`ISRN Applied MathematicsVolume 2013 (2013), Article ID 849231, 38 pageshttp://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.

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.
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.
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.
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.
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.
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.
9. J. Planas and M. Elices, “Nonlinear fracture of cohesive materials,” International Journal of Fracture, vol. 51, no. 2, pp. 139–157, 1991.
10. A. Carpinteri, “Post-peak and post-bifurcation analysis of cohesive crack propagation,” Engineering Fracture Mechanics, vol. 32, no. 2, pp. 265–278, 1989.
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.
12. A. Carpinteri, “Notch sensitivity in fracture testing of aggregative materials,” Engineering Fracture Mechanics, vol. 16, no. 4, pp. 467–481, 1982.
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.
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.
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.
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.
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.
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.
19. D. Krajcinovic and S. Mastilovic, “Some fundamental issues of damage mechanics,” Mechanics of Materials, vol. 21, no. 3, pp. 217–230, 1995.
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.
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.
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.
25. M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, Germany, 1997.
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.
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.
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.
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.
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.
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.
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.
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.
36. Z. P. Bažant, “Why continuum damage is nonlocal: micromechanics arguments,” Journal of Engineering Mechanics, vol. 117, no. 5, pp. 1070–1087, 1991.
37. G. Etse and K. Willam, “Failure analysis of elastoviscoplastic material models,” Journal of Engineering Mechanics, vol. 125, no. 1, pp. 60–68, 1999.
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.
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.
40. G. I. Barenblatt, “The mathematical theory of equilibrium cracks in brittle fracture,” Advances in Applied Mechanics, vol. 7, pp. 55–129, 1962.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
59. M. Jirásek and T. Zimmermann, “Analysis of rotating crack model,” Journal of Engineering Mechanics, vol. 124, no. 8, pp. 842–851, 1998.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
240. M. H. Aliabadi, “Boundary element formulations in fracture mechanics,” Applied Mechanics Reviews, vol. 50, no. 2, pp. 83–96, 1997.
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.
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.
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.
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.
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.
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.
248. E. Schnack, “Hybrid bem model,” International Journal for Numerical Methods in Engineering, vol. 24, no. 5, pp. 1015–1025, 1987.
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.
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.
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.
252. T. A. Cruse, “BIE fracture mechanics analysis: 25 years of developments,” Computational Mechanics, vol. 18, no. 1, pp. 1–11, 1996.
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.
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.
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.
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.
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.
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.
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.
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.
261. A. Tambat and G. Subbarayan, “Isogeometric enriched field approximations,” Computer Methods in Applied Mechanics and Engineering, vol. 245–246, pp. 1–21, 2012.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
280. D. L. Chopp, “Computing minimal surfaces via level set curvature flow,” Journal of Computational Physics, vol. 106, no. 1, pp. 77–91, 1993.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
316. M. F. Horstemeyer, “Multiscale modeling: a review,” Practical Aspects of Computational Chemistry, pp. 87–135, 2010.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.