Skip to main content
SpringerLink
Log in

Finite element modeling of quasi-brittle cracks in 2D and 3D with enhanced strain accuracy

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

This paper discusses the finite element modeling of cracking in quasi-brittle materials. The problem is addressed via a mixed strain/displacement finite element formulation and an isotropic damage constitutive model. The proposed mixed formulation is fully general and is applied in 2D and 3D. Also, it is independent of the specific finite element discretization considered; it can be equally used with triangles/tetrahedra, quadrilaterals/hexahedra and prisms. The feasibility and accuracy of the method is assessed through extensive comparison with experimental evidence. The correlation with the experimental tests shows the capacity of the mixed formulation to reproduce the experimental crack path and the force–displacement curves with remarkable accuracy. Both 2D and 3D examples produce results consistent with the documented data. Aspects related to the discrete solution, such as convergence regarding mesh resolution and mesh bias, as well as other related to the physical model, like structural size effect and the influence of Poisson’s ratio, are also investigated. The enhanced accuracy of the computed strain field leads to accurate results in terms of crack paths, failure mechanisms and force displacement curves. Spurious mesh dependency suffered by both continuous and discontinuous irreducible formulations is avoided by the mixed FE, without the need of auxiliary tracking techniques or other computational schemes that alter the continuum mechanical problem.

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

Similar content being viewed by others

References

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  3. Rots J, Nauta P, Kusters G (1984) Variable reduction factor for the shear stiffness of cracked concrete. Rep. BI-84 Ins. TNO for Build Mat. Struct. Delft 33

  4. de Borst R, Nauta P (1985) Non-orthogonal cracks in a smeared finite element model. Eng Comput 2:35–46

    Article  Google Scholar 

  5. de Borst R (1987) Smeared cracking, plasticity, creep and thermal loading: a unified approach. Comp Meth Appl Mech Eng 62(99):89–110

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

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

    Article  MATH  MathSciNet  Google Scholar 

  10. 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 Method Appl Mech Eng 294:449–485

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. 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

    Article  MathSciNet  Google Scholar 

  13. Barenblatt G (1962) The mathematical theory of equilibrium cracks in brittle faillure. Adv Appl Math 7:55–129

    Google Scholar 

  14. Ngo D, Scordelis A (1967) Finite element analysis of reinforced concrete beams. ACI J 64(14):152–163

    Google Scholar 

  15. Areias P, Msekh M, Rabczuk T (2016) Damage and fracture algorithm using the screened Poisson equation and local remeshing. Eng Fract Mech 158:116–143

    Article  Google Scholar 

  16. Areias P, Rabczuk T (2013) Finite strain fracture of plates and shells with configurational forces and edge rotations. Int J Numer Method Eng 94:1099–1122

    Article  MATH  MathSciNet  Google Scholar 

  17. Areias P, Rabczuk T, Dias-da-Costa D (2013) Element-wise fracture algorithm based on rotation of edges. Eng Fract Mech 110:113–137

    Article  MATH  Google Scholar 

  18. Areias P, Reinoso J, Camanho P, Rabczuk T (2015) A constitutive-based element-by-element crack propagation algorithm with local mesh refinement. Comput Mech 56:291–315

    Article  MATH  MathSciNet  Google Scholar 

  19. Schellekens J (1993) A non-linear finite element approach for the analysis of mode-I free edge delamination in composites. Int J Solid Struct 30(9):1239–1253

    Article  MATH  Google Scholar 

  20. Allix O, Ladevèze P (1992) Interlaminar interface modelling for the prediction of delamination. Compos Struct 22(4):235–242

    Article  Google Scholar 

  21. Bolzon G, Corigliano A (1997) A discrete formulation for elastic solids with damaging interfaces. Comp Method Appl Mech Eng 140:329–359

    Article  MATH  Google Scholar 

  22. Pandolfi A, Krysl P, Ortiz M (1999) Finite element simulation of ring expansion and fragmentation: the capturing of length and time scales through cohesive models of fracture. Int J Fract 95(1–4):279–297

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  24. Oliver J, Cervera M, Manzoli O (1999) Strong discontinuities and continuum plasticity models: the strong discontinuity approach. Int J Plast 15(3):319–351

    Article  MATH  Google Scholar 

  25. Gasser T, Holzapfel G (2003) Geometrically non-linear and consistently linearized embedded strong discontinuity models for 3D problems with an application to the dissection analysis of soft biological tissues. Comp Method Appl Mech Eng 192:5059–5098

    Article  MATH  MathSciNet  Google Scholar 

  26. Motamedi M, Weed D, Foster C (2016) Numerical simulation of mixed mode (I and II) fracture behaviour pf pre-cracked rock using the strong discontinuity approach. Int J Solid Struct 85–86:44–56

    Article  Google Scholar 

  27. 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. Comp Method Appl Mech Eng 287:335–366

    Article  MathSciNet  Google Scholar 

  28. 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 Mec Solida Sin 23(3):271–282

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  30. Gasser T, Holzapfel G (2005) Modeling 3D crack propagation in unreinforced concrete using PUFEM. Comp Method Appl Mech Eng 194:2859–2896

    Article  MATH  Google Scholar 

  31. Holl M, Rogge T, Loehnert S, Wriggers P, Rolfes R (2014) 3D multiscale crack propagation using the XFEM applied to a gas turbine blade. Comput Mech 53:173–188

    Article  MATH  MathSciNet  Google Scholar 

  32. Meschke G, Dumstorff P (2007) Energy-based modeling of cohesive and cohesion-less cracks via X-FEM. Comp Method Appl Mech Eng 196:2338–2357

    Article  MATH  Google Scholar 

  33. 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. Comp Method Appl Mech Eng 295:77–107

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  35. Rabczuk T, Belytschko T (2007) A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Comp Method Appl Mech Eng 196:2777–2799

    Article  MATH  MathSciNet  Google Scholar 

  36. Rabczuk T, Belytschko T (2005) Adaptivity for structured meshfree particle methods in 2D and 3D. Int J Numer Method Eng 63:1559–1582

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  38. Zhuang X, Augarde C, Bordas S (2011) Accurate fracture modelling using meshless methods, the visibility criterion and level sets: formulation and 2D modelling. Int J Numer Method Eng 86:249–268

    Article  MATH  Google Scholar 

  39. Zhuang X, Augarde C, Mathisen K (2012) Fracure modeling using meshless methods and level sets in 3D: framework and modeling. Int J Numer Method Eng 92:969–998

    Article  MATH  Google Scholar 

  40. Nguyen G, Nguyen C, Nguyen P, Bui H, Shen L (2016) A size-dependent constitutive modelling framework for localized faillure analysis. Comput Mech 58:257–280. doi:10.1007/s00466-016-1293-z

    Article  MATH  MathSciNet  Google Scholar 

  41. Annavarapu C, Settgast R, Vitali E, Morris J (2016) A local crack-tracking strategy to model three-dimensional crack propagation with embedded methods. Comp Method Appl Mech Eng 311:815–837

    Article  MathSciNet  Google Scholar 

  42. Dumstorffz P, Meschke G (2007) Crack propagation criteria in the framework of X-FEM-based structural analysis. Int J Numer Anal Meth Geomech 31:239–259

    Article  MATH  Google Scholar 

  43. Kim J, Armero F (2017) Three-dimensional finite elements with embedded strong discontinuities for the analysis of solids at faillure in the finite deformation range. Comput Method Appl Mech Eng doi:10.1016/j.cma.2016.12.038

  44. Riccardi F, Kishta E, Richard B (2017) A step-by-step global crack-tracking approach in E-FEM simulations of quasi-brittle materials. Eng Fract Mech 170:44–58

    Article  Google Scholar 

  45. Dias-da- Costa D, Alfaiate J, Sluys L, Júlio 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(1):97–119

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  47. Rabczuk T (2013) Computational methods for fracture in brittle and quasi-brittle solids: state-of-the-art review and future perspectives. ISRN Appl Math doi:10.1155/2013/849231

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  52. Gil A, Lee C, Bonet J, Aguirre M (2014) A stabilized Petrov–Galerkin formulation for linear tetrahedral elements in compressible, nearly incompressible and truly incompressible fast dynamics. Comp Method Appl Mech Eng 276:659–690

    Article  Google Scholar 

  53. Lafontaine N, Rossi R, Cervera M, Chiumenti M (2015) Explicit mixed strain-displacement finite element for dynamic geometrically non-linear solid mechanics. Comput Mech 55:543–559

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  55. 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

    Article  Google Scholar 

  56. Chiumenti M, Cervera M, Codina R (2014) A mixed three-field FE formulation for stress accurate analysis including the incompressible limit. Comp Method Appl Mech Eng 283:1095–1116

    Article  MathSciNet  Google Scholar 

  57. Hellinger E (1914) Die allegemeinen Ansätze der Mechanik der Kontinua, Art 30. In: Klein F, Muller C (eds) Encyclopädie der Matematischen Wissenschaften. Teubner, Leipzig, pp 654–655

    Google Scholar 

  58. Reissner E (1958) On variational principles of elasticity. Proc Symp Appl Math 8:1–6

    Article  MATH  MathSciNet  Google Scholar 

  59. Zienkiewicz O, Taylor R, Zhu Z (1989) The finite element method, vol 1, 7th edn. Elsevier, Amsterdam

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

  64. Hughes T, Feijoo G, Mazzei L, Quincy J (1998) The variational multiscale method: a paradigm for computational mechanics. Comp Method Appl Mech Eng 166:3–24

    Article  MATH  MathSciNet  Google Scholar 

  65. Hughes T, Franca L, Balestra M (1986) A new finite element formulation for computational fluid dynamics: V. circumventing the Babuska–Brezzi condition: a stable Petrov-Galerkin formulation of the stokes problem accomodating equal-order interpolations. Comp Method Appl Mech Eng 59(1):85–99

    Article  MATH  Google Scholar 

  66. Cervera M, Agelet de Saracibar C, Chiumenti M (2002) COMET: coupled mechanical and thermal analysis. Data Input manuel, version 5.0, Technical report IT-308. http://www.cimne.upc.edu

  67. GiD (2002) the personal pre and post-processor. In: CIMNE, Technical University of Catalonia. http://gid.cimne.upc.ed

  68. Winkler B (2001) Traglastuntersuchungen von unbewehrten und bewerhrten Betonskrukturen auf der Grundlage eines objektiven Werkstoffgesetzes für Beton. Ph.D. Thesis, Universität Innsbruck

  69. Trunk B (2000) Einfluss der Bauteilgrösse auf die Bruchenergie Von Beton. Aedificatio publishers, Freiburg

    Google Scholar 

  70. Gálvez J, Elices M, Guinea G, Planas J (1998) Mixed mode fracture of concrete under proportional and nonproportional loading. Int J Fract 94:267–284

    Article  Google Scholar 

  71. Cervera M, Pela L, Clemente R, Roca P (2010) A crack-tracking technique for localized damage in quasi-brittle materials. Eng Fract Mech 77:2431–2450

    Article  Google Scholar 

  72. Ingraffea A, Grigoriu M (1990) Probabilistic Fracture Mechanics: a validation of predictive capability. Tech Rep 90–8, DTIC Document

  73. Miehe C, Gürses E (2007) A robust algorithm for the configurational-force-driven brittle crack propagation with R-adaptative mesh alignment. Int J Numer Method Eng 72:127–155

    Article  MATH  Google Scholar 

  74. Bocca P, Carpinteri A, Valente S (1991) Mixed mode fracture of concrete. Int. J. Solid Struct 27(9):1139–1153

    Article  Google Scholar 

  75. Saleh A, Aliabadi M (1995) Crack growth analysis in concrete using boundary element method. Eng Fract Mech 51(4):533–545

    Article  Google Scholar 

  76. Areias P, Rabczuk T, César de Sá J (2016) A novel two-stage discrete crack method based on the screened Poisson equation and local mesh refinement. Comput Mech 58:1003–1018

    Article  MathSciNet  Google Scholar 

  77. Buchholz F, 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

    Article  Google Scholar 

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

    Article  Google Scholar 

  79. Ferte G, Massin P, Moës N (2016) 3D crack propagation with cohesive elements in the extended finite element method. Comp Method Appl Mech Eng 300:347–374

    Article  MathSciNet  Google Scholar 

Download references

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

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

Corresponding author

Correspondence to M. Cervera.

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. Finite element modeling of quasi-brittle cracks in 2D and 3D with enhanced strain accuracy. Comput Mech 60, 767–796 (2017). https://doi.org/10.1007/s00466-017-1438-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-017-1438-8

Keywords

Search

Navigation