×

Computational methods for fracture in brittle and quasi-brittle solids: state-of-the-art review and future perspectives. (English) Zbl 1298.74002

Summary: An overview of computational methods to model fracture in brittle and quasi-brittle materials is given. The overview focuses on continuum models for fracture. First, numerical difficulties related to modelling fracture for quasi-brittle materials will be discussed. Different techniques to eliminate or circumvent those difficulties will be described subsequently. In that context, regularization techniques such as nonlocal models, gradient enhanced models, viscous models, cohesive zone models, and smeared crack models will be discussed. The main focus of this paper will be on computational methods for discrete fracture (discrete cracks). Element erosion technques, inter-element separation methods, the embedded finite element method (EFEM), the extended finite element method (XFEM), meshfree methods (MMs), boundary elements (BEMs), isogeometric analysis, and the variational approach to fracture will be reviewed elucidating advantages and drawbacks of each approach. As tracking the crack path is of major concern in computational methods that preserve crack path continuity, one section will discuss different crack tracking techniques. Finally, cracking criteria will be reviewed before the paper ends with future research perspectives.

MSC:

74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74-04 Software, source code, etc. for problems pertaining to mechanics of deformable solids
74R05 Brittle damage
74R10 Brittle fracture

Software:

XFEM; Mfree2D
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Li, S.; Liu, W. K.; Rosakis, A. J.; Belytschko, T.; Hao, W., Mesh-free Galerkin simulations of dynamic shear band propagation and failure mode transition, International Journal of Solids and Structures, 39, 5, 1213-1240 (2002) · Zbl 1090.74698 · doi:10.1016/S0020-7683(01)00188-3
[2] Li, S.; Simonson, B. C., Meshfree simulation of ductile crack propagation, International Journal of Computational Engineering Science, 6, 1-25 (2005)
[3] Simonsen, B. C.; Li, S., Mesh-free simulation of ductile fracture, International Journal for Numerical Methods in Engineering, 60, 8, 1425-1450 (2004) · Zbl 1060.74673 · doi:10.1002/nme.1009
[4] Simkins, D. C.; Li, S., Meshfree simulations of thermo-mechanical ductile fracture, Computational Mechanics, 38, 3, 235-249 (2006) · Zbl 1162.74052 · doi:10.1007/s00466-005-0744-8
[5] Ren, B.; Li, S., Meshfree simulations of plugging failures in high-speed impacts, Computers and Structures, 88, 15-16, 909-923 (2010) · doi:10.1016/j.compstruc.2010.05.003
[6] Ren, B.; Qian, J.; Zeng, X.; Jha, A. K.; Xiao, S.; Li, S., Recent developments on thermo-mechanical simulations of ductile failure by meshfree method, Computer Modeling in Engineering and Sciences, 71, 3, 253-277 (2011)
[7] Ren, B.; Li, S., Modeling and simulation of large-scale ductile fracture in plates and shells, International Journal of Solids and Structures, 49, 18, 2373-2393 (2012)
[8] Gdoutos, E. E., Fracture Mechanics: An Introduction, 123 (2005), Kluwer Academic · Zbl 1094.74001
[9] Planas, J.; Elices, M., Nonlinear fracture of cohesive materials, International Journal of Fracture, 51, 2, 139-157 (1991) · doi:10.1007/BF00033975
[10] Carpinteri, A., Post-peak and post-bifurcation analysis of cohesive crack propagation, Engineering Fracture Mechanics, 32, 2, 265-278 (1989)
[11] Moës, N.; Belytschko, T., Extended finite element method for cohesive crack growth, Engineering Fracture Mechanics, 69, 7, 813-833 (2002) · doi:10.1016/S0013-7944(01)00128-X
[12] Carpinteri, A., Notch sensitivity in fracture testing of aggregative materials, Engineering Fracture Mechanics, 16, 4, 467-481 (1982)
[13] Bažant, Z. P.; Pijaudier-Cabot, G., Nonlocal continuum damage, localization instabilities and convergence, Journal of Applied Mechanics, Transactions ASME, 55, 2, 287-293 (1988) · Zbl 0663.73075
[14] Bažant, Z. P.; Jirásek, M., Nonlocal integral formulations of plasticity and damage: survey of progress, Journal of Engineering Mechanics, 128, 11, 1119-1149 (2002) · doi:10.1061/(ASCE)0733-9399(2002)128:11(1119)
[15] De Borst, R.; Pamin, J.; Geers, M. G. D., On coupled gradient-dependent plasticity and damage theories with a view to localization analysis, European Journal of Mechanics A, 18, 6, 939-962 (1999) · Zbl 0968.74007 · doi:10.1016/S0997-7538(99)00114-X
[16] de Borst, R.; Gutiérrez, M. A.; Wells, G. N.; Remmers, J. J. C.; Askes, H., Cohesive-zone models, higher-order continuum theories and reliability methods for computational failure analysis, International Journal for Numerical Methods in Engineering, 60, 1, 289-315 (2004) · Zbl 1060.74620 · doi:10.1002/nme.963
[17] De Borst, R., Fracture in quasi-brittle materials: a review of continuum damage-based approaches, Engineering Fracture Mechanics, 69, 2, 95-112 (2001) · doi:10.1016/S0013-7944(01)00082-0
[18] Fish, J.; Yu, Q.; Shek, K., Computational damage mechanics for composite materials based on mathematical homogenization, International Journal for Numerical Methods in Engineering, 45, 11, 1657-1679 (1999) · Zbl 0949.74057
[19] Krajcinovic, D.; Mastilovic, S., Some fundamental issues of damage mechanics, Mechanics of Materials, 21, 3, 217-230 (1995)
[20] Krajcinovic, D., Damage Mechanics. Damage Mechanics, North Holland Series in Applied Mathematics and Mechanics (1996), Elsevier · Zbl 1111.74491
[21] Lemaitre, J.; Lippmann, H., A Course on Damage Mechanics, 2 (1996), Berlin, Germany: Springer, Berlin, Germany
[22] Peerlings, R. H. J.; De Borst, R.; Brekelmans, W. A. M.; De Vree, J. H. P., Gradient enhanced damage for quasi-brittle materials, International Journal for Numerical Methods in Engineering, 39, 19, 3391-3403 (1996) · Zbl 0882.73057
[23] Shao, J. F.; Rudnicki, J. W., Microcrack-based continuous damage model for brittle geomaterials, Mechanics of Materials, 32, 10, 607-619 (2000) · doi:10.1016/S0167-6636(00)00024-7
[24] Simo, J. C.; Ju, J. W., Strain- and stress-based continuum damage models-I. Formulation, International Journal of Solids and Structures, 23, 7, 821-840 (1987) · Zbl 0634.73106
[25] Šilhavý, M., The Mechanics and Thermodynamics of Continuous Media (1997), Berlin, Germany: Springer, Berlin, Germany · Zbl 0870.73004
[26] Kachanov, L. M., Time of the rupture process under creep conditions, Izvestiya Akademii Nauk SSSR Otdelenie Tekniches, 8, 26-31 (1958)
[27] Hill, R., Acceleration waves in solids, Journal of the Mechanics and Physics of Solids, 10, 1-16 (1962) · Zbl 0111.37701 · doi:10.1016/0022-5096(62)90024-8
[28] Loret, B.; Prevost, J. H., Dynamic strain localization in elasto-(visco-)plastic solids, Part 1. General formulation and one-dimensional examples, Computer Methods in Applied Mechanics and Engineering, 83, 3, 247-273 (1990) · Zbl 0717.73030
[29] Prevost, J. H.; Loret, B., Dynamic strain localization in elasto-(visco-)plastic solids, part 2. plane strain examples, Computer Methods in Applied Mechanics and Engineering, 83, 3, 275-294 (1990) · Zbl 0717.73031
[30] Rudnicki, J. W.; Rice, J. R., Conditions for the localization of deformation in pressure-sensitive dilatant materials, Journal of the Mechanics and Physics of Solids, 23, 6, 371-394 (1975)
[31] Belytschko, T.; Liu, W. K.; Moran, B., Nonlinear Finite Elements for Continua and Structures (2000), Chichester, UK: John Wiley & Sons, Chichester, UK · Zbl 0959.74001
[32] Bažant, Z. P.; Belytschko, T. B., Wave propagation in a strain-softening bar: exact solution, Journal of Engineering Mechanics, 111, 3, 381-389 (1985)
[33] De Borst, R.; Muehlhaus, H. B., Gradient-dependent plasticity: formulation and algorithmic aspects, International Journal for Numerical Methods in Engineering, 35, 3, 521-539 (1992) · Zbl 0768.73019
[34] Fleck, N. A.; Hutchinson, J. W., A phenomenological theory for strain gradient effects in plasticity, Journal of the Mechanics and Physics of Solids, 41, 12, 1825-1857 (1993) · Zbl 0791.73029
[35] Peerlings, R. H. J.; de Borst, R.; Brekelmans, W. A. M.; Geers, M. G. D., Localisation issues in local and nonlocal continuum approaches to fracture, European Journal of Mechanics A, 21, 2, 175-189 (2002) · Zbl 1041.74006 · doi:10.1016/S0997-7538(02)01211-1
[36] Bažant, Z. P., Why continuum damage is nonlocal: micromechanics arguments, Journal of Engineering Mechanics, 117, 5, 1070-1087 (1991) · doi:10.1061/(ASCE)0733-9399(1991)117:5(1070)
[37] Etse, G.; Willam, K., Failure analysis of elastoviscoplastic material models, Journal of Engineering Mechanics, 125, 1, 60-68 (1999)
[38] Rabczuk, T.; Eibl, J., Simulation of high velocity concrete fragmentation using SPH/MLSPH, International Journal for Numerical Methods in Engineering, 56, 10, 1421-1444 (2003) · Zbl 1106.74428 · doi:10.1002/nme.617
[39] Dugdale, D. S., Yielding of steel sheets containing slits, Journal of the Mechanics and Physics of Solids, 8, 2, 100-104 (1960)
[40] Barenblatt, G. I., The mathematical theory of equilibrium cracks in brittle fracture, Advances in Applied Mechanics, 7, 55-129 (1962) · doi:10.1016/S0065-2156(08)70121-2
[41] Hillerborg, A.; Modéer, M.; Petersson, P. E., Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement and Concrete Research, 6, 6, 773-781 (1976)
[42] Keller, K.; Weihe, S.; Siegmund, T.; Kröplin, B., Generalized Cohesive Zone Model: incorporating triaxiality dependent failure mechanisms, Computational Materials Science, 16, 1-4, 267-274 (1999)
[43] Zhou, F.; Molinari, J. F.; Shioya, T., A rate-dependent cohesive model for simulating dynamic crack propagation in brittle materials, Engineering Fracture Mechanics, 72, 9, 1383-1410 (2005) · doi:10.1016/j.engfracmech.2004.10.011
[44] Carol, I.; Prat, P. C.; Bicanic, N.; Mang, H., A statically constrained microplane model for the smeared analysis of concrete cracking, Computer Aided Analysis and Design of Concrete Structures, 919-930 (1990), Swansea, UK: Pinedidge Press, Swansea, UK
[45] Cervenka, J., Discrete crack modeling in concrete structures [Ph.D. thesis] (1994), University of Colorado
[46] Camacho, G. T.; Ortiz, M., Computational modelling of impact damage in brittle materials, International Journal of Solids and Structures, 33, 20-22, 2899-2938 (1996) · Zbl 0929.74101
[47] Pandolfi, A.; Krysl, P.; Ortiz, M., Finite element simulation of ring expansion and fragmentation: the capturing of length and time scales through cohesive models of fracture, International Journal of Fracture, 95, 1-4, 279-297 (1999)
[48] Liu, X.; Li, S.; Sheng, N., A cohesive finite element for quasi-continua, Computational Mechanics, 42, 4, 543-553 (2008) · Zbl 1421.74097 · doi:10.1007/s00466-007-0222-6
[49] Li, S.; Zeng, X.; Ren, B.; Qian, J.; Zhang, J.; Jha, A. K., An atomistic-based interphase zone model for crystalline solids, Computer Methods in Applied Mechanics and Engineering, 229-232, 87-109 (2012) · Zbl 1253.74009
[50] Qian, J.; Li, S., Application of multiscale cohesive zone model to simulate fracture in polycrystalline solids, Journal of Engineering Materials and Technology, Transactions of the ASME, 133, 1 (2011) · doi:10.1115/1.4002647
[51] Zeng, X.; Li, S., A multiscale cohesive zone model and simulations of fractures, Computer Methods in Applied Mechanics and Engineering, 199, 9-12, 547-556 (2010) · Zbl 1227.74054 · doi:10.1016/j.cma.2009.10.008
[52] Zeng, X.; Li, S., Application of a multiscale cohesive zone method to model composite materials, International Journal of Multiscale Computational Engineering, 10, 391-405 (2012) · doi:10.1615/IntJMultCompEng.2012002926
[53] Liu, L.; Li, S., A finite temperature multiscale interphase finite element method and simulations of fracture, ASME Journal of Engineering Materials and Technology, 134, 1-12 (2012)
[54] He, M.; Li, S., An embedded atom hyperelastic constitutive model and multiscale cohesive finite element method, Computational Mechanics, 49, 3, 337-355 (2012) · Zbl 1355.74072 · doi:10.1007/s00466-011-0643-0
[55] Elices, M.; Guinea, G. V.; Gómez, J.; Planas, J., The cohesive zone model: advantages, limitations and challenges, Engineering Fracture Mechanics, 69, 2, 137-163 (2001) · doi:10.1016/S0013-7944(01)00083-2
[56] Papoulia, K. D.; Sam, C. H.; Vavasis, S. A., Time continuity in cohesive finite element modeling, International Journal for Numerical Methods in Engineering, 58, 5, 679-701 (2003) · Zbl 1032.74676 · doi:10.1002/nme.778
[57] Bažant, Z. P.; Oh, B. H., Crack band theory for fracture of concrete, Matériaux et Constructions, 16, 3, 155-177 (1983) · doi:10.1007/BF02486267
[58] Simo, J. C.; Oliver, J.; Armero, F., An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids, Computational Mechanics, 12, 5, 277-296 (1993) · Zbl 0783.73024 · doi:10.1007/BF00372173
[59] Jirásek, M.; Zimmermann, T., Analysis of rotating crack model, Journal of Engineering Mechanics, 124, 8, 842-851 (1998)
[60] ásek, M.; Zimmermann, T., Rotating crack model with transition to scalar damage, Journal of Engineering Mechanics, 124, 3, 277-284 (1998)
[61] Rabczuk, T.; Akkermann, J.; Eibl, J., A numerical model for reinforced concrete structures, International Journal of Solids and Structures, 42, 5-6, 1327-1354 (2005) · Zbl 1120.74790 · doi:10.1016/j.ijsolstr.2004.07.019
[62] Ohmenhäuser, F.; Weihe, S.; Kröplin, B., Algorithmic implementation of a generalized cohesive crack model, Computational Materials Science, 16, 1-4, 294-306 (1999)
[63] Carpinteri, A.; Chiaia, B.; Cornetti, P., A scale-invariant cohesive crack model for quasi-brittle materials, Engineering Fracture Mechanics, 69, 2, 207-217 (2001) · doi:10.1016/S0013-7944(01)00085-6
[64] François, M.; Royer-Carfagni, G., Structured deformation of damaged continua with cohesive-frictional sliding rough fractures, European Journal of Mechanics A, 24, 4, 644-660 (2005) · Zbl 1074.74051 · doi:10.1016/j.euromechsol.2004.12.005
[65] de Borst, R.; Remmers, J. J. C.; Needleman, A., Mesh-independent discrete numerical representations of cohesive-zone models, Engineering Fracture Mechanics, 73, 2, 160-177 (2006) · doi:10.1016/j.engfracmech.2005.05.007
[66] Johnson, G. R.; Stryk, R. A., Eroding interface and improved tetrahedral element algorithms for high-velocity impact computations in three dimensions, International Journal of Impact Engineering, 5, 1-4, 411-421 (1987)
[67] Belytschko, T.; Lin, J. I., A three-dimensional impact-penetration algorithm with erosion, International Journal of Impact Engineering, 5, 1-4, 111-127 (1987)
[68] Beissel, S. R.; Johnson, G. R.; Popelar, C. H., An element-failure algorithm for dynamic crack propagation in general directions, Engineering Fracture Mechanics, 61, 3-4, 407-425 (1998) · doi:10.1016/S0013-7944(98)00072-1
[69] Fan, R.; Fish, J., The \(r s\)-method for material failure simulations, International Journal for Numerical Methods in Engineering, 73, 11, 1607-1623 (2008) · Zbl 1159.74039 · doi:10.1002/nme.2134
[70] Song, J. H.; Wang, H.; Belytschko, T., A comparative study on finite element methods for dynamic fracture, Computational Mechanics, 42, 2, 239-250 (2008) · Zbl 1160.74048 · doi:10.1007/s00466-007-0210-x
[71] Pandolfi, A.; Ortiz, M., An eigenerosion approach to brittle fracture, International Journal for Numerical Methods in Engineering, 92, 8, 694-714 (2012) · Zbl 1352.74299
[72] Schmidt, B.; Fraternali, F.; Ortiz, M., Eigenfracture: an eigendeformation approach to variational fracture, SIAM Multiscale Modeling and Simulation, 7, 3, 1237-1266 (2008) · Zbl 1173.74040 · doi:10.1137/080712568
[73] Børvik, T.; Hopperstad, O. S.; Pedersen, K. O., Quasi-brittle fracture during structural impact of AA7075-T651 aluminium plates, International Journal of Impact Engineering, 37, 5, 537-551 (2010) · doi:10.1016/j.ijimpeng.2009.11.001
[74] Negri, M., A finite element approximation of the Griffith’s model in fracture mechanics, Numerische Mathematik, 95, 4, 653-687 (2003) · Zbl 1068.74080 · doi:10.1007/s00211-003-0456-y
[75] Xu, X. P.; Needleman, A., Numerical simulations of fast crack growth in brittle solids, Journal of the Mechanics and Physics of Solids, 42, 9, 1397-1434 (1994) · Zbl 0825.73579
[76] Xu, X. P.; Needleman, A., Void nucleation by inclusion debonding in a crystal matrix, Modelling and Simulation in Materials Science and Engineering, 1, 2, 111-132 (1993) · doi:10.1088/0965-0393/1/2/001
[77] Ortiz, M.; Leroy, Y.; Needleman, A., A finite element method for localized failure analysis, Computer Methods in Applied Mechanics and Engineering, 61, 2, 189-214 (1987) · Zbl 0597.73105
[78] Cirak, F.; Ortiz, M.; Pandolfi, A., A cohesive approach to thin-shell fracture and fragmentation, Computer Methods in Applied Mechanics and Engineering, 194, 21-24, 2604-2618 (2005) · Zbl 1082.74052 · doi:10.1016/j.cma.2004.07.048
[79] Pandolfi, A.; Guduru, P. R.; Ortiz, M.; Rosakis, A. J., Three dimensional cohesive-element analysis and experiments of dynamic fracture in C300 steel, International Journal of Solids and Structures, 37, 27, 3733-3760 (2000)
[80] Zhou, F.; Molinari, J. F., Dynamic crack propagation with cohesive elements: a methodology to address mesh dependency, International Journal for Numerical Methods in Engineering, 59, 1, 1-24 (2004) · Zbl 1047.74074 · doi:10.1002/nme.857
[81] Falk, M. L.; Needleman, A.; Rice, J. R., A critical evaluation of cohesive zone models of dynamic fracture, Journal of Physics, 11, Pr5-43-Pr5-50 (2001) · doi:10.1051/jp4:2001506
[82] Espinosa, H. D.; Zavattieri, P. D.; Emore, G. L., Adaptive FEM computation of geometric and material nonlinearities with application to brittle failure, Mechanics of Materials, 29, 3-4, 275-305 (1998)
[83] Knell, S., A numerical modeling approach for the transient response of solids at the mesoscale [Ph.D. thesis] (2011), Univerität der Bundeswehr München
[84] Espinosa, H. D.; Zavattieri, P. D.; Dwivedi, S. K., A finite deformation continuum/discrete model for the description of fragmentation and damage in brittle materials, Journal of the Mechanics and Physics of Solids, 46, 10, 1909-1942 (1998) · Zbl 1056.74510
[85] Roshdy, S.; Barsoum, R. S., Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements, International Journal For Numerical Methods in Engineering, 11, 1, 85-98 (1977) · Zbl 0348.73030 · doi:10.1002/nme.1620110109
[86] Barsoum, R. S., Application of quadratic isoparametric finite elements in linear fracture mechanics, International Journal of Fracture, 10, 4, 603-605 (1974) · doi:10.1007/BF00155266
[87] Barsoum, R. S., Further application of quadratic isoparametric finite elements to linear fracture mechanics of plate bending and general shells, International Journal of Fracture, 11, 1, 167-169 (1975) · doi:10.1007/BF00034724
[88] Barsoum, R. S., On the use of isoparametric finite elements in linear fracture mechanics, International Journal for Numerical Methods in Engineering, 10, 1, 25-37 (1976) · Zbl 0321.73067
[89] Liu, G. R.; Nourbakhshnia, N.; Zhang, Y. W., A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems, Engineering Fracture Mechanics, 78, 6, 863-876 (2011) · doi:10.1016/j.engfracmech.2009.11.004
[90] Liu, G. R.; Nourbakhshnia, N.; Chen, L.; Zhang, Y. W., 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, 7, 1, 191-214 (2010) · Zbl 1267.74112 · doi:10.1142/S0219876210002131
[91] Chen, L.; Liu, G. R.; Jiang, Y.; Zeng, K.; Zhang, J., A singular edge-based smoothed finite element method (ES-FEM) for crack analyses in anisotropic media, Engineering Fracture Mechanics, 78, 1, 85-109 (2011) · doi:10.1016/j.engfracmech.2010.09.018
[92] Chen, L.; Liu, G. R.; Nourbakhsh-Nia, N.; Zeng, K., A singular edge-based smoothed finite element method (ES-FEM) for bimaterial interface cracks, Computational Mechanics, 45, 2-3, 109-125 (2010) · Zbl 1398.74316 · doi:10.1007/s00466-009-0422-3
[93] Jiang, Y.; Liu, G. R.; Zhang, Y. W.; Chen, L.; Tay, T. E., A singular ES-FEM for plastic fracture mechanics, Computer Methods in Applied Mechanics and Engineering, 200, 45-46, 2943-2955 (2011) · Zbl 1230.74183 · doi:10.1016/j.cma.2011.06.001
[94] Nguyen-Xuan, H.; Liu, G. R.; Nourbakhshnia, N.; Chen, L., A novel singular es-fem for crack growth simulation, Engineering Fracture Mechanics, 84, 41-66 (2012) · doi:10.1016/j.engfracmech.2012.01.001
[95] Nourbakhshnia, N.; Liu, G. R., A quasi-static crack growth simulation based on the singular ES-FEM, International Journal for Numerical Methods in Engineering, 88, 5, 473-492 (2011) · Zbl 1242.74144 · doi:10.1002/nme.3186
[96] Belytschko, T.; Fish, J.; Engelmann, B. E., A finite element with embedded localization zones, Computer Methods in Applied Mechanics and Engineering, 70, 1, 59-89 (1988) · Zbl 0653.73032
[97] Dvorkin, E. N.; Cuitino, A. M.; Gioia, G., Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions, International Journal for Numerical Methods in Engineering, 30, 3, 541-564 (1990) · Zbl 0729.73209
[98] Jirásek, M., Comparative study on finite elements with embedded discontinuities, Computer Methods in Applied Mechanics and Engineering, 188, 1, 307-330 (2000) · Zbl 1166.74427 · doi:10.1016/S0045-7825(99)00154-1
[99] Lotfi, H. R.; Shing, P. B., Embedded representation of fracture in concrete with mixed finite elements, International Journal for Numerical Methods in Engineering, 38, 8, 1307-1325 (1995) · Zbl 0824.73070
[100] Klisinski, M.; Runesson, K.; Sture, S., Finite element with inner softening band, Journal of Engineering Mechanics ASCE, 117, 575-587 (1991) · doi:10.1061/(ASCE)0733-9399(1991)117:3(575)
[101] Samaniego, E.; Oliver, X.; Huespe, A., Contributions to the continuum modelling of strong discontinuities in two-dimensional solids [Ph.D. thesis] (2003), Barcelona, Spain: International Center for Numerical Methods in Engineering, Barcelona, Spain
[102] Linder, C.; Armero, F., Finite elements with embedded strong discontinuities for the modeling of failure in solids, International Journal for Numerical Methods in Engineering, 72, 12, 1391-1433 (2007) · Zbl 1194.74431 · doi:10.1002/nme.2042
[103] Oliver, J.; Huespe, A. E.; Pulido, M. D. G.; Samaniego, E., On the strong discontinuity approach in finite deformation settings, International Journal for Numerical Methods in Engineering, 56, 7, 1051-1082 (2003) · Zbl 1031.74010 · doi:10.1002/nme.607
[104] Oliver, J., On the discrete constitutive models induced by strong discontinuity kinematics and continuum constitutive equations, International Journal of Solids and Structures, 37, 48-50, 7207-7229 (2000) · Zbl 0994.74004 · doi:10.1016/S0020-7683(00)00196-7
[105] Oliver, J.; Cervera, M.; Manzoli, O., Strong discontinuities and continuum plasticity models: the strong discontinuity approach, International Journal of Plasticity, 15, 3, 319-351 (1999) · Zbl 1057.74512 · doi:10.1016/S0749-6419(98)00073-4
[106] Oliver, J., 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, 39, 3575-3624 (1996) · Zbl 0888.73018
[107] Linder, C.; Miehe, C., 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, 60, 5, 882-903 (2012) · doi:10.1016/j.jmps.2012.01.012
[108] Foster, C. D.; Borja, R. I.; Regueiro, R. A., Embedded strong discontinuity finite elements for fractured geomaterials with variable friction, International Journal for Numerical Methods in Engineering, 72, 5, 549-581 (2007) · Zbl 1194.74395 · doi:10.1002/nme.2020
[109] Linder, C.; Rosato, D.; Miehe, C., New finite elements with embedded strong discontinuities for the modeling of failure in electromechanical coupled solids, Computer Methods in Applied Mechanics and Engineering, 200, 1-4, 141-161 (2011) · Zbl 1225.74095 · doi:10.1016/j.cma.2010.07.021
[110] Armero, F.; Garikipati, K., 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, 33, 20-22, 2863-2885 (1996) · Zbl 0924.73084 · doi:10.1016/0020-7683(95)00257-X
[111] Linder, C.; Armero, F., Finite elements with embedded branching, Finite Elements in Analysis and Design, 45, 4, 280-293 (2009) · doi:10.1016/j.finel.2008.10.012
[112] Armero, F.; Linder, C., Numerical simulation of dynamic fracture using finite elements with embedded discontinuities, International Journal of Fracture, 160, 2, 119-141 (2009) · Zbl 1273.74422 · doi:10.1007/s10704-009-9413-9
[113] Oliver, J.; Huespe, A. E.; Sánchez, P. J., A comparative study on finite elements for capturing strong discontinuities: E-FEM vs X-FEM, Computer Methods in Applied Mechanics and Engineering, 195, 37-40, 4732-4752 (2006) · Zbl 1144.74043 · doi:10.1016/j.cma.2005.09.020
[114] Feist, C.; Hofstetter, G., Three-dimensional fracture simulations based on the SDA, International Journal for Numerical and Analytical Methods in Geomechanics, 31, 2, 189-212 (2007) · Zbl 1113.74069 · doi:10.1002/nag.542
[115] Sancho, J. M.; Planas, J.; Fathy, A. M.; Gálvez, J. C.; Cendón, D. A., Three-dimensional simulation of concrete fracture using embedded crack elements without enforcing crack path continuity, International Journal for Numerical and Analytical Methods in Geomechanics, 31, 2, 173-187 (2007) · Zbl 1158.74042 · doi:10.1002/nag.540
[116] Mosler, J.; Meschke, G., 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, 193, 30-32, 3351-3375 (2004) · Zbl 1060.74606 · doi:10.1016/j.cma.2003.09.022
[117] Belytschko, T.; Black, T., Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering, 45, 5, 601-620 (1999) · Zbl 0943.74061
[118] Moës, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering, 46, 1, 131-150 (1999) · Zbl 0955.74066
[119] Melenk, J. M.; Babuška, I., The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering, 139, 1-4, 289-314 (1996) · Zbl 0881.65099 · doi:10.1016/S0045-7825(96)01087-0
[120] Strouboulis, T.; Copps, K.; Babuška, I., The generalized finite element method: an example of its implementation and illustration of its performance, International Journal for Numerical Methods in Engineering, 47, 8, 1401-1417 (2000) · Zbl 0955.65080
[121] Strouboulis, T.; Babuška, I.; Copps, K., The design and analysis of the generalized finite element method, Computer Methods in Applied Mechanics and Engineering, 181, 1-3, 43-69 (2000) · Zbl 0983.65127 · doi:10.1016/S0045-7825(99)00072-9
[122] Chessa, J.; Belytschko, T., An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension, International Journal for Numerical Methods in Engineering, 58, 13, 2041-2064 (2003) · Zbl 1032.76591 · doi:10.1002/nme.946
[123] Chessa, J.; Belytschko, T., An extended finite element method for two-phase fluids, Journal of Applied Mechanics, Transactions ASME, 70, 1, 10-17 (2003) · Zbl 1110.74391 · doi:10.1115/1.1526599
[124] Zilian, A.; Legay, A., The enriched space-time finite element method (EST) for simultaneous solution of fluid-structure interaction, International Journal for Numerical Methods in Engineering, 75, 3, 305-334 (2008) · Zbl 1195.74212 · doi:10.1002/nme.2258
[125] Mayer, U. M.; Gerstenberger, A.; Wall, W. A., Interface handling for three-dimensional higher-order XFEM-computations in fluid-structure interaction, International Journal for Numerical Methods in Engineering, 79, 7, 846-869 (2009) · Zbl 1171.74447 · doi:10.1002/nme.2600
[126] Duddu, R.; Bordas, S.; Chopp, D.; Moran, B., A combined extended finite element and level set method for biofilm growth, International Journal for Numerical Methods in Engineering, 74, 5, 848-870 (2008) · Zbl 1195.74169 · doi:10.1002/nme.2200
[127] Rabinovich, D.; Givoli, D.; Vigdergauz, S., XFEM-based crack detection scheme using a genetic algorithm, International Journal for Numerical Methods in Engineering, 71, 9, 1051-1080 (2007) · Zbl 1194.74309 · doi:10.1002/nme.1975
[128] Rabinovich, D.; Givoli, D.; Vigdergauz, S., Crack identification by lsquoarrival timersquo using XFEM and a genetic algorithm, International Journal for Numerical Methods in Engineering, 77, 3, 337-359 (2009) · Zbl 1155.74398 · doi:10.1002/nme.2416
[129] Béchet, E.; Scherzer, M.; Kuna, M., Application of the X-FEM to the fracture of piezoelectric materials, International Journal for Numerical Methods in Engineering, 77, 11, 1535-1565 (2009) · Zbl 1158.74477 · doi:10.1002/nme.2455
[130] Verhoosel, C. V.; Remmers, J. J. C.; Gutiérrez, M. A., A partition of unity-based multiscale approach for modelling fracture in piezoelectric ceramics, International Journal for Numerical Methods in Engineering, 82, 8, 966-994 (2010) · Zbl 1188.74074 · doi:10.1002/nme.2792
[131] Duflot, M., The extended finite element method in thermoelastic fracture mechanics, International Journal for Numerical Methods in Engineering, 74, 5, 827-847 (2008) · Zbl 1195.74170 · doi:10.1002/nme.2197
[132] Areias, P. M. A.; Belytschko, T., Two-scale shear band evolution by local partition of unity, International Journal for Numerical Methods in Engineering, 66, 5, 878-910 (2006) · Zbl 1110.74841 · doi:10.1002/nme.1589
[133] Duarte, C. A.; Reno, L. G.; Simone, A., A high-order generalized FEM for through-the-thickness branched cracks, International Journal for Numerical Methods in Engineering, 72, 3, 325-351 (2007) · Zbl 1194.74385 · doi:10.1002/nme.2012
[134] Duarte, C. A.; Hamzeh, O. N.; Liszka, T. J.; Tworzydlo, W. W., A generalized finite element method for the simulation of three-dimensional dynamic crack propagation, Computer Methods in Applied Mechanics and Engineering, 190, 15-17, 2227-2262 (2001) · Zbl 1047.74056 · doi:10.1016/S0045-7825(00)00233-4
[135] Duarte, C. A.; Kim, D.-J., Analysis and applications of a generalized finite element method with global-local enrichment functions, Computer Methods in Applied Mechanics and Engineering, 197, 6-8, 487-504 (2008) · Zbl 1169.74597 · doi:10.1016/j.cma.2007.08.017
[136] Pereira, J. P.; Duarte, C. A.; Jiao, X.; Guoy, D., Generalized finite element method enrichment functions for curved singularities in 3D fracture mechanics problems, Computational Mechanics, 44, 1, 73-92 (2009) · Zbl 1162.74473 · doi:10.1007/s00466-008-0356-1
[137] Duarte, C. A.; Kim, D.-J.; Babuška, I., A global-local approach for the construction of enrichment functions for the generalized FEM and its application to three-dimensional cracks, Advances in Meshfree Techniques, 5, 1-26 (2007), Dordrecht, The Netherlands: Springer, Dordrecht, The Netherlands · Zbl 1323.74081 · doi:10.1007/978-1-4020-6095-3_1
[138] Kim, D. J.; Pereira, J. P.; Duarte, C. A., Analysis of three-dimensional fracture mechanics problems: a two-scale approach using coarse-generalized FEM meshes, International Journal for Numerical Methods in Engineering, 81, 3, 335-365 (2010) · Zbl 1183.74285 · doi:10.1002/nme.2690
[139] Kim, D.-J.; Duarte, C. A.; Sobh, N. A., Parallel simulations of three-dimensional cracks using the generalized finite element method, Computational Mechanics, 47, 3, 265-282 (2011) · Zbl 05873376 · doi:10.1007/s00466-010-0546-5
[140] Menk, A.; Bordas, S. P. A., A robust preconditioning technique for the extended finite element method, International Journal for Numerical Methods in Engineering, 85, 13, 1609-1632 (2011) · Zbl 1217.74128 · doi:10.1002/nme.3032
[141] Babuška, I.; Banerjee, U., Stable generalized finite element method (SGFEM), Computer Methods in Applied Mechanics and Engineering, 201-204, 91-111 (2012) · Zbl 1239.74093 · doi:10.1016/j.cma.2011.09.012
[142] Zi, G.; Belytschko, T., New crack-tip elements for XFEM and applications to cohesive cracks, International Journal for Numerical Methods in Engineering, 57, 15, 2221-2240 (2003) · Zbl 1062.74633 · doi:10.1002/nme.849
[143] Zi, G.; Song, J. H.; Budyn, E.; Lee, S. H.; Belytschko, T., A method for growing multiple cracks without remeshing and its application to fatigue crack growth, Modelling and Simulation in Materials Science and Engineering, 12, 5, 901-915 (2004) · doi:10.1088/0965-0393/12/5/009
[144] Budyn, E.; Zi, G.; Moes, N.; Belytschko, T., A method for multiple crack growth in brittle materials without remeshing, International Journal for Numerical Methods in Engineering, 61, 10, 1741-1770 (2004) · Zbl 1075.74638 · doi:10.1002/nme.1130
[145] Belytschko, T.; Moes, N.; Usui, S.; Parimi, C., Arbitrary discontinuities in finite elements, International Journal For Numerical Methods in Engineering, 50, 4, 993-1013 (2001) · Zbl 0981.74062
[146] Daux, C.; Moës, N.; Dolbow, J.; Sukumar, N.; Belytschko, T., Arbitrary branched and intersecting cracks with the extended finite element method, International Journal for Numerical Methods in Engineering, 48, 12, 1741-1760 (2000) · Zbl 0989.74066
[147] Areias, P. M. A.; Belytschko, T., Analysis of three-dimensional crack initiation and propagation using the extended finite element method, International Journal for Numerical Methods in Engineering, 63, 5, 760-788 (2005) · Zbl 1122.74498 · doi:10.1002/nme.1305
[148] Areias, P. M. A.; Song, J. H.; Belytschko, T., Analysis of fracture in thin shells by overlapping paired elements, Computer Methods in Applied Mechanics and Engineering, 195, 41-43, 5343-5360 (2006) · Zbl 1120.74048 · doi:10.1016/j.cma.2005.10.024
[149] Sethian, J. A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (1999), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0973.76003
[150] Osher, S.; Sethian, J. A., Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79, 1, 12-49 (1988) · Zbl 0659.65132
[151] Prabel, B.; Combescure, A.; Gravouil, A.; Marie, S., Level set X-FEM non-matching meshes: application to dynamic crack propagation in elastic-plastic media, International Journal for Numerical Methods in Engineering, 69, 8, 1553-1569 (2007) · Zbl 1194.74465 · doi:10.1002/nme.1819
[152] Laborde, P.; Pommier, J.; Renard, Y.; Salaün, M., High-order extended finite element method for cracked domains, International Journal for Numerical Methods in Engineering, 64, 3, 354-381 (2005) · Zbl 1181.74136 · doi:10.1002/nme.1370
[153] Ventura, G., On the elimination of quadrature subcells for discontinuous functions in the extended finite-element method, International Journal for Numerical Methods in Engineering, 66, 5, 761-795 (2006) · Zbl 1110.74858 · doi:10.1002/nme.1570
[154] Ventura, G.; Gracie, R.; Belytschko, T., Fast integration and weight function blending in the extended finite element method, International Journal for Numerical Methods in Engineering, 77, 1, 1-29 (2009) · Zbl 1195.74201 · doi:10.1002/nme.2387
[155] Gracie, R.; Wang, H.; Belytschko, T., Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods, International Journal for Numerical Methods in Engineering, 74, 11, 1645-1669 (2008) · Zbl 1195.74175 · doi:10.1002/nme.2217
[156] Bordas, S. P. A.; Rabczuk, T.; Hung, N. X.; Nguyen, V. P.; Natarajan, S.; Bog, T.; Quan, D. M.; Hiep, N. V., Strain smoothing in FEM and XFEM, Computers and Structures, 88, 23-24, 1419-1443 (2010) · doi:10.1016/j.compstruc.2008.07.006
[157] Cheng, K. W.; Fries, T. P., Higher-order XFEM for curved strong and weak discontinuities, International Journal for Numerical Methods in Engineering, 82, 5, 564-590 (2010) · Zbl 1188.74052 · doi:10.1002/nme.2768
[158] Nagarajan, A.; Mukherjee, S., A mapping method for numerical evaluation of two-dimensional integrals with \(1 / r\) singularity, Computational Mechanics, 12, 1-2, 19-26 (1993) · Zbl 0776.73073 · doi:10.1007/BF00370482
[159] Béchet, E.; Minnebo, H.; Moës, N.; Burgardt, B., Improved implementation and robustness study of the X-FEM for stress analysis around cracks, International Journal for Numerical Methods in Engineering, 64, 8, 1033-1056 (2005) · Zbl 1122.74499 · doi:10.1002/nme.1386
[160] Chessa, J.; Wang, H.; Belytschko, T., On the construction of blending elements for local partition of unity enriched finite elements, International Journal for Numerical Methods in Engineering, 57, 7, 1015-1038 (2003) · Zbl 1035.65122 · doi:10.1002/nme.777
[161] Stazi, F.; Budyn, E.; Chessa, J.; Belytschko, T., XFEM for fracture mechanics with quadratic elements, Computational Mechanics, 31, 38-48 (2003) · Zbl 1038.74651 · doi:10.1007/s00466-002-0391-2
[162] Fries, T.-P., A corrected XFEM approximation without problems in blending elements, International Journal for Numerical Methods in Engineering, 75, 5, 503-532 (2008) · Zbl 1195.74173 · doi:10.1002/nme.2259
[163] Tarancón, J. E.; Vercher, A.; Giner, E.; Fuenmayor, F. J., Enhanced blending elements for XFEM applied to linear elastic fracture mechanics, International Journal for Numerical Methods in Engineering, 77, 1, 126-148 (2009) · Zbl 1195.74199 · doi:10.1002/nme.2402
[164] Bellec, J.; Dolbow, J. E., A note on enrichment functions for modelling crack nucleation, Communications in Numerical Methods in Engineering, 19, 12, 921-932 (2003) · Zbl 1047.74536 · doi:10.1002/cnm.641
[165] Karihaloo, B. L.; Xiao, Q. Z., Modelling of stationary and growing cracks in FE framework without remeshing: a state-of-the-art review, Computers and Structures, 81, 3, 119-129 (2003) · doi:10.1016/S0045-7949(02)00431-5
[166] Fries, T.-P.; Belytschko, T., The extended/generalized finite element method: an overview of the method and its applications, International Journal for Numerical Methods in Engineering, 84, 3, 253-304 (2010) · Zbl 1202.74169 · doi:10.1002/nme.2914
[167] Belytschko, T.; Gracie, R.; Ventura, G., A review of extended/generalized finite element methods for material modeling, Modelling and Simulation in Materials Science and Engineering, 17, 4 (2009) · doi:10.1088/0965-0393/17/4/043001
[168] Mohammadi, S., Extended Finite Element Method for Fracture Analysis of Structures (2008), Oxford, UK: Blackwell Publishing, Oxford, UK · Zbl 1132.74001
[169] Hansbo, A.; Hansbo, P., A finite element method for the simulation of strong and weak discontinuities in solid mechanics, Computer Methods in Applied Mechanics and Engineering, 193, 33-35, 3523-3540 (2004) · Zbl 1068.74076 · doi:10.1016/j.cma.2003.12.041
[170] Song, J. H.; Areias, P. M. A.; Belytschko, T., A method for dynamic crack and shear band propagation with phantom nodes, International Journal for Numerical Methods in Engineering, 67, 6, 868-893 (2006) · Zbl 1113.74078 · doi:10.1002/nme.1652
[171] Menouillard, T.; Réthoré, J.; Combescure, A.; Bung, H., Efficient explicit time stepping for the extended finite element method (X-FEM), International Journal for Numerical Methods in Engineering, 68, 9, 911-939 (2006) · Zbl 1128.74045 · doi:10.1002/nme.1718
[172] Menouillard, T.; Réthoré, J.; Moës, N.; Combescure, A.; Bung, H., Mass lumping strategies for X-FEM explicit dynamics: application to crack propagation, International Journal for Numerical Methods in Engineering, 74, 3, 447-474 (2008) · Zbl 1159.74432 · doi:10.1002/nme.2180
[173] Talebi, H.; Samaniego, C.; Samaniego, E.; Rabczuk, T., On the numerical stability and masslumping schemes for explicit enriched meshfree methods, International Journal for Numerical Methods in Engineering, 89, 1009-1027 (2012) · Zbl 1242.74219 · doi:10.1002/nme.3275
[174] Chau-Dinh, T.; Zi, G.; Lee, P. S.; Rabczuk, T.; Song, J. H., Phantom-node method for shell models with arbitrary cracks, Computers and Structures, 92-93, 242-256 (2012) · doi:10.1016/j.compstruc.2011.10.021
[175] Rabczuk, T.; Zi, G.; Gerstenberger, A.; Wall, W. A., A new crack tip element for the phantom-node method with arbitrary cohesive cracks, International Journal for Numerical Methods in Engineering, 75, 5, 577-599 (2008) · Zbl 1195.74193 · doi:10.1002/nme.2273
[176] Mergheim, J.; Kuhl, E.; Steinmann, P., A finite element method for the computational modelling of cohesive cracks, International Journal for Numerical Methods in Engineering, 63, 2, 276-289 (2005) · Zbl 1118.74349 · doi:10.1002/nme.1286
[177] Mergheim, J.; Steinmann, P., A geometrically nonlinear FE approach for the simulation of strong and weak discontinuities, Computer Methods in Applied Mechanics and Engineering, 195, 37-40, 5037-5052 (2006) · Zbl 1126.74050 · doi:10.1016/j.cma.2005.05.057
[178] Organ, D.; Fleming, M.; Terry, T.; Belytschko, T., Continuous meshless approximations for nonconvex bodies by diffraction and transparency, Computational Mechanics, 18, 3, 225-235 (1996) · Zbl 0864.73076
[179] Belytschko, T.; Organ, D.; Tabbara, M., Numerical simulations of mixed mode dynamic fracture in concrete using element-free Galerkin methods, Proceedings of the International Conference on Environmental Systems (ICES ’95)
[180] Belytschko, T.; Lu, Y. Y.; Gu, L.; Tabbara, M., Element-free galerkin methods for static and dynamic fracture, International Journal of Solids and Structures, 32, 17-18, 2547-2570 (1995) · Zbl 0918.73268 · doi:10.1016/0020-7683(94)00282-2
[181] Belytschko, T.; Lu, Y. Y.; Gu, L., Crack propagation by element-free Galerkin methods, Engineering Fracture Mechanics, 51, 2, 295-315 (1995)
[182] Remmers, J. J. C.; De Borst, R.; Needleman, A., A cohesive segments method for the simulation of crack growth, Computational Mechanics, 31, 1-2, 69-77 (2003) · Zbl 1038.74679 · doi:10.1007/s00466-002-0394-z
[183] Remmers, J. J. C.; de Borst, R.; Needleman, A., The simulation of dynamic crack propagation using the cohesive segments method, Journal of the Mechanics and Physics of Solids, 56, 1, 70-92 (2008) · Zbl 1162.74438 · doi:10.1016/j.jmps.2007.08.003
[184] Song, J.-H.; Belytschko, T., Cracking node method for dynamic fracture with finite elements, International Journal for Numerical Methods in Engineering, 77, 3, 360-385 (2009) · Zbl 1155.74415 · doi:10.1002/nme.2415
[185] You, Y.; Chen, J. S.; Lu, H., Filters, reproducing kernel, and adaptive meshfree method, Computational Mechanics, 31, 3-4, 316-326 (2003) · Zbl 1038.74681
[186] Rabczuk, T.; Belytschko, T., Adaptivity for structured meshfree particle methods in 2D and 3D, International Journal for Numerical Methods in Engineering, 63, 11, 1559-1582 (2005) · Zbl 1145.74041 · doi:10.1002/nme.1326
[187] Rabczuk, T.; Samaniego, E., Discontinuous modelling of shear bands using adaptive meshfree methods, Computer Methods in Applied Mechanics and Engineering, 197, 6-8, 641-658 (2008) · Zbl 1169.74655 · doi:10.1016/j.cma.2007.08.027
[188] Fries, T.-P.; Byfut, A.; Alizada, A.; Cheng, K. W.; Schröder, A., Hanging nodes and XFEM, International Journal for Numerical Methods in Engineering, 86, 4-5, 404-430 (2011) · Zbl 1216.74020 · doi:10.1002/nme.3024
[189] Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics:theory and applications to non-spherical stars, Monthly Notices of the Royal Astronomical Society, 181, 375-389 (1977) · Zbl 0421.76032
[190] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, International Journal for Numerical Methods in Engineering, 37, 2, 229-256 (1994) · Zbl 0796.73077 · doi:10.1002/nme.1620370205
[191] Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, International Journal for Numerical Methods in Fluids, 20, 8-9, 1081-1106 (1995) · Zbl 0881.76072 · doi:10.1002/fld.1650200824
[192] Liu, G. R.; Gu, Y. T., A point interpolation method for two-dimensional solids, International Journal For Numerical Methods in Engineering, 50, 937-951 (2001) · Zbl 1050.74057
[193] Atluri, S. N., The Meshless Local Petrov-Galerkin (MLPG) Method (2002), Tech Science Press · Zbl 1012.65116
[194] Oñate, E.; Idelsohn, S.; Zienkiewicz, O. C.; Taylor, R. L., A finite point method in computational mechanics. Applications to convective transport and fluid flow, International Journal for Numerical Methods in Engineering, 39, 22, 3839-3866 (1996) · Zbl 0884.76068
[195] Liu, G. R.; Liu, M. B., Smoothed Particle Hydrodynamics: A Meshfree Particle Method (2003) · Zbl 1046.76001
[196] Liu, G. R., Mesh Free Methods: Moving Beyond the Finite Element Method (2002), Boca Raton, Fla, USA: CRC Press, Boca Raton, Fla, USA
[197] Liu, G. R., An Introduction to Meshfree Methods and Their Programming (2006), Springer
[198] Li, S.; Liu, W. K., Meshfree and particle methods and their applications, Applied Mechanics Reviews, 55, 1, 1-34 (2002) · doi:10.1115/1.1431547
[199] Li, S.; Liu, W. K., Meshfree Particle Methods (2004), Berlin, Germany: Springer, Berlin, Germany · Zbl 1073.65002
[200] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering, 139, 1-4, 3-47 (1996) · Zbl 0891.73075
[201] Nguyen, V. P.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: a review and computer implementation aspects, Mathematics and Computers in Simulation, 79, 3, 763-813 (2008) · Zbl 1152.74055 · doi:10.1016/j.matcom.2008.01.003
[202] Huerta, A.; Belytschko, T.; Fernandez-Mendez, S.; Rabczuk, T., Encyclopedia of Computational Mechanics (2004), John Wiley and Sons
[203] Libersky, L. D.; Randles, P. W.; Carney, T. C.; Dickinson, D. L., Recent improvements in SPH modeling of hypervelocity impact, International Journal of Impact Engineering, 20, 6-10, 525-532 (1997)
[204] Rabczuk, T.; Eibl, J., Modelling dynamic failure of concrete with meshfree methods, International Journal of Impact Engineering, 32, 11, 1878-1897 (2006) · doi:10.1016/j.ijimpeng.2005.02.008
[205] Dilts, G. A., Moving least-squares particle hydrodynamics. II. Conservation and boundaries, International Journal for Numerical Methods in Engineering, 48, 10, 1503-1524 (2000) · Zbl 0960.76068
[206] Haque, A.; Dilts, G. A., Three-dimensional boundary detection for particle methods, Journal of Computational Physics, 226, 2, 1710-1730 (2007) · Zbl 1173.76393 · doi:10.1016/j.jcp.2007.06.012
[207] Rabczuk, T.; Belytschko, T.; Xiao, S. P., Stable particle methods based on Lagrangian kernels, Computer Methods in Applied Mechanics and Engineering, 193, 12-14, 1035-1063 (2004) · Zbl 1060.74672 · doi:10.1016/j.cma.2003.12.005
[208] Maurel, B.; Combescure, A., An SPH shell formulation for plasticity and fracture analysis in explicit dynamics, International Journal for Numerical Methods in Engineering, 76, 7, 949-971 (2008) · Zbl 1195.74293 · doi:10.1002/nme.2316
[209] Combescure, A.; Maurel, B.; Potapov, S., Modelling dynamic fracture of thin shells filled with fluid: a fully SPH model, Mecanique et Industries, 9, 2, 167-174 (2008) · doi:10.1051/meca:2008022
[210] Maurel, B.; Potapov, S.; Fabis, J.; Combescure, A., Full SPH fluid-shell interaction for leakage simulation in explicit dynamics, International Journal for Numerical Methods in Engineering, 80, 2, 210-234 (2009) · Zbl 1176.76105 · doi:10.1002/nme.2629
[211] Potapov, S.; Maurel, B.; Combescure, A.; Fabis, J., Modeling accidental-type fluid-structure interaction problems with the SPH method, Computers and Structures, 87, 11-12, 721-734 (2009) · Zbl 1176.76105 · doi:10.1016/j.compstruc.2008.09.009
[212] Sulsky, D.; Chen, Z.; Schreyer, H. L., A particle method for history-dependent materials, Computer Methods in Applied Mechanics and Engineering, 118, 1-2, 179-196 (1994) · Zbl 0851.73078 · doi:10.1016/0045-7825(94)00033-6
[213] Ma, S.; Zhang, X.; Qiu, X. M., Comparison study of MPM and SPH in modeling hypervelocity impact problems, International Journal of Impact Engineering, 36, 2, 272-282 (2009) · doi:10.1016/j.ijimpeng.2008.07.001
[214] Krysl, P.; Belytschko, T., Element-free Galerkin method: convergence of the continuous and discontinuous shape functions, Computer Methods in Applied Mechanics and Engineering, 148, 3-4, 257-277 (1997) · Zbl 0918.73125 · doi:10.1016/S0045-7825(96)00007-2
[215] Rabczuk, T.; Belytschko, T., An adaptive continuum/discrete crack approach for meshfree particle methods, Latin American Journal of Solids and Structures, 1, 141-166 (2003)
[216] Sukumar, N.; Moran, B.; Black, T.; Belytschko, T., An element-free Galerkin method for three-dimensional fracture mechanics, Computational Mechanics, 20, 1-2, 170-175 (1997) · Zbl 0888.73066
[217] Duflot, M., A meshless method with enriched weight functions for three-dimensional crack propagation, International Journal for Numerical Methods in Engineering, 65, 12, 1970-2006 (2006) · Zbl 1114.74064 · doi:10.1002/nme.1530
[218] Terry, T. G., Fatigue crack propagation modeling using the element free galerkin method [M.S. thesis] (1994), Northwestern University
[219] Duarte, C. A.; Oden, J. T., An h-p adaptive method using clouds, Computer Methods in Applied Mechanics and Engineering, 139, 1-4, 237-262 (1996) · Zbl 0918.73328
[220] Belytschko, T.; Fleming, M., Smoothing, enrichment and contact in the element-free Galerkin method, Computers & Structures, 71, 2, 173-195 (1999) · doi:10.1016/S0045-7949(98)00205-3
[221] Fleming, M.; Chu, Y. A.; Moran, B.; Belytschko, T., Enriched element-free Galerkin methods for crack tip fields, International Journal for Numerical Methods in Engineering, 40, 8, 1483-1504 (1997)
[222] Fries, T. P.; Belytschko, T., The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns, International Journal for Numerical Methods in Engineering, 68, 13, 1358-1385 (2006) · Zbl 1129.74045 · doi:10.1002/nme.1761
[223] Duflot, M.; Nguyen-Dang, H., A meshless method with enriched weight functions for fatigue crack growth, International Journal for Numerical Methods in Engineering, 59, 14, 1945-1961 (2004) · Zbl 1060.74664
[224] Zamani, A.; Gracie, R.; Eslami, M. R., Higher order tip enrichment of extended finite element method in thermoelasticity, Computational Mechanics, 46, 6, 851-866 (2010) · Zbl 1344.74062 · doi:10.1007/s00466-010-0520-2
[225] Zamani, A.; Gracie, R.; Eslami, M. R., Cohesive and non-cohesive fracture by higher-order enrichment of xfem, International Journal For Numerical Methods in Engineering, 90, 452-483 (2012) · Zbl 1242.74177 · doi:10.1002/nme.3329
[226] Ventura, G.; Xu, J. X.; Belytschko, T., A vector level set method and new discontinuity approximations for crack growth by EFG, International Journal for Numerical Methods in Engineering, 54, 6, 923-944 (2002) · Zbl 1034.74053 · doi:10.1002/nme.471
[227] Rabczuk, T.; Zi, G., A meshfree method based on the local partition of unity for cohesive cracks, Computational Mechanics, 39, 6, 743-760 (2007) · Zbl 1161.74055 · doi:10.1007/s00466-006-0067-4
[228] Rabczuk, T.; Bordas, S.; Zi, G., A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics, Computational Mechanics, 40, 3, 473-495 (2007) · Zbl 1161.74054 · doi:10.1007/s00466-006-0122-1
[229] Bordas, S.; Rabczuk, T.; Zi, G., Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment, Engineering Fracture Mechanics, 75, 5, 943-960 (2008) · doi:10.1016/j.engfracmech.2007.05.010
[230] Zi, G.; Rabczuk, T.; Wall, W., Extended meshfree methods without branch enrichment for cohesive cracks, Computational Mechanics, 40, 2, 367-382 (2007) · Zbl 1162.74053 · doi:10.1007/s00466-006-0115-0
[231] Rabczuk, T.; Belytschko, T., Cracking particles: a simplified meshfree method for arbitrary evolving cracks, International Journal for Numerical Methods in Engineering, 61, 13, 2316-2343 (2004) · Zbl 1075.74703 · doi:10.1002/nme.1151
[232] Rabczuk, T.; Belytschko, T., A three-dimensional large deformation meshfree method for arbitrary evolving cracks, Computer Methods in Applied Mechanics and Engineering, 196, 29-30, 2777-2799 (2007) · Zbl 1128.74051 · doi:10.1016/j.cma.2006.06.020
[233] Rabczuk, T.; Belytschko, T., Application of particle methods to static fracture of reinforced concrete structures, International Journal of Fracture, 137, 1-4, 19-49 (2006) · Zbl 1197.74175 · doi:10.1007/s10704-005-3075-z
[234] Rabczuk, T.; Areias, P. M. A.; Belytschko, T., A simplified mesh-free method for shear bands with cohesive surfaces, International Journal for Numerical Methods in Engineering, 69, 5, 993-1021 (2007) · Zbl 1194.74536 · doi:10.1002/nme.1797
[235] Rabczuk, T.; Zi, G.; Bordas, S.; Nguyen-Xuan, H., A simple and robust three-dimensional cracking-particle method without enrichment, Computer Methods in Applied Mechanics and Engineering, 199, 37-40, 2437-2455 (2010) · Zbl 1231.74493 · doi:10.1016/j.cma.2010.03.031
[236] Chen, L.; Zhang, Y., Dynamic fracture analysis using discrete cohesive crack method, International Journal for Numerical Methods in Biomedical Engineering, 26, 11, 1493-1502 (2010) · Zbl 05828324 · doi:10.1002/cnm.1232
[237] Zhang, Y. Y., Meshless modelling of crack growth with discrete rotating crack segments, International Journal of Mechanics and Materials in Design, 4, 1, 71-77 (2008) · doi:10.1007/s10999-008-9062-6
[238] Wang, H. X.; Wang, S. X., Analysis of dynamic fracture with cohesive crack segment method, CMES. Computer Modeling in Engineering & Sciences, 35, 3, 253-274 (2008) · Zbl 1153.74373
[239] Caleyron, F.; Combescure, A.; Faucher, V.; Potapov, S., Dynamic simulation of damagefracture transition in smoothed particles hydrodynamics shells, International Journal For Numerical Methods in Engineering, 90, 707-738 (2012) · Zbl 1242.74204 · doi:10.1002/nme.3337
[240] Aliabadi, M. H., Boundary element formulations in fracture mechanics, Applied Mechanics Reviews, 50, 2, 83-96 (1997)
[241] Mi, Y.; Aliabadi, M. H., Three-dimensional crack growth simulation using BEM, Computers and Structures, 52, 5, 871-878 (1994) · Zbl 0900.73900
[242] Pan, E., A general boundary element analysis of 2-D linear elastic fracture mechanics, International Journal of Fracture, 88, 1, 41-59 (1997) · doi:10.1023/A:1007462319811
[243] Pan, E., A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids, Engineering Analysis with Boundary Elements, 23, 1, 67-76 (1999) · Zbl 1062.74639
[244] Ryoji, Y.; Sang-Bong, C., Efficient boundary element analysis of stress intensity factors for interface cracks in dissimilar materials, Engineering Fracture Mechanics, 34, 1, 179-188 (1989)
[245] Pan, E.; Yuan, F. G., Boundary element analysis of three-dimensional cracks in anisotropic solids, International Journal For Numerical Methods in Engineering, 48, 211-237 (2000) · Zbl 0977.74077
[246] Doblare, M.; Espiga, F.; Gracia, L.; Alcantud, M., Study of crack propagation in orthotropic materials by using the boundary element method, Engineering Fracture Mechanics, 37, 5, 953-967 (1990)
[247] Sfantos, G. K.; Aliabadi, M. H., Multi-scale boundary element modelling of material degradation and fracture, Computer Methods in Applied Mechanics and Engineering, 196, 7, 1310-1329 (2007) · Zbl 1173.74459 · doi:10.1016/j.cma.2006.09.004
[248] Schnack, E., Hybrid bem model, International Journal for Numerical Methods in Engineering, 24, 5, 1015-1025 (1987) · Zbl 0611.73081
[249] Sládek, J.; Sládek, V.; Bažant, Z. P., Non-local boundary integral formulation for softening damage, International Journal for Numerical Methods in Engineering, 57, 1, 103-116 (2003) · Zbl 1062.74644 · doi:10.1002/nme.673
[250] Gao, X. W.; Zhang, C.; Sladek, J.; Sladek, V., Fracture analysis of functionally graded materials by a BEM, Composites Science and Technology, 68, 5, 1209-1215 (2008) · doi:10.1016/j.compscitech.2007.08.029
[251] García-Sánchez, F.; Rojas-Díaz, R.; Sáez, A.; Zhang, C., Fracture of magnetoelectroelastic composite materials using boundary element method (BEM), Theoretical and Applied Fracture Mechanics, 47, 3, 192-204 (2007) · doi:10.1016/j.tafmec.2007.01.008
[252] Cruse, T. A., BIE fracture mechanics analysis: 25 years of developments, Computational Mechanics, 18, 1, 1-11 (1996) · Zbl 0946.74073 · doi:10.1007/s004660050125
[253] Simpson, R. N.; Bordas, S. P. A.; Trevelyan, J.; Rabczuk, T., A two-dimensional isogeometric boundary element method for elastostatic analysis, Computer Methods in Applied Mechanics and Engineering, 209/212, 87-100 (2012) · Zbl 1243.74193 · doi:10.1016/j.cma.2011.08.008
[254] Cisilino, A. P.; Aliabadi, M. H., Three-dimensional boundary element analysis of fatigue crack growth in linear and non-linear fracture problems, Engineering Fracture Mechanics, 63, 6, 713-733 (1999)
[255] Simpson, R.; Trevelyan, J., Evaluation of j 1 and j 2 integrals for curved cracks using an enriched boundary element method, Engineering Fracture Mechanics, 78, 623-637 (2011) · doi:10.1016/j.engfracmech.2010.12.006
[256] Simpson, R.; Trevelyan, J., A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics, Computer Methods in Applied Mechanics and Engineering, 200, 1-4, 1-10 (2011) · Zbl 1225.74117 · doi:10.1016/j.cma.2010.06.015
[257] Bird, G. E.; Trevelyan, J.; Augarde, C. E., A coupled BEM/scaled boundary FEM formulation for accurate computations in linear elastic fracture mechanics, Engineering Analysis with Boundary Elements, 34, 6, 599-610 (2010) · Zbl 1267.74120 · doi:10.1016/j.enganabound.2010.01.007
[258] Hughes, T. J. R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194, 39-41, 4135-4195 (2005) · Zbl 1151.74419 · doi:10.1016/j.cma.2004.10.008
[259] De Luycker, E.; Benson, D. J.; Belytschko, T.; Bazilevs, Y.; Hsu, M. C., X-FEM in isogeometric analysis for linear fracture mechanics, International Journal for Numerical Methods in Engineering, 87, 6, 541-565 (2011) · Zbl 1242.74105 · doi:10.1002/nme.3121
[260] Ghorashi, S. S.; Valizadeh, N.; Mohammadi, S., Extended isogeometric analysis for simulation of stationary and propagating cracks, International Journal For Numerical Methods in Engineering, 89, 1069-1101 (2012) · Zbl 1242.74119 · doi:10.1002/nme.3277
[261] Tambat, A.; Subbarayan, G., Isogeometric enriched field approximations, Computer Methods in Applied Mechanics and Engineering, 245-246, 1-21 (2012) · Zbl 1354.65044 · doi:10.1016/j.cma.2012.06.006
[262] Francfort, G. A.; Marigo, J.-J., Revisiting brittle fracture as an energy minimization problem, Journal of the Mechanics and Physics of Solids, 46, 8, 1319-1342 (1998) · Zbl 0966.74060 · doi:10.1016/S0022-5096(98)00034-9
[263] Bourdin, B.; Francfort, G. A.; Marigo, J.-J., Numerical experiments in revisited brittle fracture, Journal of the Mechanics and Physics of Solids, 48, 4, 797-826 (2000) · Zbl 0995.74057 · doi:10.1016/S0022-5096(99)00028-9
[264] Karma, A.; Kessler, D. A.; Levine, H., Phase-field model of mode III dynamic fracture, Physical Review Letters, 87, 4 (2001)
[265] Hakim, V.; Karma, A., Laws of crack motion and phase-field models of fracture, Journal of the Mechanics and Physics of Solids, 57, 2, 342-368 (2009) · Zbl 1421.74089 · doi:10.1016/j.jmps.2008.10.012
[266] Miehe, C.; Hofacker, M.; Welschinger, F., A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits, Computer Methods in Applied Mechanics and Engineering, 199, 45-48, 2765-2778 (2010) · Zbl 1231.74022 · doi:10.1016/j.cma.2010.04.011
[267] Miehe, C.; Welschinger, F.; Hofacker, M., Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations, International Journal for Numerical Methods in Engineering, 83, 10, 1273-1311 (2010) · Zbl 1202.74014 · doi:10.1002/nme.2861
[268] Kuhn, C.; Müller, R., A new finite element technique for a phase field model of brittle fracture, Journal of Theoretical and Applied Mechanics, 49, 1115-1133 (2011)
[269] Kuhn, C.; Müller, R., Exponential finite element shape functions for a phase field model of brittle fracture, Proceedings of the 11th International Conference on Computational Plasticity (COMPLAS ’11)
[270] Kuhn, C.; Müller, R., A continuum phase field model for fracture, Engineering Fracture Mechanics, 77, 18, 3625-3634 (2010) · doi:10.1016/j.engfracmech.2010.08.009
[271] Rabczuk, T.; Bordas, S.; Zi, G., On three-dimensional modelling of crack growth using partition of unity methods, Computers and Structures, 88, 23-24, 1391-1411 (2010) · doi:10.1016/j.compstruc.2008.08.010
[272] Jäger, P.; Steinmann, P.; Kuhl, E., On local tracking algorithms for the simulation of three-dimensional discontinuities, Computational Mechanics, 42, 3, 395-406 (2008) · Zbl 1173.74044 · doi:10.1007/s00466-008-0249-3
[273] Gasser, T. C.; Holzapfel, G. A., Modeling 3D crack propagation in unreinforced concrete using PUFEM, Computer Methods in Applied Mechanics and Engineering, 194, 25-26, 2859-2896 (2005) · Zbl 1176.74180 · doi:10.1016/j.cma.2004.07.025
[274] Gasser, T. C.; Holzapfel, G. A., 3D Crack propagation in unreinforced concrete. A two-step algorithm for tracking 3D crack paths, Computer Methods in Applied Mechanics and Engineering, 195, 37-40, 5198-5219 (2006) · Zbl 1154.74376 · doi:10.1016/j.cma.2005.10.023
[275] Krysl, P.; Belytschko, T., The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks, International Journal for Numerical Methods in Engineering, 44, 6, 767-800 (1999) · Zbl 0953.74078
[276] Oliver, J.; Huespe, A. E.; Samaniego, E.; Chaves, E. W. V., Continuum approach to the numerical simulation of material failure in concrete, International Journal for Numerical and Analytical Methods in Geomechanics, 28, 7-8, 609-632 (2004) · Zbl 1112.74493 · doi:10.1002/nag.365
[277] Jäger, P.; Steinmann, P.; Kuhl, E., Towards the treatment of boundary conditions for global crack path tracking in three-dimensional brittle fracture, Computational Mechanics, 45, 1, 91-107 (2009) · Zbl 1398.74276 · doi:10.1007/s00466-009-0417-0
[278] Ventura, G.; Budyn, E.; Belytschko, T., Vector level sets for description of propagating cracks in finite elements, International Journal for Numerical Methods in Engineering, 58, 10, 1571-1592 (2003) · Zbl 1032.74687 · doi:10.1002/nme.829
[279] Duflot, M., A study of the representation of cracks with level sets, International Journal for Numerical Methods in Engineering, 70, 11, 1261-1302 (2007) · Zbl 1194.74516 · doi:10.1002/nme.1915
[280] Chopp, D. L., Computing minimal surfaces via level set curvature flow, Journal of Computational Physics, 106, 1, 77-91 (1993) · Zbl 0786.65015 · doi:10.1006/jcph.1993.1092
[281] Chopp, D. L.; Sethian, J. A., Flow under curvature: singularity formation, minimal surfaces, and geodesics, Experimental Mathematics, 2, 4, 235-255 (1993) · Zbl 0806.53004 · doi:10.1080/10586458.1993.10504566
[282] Stolarska, M.; Chopp, D. L.; Mos, N.; Belytschko, T., Modelling crack growth by level sets in the extended finite element method, International Journal for Numerical Methods in Engineering, 51, 8, 943-960 (2001) · Zbl 1022.74049 · doi:10.1002/nme.201
[283] Fries, T.-P.; Baydoun, M., Crack propagation with the extended finite element method and a hybrid explicit-implicit crack description, International Journal for Numerical Methods in Engineering, 89, 12, 1527-1558 (2012) · Zbl 1242.74113 · doi:10.1002/nme.3299
[284] Zhuang, X.; Augarde, C.; Bordas, S., Accurate fracture modelling using meshless methods, the visibility criterion and level sets: formulation and 2D modelling, International Journal for Numerical Methods in Engineering, 86, 2, 249-268 (2011) · Zbl 1235.74346 · doi:10.1002/nme.3063
[285] Zhuang, X.; Augarde, C.; Mathisen, K. M., Fracture modeling using meshless methods and level sets in 3d: framework and modeling, International Journal for Numerical Methods in Engineering, 92, 11, 969-998 (2012) · Zbl 1352.74312 · doi:10.1002/nme.4365
[286] Mosler, J., A variationally consistent approach for crack propagation based on configurational forces, IUTAM Symposium on Progress in the Theory and Numerics of Configurational Mechanics. IUTAM Symposium on Progress in the Theory and Numerics of Configurational Mechanics, IUTAM Bookseries, 17, 239-247 (2009) · doi:10.1007/978-90-481-3447-2_22
[287] Gurtin, M. E.; Podio-Guidugli, P., Configurational forces and the basic laws for crack propagation, Journal of the Mechanics and Physics of Solids, 44, 6, 905-927 (1996) · Zbl 1054.74508 · doi:10.1016/0022-5096(96)00014-2
[288] Gurtin, M. E.; Podio-Guidugli, P., Configurational forces and a constitutive theory for crack propagation that allows for kinking and curving, Journal of the Mechanics and Physics of Solids, 46, 8, 1343-1378 (1998) · Zbl 0955.74004 · doi:10.1016/S0022-5096(98)00002-7
[289] Miehe, C.; Gürses, E., A robust algorithm for configurational-force-driven brittle crack propagation with r-adaptive mesh alignment, International Journal for Numerical Methods in Engineering, 72, 2, 127-155 (2007) · Zbl 1194.74444 · doi:10.1002/nme.1999
[290] Sih, G. C., Strain-energy-density factor applied to mixed mode crack problems, International Journal of Fracture, 10, 3, 305-321 (1974) · doi:10.1007/BF00035493
[291] Wu, C. H., Fracture under combined loads by maximum energy release rate criterion, Journal of Applied Mechanics, Transactions ASME, 45, 3, 553-558 (1978) · Zbl 0386.73080
[292] Goldstein, R. V.; Salganik, R. L., Brittle fracture of solids with arbitrary cracks, International Journal of Fracture, 10, 4, 507-523 (1974) · doi:10.1007/BF00155254
[293] Shen, B.; Stephansson, O., Modification of the G-criterion for crack propagation subjected to compression, Engineering Fracture Mechanics, 47, 2, 177-189 (1994)
[294] Wells, G. N.; Sluys, L. J., A new method for modelling cohesive cracks using finite elements, International Journal for Numerical Methods in Engineering, 50, 12, 2667-2682 (2001) · Zbl 1013.74074 · doi:10.1002/nme.143
[295] Mariani, S.; Perego, U., Extended finite element method for quasi-brittle fracture, International Journal for Numerical Methods in Engineering, 58, 1, 103-126 (2003) · Zbl 1032.74673 · doi:10.1002/nme.761
[296] Marsden, J. E.; Hughes, T. J. R., Mathematical Foundations of Elasticity (1983), New York, NY, USA: Dover Publications, New York, NY, USA · Zbl 0545.73031
[297] Ogden, R. W., Non-Linear Elastic Deformations (1984), New York, NY, USA: Halsted Press, New York, NY, USA · Zbl 0541.73044
[298] Oliver, J.; Linero, D. L.; Huespe, A. E.; Manzoli, O. L., Two-dimensional modeling of material failure in reinforced concrete by means of a continuum strong discontinuity approach, Computer Methods in Applied Mechanics and Engineering, 197, 5, 332-348 (2008) · Zbl 1169.74566 · doi:10.1016/j.cma.2007.05.017
[299] Belytschko, T.; Loehnert, S.; Song, J.-H., Multiscale aggregating discontinuities: a method for circumventing loss of material stability, International Journal for Numerical Methods in Engineering, 73, 6, 869-894 (2008) · Zbl 1195.74008 · doi:10.1002/nme.2156
[300] Meschke, G.; Dumstorff, P., Energy-based modeling of cohesive and cohesionless cracks via X-FEM, Computer Methods in Applied Mechanics and Engineering, 196, 21-24, 2338-2357 (2007) · Zbl 1173.74384 · doi:10.1016/j.cma.2006.11.016
[301] Dummerstorf, P.; Meschke, G., Crack propagation criteria in the framework of X-FEM-based structural analyses, International Journal for Numerical and Analytical Methods in Geomechanics, 31, 2, 239-259 (2007) · Zbl 1159.74038 · doi:10.1002/nag.560
[302] Belytschko, T.; Chen, H.; Xu, J.; Zi, G., Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment, International Journal for Numerical Methods in Engineering, 58, 12, 1873-1905 (2003) · Zbl 1032.74662 · doi:10.1002/nme.941
[303] Cundall, P. A.; Hart, R. D., Development of generalized 2-d and 3-d distinct element programs for modeling jointed rock, Misc. Paper, SL-85-1 (1985), US Army Corps of Engineers
[304] Cundall, P. A.; Strack, O. D. L., A discrete numerical model for granular assemblies, Geotechnique, 29, 1, 47-65 (1979)
[305] Shi, G. H.; Goodman, R. E., Two dimensional discontinuous deformation analysis, International Journal for Numerical & Analytical Methods in Geomechanics, 9, 6, 541-556 (1985) · Zbl 0573.73106
[306] Shi, G.-H.; Goodman, R. E., Generalization of two-dimensional discontinuous deformation analysis for forward modelling, International Journal for Numerical & Analytical Methods in Geomechanics, 13, 4, 359-380 (1989) · Zbl 0711.73202
[307] Cundall, P. A.; Konietzky, H., Pfc-ein neues werkzeug für numerische modellierungen, Bautechnik, 73, 8 (1996)
[308] Macek, R. W.; Silling, S. A., Peridynamics via finite element analysis, Finite Elements in Analysis and Design, 43, 15, 1169-1178 (2007) · doi:10.1016/j.finel.2007.08.012
[309] Liu, W. K.; Qian, D.; Gonella, S.; Li, S.; Chen, W.; Chirputkar, S., Multiscale methods for mechanical science of complex materials: bridging from quantum to stochastic multiresolution continuum, International Journal for Numerical Methods in Engineering, 83, 8-9, 1039-1080 (2010) · Zbl 1197.74007 · doi:10.1002/nme.2915
[310] Nouy, A.; Clément, A.; Schoefs, F.; Moës, N., An extended stochastic finite element method for solving stochastic partial differential equations on random domains, Computer Methods in Applied Mechanics and Engineering, 197, 51-52, 4663-4682 (2008) · Zbl 1194.74457 · doi:10.1016/j.cma.2008.06.010
[311] Grasa, J.; Bea, J. A.; Doblaré, M., A probabilistic extended finite element approach: application to the prediction of bone crack propagation, Key Engineering Materials, 348-349, 77-80 (2007)
[312] Grasa, J.; Bea, J. A.; Rodríguez, J. F.; Doblaré, M., The perturbation method and the extended finite element method. An application to fracture mechanics problems, Fatigue and Fracture of Engineering Materials and Structures, 29, 8, 581-587 (2006) · doi:10.1111/j.1460-2695.2006.01028.x
[313] Arias, I.; Serebrinsky, S.; Ortiz, M., A cohesive model of fatigue of ferroelectric materials under electro-mechanical cyclic loading, Smart Structures and Materials 2004, Active Materials: Behaviour and Mechanics · doi:10.1117/12.540097
[314] Lucas, L. J.; Owhadi, H.; Ortiz, M., Rigorous verification, validation, uncertainty quantification and certification through concentration-of-measure inequalities, Computer Methods in Applied Mechanics and Engineering, 197, 51-52, 4591-4609 (2008) · Zbl 1194.74550 · doi:10.1016/j.cma.2008.06.008
[315] Belytschko, T.; Song, J. H., Coarse-graining of multiscale crack propagation, International Journal for Numerical Methods in Engineering, 81, 5, 537-563 (2010) · Zbl 1183.74260 · doi:10.1002/nme.2694
[316] Horstemeyer, M. F., Multiscale modeling: a review, Practical Aspects of Computational Chemistry, 87-135 (2010) · doi:10.1007/978-90-481-2687-3_4
[317] Fish, J.; Yuan, Z., Multiscale enrichment based on partition of unity, International Journal for Numerical Methods in Engineering, 62, 10, 1341-1359 (2005) · Zbl 1078.74637 · doi:10.1002/nme.1230
[318] Gracie, R.; Belytschko, T., Concurrently coupled atomistic and XFEM models for dislocations and cracks, International Journal for Numerical Methods in Engineering, 78, 3, 354-378 (2009) · Zbl 1183.74278 · doi:10.1002/nme.2488
[319] Kouznetsova, V., Computational homogenization for the multi-scale analysis of multi-phase materials [Ph.D. thesis] (2002), Amsterdam, The Netherlands: Netherlands Institute for Metals Research, Amsterdam, The Netherlands
[320] Nemat-Nasser, S.; Hori, M., Micromechanics: Overall Properties of Heterogeneous Materials (1993), Amsterdam, The Netherlands: Elsevier, Amsterdam, The Netherlands · Zbl 0924.73006
[321] Gracie, R.; Oswald, J.; Belytschko, T., On a new extended finite element method for dislocations: core enrichment and nonlinear formulation, Journal of the Mechanics and Physics of Solids, 56, 1, 200-214 (2008) · Zbl 1162.74464 · doi:10.1016/j.jmps.2007.07.010
[322] Gracie, R.; Ventura, G.; Belytschko, T., A new fast finite element method for dislocations based on interior discontinuities, International Journal for Numerical Methods in Engineering, 69, 2, 423-441 (2007) · Zbl 1194.74402 · doi:10.1002/nme.1896
[323] Belytschko, T.; Gracie, R., On XFEM applications to dislocations and interfaces, International Journal of Plasticity, 23, 10-11, 1721-1738 (2007) · Zbl 1126.74046 · doi:10.1016/j.ijplas.2007.03.003
[324] Xu, M.; Gracie, R.; Belytschko, T.; Fish, J., Multiscale modeling with extended bridging domain method, Bridging the Scales in Science and Engineering (2002), Oxford University Press
[325] Feyel, F.; Chaboche, J. L., FE 2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials, Computer Methods in Applied Mechanics and Engineering, 183, 3-4, 309-330 (2000) · Zbl 0993.74062
[326] Kouznetsova, V.; Geers, M. G. D.; Brekelmans, W. A. M., Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme, International Journal for Numerical Methods in Engineering, 54, 8, 1235-1260 (2002) · Zbl 1058.74070 · doi:10.1002/nme.541
[327] Nguyen, V. P.; Lloberas-Valls, O.; Stroeven, M.; Sluys, L. J., Homogenization-based multiscale crack modelling: from micro-diffusive damage to macro-cracks, Computer Methods in Applied Mechanics and Engineering, 200, 9-12, 1220-1236 (2011) · Zbl 1225.74070 · doi:10.1016/j.cma.2010.10.013
[328] Verhoosel, C. V.; Remmers, J. J. C.; Gutiérrez, M. A.; de Borst, R., Computational homogenization for adhesive and cohesive failure in quasi-brittle solids, International Journal for Numerical Methods in Engineering, 83, 8-9, 1155-1179 (2010) · Zbl 1197.74139 · doi:10.1002/nme.2854
[329] Tadmor, E. B.; Ortiz, M.; Phillips, R., Quasicontinuum analysis of defects in solids, Philosophical Magazine A, 73, 6, 1529-1563 (1996)
[330] Miller, R. E.; Tadmor, E. B., The Quasicontinuum method: overview, applications and current directions, Journal of Computer-Aided Materials Design, 9, 3, 203-239 (2002) · doi:10.1023/A:1026098010127
[331] Dhia, H. B., The arlequin method: a partition of models for concurrent multiscale analyses, Proceedings of the Challenges in Computational Mechanics Workshop
[332] Abraham, F. F.; Broughton, J. Q.; Bernstein, N.; Kaxiras, E., Spanning the length scales in dynamic simulation, Computational Physics, 12, 538-546 (1998)
[333] Loehnert, S.; Belytschko, T., A multiscale projection method for macro/microcrack simulations, International Journal for Numerical Methods in Engineering, 71, 12, 1466-1482 (2007) · Zbl 1194.74436 · doi:10.1002/nme.2001
[334] Guidault, P. A.; Allix, O.; Champaney, L.; Cornuault, C., A multiscale extended finite element method for crack propagation, Computer Methods in Applied Mechanics and Engineering, 197, 5, 381-399 (2008) · Zbl 1169.74604 · doi:10.1016/j.cma.2007.07.023
[335] Aubertin, P.; Réthoré, J.; de Borst, R., Energy conservation of atomistic/continuum coupling, International Journal for Numerical Methods in Engineering, 78, 11, 1365-1386 (2009) · Zbl 1183.74011 · doi:10.1002/nme.2542
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.