. Krysl, and M. Ortiz, “Finite element simulation of ring expansion and fragmentation: the capturing of length and time scales through cohesive models of fracture,” International Journal of Fracture, vol. 95, no. 1–4, pp. 279–297, 1999. View at Scopus
  48. X. Liu, S. Li, and N. Sheng, “A cohesive finite element for quasi-continua,” Computational Mechanics, vol. 42, no. 4, pp. 543–553, 2008. View at Publisher · View at Google Scholar · View at Scopus
  49. S. Li, X. Zeng, B. Ren, J. Qian, J. Zhang, and A. K. Jha, “An atomistic-based interphase zone model for crystalline solids,” Computer Methods in Applied Mechanics and Engineering, vol. 229–232, pp. 87–109, 2012.
  50. J. Qian and S. Li, “Application of multiscale cohesive zone model to simulate fracture in polycrystalline solids,” Journal of Engineering Materials and Technology, Transactions of the ASME, vol. 133, no. 1, Article ID 011010, 2011. View at Publisher · View at Google Scholar · View at Scopus
  51. X. Zeng and S. Li, “A multiscale cohesive zone model and simulations of fractures,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 9–12, pp. 547–556, 2010. View at Publisher · View at Google Scholar · View at Scopus
  52. X. Zeng and S. Li, “Application of a multiscale cohesive zone method to model composite materials,” International Journal of Multiscale Computational Engineering, vol. 10, pp. 391–405, 2012. View at Publisher · View at Google Scholar
  53. L. Liu and S. Li, “A finite temperature multiscale interphase finite element method and simulations of fracture,” ASME Journal of Engineering Materials and Technology, vol. 134, Article ID 03014, pp. 1–12, 2012.
  54. M. He and S. Li, “An embedded atom hyperelastic constitutive model and multiscale cohesive finite element method,” Computational Mechanics, vol. 49, no. 3, pp. 337–355, 2012. View at Publisher · View at Google Scholar
  55. M. Elices, G. V. Guinea, J. Gómez, and J. Planas, “The cohesive zone model: advantages, limitations and challenges,” Engineering Fracture Mechanics, vol. 69, no. 2, pp. 137–163, 2001. View at Publisher · View at Google Scholar · View at Scopus
  56. K. D. Papoulia, C. H. Sam, and S. A. Vavasis, “Time continuity in cohesive finite element modeling,” International Journal for Numerical Methods in Engineering, vol. 58, no. 5, pp. 679–701, 2003. View at Publisher · View at Google Scholar · View at Scopus
  57. Z. P. Bažant and B. H. Oh, “Crack band theory for fracture of concrete,” Matériaux et Constructions, vol. 16, no. 3, pp. 155–177, 1983. View at Publisher · View at Google Scholar · View at Scopus
  58. J. C. Simo, J. Oliver, and F. Armero, “An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids,” Computational Mechanics, vol. 12, no. 5, pp. 277–296, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  59. M. Jirásek and T. Zimmermann, “Analysis of rotating crack model,” Journal of Engineering Mechanics, vol. 124, no. 8, pp. 842–851, 1998. View at Scopus
  60. M. ásek and T. Zimmermann, “Rotating crack model with transition to scalar damage,” Journal of Engineering Mechanics, vol. 124, no. 3, pp. 277–284, 1998. View at Scopus
  61. T. Rabczuk, J. Akkermann, and J. Eibl, “A numerical model for reinforced concrete structures,” International Journal of Solids and Structures, vol. 42, no. 5-6, pp. 1327–1354, 2005. View at Publisher · View at Google Scholar · View at Scopus
  62. F. Ohmenhäuser, S. Weihe, and B. Kröplin, “Algorithmic implementation of a generalized cohesive crack model,” Computational Materials Science, vol. 16, no. 1–4, pp. 294–306, 1999. View at Scopus
  63. A. Carpinteri, B. Chiaia, and P. Cornetti, “A scale-invariant cohesive crack model for quasi-brittle materials,” Engineering Fracture Mechanics, vol. 69, no. 2, pp. 207–217, 2001. View at Publisher · View at Google Scholar · View at Scopus
  64. M. François and G. Royer-Carfagni, “Structured deformation of damaged continua with cohesive-frictional sliding rough fractures,” European Journal of Mechanics A, vol. 24, no. 4, pp. 644–660, 2005. View at Publisher · View at Google Scholar · View at Scopus
  65. R. de Borst, J. J. C. Remmers, and A. Needleman, “Mesh-independent discrete numerical representations of cohesive-zone models,” Engineering Fracture Mechanics, vol. 73, no. 2, pp. 160–177, 2006. View at Publisher · View at Google Scholar · View at Scopus
  66. G. R. Johnson and R. A. Stryk, “Eroding interface and improved tetrahedral element algorithms for high-velocity impact computations in three dimensions,” International Journal of Impact Engineering, vol. 5, no. 1–4, pp. 411–421, 1987. View at Scopus
  67. T. Belytschko and J. I. Lin, “A three-dimensional impact-penetration algorithm with erosion,” International Journal of Impact Engineering, vol. 5, no. 1–4, pp. 111–127, 1987. View at Scopus
  68. S. R. Beissel, G. R. Johnson, and C. H. Popelar, “An element-failure algorithm for dynamic crack propagation in general directions,” Engineering Fracture Mechanics, vol. 61, no. 3-4, pp. 407–425, 1998. View at Publisher · View at Google Scholar · View at Scopus
  69. R. Fan and J. Fish, “The rs-method for material failure simulations,” International Journal for Numerical Methods in Engineering, vol. 73, no. 11, pp. 1607–1623, 2008. View at Publisher · View at Google Scholar
  70. J. H. Song, H. Wang, and T. Belytschko, “A comparative study on finite element methods for dynamic fracture,” Computational Mechanics, vol. 42, no. 2, pp. 239–250, 2008. View at Publisher · View at Google Scholar · View at Scopus
  71. A. Pandolfi and M. Ortiz, “An eigenerosion approach to brittle fracture,” International Journal for Numerical Methods in Engineering, vol. 92, no. 8, pp. 694–714, 2012.
  72. B. Schmidt, F. Fraternali, and M. Ortiz, “Eigenfracture: an eigendeformation approach to variational fracture,” SIAM Multiscale Modeling and Simulation, vol. 7, no. 3, pp. 1237–1266, 2008. View at Publisher · View at Google Scholar
  73. T. Børvik, O. S. Hopperstad, and K. O. Pedersen, “Quasi-brittle fracture during structural impact of AA7075-T651 aluminium plates,” International Journal of Impact Engineering, vol. 37, no. 5, pp. 537–551, 2010. View at Publisher · View at Google Scholar · View at Scopus
  74. M. Negri, “A finite element approximation of the Griffith's model in fracture mechanics,” Numerische Mathematik, vol. 95, no. 4, pp. 653–687, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  75. X. P. Xu and A. Needleman, “Numerical simulations of fast crack growth in brittle solids,” Journal of the Mechanics and Physics of Solids, vol. 42, no. 9, pp. 1397–1434, 1994. View at Scopus
  76. X. P. Xu and A. Needleman, “Void nucleation by inclusion debonding in a crystal matrix,” Modelling and Simulation in Materials Science and Engineering, vol. 1, no. 2, pp. 111–132, 1993. View at Publisher · View at Google Scholar · View at Scopus
  77. M. Ortiz, Y. Leroy, and A. Needleman, “A finite element method for localized failure analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 61, no. 2, pp. 189–214, 1987. View at Scopus
  78. F. Cirak, M. Ortiz, and A. Pandolfi, “A cohesive approach to thin-shell fracture and fragmentation,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 21–24, pp. 2604–2618, 2005. View at Publisher · View at Google Scholar · View at Scopus
  79. A. Pandolfi, P. R. Guduru, M. Ortiz, and A. J. Rosakis, “Three dimensional cohesive-element analysis and experiments of dynamic fracture in C300 steel,” International Journal of Solids and Structures, vol. 37, no. 27, pp. 3733–3760, 2000. View at Scopus
  80. F. Zhou and J. F. Molinari, “Dynamic crack propagation with cohesive elements: a methodology to address mesh dependency,” International Journal for Numerical Methods in Engineering, vol. 59, no. 1, pp. 1–24, 2004. View at Publisher · View at Google Scholar · View at Scopus
  81. M. L. Falk, A. Needleman, and J. R. Rice, “A critical evaluation of cohesive zone models of dynamic fracture,” Journal of Physics, vol. 11, pp. Pr5-43–Pr5-50, 2001. View at Publisher · View at Google Scholar
  82. H. D. Espinosa, P. D. Zavattieri, and G. L. Emore, “Adaptive FEM computation of geometric and material nonlinearities with application to brittle failure,” Mechanics of Materials, vol. 29, no. 3-4, pp. 275–305, 1998. View at Scopus
  83. S. Knell, A numerical modeling approach for the transient response of solids at the mesoscale [Ph.D. thesis], Univerität der Bundeswehr München, 2011.
  84. H. D. Espinosa, P. D. Zavattieri, and S. K. Dwivedi, “A finite deformation continuum/discrete model for the description of fragmentation and damage in brittle materials,” Journal of the Mechanics and Physics of Solids, vol. 46, no. 10, pp. 1909–1942, 1998. View at Scopus
  85. S. Roshdy and R. S. Barsoum, “Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements,” International Journal For Numerical Methods in Engineering, vol. 11, no. 1, pp. 85–98, 1977. View at Publisher · View at Google Scholar
  86. R. S. Barsoum, “Application of quadratic isoparametric finite elements in linear fracture mechanics,” International Journal of Fracture, vol. 10, no. 4, pp. 603–605, 1974. View at Publisher · View at Google Scholar · View at Scopus
  87. R. S. Barsoum, “Further application of quadratic isoparametric finite elements to linear fracture mechanics of plate bending and general shells,” International Journal of Fracture, vol. 11, no. 1, pp. 167–169, 1975. View at Publisher · View at Google Scholar · View at Scopus
  88. R. S. Barsoum, “On the use of isoparametric finite elements in linear fracture mechanics,” International Journal for Numerical Methods in Engineering, vol. 10, no. 1, pp. 25–37, 1976. View at Scopus
  89. G. R. Liu, N. Nourbakhshnia, and Y. W. Zhang, “A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems,” Engineering Fracture Mechanics, vol. 78, no. 6, pp. 863–876, 2011. View at Publisher · View at Google Scholar · View at Scopus
  90. G. R. Liu, N. Nourbakhshnia, L. Chen, and Y. W. Zhang, “A novel general formulation for singular stress field using the ES-FEM method for the analysis of mixed-mode cracks,” International Journal of Computational Methods, vol. 7, no. 1, pp. 191–214, 2010. View at Publisher · View at Google Scholar
  91. L. Chen, G. R. Liu, Y. Jiang, K. Zeng, and J. Zhang, “A singular edge-based smoothed finite element method (ES-FEM) for crack analyses in anisotropic media,” Engineering Fracture Mechanics, vol. 78, no. 1, pp. 85–109, 2011. View at Publisher · View at Google Scholar · View at Scopus
  92. L. Chen, G. R. Liu, N. Nourbakhsh-Nia, and K. Zeng, “A singular edge-based smoothed finite element method (ES-FEM) for bimaterial interface cracks,” Computational Mechanics, vol. 45, no. 2-3, pp. 109–125, 2010. View at Publisher · View at Google Scholar · View at Scopus
  93. Y. Jiang, G. R. Liu, Y. W. Zhang, L. Chen, and T. E. Tay, “A singular ES-FEM for plastic fracture mechanics,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 45-46, pp. 2943–2955, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  94. H. Nguyen-Xuan, G. R. Liu, N. Nourbakhshnia, and L. Chen, “A novel singular es-fem for crack growth simulation,” Engineering Fracture Mechanics, vol. 84, pp. 41–66, 2012. View at Publisher · View at Google Scholar
  95. N. Nourbakhshnia and G. R. Liu, “A quasi-static crack growth simulation based on the singular ES-FEM,” International Journal for Numerical Methods in Engineering, vol. 88, no. 5, pp. 473–492, 2011. View at Publisher · View at Google Scholar · View at Scopus
  96. T. Belytschko, J. Fish, and B. E. Engelmann, “A finite element with embedded localization zones,” Computer Methods in Applied Mechanics and Engineering, vol. 70, no. 1, pp. 59–89, 1988. View at Scopus
  97. E. N. Dvorkin, A. M. Cuitino, and G. Gioia, “Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions,” International Journal for Numerical Methods in Engineering, vol. 30, no. 3, pp. 541–564, 1990. View at Scopus
  98. M. Jirásek, “Comparative study on finite elements with embedded discontinuities,” Computer Methods in Applied Mechanics and Engineering, vol. 188, no. 1, pp. 307–330, 2000. View at Publisher · View at Google Scholar · View at Scopus
  99. H. R. Lotfi and P. B. Shing, “Embedded representation of fracture in concrete with mixed finite elements,” International Journal for Numerical Methods in Engineering, vol. 38, no. 8, pp. 1307–1325, 1995. View at Scopus
  100. M. Klisinski, K. Runesson, and S. Sture, “Finite element with inner softening band,” Journal of Engineering Mechanics ASCE, vol. 117, pp. 575–587, 1991. View at Publisher · View at Google Scholar
  101. E. Samaniego, X. Oliver, and A. Huespe, Contributions to the continuum modelling of strong discontinuities in two-dimensional solids [Ph.D. thesis], International Center for Numerical Methods in Engineering, Barcelona, Spain, 2003, Monograph CIMNE No. 72.
  102. C. Linder and F. Armero, “Finite elements with embedded strong discontinuities for the modeling of failure in solids,” International Journal for Numerical Methods in Engineering, vol. 72, no. 12, pp. 1391–1433, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  103. J. Oliver, A. E. Huespe, M. D. G. Pulido, and E. Samaniego, “On the strong discontinuity approach in finite deformation settings,” International Journal for Numerical Methods in Engineering, vol. 56, no. 7, pp. 1051–1082, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  104. J. Oliver, “On the discrete constitutive models induced by strong discontinuity kinematics and continuum constitutive equations,” International Journal of Solids and Structures, vol. 37, no. 48–50, pp. 7207–7229, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  105. J. Oliver, M. Cervera, and O. Manzoli, “Strong discontinuities and continuum plasticity models: the strong discontinuity approach,” International Journal of Plasticity, vol. 15, no. 3, pp. 319–351, 1999. View at Publisher · View at Google Scholar · View at Scopus
  106. J. Oliver, “Modelling strong discontinuities in solid mechanics via strain softening constituitive equations, part 1: fundamentals. part 2: numerical simulation,” International Journal For Numerical Methods in Engineering, vol. 39, pp. 3575–3624, 1996.
  107. C. Linder and C. Miehe, “Effect of electric displacement saturation on the hysteretic behavior of ferroelectric ceramics and the initiation and propagation of cracks in piezoelectric ceramics,” Journal of the Mechanics and Physics of Solids, vol. 60, no. 5, pp. 882–903, 2012. View at Publisher · View at Google Scholar
  108. C. D. Foster, R. I. Borja, and R. A. Regueiro, “Embedded strong discontinuity finite elements for fractured geomaterials with variable friction,” International Journal for Numerical Methods in Engineering, vol. 72, no. 5, pp. 549–581, 2007. View at Publisher · View at Google Scholar · View at Scopus
  109. C. Linder, D. Rosato, and C. Miehe, “New finite elements with embedded strong discontinuities for the modeling of failure in electromechanical coupled solids,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 1–4, pp. 141–161, 2011. View at Publisher · View at Google Scholar
  110. F. Armero and K. Garikipati, “An analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids,” International Journal of Solids and Structures, vol. 33, no. 20–22, pp. 2863–2885, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  111. C. Linder and F. Armero, “Finite elements with embedded branching,” Finite Elements in Analysis and Design, vol. 45, no. 4, pp. 280–293, 2009. View at Publisher · View at Google Scholar
  112. F. Armero and C. Linder, “Numerical simulation of dynamic fracture using finite elements with embedded discontinuities,” International Journal of Fracture, vol. 160, no. 2, pp. 119–141, 2009. View at Publisher · View at Google Scholar · View at Scopus
  113. J. Oliver, A. E. Huespe, and P. J. Sánchez, “A comparative study on finite elements for capturing strong discontinuities: E-FEM vs X-FEM,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 37–40, pp. 4732–4752, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  114. C. Feist and G. Hofstetter, “Three-dimensional fracture simulations based on the SDA,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 31, no. 2, pp. 189–212, 2007. View at Publisher · View at Google Scholar · View at Scopus
  115. J. M. Sancho, J. Planas, A. M. Fathy, J. C. Gálvez, and D. A. Cendón, “Three-dimensional simulation of concrete fracture using embedded crack elements without enforcing crack path continuity,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 31, no. 2, pp. 173–187, 2007. View at Publisher · View at Google Scholar · View at Scopus
  116. J. Mosler and G. Meschke, “Embedded crack vs. smeared crack models: a comparison of elementwise discontinuous crack path approaches with emphasis on mesh bias,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 30–32, pp. 3351–3375, 2004. View at Publisher · View at Google Scholar · View at Scopus
  117. 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 Scopus
  118. N. Moës, J. Dolbow, and T. Belytschko, “A finite element method for crack growth without remeshing,” International Journal for Numerical Methods in Engineering, vol. 46, no. 1, pp. 131–150, 1999. View at Scopus
  119. J. M. Melenk and I. Babuška, “The partition of unity finite element method: basic theory and applications,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1–4, pp. 289–314, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  120. T. Strouboulis, K. Copps, and I. Babuška, “The generalized finite element method: an example of its implementation and illustration of its performance,” International Journal for Numerical Methods in Engineering, vol. 47, no. 8, pp. 1401–1417, 2000. View at Zentralblatt MATH
  121. T. Strouboulis, I. Babuška, and K. Copps, “The design and analysis of the generalized finite element method,” Computer Methods in Applied Mechanics and Engineering, vol. 181, no. 1–3, pp. 43–69, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  122. J. Chessa and T. Belytschko, “An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension,” International Journal for Numerical Methods in Engineering, vol. 58, no. 13, pp. 2041–2064, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  123. J. Chessa and T. Belytschko, “An extended finite element method for two-phase fluids,” Journal of Applied Mechanics, Transactions ASME, vol. 70, no. 1, pp. 10–17, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  124. A. Zilian and A. Legay, “The enriched space-time finite element method (EST) for simultaneous solution of fluid-structure interaction,” International Journal for Numerical Methods in Engineering, vol. 75, no. 3, pp. 305–334, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  125. U. M. Mayer, A. Gerstenberger, and W. A. Wall, “Interface handling for three-dimensional higher-order XFEM-computations in fluid-structure interaction,” International Journal for Numerical Methods in Engineering, vol. 79, no. 7, pp. 846–869, 2009. View at Publisher · View at Google Scholar · View at Scopus
  126. R. Duddu, S. Bordas, D. Chopp, and B. Moran, “A combined extended finite element and level set method for biofilm growth,” International Journal for Numerical Methods in Engineering, vol. 74, no. 5, pp. 848–870, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  127. D. Rabinovich, D. Givoli, and S. Vigdergauz, “XFEM-based crack detection scheme using a genetic algorithm,” International Journal for Numerical Methods in Engineering, vol. 71, no. 9, pp. 1051–1080, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  128. D. Rabinovich, D. Givoli, and S. Vigdergauz, “Crack identification by lsquoarrival timersquo using XFEM and a genetic algorithm,” International Journal for Numerical Methods in Engineering, vol. 77, no. 3, pp. 337–359, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  129. E. Béchet, M. Scherzer, and M. Kuna, “Application of the X-FEM to the fracture of piezoelectric materials,” International Journal for Numerical Methods in Engineering, vol. 77, no. 11, pp. 1535–1565, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  130. C. V. Verhoosel, J. J. C. Remmers, and M. A. Gutiérrez, “A partition of unity-based multiscale approach for modelling fracture in piezoelectric ceramics,” International Journal for Numerical Methods in Engineering, vol. 82, no. 8, pp. 966–994, 2010. View at Publisher · View at Google Scholar · View at Scopus
  131. M. Duflot, “The extended finite element method in thermoelastic fracture mechanics,” International Journal for Numerical Methods in Engineering, vol. 74, no. 5, pp. 827–847, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  132. P. M. A. Areias and T. Belytschko, “Two-scale shear band evolution by local partition of unity,” International Journal for Numerical Methods in Engineering, vol. 66, no. 5, pp. 878–910, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  133. C. A. Duarte, L. G. Reno, and A. Simone, “A high-order generalized FEM for through-the-thickness branched cracks,” International Journal for Numerical Methods in Engineering, vol. 72, no. 3, pp. 325–351, 2007. View at Publisher · View at Google Scholar · View at Scopus
  134. C. A. Duarte, O. N. Hamzeh, T. J. Liszka, and W. W. Tworzydlo, “A generalized finite element method for the simulation of three-dimensional dynamic crack propagation,” Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 15–17, pp. 2227–2262, 2001. View at Publisher · View at Google Scholar · View at Scopus
  135. C. A. Duarte and D.-J. Kim, “Analysis and applications of a generalized finite element method with global-local enrichment functions,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 6–8, pp. 487–504, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  136. J. P. Pereira, C. A. Duarte, X. Jiao, and D. Guoy, “Generalized finite element method enrichment functions for curved singularities in 3D fracture mechanics problems,” Computational Mechanics, vol. 44, no. 1, pp. 73–92, 2009. View at Publisher · View at Google Scholar · View at Scopus
  137. C. A. Duarte, D.-J. Kim, and I. Babuška, “A global-local approach for the construction of enrichment functions for the generalized FEM and its application to three-dimensional cracks,” in Advances in Meshfree Techniques, vol. 5, pp. 1–26, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar
  138. D. J. Kim, J. P. Pereira, and C. A. Duarte, “Analysis of three-dimensional fracture mechanics problems: a two-scale approach using coarse-generalized FEM meshes,” International Journal for Numerical Methods in Engineering, vol. 81, no. 3, pp. 335–365, 2010. View at Publisher · View at Google Scholar · View at Scopus
  139. D.-J. Kim, C. A. Duarte, and N. A. Sobh, “Parallel simulations of three-dimensional cracks using the generalized finite element method,” Computational Mechanics, vol. 47, no. 3, pp. 265–282, 2011. View at Publisher · View at Google Scholar
  140. A. Menk and S. P. A. Bordas, “A robust preconditioning technique for the extended finite element method,” International Journal for Numerical Methods in Engineering, vol. 85, no. 13, pp. 1609–1632, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  141. I. Babuška and U. Banerjee, “Stable generalized finite element method (SGFEM),” Computer Methods in Applied Mechanics and Engineering, vol. 201–204, pp. 91–111, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  142. G. Zi and T. Belytschko, “New crack-tip elements for XFEM and applications to cohesive cracks,” International Journal for Numerical Methods in Engineering, vol. 57, no. 15, pp. 2221–2240, 2003. View at Publisher · View at Google Scholar · View at Scopus
  143. G. Zi, J. H. Song, E. Budyn, S. H. Lee, and T. Belytschko, “A method for growing multiple cracks without remeshing and its application to fatigue crack growth,” Modelling and Simulation in Materials Science and Engineering, vol. 12, no. 5, pp. 901–915, 2004. View at Publisher · View at Google Scholar · View at Scopus
  144. E. Budyn, G. Zi, N. Moes, and T. Belytschko, “A method for multiple crack growth in brittle materials without remeshing,” International Journal for Numerical Methods in Engineering, vol. 61, no. 10, pp. 1741–1770, 2004. View at Publisher · View at Google Scholar · View at Scopus
  145. T. Belytschko, N. Moes, S. Usui, and C. Parimi, “Arbitrary discontinuities in finite elements,” International Journal For Numerical Methods in Engineering, vol. 50, no. 4, pp. 993–1013, 2001.
  146. C. Daux, N. Moës, J. Dolbow, N. Sukumar, and T. Belytschko, “Arbitrary branched and intersecting cracks with the extended finite element method,” International Journal for Numerical Methods in Engineering, vol. 48, no. 12, pp. 1741–1760, 2000. View at Scopus
  147. P. M. A. Areias and T. Belytschko, “Analysis of three-dimensional crack initiation and propagation using the extended finite element method,” International Journal for Numerical Methods in Engineering, vol. 63, no. 5, pp. 760–788, 2005. View at Publisher · View at Google Scholar · View at Scopus
  148. P. M. A. Areias, J. H. Song, and T. Belytschko, “Analysis of fracture in thin shells by overlapping paired elements,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 41–43, pp. 5343–5360, 2006. View at Publisher · View at Google Scholar · View at Scopus
  149. J. A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge, UK, 1999.
  150. S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” Journal of Computational Physics, vol. 79, no. 1, pp. 12–49, 1988. View at Scopus
  151. B. Prabel, A. Combescure, A. Gravouil, and S. Marie, “Level set X-FEM non-matching meshes: application to dynamic crack propagation in elastic-plastic media,” International Journal for Numerical Methods in Engineering, vol. 69, no. 8, pp. 1553–1569, 2007. View at Publisher · View at Google Scholar · View at Scopus
  152. 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
  153. G. Ventura, “On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method,” International Journal for Numerical Methods in Engineering, vol. 66, no. 5, pp. 761–795, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  154. 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
  155. 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
  156. 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
  157. K. W. Cheng and T. P. Fries, “Higher-order XFEM for curved strong and weak discontinuities,” International Journal for Numerical Methods in Engineering, vol. 82, no. 5, pp. 564–590, 2010. View at Publisher · View at Google Scholar · View at Scopus
  158. A. Nagarajan and S. Mukherjee, “A mapping method for numerical evaluation of two-dimensional integrals with 1/r singularity,” Computational Mechanics, vol. 12, no. 1-2, pp. 19–26, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  159. 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
  160. J. Chessa, H. Wang, and T. Belytschko, “On the construction of blending elements for local partition of unity enriched finite elements,” International Journal for Numerical Methods in Engineering, vol. 57, no. 7, pp. 1015–1038, 2003. View at Publisher · View at Google Scholar · View at Scopus
  161. F. Stazi, E. Budyn, J. Chessa, and T. Belytschko, “XFEM for fracture mechanics with quadratic elements,” Computational Mechanics, vol. 31, pp. 38–48, 2003. View at Publisher · View at Google Scholar
  162. T.-P. Fries, “A corrected XFEM approximation without problems in blending elements,” International Journal for Numerical Methods in Engineering, vol. 75, no. 5, pp. 503–532, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  163. J. E. Tarancón, A. Vercher, E. Giner, and F. J. Fuenmayor, “Enhanced blending elements for XFEM applied to linear elastic fracture mechanics,” International Journal for Numerical Methods in Engineering, vol. 77, no. 1, pp. 126–148, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  164. J. Bellec and J. E. Dolbow, “A note on enrichment functions for modelling crack nucleation,” Communications in Numerical Methods in Engineering, vol. 19, no. 12, pp. 921–932, 2003. View at Publisher · View at Google Scholar · View at Scopus
  165. B. L. Karihaloo and Q. Z. Xiao, “Modelling of stationary and growing cracks in FE framework without remeshing: a state-of-the-art review,” Computers and Structures, vol. 81, no. 3, pp. 119–129, 2003. View at Publisher · View at Google Scholar · View at Scopus
  166. T.-P. Fries and T. Belytschko, “The extended/generalized finite element method: an overview of the method and its applications,” International Journal for Numerical Methods in Engineering, vol. 84, no. 3, pp. 253–304, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  167. T. Belytschko, R. Gracie, and G. Ventura, “A review of extended/generalized finite element methods for material modeling,” Modelling and Simulation in Materials Science and Engineering, vol. 17, no. 4, Article ID 043001, 2009. View at Publisher · View at Google Scholar · View at Scopus
  168. S. Mohammadi, Extended Finite Element Method for Fracture Analysis of Structures, Blackwell Publishing, Oxford, UK, 2008.
  169. 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
  170. 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
  171. 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
  172. 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 Zentralblatt MATH
  173. H. Talebi, C. Samaniego, E. Samaniego, and T. Rabczuk, “On the numerical stability and masslumping schemes for explicit enriched meshfree methods,” International Journal for Numerical Methods in Engineering, vol. 89, pp. 1009–1027, 2012. View at Publisher · View at Google Scholar
  174. 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. 242–256, 2012. View at Publisher · View at Google Scholar
  175. 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
  176. 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
  177. J. Mergheim and P. Steinmann, “A geometrically nonlinear FE approach for the simulation of strong and weak discontinuities,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 37–40, pp. 5037–5052, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  178. D. Organ, M. Fleming, T. Terry, and T. Belytschko, “Continuous meshless approximations for nonconvex bodies by diffraction and transparency,” Computational Mechanics, vol. 18, no. 3, pp. 225–235, 1996. View at Scopus
  179. T. Belytschko, D. Organ, and M. Tabbara, “Numerical simulations of mixed mode dynamic fracture in concrete using element-free Galerkin methods,” in Proceedings of the International Conference on Environmental Systems (ICES '95), 1995.
  180. T. Belytschko, Y. Y. Lu, L. Gu, and M. Tabbara, “Element-free galerkin methods for static and dynamic fracture,” International Journal of Solids and Structures, vol. 32, no. 17-18, pp. 2547–2570, 1995. View at Publisher · View at Google Scholar · View at Scopus
  181. T. Belytschko, Y. Y. Lu, and L. Gu, “Crack propagation by element-free Galerkin methods,” Engineering Fracture Mechanics, vol. 51, no. 2, pp. 295–315, 1995. View at Scopus
  182. J. J. C. Remmers, R. De Borst, and A. Needleman, “A cohesive segments method for the simulation of crack growth,” Computational Mechanics, vol. 31, no. 1-2, pp. 69–77, 2003. View at Publisher · View at Google Scholar · View at Scopus
  183. J. J. C. Remmers, R. de Borst, and A. Needleman, “The simulation of dynamic crack propagation using the cohesive segments method,” Journal of the Mechanics and Physics of Solids, vol. 56, no. 1, pp. 70–92, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  184. J.-H. Song and T. Belytschko, “Cracking node method for dynamic fracture with finite elements,” International Journal for Numerical Methods in Engineering, vol. 77, no. 3, pp. 360–385, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  185. Y. You, J. S. Chen, and H. Lu, “Filters, reproducing kernel, and adaptive meshfree method,” Computational Mechanics, vol. 31, no. 3-4, pp. 316–326, 2003. View at Scopus
  186. T. Rabczuk and T. Belytschko, “Adaptivity for structured meshfree particle methods in 2D and 3D,” International Journal for Numerical Methods in Engineering, vol. 63, no. 11, pp. 1559–1582, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  187. T. Rabczuk and E. Samaniego, “Discontinuous modelling of shear bands using adaptive meshfree methods,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 6–8, pp. 641–658, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  188. T.-P. Fries, A. Byfut, A. Alizada, K. W. Cheng, and A. Schröder, “Hanging nodes and XFEM,” International Journal for Numerical Methods in Engineering, vol. 86, no. 4-5, pp. 404–430, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  189. R. A. Gingold and J. J. Monaghan, “Smoothed particle hydrodynamics:theory and applications to non-spherical stars,” Monthly Notices of the Royal Astronomical Society, vol. 181, pp. 375–389, 1977.
  190. T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” International Journal for Numerical Methods in Engineering, vol. 37, no. 2, pp. 229–256, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  191. W. K. Liu, S. Jun, and Y. F. Zhang, “Reproducing kernel particle methods,” International Journal for Numerical Methods in Fluids, vol. 20, no. 8-9, pp. 1081–1106, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  192. G. R. Liu and Y. T. Gu, “A point interpolation method for two-dimensional solids,” International Journal For Numerical Methods in Engineering, vol. 50, pp. 937–951, 2001.
  193. S. N. Atluri, The Meshless Local Petrov-Galerkin (MLPG) Method, Tech Science Press, 2002.
  194. E. Oñate, S. Idelsohn, O. C. Zienkiewicz, and R. L. Taylor, “A finite point method in computational mechanics. Applications to convective transport and fluid flow,” International Journal for Numerical Methods in Engineering, vol. 39, no. 22, pp. 3839–3866, 1996. View at Zentralblatt MATH
  195. G. R. Liu and M. B. Liu, Smoothed Particle Hydrodynamics: A Meshfree Particle Method, 2003.
  196. G. R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, Boca Raton, Fla, USA, 2002.
  197. G. R. Liu, An Introduction to Meshfree Methods and Their Programming, Springer, 2006.
  198. S. Li and W. K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Reviews, vol. 55, no. 1, pp. 1–34, 2002. View at Publisher · View at Google Scholar · View at Scopus
  199. S. Li and W. K. Liu, Meshfree Particle Methods, Springer, Berlin, Germany, 2004.
  200. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Meshless methods: an overview and recent developments,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1–4, pp. 3–47, 1996. View at Scopus
  201. V. P. Nguyen, T. Rabczuk, S. Bordas, and M. Duflot, “Meshless methods: a review and computer implementation aspects,” Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 763–813, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  202. A. Huerta, T. Belytschko, S. Fernandez-Mendez, and T. Rabczuk, Encyclopedia of Computational Mechanics, chapter Meshfree Methods, John Wiley and Sons, 2004.
  203. L. D. Libersky, P. W. Randles, T. C. Carney, and D. L. Dickinson, “Recent improvements in SPH modeling of hypervelocity impact,” International Journal of Impact Engineering, vol. 20, no. 6–10, pp. 525–532, 1997. View at Scopus
  204. T. Rabczuk and J. Eibl, “Modelling dynamic failure of concrete with meshfree methods,” International Journal of Impact Engineering, vol. 32, no. 11, pp. 1878–1897, 2006. View at Publisher · View at Google Scholar · View at Scopus
  205. G. A. Dilts, “Moving least-squares particle hydrodynamics. II. Conservation and boundaries,” International Journal for Numerical Methods in Engineering, vol. 48, no. 10, pp. 1503–1524, 2000. View at Zentralblatt MATH
  206. A. Haque and G. A. Dilts, “Three-dimensional boundary detection for particle methods,” Journal of Computational Physics, vol. 226, no. 2, pp. 1710–1730, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  207. T. Rabczuk, T. Belytschko, and S. P. Xiao, “Stable particle methods based on Lagrangian kernels,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 12–14, pp. 1035–1063, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  208. B. Maurel and A. Combescure, “An SPH shell formulation for plasticity and fracture analysis in explicit dynamics,” International Journal for Numerical Methods in Engineering, vol. 76, no. 7, pp. 949–971, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  209. A. Combescure, B. Maurel, and S. Potapov, “Modelling dynamic fracture of thin shells filled with fluid: a fully SPH model,” Mecanique et Industries, vol. 9, no. 2, pp. 167–174, 2008. View at Publisher · View at Google Scholar · View at Scopus
  210. B. Maurel, S. Potapov, J. Fabis, and A. Combescure, “Full SPH fluid-shell interaction for leakage simulation in explicit dynamics,” International Journal for Numerical Methods in Engineering, vol. 80, no. 2, pp. 210–234, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  211. S. Potapov, B. Maurel, A. Combescure, and J. Fabis, “Modeling accidental-type fluid-structure interaction problems with the SPH method,” Computers and Structures, vol. 87, no. 11-12, pp. 721–734, 2009. View at Publisher · View at Google Scholar · View at Scopus
  212. D. Sulsky, Z. Chen, and H. L. Schreyer, “A particle method for history-dependent materials,” Computer Methods in Applied Mechanics and Engineering, vol. 118, no. 1-2, pp. 179–196, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  213. S. Ma, X. Zhang, and X. M. Qiu, “Comparison study of MPM and SPH in modeling hypervelocity impact problems,” International Journal of Impact Engineering, vol. 36, no. 2, pp. 272–282, 2009. View at Publisher · View at Google Scholar · View at Scopus
  214. P. Krysl and T. Belytschko, “Element-free Galerkin method: convergence of the continuous and discontinuous shape functions,” Computer Methods in Applied Mechanics and Engineering, vol. 148, no. 3-4, pp. 257–277, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  215. T. Rabczuk and T. Belytschko, “An adaptive continuum/discrete crack approach for meshfree particle methods,” Latin American Journal of Solids and Structures, vol. 1, pp. 141–166, 2003.
  216. N. Sukumar, B. Moran, T. Black, and T. Belytschko, “An element-free Galerkin method for three-dimensional fracture mechanics,” Computational Mechanics, vol. 20, no. 1-2, pp. 170–175, 1997. View at Scopus
  217. M. Duflot, “A meshless method with enriched weight functions for three-dimensional crack propagation,” International Journal for Numerical Methods in Engineering, vol. 65, no. 12, pp. 1970–2006, 2006. View at Publisher · View at Google Scholar · View at Scopus
  218. T. G. Terry, Fatigue crack propagation modeling using the element free galerkin method [M.S. thesis], Northwestern University, 1994.
  219. C. A. Duarte and J. T. Oden, “An h-p adaptive method using clouds,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1–4, pp. 237–262, 1996. View at Scopus
  220. T. Belytschko and M. Fleming, “Smoothing, enrichment and contact in the element-free Galerkin method,” Computers & Structures, vol. 71, no. 2, pp. 173–195, 1999. View at Publisher · View at Google Scholar
  221. M. Fleming, Y. A. Chu, B. Moran, and T. Belytschko, “Enriched element-free Galerkin methods for crack tip fields,” International Journal for Numerical Methods in Engineering, vol. 40, no. 8, pp. 1483–1504, 1997.
  222. T. P. Fries and T. Belytschko, “The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns,” International Journal for Numerical Methods in Engineering, vol. 68, no. 13, pp. 1358–1385, 2006. View at Publisher · View at Google Scholar · View at Scopus
  223. M. Duflot and H. Nguyen-Dang, “A meshless method with enriched weight functions for fatigue crack growth,” International Journal for Numerical Methods in Engineering, vol. 59, no. 14, pp. 1945–1961, 2004. View at Scopus
  224. A. Zamani, R. Gracie, and M. R. Eslami, “Higher order tip enrichment of extended finite element method in thermoelasticity,” Computational Mechanics, vol. 46, no. 6, pp. 851–866, 2010. View at Publisher · View at Google Scholar
  225. A. Zamani, R. Gracie, and M. R. Eslami, “Cohesive and non-cohesive fracture by higher-order enrichment of xfem,” International Journal For Numerical Methods in Engineering, vol. 90, pp. 452–483, 2012. View at Publisher · View at Google Scholar
  226. G. Ventura, J. X. Xu, and T. Belytschko, “A vector level set method and new discontinuity approximations for crack growth by EFG,” International Journal for Numerical Methods in Engineering, vol. 54, no. 6, pp. 923–944, 2002. View at Publisher · View at Google Scholar · View at Scopus
  227. T. Rabczuk and G. Zi, “A meshfree method based on the local partition of unity for cohesive cracks,” Computational Mechanics, vol. 39, no. 6, pp. 743–760, 2007. View at Publisher · View at Google Scholar · View at Scopus
  228. T. Rabczuk, S. Bordas, and G. Zi, “A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics,” Computational Mechanics, vol. 40, no. 3, pp. 473–495, 2007. View at Publisher · View at Google Scholar · View at Scopus
  229. S. Bordas, T. Rabczuk, and G. Zi, “Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment,” Engineering Fracture Mechanics, vol. 75, no. 5, pp. 943–960, 2008. View at Publisher · View at Google Scholar · View at Scopus
  230. G. Zi, T. Rabczuk, and W. Wall, “Extended meshfree methods without branch enrichment for cohesive cracks,” Computational Mechanics, vol. 40, no. 2, pp. 367–382, 2007. View at Publisher · View at Google Scholar · View at Scopus
  231. T. Rabczuk and T. Belytschko, “Cracking particles: a simplified meshfree method for arbitrary evolving cracks,” International Journal for Numerical Methods in Engineering, vol. 61, no. 13, pp. 2316–2343, 2004. View at Publisher · View at Google Scholar · View at Scopus
  232. 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
  233. 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
  234. T. Rabczuk, P. M. A. Areias, and T. Belytschko, “A simplified mesh-free method for shear bands with cohesive surfaces,” International Journal for Numerical Methods in Engineering, vol. 69, no. 5, pp. 993–1021, 2007. View at Publisher · View at Google Scholar · View at Scopus
  235. T. Rabczuk, G. Zi, S. Bordas, and H. Nguyen-Xuan, “A simple and robust three-dimensional cracking-particle method without enrichment,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 37–40, pp. 2437–2455, 2010. View at Publisher · View at Google Scholar · View at Scopus
  236. L. Chen and Y. Zhang, “Dynamic fracture analysis using discrete cohesive crack method,” International Journal for Numerical Methods in Biomedical Engineering, vol. 26, no. 11, pp. 1493–1502, 2010. View at Publisher · View at Google Scholar · View at Scopus
  237. Y. Y. Zhang, “Meshless modelling of crack growth with discrete rotating crack segments,” International Journal of Mechanics and Materials in Design, vol. 4, no. 1, pp. 71–77, 2008. View at Publisher · View at Google Scholar · View at Scopus
  238. H. X. Wang and S. X. Wang, “Analysis of dynamic fracture with cohesive crack segment method,” CMES. Computer Modeling in Engineering & Sciences, vol. 35, no. 3, pp. 253–274, 2008. View at Zentralblatt MATH
  239. F. Caleyron, A. Combescure, V. Faucher, and S. Potapov, “Dynamic simulation of damagefracture transition in smoothed particles hydrodynamics shells,” International Journal For Numerical Methods in Engineering, vol. 90, pp. 707–738, 2012. View at Publisher · View at Google Scholar
  240. M. H. Aliabadi, “Boundary element formulations in fracture mechanics,” Applied Mechanics Reviews, vol. 50, no. 2, pp. 83–96, 1997. View at Scopus
  241. Y. Mi and M. H. Aliabadi, “Three-dimensional crack growth simulation using BEM,” Computers and Structures, vol. 52, no. 5, pp. 871–878, 1994. View at Scopus
  242. E. Pan, “A general boundary element analysis of 2-D linear elastic fracture mechanics,” International Journal of Fracture, vol. 88, no. 1, pp. 41–59, 1997. View at Publisher · View at Google Scholar · View at Scopus
  243. E. Pan, “A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids,” Engineering Analysis with Boundary Elements, vol. 23, no. 1, pp. 67–76, 1999. View at Scopus
  244. Y. Ryoji and C. Sang-Bong, “Efficient boundary element analysis of stress intensity factors for interface cracks in dissimilar materials,” Engineering Fracture Mechanics, vol. 34, no. 1, pp. 179–188, 1989. View at Scopus
  245. E. Pan and F. G. Yuan, “Boundary element analysis of three-dimensional cracks in anisotropic solids,” International Journal For Numerical Methods in Engineering, vol. 48, pp. 211–237, 2000.
  246. M. Doblare, F. Espiga, L. Gracia, and M. Alcantud, “Study of crack propagation in orthotropic materials by using the boundary element method,” Engineering Fracture Mechanics, vol. 37, no. 5, pp. 953–967, 1990. View at Scopus
  247. G. K. Sfantos and M. H. Aliabadi, “Multi-scale boundary element modelling of material degradation and fracture,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 7, pp. 1310–1329, 2007. View at Publisher · View at Google Scholar · View at Scopus
  248. E. Schnack, “Hybrid bem model,” International Journal for Numerical Methods in Engineering, vol. 24, no. 5, pp. 1015–1025, 1987. View at Scopus
  249. J. Sládek, V. Sládek, and Z. P. Bažant, “Non-local boundary integral formulation for softening damage,” International Journal for Numerical Methods in Engineering, vol. 57, no. 1, pp. 103–116, 2003. View at Publisher · View at Google Scholar · View at Scopus
  250. X. W. Gao, C. Zhang, J. Sladek, and V. Sladek, “Fracture analysis of functionally graded materials by a BEM,” Composites Science and Technology, vol. 68, no. 5, pp. 1209–1215, 2008. View at Publisher · View at Google Scholar · View at Scopus
  251. F. García-Sánchez, R. Rojas-Díaz, A. Sáez, and C. Zhang, “Fracture of magnetoelectroelastic composite materials using boundary element method (BEM),” Theoretical and Applied Fracture Mechanics, vol. 47, no. 3, pp. 192–204, 2007. View at Publisher · View at Google Scholar · View at Scopus
  252. T. A. Cruse, “BIE fracture mechanics analysis: 25 years of developments,” Computational Mechanics, vol. 18, no. 1, pp. 1–11, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  253. R. N. Simpson, S. P. A. Bordas, J. Trevelyan, and T. Rabczuk, “A two-dimensional isogeometric boundary element method for elastostatic analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 209/212, pp. 87–100, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  254. A. P. Cisilino and M. H. Aliabadi, “Three-dimensional boundary element analysis of fatigue crack growth in linear and non-linear fracture problems,” Engineering Fracture Mechanics, vol. 63, no. 6, pp. 713–733, 1999. View at Scopus
  255. R. Simpson and J. Trevelyan, “Evaluation of j 1 and j 2 integrals for curved cracks using an enriched boundary element method,” Engineering Fracture Mechanics, vol. 78, pp. 623–637, 2011. View at Publisher · View at Google Scholar
  256. R. Simpson and J. Trevelyan, “A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 1–4, pp. 1–10, 2011. View at Publisher · View at Google Scholar
  257. G. E. Bird, J. Trevelyan, and C. E. Augarde, “A coupled BEM/scaled boundary FEM formulation for accurate computations in linear elastic fracture mechanics,” Engineering Analysis with Boundary Elements, vol. 34, no. 6, pp. 599–610, 2010. View at Publisher · View at Google Scholar · View at Scopus
  258. T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, “Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 39–41, pp. 4135–4195, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  259. E. De Luycker, D. J. Benson, T. Belytschko, Y. Bazilevs, and M. C. Hsu, “X-FEM in isogeometric analysis for linear fracture mechanics,” International Journal for Numerical Methods in Engineering, vol. 87, no. 6, pp. 541–565, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  260. S. S. Ghorashi, N. Valizadeh, and S. Mohammadi, “Extended isogeometric analysis for simulation of stationary and propagating cracks,” International Journal For Numerical Methods in Engineering, vol. 89, pp. 1069–1101, 2012. View at Publisher · View at Google Scholar
  261. A. Tambat and G. Subbarayan, “Isogeometric enriched field approximations,” Computer Methods in Applied Mechanics and Engineering, vol. 245–246, pp. 1–21, 2012. View at Publisher · View at Google Scholar
  262. G. A. Francfort and J.-J. Marigo, “Revisiting brittle fracture as an energy minimization problem,” Journal of the Mechanics and Physics of Solids, vol. 46, no. 8, pp. 1319–1342, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  263. B. Bourdin, G. A. Francfort, and J.-J. Marigo, “Numerical experiments in revisited brittle fracture,” Journal of the Mechanics and Physics of Solids, vol. 48, no. 4, pp. 797–826, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  264. A. Karma, D. A. Kessler, and H. Levine, “Phase-field model of mode III dynamic fracture,” Physical Review Letters, vol. 87, no. 4, Article ID 045501, 4 pages, 2001. View at Scopus
  265. V. Hakim and A. Karma, “Laws of crack motion and phase-field models of fracture,” Journal of the Mechanics and Physics of Solids, vol. 57, no. 2, pp. 342–368, 2009. View at Publisher · View at Google Scholar · View at Scopus
  266. C. Miehe, M. Hofacker, and F. Welschinger, “A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 45–48, pp. 2765–2778, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  267. C. Miehe, F. Welschinger, and M. Hofacker, “Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations,” International Journal for Numerical Methods in Engineering, vol. 83, no. 10, pp. 1273–1311, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  268. C. Kuhn and R. Müller, “A new finite element technique for a phase field model of brittle fracture,” Journal of Theoretical and Applied Mechanics, vol. 49, pp. 1115–1133, 2011.
  269. C. Kuhn and R. Müller, “Exponential finite element shape functions for a phase field model of brittle fracture,” in Proceedings of the 11th International Conference on Computational Plasticity (COMPLAS '11), pp. 478–489, 2011.
  270. C. Kuhn and R. Müller, “A continuum phase field model for fracture,” Engineering Fracture Mechanics, vol. 77, no. 18, pp. 3625–3634, 2010. View at Publisher · View at Google Scholar · View at Scopus
  271. T. Rabczuk, S. Bordas, and G. Zi, “On three-dimensional modelling of crack growth using partition of unity methods,” Computers and Structures, vol. 88, no. 23-24, pp. 1391–1411, 2010. View at Publisher · View at Google Scholar · View at Scopus
  272. P. Jäger, P. Steinmann, and E. Kuhl, “On local tracking algorithms for the simulation of three-dimensional discontinuities,” Computational Mechanics, vol. 42, no. 3, pp. 395–406, 2008. View at Publisher · View at Google Scholar · View at Scopus
  273. T. C. Gasser and G. A. Holzapfel, “Modeling 3D crack propagation in unreinforced concrete using PUFEM,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 25-26, pp. 2859–2896, 2005. View at Publisher · View at Google Scholar · View at Scopus
  274. T. C. Gasser and G. A. Holzapfel, “3D Crack propagation in unreinforced concrete. A two-step algorithm for tracking 3D crack paths,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 37–40, pp. 5198–5219, 2006. View at Publisher · View at Google Scholar · View at Scopus
  275. P. Krysl and T. Belytschko, “The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks,” International Journal for Numerical Methods in Engineering, vol. 44, no. 6, pp. 767–800, 1999. View at Scopus
  276. J. Oliver, A. E. Huespe, E. Samaniego, and E. W. V. Chaves, “Continuum approach to the numerical simulation of material failure in concrete,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 28, no. 7-8, pp. 609–632, 2004. View at Publisher · View at Google Scholar · View at Scopus
  277. P. Jäger, P. Steinmann, and E. Kuhl, “Towards the treatment of boundary conditions for global crack path tracking in three-dimensional brittle fracture,” Computational Mechanics, vol. 45, no. 1, pp. 91–107, 2009. View at Publisher · View at Google Scholar
  278. G. Ventura, E. Budyn, and T. Belytschko, “Vector level sets for description of propagating cracks in finite elements,” International Journal for Numerical Methods in Engineering, vol. 58, no. 10, pp. 1571–1592, 2003. View at Publisher · View at Google Scholar · View at Scopus
  279. M. Duflot, “A study of the representation of cracks with level sets,” International Journal for Numerical Methods in Engineering, vol. 70, no. 11, pp. 1261–1302, 2007. View at Publisher · View at Google Scholar · View at Scopus
  280. D. L. Chopp, “Computing minimal surfaces via level set curvature flow,” Journal of Computational Physics, vol. 106, no. 1, pp. 77–91, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  281. D. L. Chopp and J. A. Sethian, “Flow under curvature: singularity formation, minimal surfaces, and geodesics,” Experimental Mathematics, vol. 2, no. 4, pp. 235–255, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  282. M. Stolarska, D. L. Chopp, N. Mos, and T. Belytschko, “Modelling crack growth by level sets in the extended finite element method,” International Journal for Numerical Methods in Engineering, vol. 51, no. 8, pp. 943–960, 2001. View at Publisher · View at Google Scholar · View at Scopus
  283. T.-P. Fries and M. Baydoun, “Crack propagation with the extended finite element method and a hybrid explicit-implicit crack description,” International Journal for Numerical Methods in Engineering, vol. 89, no. 12, pp. 1527–1558, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  284. X. Zhuang, C. Augarde, and S. Bordas, “Accurate fracture modelling using meshless methods, the visibility criterion and level sets: formulation and 2D modelling,” International Journal for Numerical Methods in Engineering, vol. 86, no. 2, pp. 249–268, 2011. View at Publisher · View at Google Scholar · View at Scopus
  285. X. Zhuang, C. Augarde, and K. M. Mathisen, “Fracture modeling using meshless methods and level sets in 3d: framework and modeling,” International Journal for Numerical Methods in Engineering, vol. 92, no. 11, pp. 969–998, 2012. View at Publisher · View at Google Scholar
  286. J. Mosler, “A variationally consistent approach for crack propagation based on configurational forces,” in IUTAM Symposium on Progress in the Theory and Numerics of Configurational Mechanics, vol. 17 of IUTAM Bookseries, pp. 239–247, 2009. View at Publisher · View at Google Scholar
  287. M. E. Gurtin and P. Podio-Guidugli, “Configurational forces and the basic laws for crack propagation,” Journal of the Mechanics and Physics of Solids, vol. 44, no. 6, pp. 905–927, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  288. M. E. Gurtin and P. Podio-Guidugli, “Configurational forces and a constitutive theory for crack propagation that allows for kinking and curving,” Journal of the Mechanics and Physics of Solids, vol. 46, no. 8, pp. 1343–1378, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  289. C. Miehe and E. Gürses, “A robust algorithm for configurational-force-driven brittle crack propagation with r-adaptive mesh alignment,” International Journal for Numerical Methods in Engineering, vol. 72, no. 2, pp. 127–155, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  290. G. C. Sih, “Strain-energy-density factor applied to mixed mode crack problems,” International Journal of Fracture, vol. 10, no. 3, pp. 305–321, 1974. View at Publisher · View at Google Scholar · View at Scopus
  291. C. H. Wu, “Fracture under combined loads by maximum energy release rate criterion,” Journal of Applied Mechanics, Transactions ASME, vol. 45, no. 3, pp. 553–558, 1978. View at Scopus
  292. R. V. Goldstein and R. L. Salganik, “Brittle fracture of solids with arbitrary cracks,” International Journal of Fracture, vol. 10, no. 4, pp. 507–523, 1974. View at Publisher · View at Google Scholar · View at Scopus
  293. B. Shen and O. Stephansson, “Modification of the G-criterion for crack propagation subjected to compression,” Engineering Fracture Mechanics, vol. 47, no. 2, pp. 177–189, 1994. View at Scopus
  294. G. N. Wells and L. J. Sluys, “A new method for modelling cohesive cracks using finite elements,” International Journal for Numerical Methods in Engineering, vol. 50, no. 12, pp. 2667–2682, 2001. View at Publisher · View at Google Scholar · View at Scopus
  295. S. Mariani and U. Perego, “Extended finite element method for quasi-brittle fracture,” International Journal for Numerical Methods in Engineering, vol. 58, no. 1, pp. 103–126, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  296. J. E. Marsden and T. J. R. Hughes, Mathematical Foundations of Elasticity, Dover Publications, New York, NY, USA, 1983.
  297. R. W. Ogden, Non-Linear Elastic Deformations, Halsted Press, New York, NY, USA, 1984.
  298. J. Oliver, D. L. Linero, A. E. Huespe, and O. L. Manzoli, “Two-dimensional modeling of material failure in reinforced concrete by means of a continuum strong discontinuity approach,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 5, pp. 332–348, 2008. View at Publisher · View at Google Scholar · View at Scopus
  299. T. Belytschko, S. Loehnert, and J.-H. Song, “Multiscale aggregating discontinuities: a method for circumventing loss of material stability,” International Journal for Numerical Methods in Engineering, vol. 73, no. 6, pp. 869–894, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  300. G. Meschke and P. Dumstorff, “Energy-based modeling of cohesive and cohesionless cracks via X-FEM,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 21–24, pp. 2338–2357, 2007. View at Publisher · View at Google Scholar · View at Scopus
  301. P. Dummerstorf and G. Meschke, “Crack propagation criteria in the framework of X-FEM-based structural analyses,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 31, no. 2, pp. 239–259, 2007. View at Publisher · View at Google Scholar · View at Scopus
  302. T. Belytschko, H. Chen, J. Xu, and G. Zi, “Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment,” International Journal for Numerical Methods in Engineering, vol. 58, no. 12, pp. 1873–1905, 2003. View at Publisher · View at Google Scholar · View at Scopus
  303. P. A. Cundall and R. D. Hart, “Development of generalized 2-d and 3-d distinct element programs for modeling jointed rock,” Misc. Paper SL-85-1, US Army Corps of Engineers, 1985.
  304. P. A. Cundall and O. D. L. Strack, “A discrete numerical model for granular assemblies,” Geotechnique, vol. 29, no. 1, pp. 47–65, 1979. View at Scopus
  305. G. H. Shi and R. E. Goodman, “Two dimensional discontinuous deformation analysis,” International Journal for Numerical & Analytical Methods in Geomechanics, vol. 9, no. 6, pp. 541–556, 1985. View at Scopus
  306. G.-H. Shi and R. E. Goodman, “Generalization of two-dimensional discontinuous deformation analysis for forward modelling,” International Journal for Numerical & Analytical Methods in Geomechanics, vol. 13, no. 4, pp. 359–380, 1989. View at Scopus
  307. P. A. Cundall and H. Konietzky, “Pfc-ein neues werkzeug für numerische modellierungen,” Bautechnik, vol. 73, no. 8, 1996.
  308. R. W. Macek and S. A. Silling, “Peridynamics via finite element analysis,” Finite Elements in Analysis and Design, vol. 43, no. 15, pp. 1169–1178, 2007. View at Publisher · View at Google Scholar
  309. W. K. Liu, D. Qian, S. Gonella, S. Li, W. Chen, and S. Chirputkar, “Multiscale methods for mechanical science of complex materials: bridging from quantum to stochastic multiresolution continuum,” International Journal for Numerical Methods in Engineering, vol. 83, no. 8-9, pp. 1039–1080, 2010. View at Publisher · View at Google Scholar · View at Scopus
  310. A. Nouy, A. Clément, F. Schoefs, and N. Moës, “An extended stochastic finite element method for solving stochastic partial differential equations on random domains,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 51-52, pp. 4663–4682, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  311. J. Grasa, J. A. Bea, and M. Doblaré, “A probabilistic extended finite element approach: application to the prediction of bone crack propagation,” Key Engineering Materials, vol. 348-349, pp. 77–80, 2007. View at Scopus
  312. J. Grasa, J. A. Bea, J. F. Rodríguez, and M. Doblaré, “The perturbation method and the extended finite element method. An application to fracture mechanics problems,” Fatigue and Fracture of Engineering Materials and Structures, vol. 29, no. 8, pp. 581–587, 2006. View at Publisher · View at Google Scholar · View at Scopus
  313. I. Arias, S. Serebrinsky, and M. Ortiz, “A cohesive model of fatigue of ferroelectric materials under electro-mechanical cyclic loading,” in Smart Structures and Materials 2004, Active Materials: Behaviour and Mechanics, Proceedings of SPIE, pp. 371–378, San Diego, Calif, USA, March 2004. View at Publisher · View at Google Scholar · View at Scopus
  314. L. J. Lucas, H. Owhadi, and M. Ortiz, “Rigorous verification, validation, uncertainty quantification and certification through concentration-of-measure inequalities,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 51-52, pp. 4591–4609, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  315. T. Belytschko and J. H. Song, “Coarse-graining of multiscale crack propagation,” International Journal for Numerical Methods in Engineering, vol. 81, no. 5, pp. 537–563, 2010. View at Publisher · View at Google Scholar · View at Scopus
  316. M. F. Horstemeyer, “Multiscale modeling: a review,” Practical Aspects of Computational Chemistry, pp. 87–135, 2010. View at Publisher · View at Google Scholar
  317. J. Fish and Z. Yuan, “Multiscale enrichment based on partition of unity,” International Journal for Numerical Methods in Engineering, vol. 62, no. 10, pp. 1341–1359, 2005. View at Publisher · View at Google Scholar · View at Scopus
  318. R. Gracie and T. Belytschko, “Concurrently coupled atomistic and XFEM models for dislocations and cracks,” International Journal for Numerical Methods in Engineering, vol. 78, no. 3, pp. 354–378, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  319. V. Kouznetsova, Computational homogenization for the multi-scale analysis of multi-phase materials [Ph.D. thesis], Netherlands Institute for Metals Research, Amsterdam, The Netherlands, 2002.
  320. S. Nemat-Nasser and M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, Elsevier, Amsterdam, The Netherlands, 1993.
  321. R. Gracie, J. Oswald, and T. Belytschko, “On a new extended finite element method for dislocations: core enrichment and nonlinear formulation,” Journal of the Mechanics and Physics of Solids, vol. 56, no. 1, pp. 200–214, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  322. R. Gracie, G. Ventura, and T. Belytschko, “A new fast finite element method for dislocations based on interior discontinuities,” International Journal for Numerical Methods in Engineering, vol. 69, no. 2, pp. 423–441, 2007. View at Publisher · View at Google Scholar · View at Scopus
  323. T. Belytschko and R. Gracie, “On XFEM applications to dislocations and interfaces,” International Journal of Plasticity, vol. 23, no. 10-11, pp. 1721–1738, 2007. View at Publisher · View at Google Scholar · View at Scopus
  324. M. Xu, R. Gracie, and T. Belytschko, “Multiscale modeling with extended bridging domain method,” in Bridging the Scales in Science and Engineering, J. Fish, Ed., Oxford University Press, 2002.
  325. F. Feyel and J. L. Chaboche, “FE 2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials,” Computer Methods in Applied Mechanics and Engineering, vol. 183, no. 3-4, pp. 309–330, 2000. View at Scopus
  326. V. Kouznetsova, M. G. D. Geers, and W. A. M. Brekelmans, “Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme,” International Journal for Numerical Methods in Engineering, vol. 54, no. 8, pp. 1235–1260, 2002. View at Publisher · View at Google Scholar · View at Scopus
  327. V. P. Nguyen, O. Lloberas-Valls, M. Stroeven, and L. J. Sluys, “Homogenization-based multiscale crack modelling: from micro-diffusive damage to macro-cracks,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 9–12, pp. 1220–1236, 2011. View at Publisher · View at Google Scholar · View at Scopus
  328. C. V. Verhoosel, J. J. C. Remmers, M. A. Gutiérrez, and R. de Borst, “Computational homogenization for adhesive and cohesive failure in quasi-brittle solids,” International Journal for Numerical Methods in Engineering, vol. 83, no. 8-9, pp. 1155–1179, 2010. View at Publisher · View at Google Scholar · View at Scopus
  329. E. B. Tadmor, M. Ortiz, and R. Phillips, “Quasicontinuum analysis of defects in solids,” Philosophical Magazine A, vol. 73, no. 6, pp. 1529–1563, 1996. View at Scopus
  330. R. E. Miller and E. B. Tadmor, “The Quasicontinuum method: overview, applications and current directions,” Journal of Computer-Aided Materials Design, vol. 9, no. 3, pp. 203–239, 2002. View at Publisher · View at Google Scholar · View at Scopus
  331. H. B. Dhia, “The arlequin method: a partition of models for concurrent multiscale analyses,” in Proceedings of the Challenges in Computational Mechanics Workshop, 2006.
  332. F. F. Abraham, J. Q. Broughton, N. Bernstein, and E. Kaxiras, “Spanning the length scales in dynamic simulation,” Computational Physics, vol. 12, pp. 538–546, 1998.
  333. S. Loehnert and T. Belytschko, “A multiscale projection method for macro/microcrack simulations,” International Journal for Numerical Methods in Engineering, vol. 71, no. 12, pp. 1466–1482, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  334. P. A. Guidault, O. Allix, L. Champaney, and C. Cornuault, “A multiscale extended finite element method for crack propagation,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 5, pp. 381–399, 2008. View at Publisher · View at Google Scholar · View at Scopus
  335. P. Aubertin, J. Réthoré, and R. de Borst, “Energy conservation of atomistic/continuum coupling,” International Journal for Numerical Methods in Engineering, vol. 78, no. 11, pp. 1365–1386, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH