Skip to main content
SpringerLink
Log in

A Comparative Review of XFEM, Mixed FEM and Phase-Field Models for Quasi-brittle Cracking

  • Review article
  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

In this work, a critical comparison between three different numerical approaches for the computational modelling of quasi-brittle structural failure is presented. Among the many finite element approaches devised to solve the problem, both using continuous and discontinuous methods, the present study examines the relative performance of the XFEM, the mixed strain/displacement FE and phase-field models. These numerical techniques are selected as the current representatives of embedded, smeared and regularized models for analyzing the phenomenon of fracture with different mathematical descriptions for the cracking induced discontinuities in the displacement and strain fields. The present investigation focusses on the main differences of the formulation of these models both at the continuum and discrete level and discusses the main assets and burdens that ensue in their practical application. The relative advantages and difficulties related to their use in the computation of localized structural failure in engineering practice are evaluated against a 10-point checklist that cover the main challenges met by these models. The paper includes an extensive comparison of selected numerical benchmark problems analyzed with the three examined methods. Relative performance is assessed in terms of load capacity, force–displacement curves, crack paths, collapse mechanisms, cost-efficiency and other key issues.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36
Fig. 37:
Fig. 38
Fig. 39
Fig. 40
Fig. 41
Fig. 42
Fig. 43
Fig. 44
Fig. 45
Fig. 46
Fig. 47
Fig. 48
Fig. 49
Fig. 50
Fig. 51
Fig. 52
Fig. 53
Fig. 54
Fig. 55
Fig. 56
Fig. 57
Fig. 58
Fig. 59
Fig. 60
Fig. 61
Fig. 62
Fig. 63
Fig. 64
Fig. 65
Fig. 66
Fig. 67
Fig. 68
Fig. 69
Fig. 70
Fig. 71
Fig. 72
Fig. 73
Fig. 74
Fig. 75
Fig. 76
Fig. 77
Fig. 78
Fig. 79
Fig. 80
Fig. 81
Fig. 82
Fig. 83
Fig. 84
Fig. 85

Similar content being viewed by others

References

  1. Rots J, Nauta P, Kusters G, Blauwendraad J (1985) Smeared crack approach and fracture localization in concrete. Heron 30:1–48

    Google Scholar 

  2. Rots J (1988) Computational modeling of concrete fracture. Ph.D. thesis, Delft University of Technology

  3. Oliver J, Cervera M, Oller S, Lubliner J (1990) Isotropic damage models and smeared crack analysis of concrete. In: II International conference on computer aided analysis and design of concrete, pp 945–957

  4. Sluys L (1992) Wave propagation localisation and dispersion in softening solids. Ph.D. thesis, Delft University of Technology

  5. Hillerborg A, Modeer M, Petersson P-E (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 6(6):773–781

    Google Scholar 

  6. Hillerborg A (1978) A model for fracture analysis, report TVBM-3005, Division of Building Materials, Lund Institute of Technology, Sweden

  7. Modeer M (1979) A fracture mechanics approach to failure analyses of concrete materials, report TVBM-1001, Division of Building Materials, University of Lund, Sweden

  8. Petersson P-E (1981) Crack growth and development of fracture zones in plain concrete and similar materials, report TVBM-1006, Division of Building Materials, Lund Institute of Technology, Sweden

  9. Gustafsson P-J (1985) Fracture mechanics studies of non-yielding materials like concrete: modelling of tensile fracture and applied strength analyses, report TVBM-1007, Division of Building Materials, Lund University, Sweden

  10. Rashid Y (1968) Ultimate strength analysis of prestressed concrete pressure vessels. Nucl Eng Des 7(4):334–344

    Google Scholar 

  11. Dugdale D (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8(2):100–104

    Google Scholar 

  12. Barenblatt G (1962) The mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech 7:55–129

    MathSciNet  Google Scholar 

  13. Cervera M, Wu J-Y (2015) On the conformity of strong, regularized, embedded and smeared discontinuity approaches for the modelling of localized faillure in solids. Int J Solids Struct 71:19–38

    Google Scholar 

  14. Wu J, Cervera M (2015) On the equivalence between traction- and stress-based approaches for the modeling of localized failure in solids. J Mech Phys Solids 82:137–163

    MathSciNet  Google Scholar 

  15. Ortiz M, Leroy Y, Needleman A (1987) A finite element method for localized failure analysis. Comput Methods Appl Mech Eng 61(2):189–214

    MATH  Google Scholar 

  16. Belytschko T, Fish J, Engelmann B (1988) A finite element with embedded localization zones. Comput Methods Appl Mech Eng 70(1):59–89

    MATH  Google Scholar 

  17. Dvorkin E, Cuitino A, Gioia G (1990) Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distorsions. Int J Numer Methods Eng 30:541–564

    MATH  Google Scholar 

  18. Oliver J, Huespe A, Pulido M, Chaves E (2002) From continuum mechanics to fracture mechanics: the strong discontinuity approach. Eng Fract Mech 69:113–136

    Google Scholar 

  19. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150

    MathSciNet  MATH  Google Scholar 

  20. Dolbow J, Moës N, Belytschko T (2000) Discontinuous enrichment in finite elements with a partition of unity method. Finite Elem Anal Des 36(3–4):235–260

    MATH  Google Scholar 

  21. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620

    MATH  Google Scholar 

  22. Meschke G, Dumstorff P (2007) Energy-based modeling of cohesive and cohesionless cracks via X-FEM. Comput Methods Appl Mech Eng 196:2338–2357

    MATH  Google Scholar 

  23. Bazant Z, Oh B (1983) Crack band theory for fracture of concrete. Matér Constr 16(3):155–177

    Google Scholar 

  24. Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finite element methods in nonlinear solid mechanics. Part I: formulation. Comput Methods Appl Mech Eng 199(37–40):2559–2570

    MathSciNet  MATH  Google Scholar 

  25. Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finite element methods in nonlinear solid mechanics. Part II: strain localization. Comput Methods Appl Mech Eng 199(37–40):2571–2589

    MathSciNet  MATH  Google Scholar 

  26. Cervera M, Chiumenti M (2006) Smeared crack approach: back to the original track. Int J Numer Anal Methods Geomech 30:1173–1199

    MATH  Google Scholar 

  27. Cervera M, Chiumenti M (2006) Mesh objective tensile cracking via a local continuum damage model and a crack tracking technique. Computer Methods Appl Mech Eng 196(1–3):304–320

    MATH  Google Scholar 

  28. Cervera M, Barbat G, Chiumenti M (2017) Finite element modelling of quasi-brittle cracks in 2D and 3D with enhanced strain accuracy. Comput Mech 60(5):767–796

    MathSciNet  MATH  Google Scholar 

  29. Barbat G, Cervera M, Chiumenti M (2018) Appraisement of planar, bending and twisting cracks in 3D with isotropic and orthotropic damage models. Int J Fract 210(1–2):45–79

    Google Scholar 

  30. Vlachakis G, Cervera M, Barbat G, Saloustros S (2019) Out-of-plane seismic response and faillure mechanism of masonry structures using finite elements with enhanced strain accuracy. Eng Fail Anal 97:534–555

    Google Scholar 

  31. Cervera M, Barbat G, Chiumenti M (2020) Architecture of a multi-crack model with full closing, reopening and sliding capabilities. Comput Mech 65(6):1593–1620

  32. Barbat G, Cervera M, Chiumenti M, Espinoza E (2020) Structural size effect: experimental, theoretical and accurate computational assessment. Eng Struct 213:110555

  33. Cervera M, Chiumenti M, Codina R (2011) Mesh objective modeling of cracks using continuous linear strain and displacement interpolations. Int J Numer Methods Eng 87(10):962–987

    MathSciNet  MATH  Google Scholar 

  34. Pijaudier-Cabot G, Bazant Z (1987) Nonlocal damage theory. J Eng Mech 113(10):1512–1533

    MATH  Google Scholar 

  35. Bazant Z, Pijaudier-Cabot G (1988) Nonlocal continum damage, localization instabilities and convergence. J Eng Mech 55:287–293

    MATH  Google Scholar 

  36. Bazant Z, Lin F-B (1988) Nonlocal smeared cracking model for concrete fracture. J Struct Eng 114(11):2493–2510

    Google Scholar 

  37. Jirasek M, Zimmermann T (1998) Rotating crack model with transition to scalar damage. ASCE J Eng Mech 124(3):277–284

    Google Scholar 

  38. Giry C, Dufour F, Mazars J (2011) Stress-based nonlocal damage model. Int J Solids Struct 48:3431–3443

    Google Scholar 

  39. Peerlings R, de Borst R, Brekelmans W, de Vree J (1996) Gradient enhanced damage for quasi brittle materials. Int J Numer Methods Eng 39:3391–3403

    MATH  Google Scholar 

  40. Peerlings R, de Borst R, Brekelmans W, Geers M (1998) Gradient-enhanced damage modelling of concrete fracture. Mech Cohesive Frict Mater 3:323–342

    Google Scholar 

  41. Geers M, de Borst R, Brekelmans W, Peerlings R (1998) Strain-based transient-gradient damage model for failure analyses. Comput Methods Appl Mech Eng 160:133–153

    MATH  Google Scholar 

  42. Geers M, de Borst R, Brekelmans W, Peerlings R (1999) Validation and internal length scale determination for a gradient damage model: application to short glass-fibre-reinforced polypropylene. Int J Solids Struct 36:2557–2583

    MATH  Google Scholar 

  43. Comi C (1999) Computational modelling of gradient-enhanced damage in quasi-brittle materials. Mech Cohesive Frict Mater 4:17–36

    Google Scholar 

  44. Saroukhani S, Vafadari R, Simone A (2013) A simplifiend implementation of a gradient-enhanced damage model with transient length scale effects. Comput Mech 51:899–909

    MathSciNet  MATH  Google Scholar 

  45. Bourdin B, Francfort G, Marigo J-J (2000) Numerical experiments in revisited brittle fracture. J Mech Phys Solids 48(4):797–826

    MathSciNet  MATH  Google Scholar 

  46. Bourdin B, Francfort G, Marigo J (2008) The variational approach to fracture. J Elast 91:5–148

    MathSciNet  MATH  Google Scholar 

  47. Miehe C, Hofacker M, Welschinger F (2010) A phase field model for rate-independent crack propagation: robust algorithmic implementation based on operator splits. Comput Methods Appl Mech Eng 199:2765–2778

    MathSciNet  MATH  Google Scholar 

  48. Miehe C, Welschinger F, Hofacker M (2010) Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int J Numer Methods Eng 83:1273–1311

    MathSciNet  MATH  Google Scholar 

  49. Borden M, Verhoosel C, Scott M, Hughes T, Landis C (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95

    MathSciNet  MATH  Google Scholar 

  50. Miehe C, Schänzel L-M, Ulmer H (2015) Phase field modeling of fracture in multi-physics problems. Part I. Balance of cracks surface and failure criteria for brittle crack propagation in thermo-elastic solids. Comput Methods Appl Mech Eng 294:449–485

    MathSciNet  MATH  Google Scholar 

  51. Griffith A (1920) The phenomena of rupture and flow in solids. Philos Trans R Soc CCXXI-A:163–198

    MATH  Google Scholar 

  52. Chambolle A, Francfort G, Marigo J-J (2010) Revisiting energy release rates in brittle fracture. J Nonlinear Sci 20:395–424

    MathSciNet  MATH  Google Scholar 

  53. Braides A (1998) Approximation of free-discontinuity problems. Springer, Berlin

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  55. Benedetti L, Cervera M, Chiumenti M (2016) High-fidelity prediction of crack formation in 2D and 3D pullout tests. Comput Struct 172:93–109

    Google Scholar 

  56. Wu J-Y (2011) Unified analysis of enriched finite elements for modeling cohesive cracks. Comput Methods Appl Mech Eng 200(45–46):3031–3050

    MathSciNet  MATH  Google Scholar 

  57. Patzak B, Jirasek M (2003) Process zone resolution by extended finite elements. Eng Fract Mech 70(7–8):957–977

    Google Scholar 

  58. Benvenuti E (2008) A regularized XFEM framework for embedded cohesive interfaces. Comput Methods Appl Mech Eng 197(49–50):4367–4378

    MATH  Google Scholar 

  59. Cervera M (2008) An orthotropic mesh corrected crack model. Comput Methods Appl Mech Eng 197(17–18):1603–1619

    MATH  Google Scholar 

  60. Cervera M (2008) A smeared-embedded mesh corrected damage model for tensile cracking. Int J Numer Methods Eng 76(12):1930–1954

    MathSciNet  MATH  Google Scholar 

  61. Jirasek M (2000) Comparative study on finite elements with embedded discontinuities. Comput Methods Appl Mech Eng 188:307–330

    MATH  Google Scholar 

  62. Faria R, Oliver J, Cervera M (2004) Modeling material failure in concrete structures under cyclic actions. J Struct Eng 130(12):1997–2005

  63. Cornelissen H, Hordijk D, Reinhardt H (1986) Experimental determination of crack softening characteristics of normalweight and lightweight concrete. Heron 31(2):45–56

    Google Scholar 

  64. Wittmann F, Rokugo K, Bruhwiler E, Mihashi H, Simonin P (1988) Fracture energy and strain softening of concrete as determined by means of compact tension specimens. Mater Struct 21:21–32

    Google Scholar 

  65. Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer Methods Eng 44:1267–1282

    MATH  Google Scholar 

  66. Teichtmeister S, Kienle D, Aldakheel F, Keip M-A (2017) Phase field modeling of fracture in anisotropic brittle solids. Int J Non-Linear Mech 97:1–21

    Google Scholar 

  67. Li B, Maurini C (2019) Crack kinking in a variational phase-field model of brittle fracture with strongly anisotropic surface energy. J Mech Phys Solids 125:502–522

    MathSciNet  MATH  Google Scholar 

  68. Bleyer J, Alessi R (2018) Phase-field modeling of anisotropic brittle fracture including several damage mechanisms. Comput Methods Appl Mech Eng 336:213–236

    MathSciNet  MATH  Google Scholar 

  69. Vignollet J, May S, de Borst R, Verhoosel C (2014) Phase-field models for brittle and cohesive fracture. Meccanica 49:2587–2601

    MathSciNet  Google Scholar 

  70. Kumar A, Bourdin B, Francfort G, Lopez-Pamies O (2020) Revisiting nucleation in the phase-field approach to brittle fracture. J Mech Phys Solids 142:104027

    MathSciNet  Google Scholar 

  71. Wu J-Y, Cervera M (2017) Strain localization analysis of elastic-damaging frictional-cohesive materials: analytical results and numerical verification. Materials 10(4):434

    Google Scholar 

  72. Cervera M, Wu J-Y, Chiumenti M, Kim S (2019) Strain localization analysis of Hill’s orthotropic elastoplasticity: analytical results and numerical verification. Comput Mech 65:533–554

    MathSciNet  MATH  Google Scholar 

  73. Wu J-Y, Cervera M (2016) A thermodynamically consistent plastic-damage framework for localized failure in quasi-brittle solids: material model and strain localization analysis. Int J Solids Struct 88:227–247

    Google Scholar 

  74. Mohr O (1900) Umstande bedingen die Elastizitasgrenze und den Bruch eines Materials? Zeitschrift des Vereines Deutscher Ingenieure 44:1524–1530

    MATH  Google Scholar 

  75. Weihe S, Ohmenhauser F, Kroplin B (1998) A phenomenological approach to decompose geometrical and constitutive aspects of failure induced anisotropy. Comput Mater Sci 13(1–3):177–194

    Google Scholar 

  76. Keller K, Weihe S, Siegmund T, Kroplin B (1999) Generalized cohesive zone model: incorporating triaxiality dependent failure mechanisms. Comput Mater Sci 16(1–4):267–274

    Google Scholar 

  77. Ohmenhauser F, Weihe S, Kroplin B (1999) Algorithmic implementation of a generalized cohesive crack model. Comput Mater Sci 16(1–4):294–306

    Google Scholar 

  78. Oliver J, Huespe A, Samaniego E, Chaves E (2004) Continuum approach to the numerical simulation of material failure in concrete. Int J Numer Anal Methods Geomech 28:609–632

    MATH  Google Scholar 

  79. Oliver J, Huespe A (2004) Continuum approach to material failure in strong discontinuity settings. Comput Methods Appl Mech Eng 193:3195–3220

    MathSciNet  MATH  Google Scholar 

  80. Mosler J, Meschke G (2004) Embedded crack vs. smeared crack models: a comparison of elementwise discontinuous crack path approaches with emphasis on mesh bias. Comput Methods Appl Mech Eng 193(30–32):3351–3375

    MATH  Google Scholar 

  81. Feist C, Hofstetter G (2006) An embedded strong discontinuity model for cracking of plain concrete. Comput Methods Appl Mech Eng 195:7115–7138

    MathSciNet  MATH  Google Scholar 

  82. Saloustros S, Cervera M, Pela L (2019) Challenges, tools and applications of tracking algorithms in the numerical modelling of cracks in concrete and masonry structures. Arch Comput Methods Eng 26:961–1005

    MathSciNet  Google Scholar 

  83. Saloustros S, Cervera M, Pela L, Roca P (2018) Tracking multi-directional intersecting cracks in masonry shear walls under cyclic loading. Meccanica 53:1757–1776

    MathSciNet  Google Scholar 

  84. Saloustros S, Pela L, Cervera M, Roca P (2017) Finite element modelling of internal and multiple localized cracks. Comput Mech 59:299–316

    Google Scholar 

  85. Saloustros S, Pela L, Cervera M (2015) A crack-tracking technique for localized cohesive-frictional damage. Eng Fract Mech 150:96–114

    Google Scholar 

  86. Jirasek M, Zimmermann T (2001) Embedded crack model. Part II: combination with smeared cracks. Int J Numer Methods Eng 50:1291–1305

    MATH  Google Scholar 

  87. Bazant Z, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca Raton

    Google Scholar 

  88. ACI, Building Code Requirements for Structural Concrete (ACI 318-19) (2019)

  89. Bazant Z, Zi G, McClung D (2003) Size effect law and fracture mechanics of the triggering of dry snow slab avalanches. J Geophys Res 108(B2):2119

  90. Bazant Z, Guo Z (2002) Size effect on strength of floating sea ice under vertical line load. J Eng Mech 128(3):254–263

    Google Scholar 

  91. Bazant Z (2000) Size effect. Int J Solids Struct 37:69–80

    MathSciNet  MATH  Google Scholar 

  92. Rasoolinejad M, Donmez A, Bazant Z (2020) Fracture and size effect suppression by mesh reinforcement of concrete and justification of empirical shrinkage and temperature reinforcement in design codes. J Eng Mech 146(10):04020120

    Google Scholar 

  93. Bazant Z (1999) Size effect on structural strength: a review. Arch Appl Mech 69:703–725

    MATH  Google Scholar 

  94. Bazant Z (1985) Fracture in concrete and reinforced concrete. In: Mechanics of geomaterials, pp 259–303

  95. Bazant Z, Chen E-P (1997) Scaling of structural failure. Sandia report SAND96-2948, Albuquerque, New Mexico

  96. Cervera M, Chiumenti M (2009) Size effect and localization in J2 plasticity. Int J Solinds Struct 46:3301–3312

    MATH  Google Scholar 

  97. Bazant Z, Ozbolt J, Eligehausen R (1994) Fracture size effect: review of evidence for concrete structures. J Struct Eng 120(8):2377–2398

    Google Scholar 

  98. Bazant Z (2002) Concrete fracture models: testing and practice. Eng Fract Mech 69:165–205

    Google Scholar 

  99. Bazant Z, Yu Q (2009) Universal size effect law and effect of crack depth on quasi-brittle structure strength. J Eng Mech 135(2):78–84

    Google Scholar 

  100. Jones N (1989) Structural impact. Cambridge University Press, Cambridge

    Google Scholar 

  101. Oshiro R, Alves M (2004) Scaling impacted structures. Arch Appl Mech 74:130–145

    MATH  Google Scholar 

  102. Alves M, Oshiro R, Calle M, Mazzariol L (2017) Scaling and structural impact. Procedia Eng 173:391–396

    Google Scholar 

  103. Moes N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fract Mech 69:813–833

    Google Scholar 

  104. Melenk J, Babuska I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139(1–4):289–314

    MathSciNet  MATH  Google Scholar 

  105. Babuska I, Melenk J (1997) The partition of unity method. Int J Numer Methods Eng 40(4):727–758

    MathSciNet  MATH  Google Scholar 

  106. Strouboulis T, Babuska I, Copps K (2000) The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 181:43–69

    MathSciNet  MATH  Google Scholar 

  107. Duarte C, Babuska I, Oden J (2000) Generalized finite element methods for three-dimensional structural mechanics problems. Comput Struct 77(2):215–232

    MathSciNet  Google Scholar 

  108. Strouboulis T, Copps K, Babuska I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190(32–33):4081–4193

    MathSciNet  MATH  Google Scholar 

  109. Belytschko T, Gracie R, Ventura G (2009) A review of extended/generalized finite element methods for material modelling. Model Simul Mater Sci Eng 17(4):043001

    Google Scholar 

  110. Fries T, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304

    MathSciNet  MATH  Google Scholar 

  111. Babuska I, Banerjee U (2011) Stable generalized finite element method (SGFEM). Comput Methods Appl Mech Eng 201–204:91–111

    MathSciNet  MATH  Google Scholar 

  112. Dias-da-Costa D, Alfaiate J, Sluys L, Julio E (2010) A comparative study on the modelling of discontinuous fracture by means of enriched nodal and element techniques and interface elements. Int J Fract 161:97–119

    MATH  Google Scholar 

  113. Sukumar N, Dolbow J, Moes N (2015) Extended finite element method in computational fracture mechanics: a retrospective examination. Int J Fract 196:189–206

    Google Scholar 

  114. Egger A, Pillai U, Agathos K, Kakouris E, Chatzi E, Aschroft I, Triantafyllou S (2019) Discrete and phase field methods for linear elastic fracture mechanics: a comparative study and state-of-the-art review. Appl Sci 9(12):2436

    Google Scholar 

  115. Clough R (1962) The stress distribution of Norfork Dam. Struct Mater Res 100(19)

  116. Ngo D, Scordelis A (1967) Finite element analysis of reinforced concrete beams. Am Concr Inst J 64(3):152–163

    Google Scholar 

  117. Nilson A (1968) Nonlinear analysis of reinforced concrete by the finite element method. Am Concr Inst J 65(9):757–776

    Google Scholar 

  118. Ingraffea A (1977) Discrete fracture propagation in rock: laboratory tests and finite element analysis. Ph.D. thesis, University of Colorado, Boulder

  119. Saouma V (1981) Interactive finite element analysis of reinforced concrete: a fracture mechanics approach. Ph.D. thesis, Cornell University, NY

  120. Shephard M, Yehia N, Burd G, Weidner T (1985) Automatic crack propagation tracking. Comput Struct 20(1–3):211–223

    Google Scholar 

  121. Wawrzynek P, Ingraffea A (1987) Interactive finite element analysis of fracture processes: an integrated approach. Theor Appl Fract Mech 8(2):137–150

    Google Scholar 

  122. Camacho G, Ortiz M (1996) Computational modelling of impact damage in brittle materials. Int J Solids Struct 33(20–22):2899–2938

    MATH  Google Scholar 

  123. Xu X, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42(9):1397–1434

    MATH  Google Scholar 

  124. Zavattieri P (2006) Modeling of crack propagation in thin-walled structures using a cohesive model for shell elements. J Appl Mech 73(6):948–958

    MATH  Google Scholar 

  125. Tijssens M, Sluys B, van der Giessen E (2000) Numerical simulation of quasi-brittle fracture using damaging cohesive surfaces. Eur J Mech A/Solids 19(5):761–779

    MATH  Google Scholar 

  126. Falk M, Needleman A, Rice J (2001) A critical evaluation of cohesive zone models of dynamic fracture. J Phys IV 11(PR5):43–50

    Google Scholar 

  127. Nguyen-Xuan H, Liu G, Nourbakhshnia N, Chen L (2012) A novel singular ES-FEM for crack growth simulation. Eng Fract Mech 84:41–66

    Google Scholar 

  128. Nourbakhshnia N, Liu G (2011) A quasi-static growth simulation based on the singular ES-FEM. Int J Numer Methods Eng 88(5):473–492

    MathSciNet  MATH  Google Scholar 

  129. Rabczuk T (2013) Computational methods for fracture in brittle and quasi-brittle solids: state-of-the-art review and future perspectives. ISRN Appl Math 2013:38

    MathSciNet  MATH  Google Scholar 

  130. ACI Committee 446 (1997) Finite element analysis of fracture in concrete structures: state-of-the-art (ACI 446.3R-97). American Concrete Institute

  131. Jirasek M, Zimmermann T (2001) Embedded crack model: I. Basic formulation. Int J Numer Methods Eng 50:1269–1290

    MATH  Google Scholar 

  132. Zhang Y, Lackner R, Zeiml M, Mang H (2015) Strong discontinuity embedded approach with standard SOS formulation: element formulation, energy-based crack-tracking strategy, and validations. Comput Methods Appl Mech Eng 287:335–366

    MathSciNet  MATH  Google Scholar 

  133. Zhang Y, Zhuang X (2018) Cracking elements: a self-propagating strong discontinuity embedded approach for quasi-brittle fracture. Finite Elem Anal Des 144:84–100

    MathSciNet  Google Scholar 

  134. Oliver J, Huespe A, Sanchez P (2006) A comparative study on finite elements for capturing strong discontinuities: E-FEM vs X-FEM. Comput Methods Appl Mech Eng 195(37–40):4732–4752

    MathSciNet  MATH  Google Scholar 

  135. Dias-da-Costa D, Alfaiate J, Sluys L, Areias P, Julio E (2013) An embedded formulation with conforming finite elements to capture strong discontinuities. Int J Numer Methods Eng 93(2):224–244

    MathSciNet  MATH  Google Scholar 

  136. Wu J-Y, Li F-B (2015) An improved stable X-FEM (Is-FEM) with a novel enrichment function for the computational modeling of cohesive cracks. Comput Methods Appl Mech Eng 295:77–107

    MATH  Google Scholar 

  137. Li W, Wu J-Y (2018) A consistent and efficient localized damage model for concrete. Int J Damage Mech 27(4):541–567

    Google Scholar 

  138. Wu J-Y, Qiu J-F, Nguyen V, Mandal T, Zhuang L-J (2019) Computational modeling of localized failure in solids: XFEM vs PF-CZM. Comput Methods Appl Mech Eng 345:618–643

    MathSciNet  MATH  Google Scholar 

  139. Wu J-Y, Li F-B, Xu S-L (2015) Extended embedded finite elements with continuous displacement jumps for the modeling of localized failure in solids. Comput Methods Appl Mech Eng 285:346–378

    MathSciNet  MATH  Google Scholar 

  140. Simo J, Oliver J, Armero F (1993) An analysis of strong discontinuities induced by strain softening in rate-independnt inelastic solids. Comput Mech 12:277–296

    MathSciNet  MATH  Google Scholar 

  141. Oliver J, Cervera M, Manzoli O (1999) Strong discontinuities and continuum plasticity models: the strong discontinuity approach. Int J Plast 15:319–351

    MATH  Google Scholar 

  142. Hughes T (1995) Multiscale phenomena: Green’s function, Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized formulations. Comput Methods Appl Mech Eng 127(1–4):387–401

    MATH  Google Scholar 

  143. Hughes T, Feijoo G, Mazzei L, Quincy J-B (1998) The variational multiscale method—a paradigm for computational mechanics. Comput Methods Appl Mech Eng 166(1–2):3–24

    MathSciNet  MATH  Google Scholar 

  144. Vagbharathi A, Gopalakrishnan S (2004) An extended finite-element model coupled with level set method for analysis of growth of corrosion pits in metallic structures. Proc R Soc A Math Phys Eng Sci 470:20140001

    Google Scholar 

  145. Cazes F, Meschke G, Zhou M-M (2016) Strong discontinuity approaches: An algorithm for robust performance and comparative assessment of accuracy. Int J Solids Struct 96:355–379

    Google Scholar 

  146. Simone A (2007) Partition of unity-based discontinuous finite elements: GFEM, PUFEM, XFEM. Revue Europeene de Genie Civil 11(7–8):1045–1068

    Google Scholar 

  147. Armero F, Kim J (2012) Three-dimensional finite elements with embedded strong discontinuities to model material failure in the infinitesimal range. Int J Numer Methods Eng 91(12):1291–1330

    MathSciNet  Google Scholar 

  148. Wu J-Y, Cervera M (2016) Thermodynamically consistent plastic-damage framework for localized failure in quasi-brittle solids: material model and strain localization analysis. Int J Solids Struct 88–89:227–247

    Google Scholar 

  149. Bordas S, Rabczuk T, Hung N, Nguyen V, Natarajan S, Bog T, Quan D, Hiep N (2010) Strain smoothing in FEM and XFEM. Comput Struct 88(23–24):1419–1443

    Google Scholar 

  150. Vu-Bac N, Nguyen-Xuan H, Chen L, Bordas S, Kerfriden P, Simpson R, Liu G, Rabczuk T (2011) A node-based smoothed extended finite element method (NS-XFEM) for fracture analysis. Comput Model Eng Sci 73(4):331–355

    MathSciNet  MATH  Google Scholar 

  151. Moes N, Gravouil A, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets—part I: mechanical model. Int J Numer Methods Eng 53:2549–2568

    MATH  Google Scholar 

  152. Bechet E (2005) Improved implementation and robustness study of the X-FEM for stress analysis around cracks. Int J Numer Methods Eng 64(8):1033–1056

    MATH  Google Scholar 

  153. Ventura G (2006) On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite-Element Method. Int J Numer Methods Eng 66:761–795

    MathSciNet  MATH  Google Scholar 

  154. Holdych D, Noble D, Secor R (2008) Quadrature rules for triangular and tetrahedral elements with generalized functions. Int J Numer Methods Eng 73:1310–1327

    MathSciNet  MATH  Google Scholar 

  155. Ventura G, Benvenuti E (2015) Equivalent polynomials for quadrature in Heaviside function enriched elements. Int J Numer Methods Eng 102(3–4):688–710

    MathSciNet  MATH  Google Scholar 

  156. Simone A (2004) Partition of unity-based discontinuous elements for interface phenomena: computational issues. Commun Numer Methods Eng 20:465–478

    MATH  Google Scholar 

  157. Fries T-P (2018) Extended finite element methods (XFEM). In: Encyclopedia of continuum mechanics. Springer, Berlin, pp 1–10

  158. Benvenuti E, Tralli A (2012) Simulation of finite-width process zone in concrete-like materials by means of a regularized extended finite element model. Comput Mech 50:479–497

    MATH  Google Scholar 

  159. Benvenuti E, Ventura G, Ponara N, Tralli A (2013) Variationally consistent eXtended FE model for 3D planar and curved imperfect interfaces. Comput Methods Appl Mech Eng 267:434–457

    MathSciNet  MATH  Google Scholar 

  160. Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193(33–35):3523–3540

    MathSciNet  MATH  Google Scholar 

  161. Song J-H, Areias P, Belytschko T (2006) A method for dynamic crack and shear band propagation with phantom nodes. Int J Numer Methods Eng 67(6):868–893

    MATH  Google Scholar 

  162. Rabczuk T, Zi G, Gerstenberger A, Wall W (2008) A new crack tip element for the phantom-node method with arbitrary cohesive cracks. Int J Numer Methods Eng 75(5):577–599

    MATH  Google Scholar 

  163. Chau-Dinh T, Zi G, Lee P-S, Rabczuk T, Song J-H (2012) Phantom-node method for shell models with arbitrary cracks. Comput Struct 92–93:242–256

    Google Scholar 

  164. Vu-Bac N, Nguyen-Xuan H, Chen L, Lee C, Zi G, Zhuang X, Liu G, Rabczuk T (2013) A phantom-node method with edge-based strain smoothing for linear elastic fracture mechanics. J Appl Math 2013:978026

    MathSciNet  MATH  Google Scholar 

  165. Xu D, Liu Z, Liu X, Zeng Q, Zhuang Z (2014) Modeling of dynamic crack branching by enhanced extended finite element method. Comput Mech 54:489–502

    MathSciNet  MATH  Google Scholar 

  166. Zhao J, Tang H, Xue S (2018) Peridynamics versus XFEM: a comparative study for quasi-static crack problems. Front Struct Civ Eng 12:548–557

    Google Scholar 

  167. Richardson C, Hegemann J, Sifakis E, Hellrung J, Teran J (2011) An XFEM method for modeling geometrically elaborate crack propagation in brittle materials. Int J Numer Methods Eng 88(10):1042–1065

    MathSciNet  MATH  Google Scholar 

  168. Daux C, Moes N, Dolbow J, Sukumar N, Belytschko T (2000) Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Methods Eng 48:1741–1760

    MATH  Google Scholar 

  169. Budyn E, Zi G, Moes N, Belytschko T (2004) A method for multiple crack growth in brittle materials without remeshing. Int J Numer Methods Eng 61:1741–1770

    MATH  Google Scholar 

  170. Zi G, Song J, Budyn E, Lee S, Belytschko T (2004) A method for growing multiple cracks without remeshing and its application to fatigue crack growth. Modell Simul Mater Sci Eng 12(5):901–915

    Google Scholar 

  171. Sukumar N, Moes N, Moran B, Belytschko T (2000) Extended finite element method for three-dimensional crack modelling. Int J Numer Methods Eng 48(11):1549–1570

    MATH  Google Scholar 

  172. Ahmed A (2009) eXtended Finite Element Method (XFEM)—modeling arbitrary discontinuities and failure analysis. Ph.D. thesis, Universita degli Studi di Pavia

  173. Jager P, Steinmann P, Kuhl E (2008) Modeling three-dimensional crack propagation—a comparison of crack path tracking strategies. Int J Numer Methods Eng 76:1328–1352

    MathSciNet  MATH  Google Scholar 

  174. Gasser T, Holapfel G (2006) 3D Crack propagation in unreinforced concrete. A two-step algorithm for tracking 3D crack paths. Comput Methods Appl Mech Eng 195(37–40):5198–5219

    MathSciNet  MATH  Google Scholar 

  175. Reissner E (1950) On a variational theorem in elasticity. J Math Phys 29(1–4):90–95

    MathSciNet  MATH  Google Scholar 

  176. Fraeijs de Veubeke B (1965) Displacement and equilibrium models in the finite element method. In: Zienkiewicz OZ, Holister GS (eds) Stress analysis. Wiley, London, pp 145–197

    Google Scholar 

  177. Herrmann L (1965) Elasticity equations for incompressible and nearly incompressible materials by a variational theorem. Am Inst Aeronaut Astronaut J 3(10):1896–1900

    MathSciNet  Google Scholar 

  178. Dunham R, Pister K (1968) A finite element application of the Hellinger-Reissner variational theorem. In: Proceedings of the second conference on matrix methods in structural mechanics, pp 471–487

  179. Prato C (1968) A mixed finite element method for thin shell analysis. Ph.D. thesis, MIT

  180. Prato C (1969) Shell finite element method via Reissner’s principle. Int J Solids Struct 5:1119–1133

    MATH  Google Scholar 

  181. Herrmann L, Campbell D (1968) A finite-element analysis for thin shells. Am Inst Aeronaut Astronaut J 6(10):1842–1847

    MATH  Google Scholar 

  182. Herrmann L, Mason W (1970) Mixed formulations for finite element shell analyses. In: Conference on computer oriented analysis of shell structures, Palo Alto, California

  183. Taylor R, Pister K, Herrmann L (1968) On a variational theorem for incompressible and nearly-incompressible orthotropic elasticity. Int J Solids Struct 4:875–883

    MATH  Google Scholar 

  184. Pian T, Tong P (1969) Basis of finite element methods for solid continua. Int J Numer Methods Eng 1:3–28

    MATH  Google Scholar 

  185. Zienkiewicz O (1973) Finite elements—the background story. In: Mathematics of finite elements and applications. Academic Press, New York, pp 1–35

  186. Misra D, Nieber J (2018) A comparative analysis of mixed finite element and conventional finite element methods for one-dimensional steady heterogeneous Darcy flow. In: Flow and transport in subsurface environment. Springer, pp 141–187

  187. Chiumenti M, Valverde Q, Agelet de Saracibar C, Cervera M (2002) A stabilized formulation for incompressible elasticity using linear displacement and pressure interpolations. Comput Methods Appl Mech Eng 191(46):5253–5264

    MathSciNet  MATH  Google Scholar 

  188. Chiumenti M, Valverde Q, Agelet de Saracibar C, Cervera M (2004) A stabilized formulation for incompressible plasticity using linear triangles and tetrahedra. Int J Plast 20(8–9):1487–1504

    MATH  Google Scholar 

  189. Chiumenti M, Cervera M, Codina R (2015) A mixed three-field FE formulation for stress accurate analysis including the incompressible limit. Comput Methods Appl Mech Eng 286:1095–1116

    MathSciNet  MATH  Google Scholar 

  190. Allen M, Ewing R, Koebbe J (1985) Mixed finite element methods for computing groundwater velocities. Numer Methods Partial Differ Equ 3:195–207

    MATH  Google Scholar 

  191. Oden J, Wellford L (1972) Analysis of Flow of viscous fluids by the finite-element method. Am Inst Aeronaut Astronaut 10(12):1590–1599

    MATH  Google Scholar 

  192. Kawahara M, Takeuchi N (1977) Mixed finite element method for analysis of viscoelastic fluid flow. Comput Fluids 5(1):33–45

    MATH  Google Scholar 

  193. Herrmann L (1967) Finite-element bending analysis for plates. J Eng Mech Div 93(5):13–26

    Google Scholar 

  194. Poceski A (1992) Mixed finite element method. Springer, Berlin

    MATH  Google Scholar 

  195. Cervera M, Chiumenti M, Benedetti L, Codina R (2015) Mixed stabilized finite element methods in nonlinear solid mechanics. Part III: compressible and incompressible plasticity. Comput Methods Appl Mech Eng 285:752–775

    MathSciNet  MATH  Google Scholar 

  196. Benedetti L, Cervera M, Chiumenti M (2017) 3D modelling of twisting cracks under bending and torsion skew notched beams. Eng Fract Mech 176:235–256

  197. Saloustros S, Cervera M, Kim S, Chiumenti M (2021) Accurate and locking-free analysis of beams, plates and shells using solid elements. Comput Mech 67:883–914

  198. Ladyzhenskaya O (1963) The mathematical theory of viscous incompressible flows. Gordon and Breach, New York

    MATH  Google Scholar 

  199. Babuska I, Oden J, Lee J (1977) Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems. Comput Methods Appl Mech Eng 11:175–206

    MathSciNet  MATH  Google Scholar 

  200. Brezzi F (1974) On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. ESAIM Math Model Numer Anal Modél Math Anal Numér 8(R2):129–151

    MathSciNet  MATH  Google Scholar 

  201. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New York

    MATH  Google Scholar 

  202. Brezzi F, Bathe K (1990) A discourse on the stability conditions for mixed finite element formulations. Comput Methods Appl Mech Eng 82:27–59

    MathSciNet  MATH  Google Scholar 

  203. Babuska I (1971) Error-bounds for finite element method. Numer Math 16:322–333

    MathSciNet  MATH  Google Scholar 

  204. Boffi D, Brezzi F, Fortin M (2013) Mixed finite element methods and applications. Springer, Berlin

    MATH  Google Scholar 

  205. Codina R (2000) Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput Methods Appl Mech Eng 190:1579–1599

    MathSciNet  MATH  Google Scholar 

  206. Codina R (2008) Analysis of a stabilized finite element approximation of the Oseen equations using orthogonal subscales. Appl Numer Math 58:264–283

    MathSciNet  MATH  Google Scholar 

  207. Codina R (2009) Finite element approximation of the three field formulation of the Stokes problem using arbitrary interpolations. Soc Ind Appl Math J Numer Anal 47:699–718

    MathSciNet  MATH  Google Scholar 

  208. Badia S, Codina R (2009) Unified stabilized finite element formulations for the Stokes and the Darcy problems. SIAM J Numer Anal 47:1917–2000

    MathSciNet  MATH  Google Scholar 

  209. Benedetti L, Cervera M, Chiumenti M (2015) Stress-accurate mixed FEM for soil failure under shallow foundations involving strain localization in plasticity. Comput Geotech 64:32–47

    Google Scholar 

  210. E Gdoutos (1984) Problems of mixed mode crack propagation. In: Engineering application of fracture mechanics, vol II. Martinus Nijhoff, Netherlands

  211. Spatschek R, Brener E, Karma A (2011) Phase field modeling of crack propagation. Philos Mag 91:75–95

    Google Scholar 

  212. Francfort G, Marigo J (1998) Revisiting brittle fracture as an energy minimization problem. J Mech Phys Solids 48(8):1319–1342

    MathSciNet  MATH  Google Scholar 

  213. Chiumenti M, Cervera M, Moreira C, Barbat G (2020) Stress, strain and dissipation accurate 3-field formulation for inelastic isochoric deformation. Finite Elem Anal Des 192:103534

  214. Jirasek M (2007) Nonlocal damage mechanics. Rev Eur Génie Civ 11(7–8):993–1021

    Google Scholar 

  215. Bazant Z, Jirasek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128(11):1119–1149

    Google Scholar 

  216. Simone A, Askes H, Sluys L (2004) Incorrect initiation and propagation of failure in non-local and gradient-enhanced media. Int J Solids Struct 41:351–363

    MATH  Google Scholar 

  217. Rastiello G, Giry C, Gatuingt F, Thierry F, Desmorat R (2018) Nonlocal damage formulation with evolving integral length: the eikonal nonlocal approach. In: Computational modelling of concrete structures, Euro-C 2018, Bad Hofgastein, Austria

  218. Poh L, Sun G (2017) Localizing gradient damage model with decreasing interactions. Int J Numer Methods Eng 110(6):503–522

    MathSciNet  Google Scholar 

  219. Di Luzio G, Bazant Z (2005) Spectral analysis of localization in nonlocal and over-nonlocal materials with softening plasticity or damage. Int J Solids Struct 42:6071–6100

    MATH  Google Scholar 

  220. Nguyen G (2011) A damage model with evolving nonlocal interactions. Int J Solids Struct 48:1544–1559

    MATH  Google Scholar 

  221. Bui Q (2010) Initiation of damage with implicit gradient-enhanced damage models. Int J Solids Struct 47:2425–2435

    MATH  Google Scholar 

  222. Poh L, Swaddiwudhipong S (2009) Over-nonlocal gradient enhanced plastic-damage model for concrete. Int J Solids Struct 46(25–26):4369–4378

    MATH  Google Scholar 

  223. Pijaudier-Cabot G, Haidar K, Dube J-F (2004) Non-local damage model with evolving internal length. Int J Numer Anal Methods Geomech 28(7–8):633–652

    MATH  Google Scholar 

  224. Cogswell D (2010) A phase-field study of ternary multiphase microstructures. Ph.D. thesis at the Massachusetts Institute of Technology

  225. Wu J-Y (2017) A unified phase-field theory for the mechanics of damage and quasi-brittle failure. J Mech Phys Solids 103:72–99

    MathSciNet  Google Scholar 

  226. Wu J-Y, Nguyen V (2018) A length scale insensitive phase-field damage model for brittle fracture. J Mech Phys Solids 119:20–42

    MathSciNet  Google Scholar 

  227. Feng D-C, Wu J-Y (2018) Phase-field regularized cohesive zone model (CZM) and size effect of concrete. Eng Fract Mech 197:66–79

    Google Scholar 

  228. Mandal T, Nguyen V, Wu J-Y (2019) Length scale and mesh bias sensitivity of phase-field models for brittle and cohesive fracture. Eng Fract Mech 217:106532

    Google Scholar 

  229. Nguyen V, Wu J-Y (2018) Modeling dynamic fracture of solids with a phase-field regularized cohesive zone model. Comput Methods Appl Mech Eng 340:1000–1022

    MathSciNet  MATH  Google Scholar 

  230. Wu J-Y (2018) Robust numerical implementation of non-standard phase-field damage models for failure in solids. Comput Methods Appl Mech Eng 340:767–797

    MathSciNet  MATH  Google Scholar 

  231. Wu J-Y, Nguyen V, Zhou H, Huang Y (2020) A variationally consistent phase-field anisotropic damage model for fracture. Comput Methods Appl Mech Eng 358:112629

    MathSciNet  MATH  Google Scholar 

  232. Wu J-Y, Huang Y, Nguyen V (2020) On the BFGS monolithic algorithm for the unified phase field damage theory. Comput Methods Appl Mech Eng 360:112704

    MathSciNet  MATH  Google Scholar 

  233. Wu J (2017) A geometrically regularized gradient-damage model with energetic equivalence. Comput Methods Appl Mech Eng 328:612–637

  234. Yang Z-J, Li B-B, Wu J-Y (2019) X-ray computed tomography images based phase-field modeling of mesoscopicfailure in concrete. Eng Fract Mech 208:151–170

    Google Scholar 

  235. Wu J-Y, Mandal T, Nguyen V (2020) A phase-field regularized cohesive zone model for hydrogen assisted cracking. Comput Methods Appl Mech Eng 358:112614

    MathSciNet  MATH  Google Scholar 

  236. Mandal T, Nguyen V, Wu J-Y (2020) Evaluation of variational phase-field models for dynamic brittle fracture. Eng Fract Mech 235:107169

    Google Scholar 

  237. de Borst R, Verhoosel C (2016) Gradient damage vs phase-field approaches for fracture: Similarities and differences. Comput Methods Appl Mech Eng. https://doi.org/10.1016/j.cma.2016.05.015

    Article  MathSciNet  MATH  Google Scholar 

  238. Steinke C, Zreid I, Kaliske M (2017) On the relation between phase-field crack approximation and gradient damage modelling. Comput Mech 59:717–735

    MathSciNet  Google Scholar 

  239. Mandal T, Nguyen V, Heidarpour A (2019) Phase field and gradient enhanced damage models for quasi-brittle failure: a numerical comparative study. Eng Fract Mech 207:48–67

    Google Scholar 

  240. Wu JY, Nguyen V, Nguyen C, Sutula D, Sinaie S, Bordas S (2020) Phase-field modeling of fracture. Adv Appl Mech 53:1–183.

  241. May S, Vignollet J, de Borst R (2015) A numerical assessment of phase-field models for brittle and cohesive fracture: Γ-convergence and stress oscillations. Eur J Mech Solids 52:72–84

    MathSciNet  MATH  Google Scholar 

  242. Sargado J, Keilegavlen E, Berre I, Nordbotten J. A combined finite element-finite volume framework for phase-field fracture. arXiv preprint arXiv:1904.12395

  243. Wu J-Y, Huang Y (2020) Comprehensive implementations of phase-field damage models in Abaqus. Theor Appl Fract Mech 106:102440

    Google Scholar 

  244. Jirasek M, Patzak B (2000) Adaptive technique for nonlocal models. In: Proceedings of: European congress on computational methods in applied sciences and engineering, Barcelona, CIMNE

  245. Pisano A, Fuschi P (2018) Stress evaluation in displacement-based 2D nonlocal finite element method. Curved Layer Struct 5:136–145

    Google Scholar 

  246. Peerlings R (1999) Enhanced damage modelling for fracture and fatigue. Ph.D. thesis, Technische Universiteit Eindhoven, Eindhoven

  247. Simone A, Askes H, Peerlings R, Sluys L (2003) Interpolation requirements for implicit gradient-enhanced continuum damage models. Commun Numer Methods Eng 19(7):563–572

    MATH  Google Scholar 

  248. Bui Q, Lani F (2011) Non-local implicit gradient damage models with low-order finite elements. Int J Numer Methods Biomed Eng 27(6):962–976

    MATH  Google Scholar 

  249. Nguyen T, Bui T, Hirose S (2018) Smoothing gradient damage model with evolving anisotropic nonlocal interactions tailored to low-order finite elements. Comput Methods Appl Mech Eng 328:498–541

    MathSciNet  MATH  Google Scholar 

  250. Verhoosel C, de Borst R (2013) A phase-field model for cohesive fracture. Int J Numer Methods Eng 96(1):43–62

    MathSciNet  MATH  Google Scholar 

  251. Bui Q, Nguyen H, Doan H, Nguyen D, Hirose S (2016) Numerical failure simulation of quasi-brittle materials using a second-order implicit gradient damage theory. In: Proceedings of the 4th international conference on engineering mechanics and automation, Hanoi, Vietnam, pp 1–6

  252. Zhang Y (2013) Simulation methods for durability assessment of concrete structures: multifield framework and strong discontinuity embedded approach. Ph.D. thesis at the Viena University of Technology

  253. Amor H, Marigo J, Maurini C (2009) Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J Mech Phys Solids 57(8):1209–1229

    MATH  Google Scholar 

  254. Ambati M, Gerasimov T, De Lorenzis L (2015) A review on phase-field models of brittle fracture and a new fast hybrid formulation. Comput Mech 55:383–405

    MathSciNet  MATH  Google Scholar 

  255. Muixi A, Fernandez-Mendez S, Rodriguez-Ferran A (2020) Adaptive refinement for phase-field models of brittle fracture based on Nitsche’s method. Comput Mech 66:69–85

    MathSciNet  MATH  Google Scholar 

  256. Wu J-Y, Xu S-L (2013) Reconsideration on the elastic damage/degradation theory for the modeling of microcrack closure-reopening (MCR) effects. Int J Solids Struct 50(5):795–805

    Google Scholar 

  257. Ambati M, Gerasimov T, De Lorenzis L (2015) Phase-field modeling of ductile fracture. Comput Mech 55:1017–1040

    MathSciNet  MATH  Google Scholar 

  258. Kuhn C, Noll T, Muller R (2016) On phase field modeling of ductile fracture. GAMM-Mitteilungen 39(1):35–54

    MathSciNet  MATH  Google Scholar 

  259. Wu J-Y, Huang Y, Zhou H, Nguyen V (2020) Three-dimensional phase-field modeling of mode I+II/III failure in solids. Comput Methods Appl Mech Eng 373:113537

  260. Bhowmick S, Liu G-R (2018) Three dimensional CS-FEM phase-field modeling technique for brittle fracture in elastic solids. Appl Sci 8(12):2488

    Google Scholar 

  261. Kiendl J, Ambati M, De Lorenzis L, Gomez H, Reali A (2016) Phase-field description of brittle fracture in plates and shells. Comput Methods Appl Mech Eng 312:374–394

    MathSciNet  MATH  Google Scholar 

  262. Nguyen T, Yvonnet J, Zhu Q-Z, Bornert M, Chateau C (2015) A phase field method to simulate crack nucleation and propagation in strongly heterogeneous materials from direct imaging of their microstructure. Eng Fract Mech 139:18–39

    Google Scholar 

  263. Zhou S, Zhuang X, Zhu H, Rabczuk T (2018) Phase field modelling of crack propagation, branching and coalescence in rocks. Theor Appl Fract Mech 96:174–192

    Google Scholar 

  264. Zhou S, Zhuang X, Rabczuk T (2019) Phase field modeling of brittle compressive-shear fractures in rock-like materials: a new driving force and a hybrid formulation. Comput Methods Appl Mech Eng 355:729–752

    MathSciNet  MATH  Google Scholar 

  265. Hirshikesh, Natarajan S, Annabattula R, Martinez-Paneda E (2019) Phase field modelling of crack propagation in functionally graded materials. Compos Part B Eng 169:239–248

    Google Scholar 

  266. Noll T, Kuhn C, Olesch D, Muller R (2020) 3D phase field simulations of ductile fracture. GAMM-Mitteilungen 43(2):e202000008

    MathSciNet  Google Scholar 

  267. Hirshikesh, Jansari C, Kannan K, Annabattula R, Natarajan S (2019) Adaptive phase field method for quasi-static brittle fracture based on recovery based error indicator and quadtree decomposition. Eng Fract Mech 220:106599

    Google Scholar 

  268. Heister T, Wheeler M, Wick T (2015) A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach. Comput Methods Appl Mech Eng 290:466–495

    MathSciNet  MATH  Google Scholar 

  269. Zhou S, Zhuang X (2018) Adaptive phase field simulation of quasi-static crack propagation in rocks. Undergr Space 3(3):190–205

    Google Scholar 

  270. Ferro N, Micheletti S, Perotto S (2018) Anisotropic mesh adaptation for crack propagation induced by a thermal shock in 2D. Comput Methods Appl Mech Eng 331:138–158

    MathSciNet  MATH  Google Scholar 

  271. Cervera M, Agelet de Saracibar C, Chiumenti M (2002) COMET: coupled mechanical and thermal analysis. Data input manual, Version 5.0, technical report IT-308. http://www.cimne.upc.edu

  272. Coll A, Ribo R, Pasenau M, Escolano E, Perez JS, Melendo A, Monros A, Garate J (2002) GiD: the personal pre and post-processor user manual. CIMNE, Technical University of Catalonia. http://gid.cimne.upc.edu

  273. Trunk B (2000) Einfluss der Bauteilgrösse auf die Bruchenergie von Beton. Aedificatio Publishers, Freiburg

    Google Scholar 

  274. Chen J (2001) A nonlocal damage model for elasto-plastic materials based on gradient plasticity theory. Ph.D. thesis, Swiss Federal Institute of Technology, Zurich

  275. Kim J (2013) New finite elements with embedded strong discontinuities to model failure of three-dimensional continua. Ph.D. thesis, University of California, Berkeley

  276. Kim J, Armero F (2017) Three-dimensional finite elements with embedded strong discontinuities for the analysis of solids at failure in the finite deformation range. Comput Methods Appl Mech Eng 317:890–926

    MathSciNet  MATH  Google Scholar 

  277. Su X, Yang Z, Liu G (2010) Finite element modelling of complex 3D static and dynamic crack propagation by embedding cohesive elements in Abaqus. Acta Mech Solida Sin 23(3):271–282

    Google Scholar 

  278. Ni T, Zaccariotto M, Zhu Q-Z, Galvanetto U (2019) Coupling of FEM and ordinary state-based peridynamics for brittle failure analysis in 3D. Mech Adv Mater Struct. https://doi.org/10.1080/15376494.2019.1602237

    Article  Google Scholar 

  279. Arrea M, Ingraffea A (1982) Mixed-mode crack propagation in mortar and concrete. Technical report. Report no. 81-13, Department of Structural Engineering, Cornell University, New York

  280. Rots J, de Borst R (1987) Analysis of mixed-mode fracture in concrete. J Eng Mech 113(11):1739–1758

    Google Scholar 

  281. Gerstle W, Xie M (1992) FEM modeling of fictitious crack propagation in concrete. J Eng Mech 118(2):416–434

    Google Scholar 

  282. Galvez J, Cendon D (2002) Simulacion de la fractura del hormigon en modo mixto. Revista Internacional de Metodos Numericos para Calculo y Diseno en Ingenieria 18(1):31–58

    Google Scholar 

  283. Jirasek M, Patzak B (2002) Consistent tangent stiffness for nonlocal damage models. Comput Struct 80:1279–1293

    Google Scholar 

  284. Linder C, Armero F (2007) Finite elements with embedded strong discontinuities for the modeling of failure in solids. Int J Numer Methods Eng 72:1391–1433

    MathSciNet  MATH  Google Scholar 

  285. Galvez J, Planas J, Sancho J, Reyes E, Cendon D, Casati M (2013) An embedded cohesive crack model for finite element analysis of quasi-brittle materials. Eng Fract Mech 109:369–386

    Google Scholar 

  286. Bhattacharjee S, Leger P (1994) Application of NLFM models to predict cracking in concrete gravity dams. J Struct Eng 120(4):1255–1271

    Google Scholar 

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

    MATH  Google Scholar 

  288. Rabczuk T, Belytschko T (2007) A three dimensional large deformation meshfree method for arbitrary evolving cracks. Comput Methods Appl Mech Eng 196(29–30):2777–2799

    MathSciNet  MATH  Google Scholar 

  289. Monteiro Azevedo N, Lemos J (2006) Hybrid discrete element/finite element method for fracture analysis. Comput Methods Appl Mech Eng 195(33–36):4579–4593

    MATH  Google Scholar 

  290. Roth S, Leger P, Soulaimani A (2015) A combined XFEM—damage mechanics approach for concrete crack propagation. Comput Methods Appl Mech Eng 283:923–955

    MathSciNet  MATH  Google Scholar 

  291. Schlangen E (1993) Experimental and numerical analysis of fracture processes in concrete. Heron 38(2):1–117

    Google Scholar 

  292. Wells G, Sluys L (2001) A new method for modelling cohesive cracks using finite elements. Int J Numer Methods Eng 50:2667–2682

    MATH  Google Scholar 

  293. Garcia-Alvarez V, Gettu R, Carol I (2012) Analysis of mixed-mode fracture in concrete using interface elements and a cohesive crack model. Sadhana 37(1):187–205

    MathSciNet  Google Scholar 

  294. Chopra A, Chakrabarti P (1973) The Koyna earthquake and the damage to Koyna dam. Bull Seismol Soc Am 63(2):381–397

    Google Scholar 

  295. Bhattacharjee S, Leger P (1993) Seismic cracking and energy dissipation in concrete gravity dams. Earthq Eng Struct Dyn 22:991–1007

    Google Scholar 

  296. Zhang H, Ohmachi T (1998) 2 dimensional analysis of seismic cracking in concrete gravity dams. J Jpn Soc Dam Eng 8(2):93–101

    Google Scholar 

  297. Lee J, Fenves G (1998) A plastic-damage concrete model for earthquake analysis of dams. Earthq Eng Struct Dyn 27:937–956

    Google Scholar 

  298. Calayir Y, Karaton M (2005) Seismic fracture analysis of concrete gravity dams including dam–reservoir interaction. Comput Struct 83:1595–1606

    Google Scholar 

  299. Wu J-Y, Li J (2007) Unified plastic-damage model for concrete and its applications to dynamic nonlinear analysis of structures. Struct Eng Mech 25(5):519–540

    Google Scholar 

  300. Sarkar R, Paul D, Stempniewski L (2007) Influence of reservoir and foundation on the nonlinear dynamic response of concrete gravity dams. ISET J Earthq Technol 44(2):377–389

    Google Scholar 

  301. Zhang S, Wang G, Sa W (2013) Damage evaluation of concrete gravity dams under mainshock–aftershock seismic sequences. Soil Dyn Earthq Eng 50:16–27

    Google Scholar 

  302. Wang G, Wang Y, Lu W, Zhou C, Chen M, Yan P (2015) XFEM based seismic potential failure mode analysis of concrete gravity dam-water-foundation systems through incremental dynamic analysis. Eng Struct 98:81–94

    Google Scholar 

  303. Pan J, Zhang C, Xu Y, Jin F (2011) A comparative study of the different procedures for seismic cracking analysis of concrete dams. Soil Dyn Earthq Eng 31(11):1594–1606

    Google Scholar 

  304. Gioia G, Bazant Z, Pohl B (1992) Is no-tension dam design always safe? a numerical study. Dam Eng 3(1):23–34

    Google Scholar 

  305. Ghrib F, Tinawi R (1995) Nonlinear behavior of concrete dams using damage mechanics. J Eng Mech 121(4):513–527

    Google Scholar 

  306. Shi M, Zhong H, Ooi E, Zhang C, Song C (2013) Modelling of crack propagation of gravity dams by scaled boundary polygons and cohesive crack model. Int J Fract 183:29–48

    Google Scholar 

  307. Santillan D, Mosquera J, Cueto-Felgueroso L (2017) Phase-field model for brittle fracture. Validation with experimental results and extension to dam engineering problems. Eng Fract Mech 178:109–125

    Google Scholar 

  308. Buchholz F-G, Chergui A, Richard H (2004) Fracture analyses and experimental results of crack growth under general mixed mode loading conditions. Eng Fract Mech 71:455–468

    Google Scholar 

  309. Buchholz F-G, Just V, Richard H (2005) Computational simulation and experimental findings of three-dimensional fatigue crack growth in a single-edge notched specimen under torsion loading. Fatigue Fract Eng Mater Struct 28(1–2):127–134

    Google Scholar 

  310. Lazarus V, Buchholz F-G, Fulland M, Wiebesiek J (2008) Comparison of predictions by mode II or mode III criteria on crack front twisting in three or four point bending experiments. Int J Fract 153:141–151

    Google Scholar 

  311. Citarella R, Buchholz F-G (2008) Comparison of crack growth simulation by DBEM and FEM for SEN-specimens undergoing torsion or bending loading. Eng Fract Mech 75:489–509

    Google Scholar 

  312. Colombo D, Massin P (2011) Fast and robust level set update for 3D non-planar X-FEM crack propagation modelling. Comput Methods Appl Mech Eng 200(25–28):2160–2180

    MATH  Google Scholar 

  313. Geniaut S, Galenne E (2012) A simple method for crack growth in mixed mode with X-FEM. Int J Solids Struct 49(15–16):2094–2106

    Google Scholar 

  314. Ferte G, Massin P, Moes N (2016) 3D crack propagation with cohesive elements in the extended finite element method. Comput Methods Appl Mech Eng 300:347–374

    MathSciNet  MATH  Google Scholar 

  315. Sadeghirad A, Chopp D, Ren X, Fang E, Lua J (2016) A novel hybrid approach for level set characterization and tracking of non-planar 3D cracks in the extended finite element method. Eng Fract Mech 160:1–14

    Google Scholar 

  316. Agathos K, Ventura G, Chatzi E, Bordas S (2018) Stable 3D XFEM/vector level sets for non-planar 3D crack propagation and comparison of enrichment schemes. Int J Numer Methods Eng 113(2):252–276

    MathSciNet  Google Scholar 

  317. Pandolfi A, Ortiz M (2012) An eigenerosion approach to brittle fracture. Int J Numer Methods Eng 92(8):694–714

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Financial support from the Spanish Ministry of Economy and Business via the ADaMANT (Computational Framework for Additive Manufacturing of Titanium Alloy) project (Proyectos de I + D (Excelencia) DPI2017-85998-P) is gratefully acknowledged. The support provided by the Spanish Ministry of Education to Mr. Gabriel Barbat via the FPU program is also acknowledged. The authors also acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2018-000797-S).

Author information

Authors and Affiliations

  1. International Center for Numerical Methods in Engineering (CIMNE), Technical University of Catalonia – BarcelonaTECH, Edificio C1, Campus Norte, Jordi Girona 1-3, 08034, Barcelona, Spain

    M. Cervera, G. B. Barbat & M. Chiumenti

  2. State Key Laboratory of Subtropic Building Science, South China University of Technology, Guangzhou, 510641, China

    J.-Y. Wu

Authors
  1. M. Cervera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. B. Barbat
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Chiumenti
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J.-Y. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. B. Barbat.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cervera, M., Barbat, G.B., Chiumenti, M. et al. A Comparative Review of XFEM, Mixed FEM and Phase-Field Models for Quasi-brittle Cracking. Arch Computat Methods Eng 29, 1009–1083 (2022). https://doi.org/10.1007/s11831-021-09604-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-021-09604-8

Search

Navigation