Computational Mechanics

, Volume 59, Issue 2, pp 299–316 | Cite as

Finite element modelling of internal and multiple localized cracks

  • Savvas Saloustros
  • Luca Pelà
  • Miguel Cervera
  • Pere Roca
Original Paper


Tracking algorithms constitute an efficient numerical technique for modelling fracture in quasi-brittle materials. They succeed in representing localized cracks in the numerical model without mesh-induced directional bias. Currently available tracking algorithms have an important limitation: cracking originates either from the boundary of the discretized domain or from predefined “crack-root” elements and then propagates along one orientation. This paper aims to circumvent this drawback by proposing a novel tracking algorithm that can simulate cracking starting at any point of the mesh and propagating along one or two orientations. This enhancement allows the simulation of structural case-studies experiencing multiple cracking. The proposed approach is validated through the simulation of a benchmark example and an experimentally tested structural frame under in-plane loading. Mesh-bias independency of the numerical solution, computational cost and predicted collapse mechanisms with and without the tracking algorithm are discussed.


Continuum damage mechanics Crack-tracking Damage localization Quasi-brittle materials Shear/flexural/tensile cracks 



This research has received the financial support from the MINECO (Ministerio de Economia y Competitividad of the Spanish Government) and the ERDF (European Regional Development Fund) through the the MULTIMAS project (Multiscale techniques for the experimental and numerical analysis of the reliability of masonry structures, ref. num. BIA2015-63882-P) and the EACY project (Enhanced accuracy computational and experimental framework for strain localization and failure mechanisms, ref. MAT 2013-48624-C2-1-P). The authors gratefully acknowledge Dr. Fulvio Parisi for providing information regarding the experimental data.


  1. 1.
    Ngo D, Scordelis C (1967) Finite element analysis of reinforced concrete beams. ACI J 64(3):152–163Google Scholar
  2. 2.
    Rashid Y (1968) Ultimate strength analysis of prestressed concrete pressure vessels. Nucl Eng Des 7:334–344CrossRefGoogle Scholar
  3. 3.
    Mosler J, Meschke G (2004) Embedded crack versus smeared crack models: a comparison of elementwise discontinuous crack path approaches with emphasis on mesh bias. Comput Methods Appl Mech Eng 193(30–32):3351–3375CrossRefzbMATHGoogle Scholar
  4. 4.
    Peerlings RHJ, De Borst R, Brekelmans WAM, De Vree JHP (1996) Gradient enhanced damage for quasi-brittle materials. Int J Numer Methods Eng 39(19):3391–3403CrossRefzbMATHGoogle Scholar
  5. 5.
    Simone A, Wells GN, Sluys LJ (2003) From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Comput Methods Appl Mech Eng 192:4581–4607CrossRefzbMATHGoogle Scholar
  6. 6.
    Bažant ZP, Lin FB (1988) Nonlocal smeared cracking model for concrete fracture. J Struct Eng 114:2493–2510CrossRefGoogle Scholar
  7. 7.
    de Vree J, Brekelmans W, van Gils M (1995) Comparison of nonlocal approaches in continuum damage mechanics. Comput Struct 55:581–588CrossRefzbMATHGoogle Scholar
  8. 8.
    De Borst R (1991) Simulation of strain localization: a reppraisal of the cosserat continuum. Eng Comput 8:317–332CrossRefGoogle Scholar
  9. 9.
    De Borst R, Sluys L, Mühlhaus H-B, Pamin J (1993) Fundamental issues in finite element analyses of localization of deformation. Eng Comput 10(2):99–121CrossRefGoogle Scholar
  10. 10.
    Benedetti L, Cervera M, Chiumenti M (2015) Stress-accurate mixed FEM for soil failure under shallow foundations involving strain localization in plasticity. Comput Geotech 64:32–47CrossRefGoogle Scholar
  11. 11.
    Jirásek M, Zimmermann T (2001) Embedded crack model. Part II. Combination with smeared cracks. Int J Numer Methods Eng 50(6):1291–1305CrossRefzbMATHGoogle Scholar
  12. 12.
    Wells GN, Sluys LJ (2001) A new method for modelling cohesive cracks using finite elements. Int J Numer Methods Eng 50(12):2667–2682CrossRefzbMATHGoogle Scholar
  13. 13.
    Moës N, Belytschko T (2002) Extended finite element method for cohesive crack growth. Eng Fract Mech 69:813–833CrossRefGoogle Scholar
  14. 14.
    Dumstorff P, Meschke G (2007) Crack propagation criteria in the framework of X-FEM-based structural analyses. Int J Numer Anal Methods Geomech 31:239–259CrossRefzbMATHGoogle Scholar
  15. 15.
    Cervera M, Chiumenti M (2006) Mesh objective tensile cracking via a local continuum damage model and a crack tracking technique. Comput Methods Appl Mech Eng 196(1–3):304–320CrossRefzbMATHGoogle Scholar
  16. 16.
    Cervera M, Pelà L, Clemente R, Roca P (2010) A crack-tracking technique for localized damage in quasi-brittle materials. Eng Fract Mech 77(13):2431–2450CrossRefGoogle Scholar
  17. 17.
    Slobbe A, Hendriks M, Rots J (2014) Smoothing the propagation of smeared cracks. Eng Fract Mech 132:147–168CrossRefGoogle Scholar
  18. 18.
    Jirásek M, Grassl P (2008) Evaluation of directional mesh bias in concrete fracture simulations using continuum damage models. Eng Fract Mech 75(8):1921–1943CrossRefGoogle Scholar
  19. 19.
    De Borst R (2001) Fracture in quasi-brittle materials: a review of continuum damage-based approaches. Eng Fract Mech 69:95–112CrossRefGoogle Scholar
  20. 20.
    Rabczuk T (2012) Computational methods for fracture in brittle and quasi-brittle solids: state-of-the-art review and future perspectives. ISRN Appl Math 2013:1–61CrossRefGoogle Scholar
  21. 21.
    Chen W-F (1982) Plasticity in reinforced concrete. McGraw-Hill, New YorkGoogle Scholar
  22. 22.
    Chen W-F (1994) Constitutive equations for engineering materials, vol 2 plasticity and modelling. Elsevier, AmsterdamGoogle Scholar
  23. 23.
    Feenstra PH, De Borst R (1996) A composite plasticity model for concrete. Int J Solids Struct 33:707–730CrossRefzbMATHGoogle Scholar
  24. 24.
    Mazars J, Pijaudier-Cabot G (1989) Continuum damage theory—application to concrete. J Eng Mech 115(2):345–365CrossRefGoogle Scholar
  25. 25.
    Cervera M, Oliver J, Faria R (1995) Seismic evaluation of concrete dams via continuum damage models. Earthq Eng Struct Dyn 24(9):1225–1245CrossRefGoogle Scholar
  26. 26.
    Lubliner J, Oliver J, Oller S, Oñate E (1989) A plastic-damage model for concrete. Int J Solids Struct 25(3):299–326CrossRefGoogle Scholar
  27. 27.
    Lee G, Fenves GL (1998) Plastic-damage model for cyclic loading of concrete structures. J Eng Mech 124(8):892–900CrossRefGoogle Scholar
  28. 28.
    Wu JY, Li J, Faria R (2006) An energy release rate-based plastic-damage model for concrete. Int J Solids Struct 43(3–4):583–612CrossRefzbMATHGoogle Scholar
  29. 29.
    Papa E (1996) A unilateral damage model for masonry based on a homogenisation procedure. Mech Cohes Frict Mater 1(February):349–366CrossRefGoogle Scholar
  30. 30.
    Lourenço PB (2000) Anisotropic softening model for masonry plates and shells. J Struct Eng 126(9):1008–1016CrossRefGoogle Scholar
  31. 31.
    Pelà L, Cervera M, Roca P (2013) An orthotropic damage model for the analysis of masonry structures. Constr Build Mater 41:957–967CrossRefGoogle Scholar
  32. 32.
    Lopez J, Oller S, Oñate E, Lubliner J (1999) A homogeneous constitutive model for masonry. Int J Numer Methods Eng 46(10):1651–1671MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kouznetsova V, Geers MGD, Brekelmans WAM (2002) Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Methods Eng 54:1235–1260CrossRefzbMATHGoogle Scholar
  34. 34.
    Zucchini A, Louren PB (2002) A micro-mechanical model for the homogenisation of masonry. Int J Solids Struct 39:3233–3255CrossRefzbMATHGoogle Scholar
  35. 35.
    Lourenço PB, Milani G, Tralli A, Zucchini A (2007) Analysis of masonry structures: review of and recent trends in homogenization techniques. Can J Civ Eng 34(11):1443–1457CrossRefGoogle Scholar
  36. 36.
    Calderini C, Lagomarsino S (2008) Continuum model for in-plane anisotropic inelastic behavior of masonry. J Struct Eng 134(2):209–220CrossRefGoogle Scholar
  37. 37.
    Oliver J, Caicedo M, Roubin E, Huespe A, Hernández J (2015) Continuum approach to computational multiscale modeling of propagating fracture. Comput Methods Appl Mech Eng 294:384–427MathSciNetCrossRefGoogle Scholar
  38. 38.
    Petracca M, Pelà L, Rossi R, Oller S, Camata G, Spacone E (2015) Regularization of first order computational homogenization for multiscale analysis of masonry structures. Comput Mech 57:257–276MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Lourenço PB, Rots JG (1997) Multisurface interface model for analysis of masonry structures. J Eng Mech 123(7):660–668CrossRefGoogle Scholar
  40. 40.
    Macorini L, Izzuddin BA (2011) A non-linear interface element for 3D mesoscale analysis of brick-masonry structures. Int J Numer Methods Eng 85:1584–1608CrossRefzbMATHGoogle Scholar
  41. 41.
    Oliveira S, Faria R (2006) Numerical simulation of collapse scenarios in reduced scale tests of arch dams. Eng Struct 28(10):1430–1439CrossRefGoogle Scholar
  42. 42.
    Roca P, Cervera M, Gariup G, Pelà L (2010) Structural analysis of masonry historical constructions. Classical and advanced approaches. Arch Comput Methods Eng 17:299–325CrossRefzbMATHGoogle Scholar
  43. 43.
    Carvalho J, Ortega J, Lourenço PB, Ramos LF, Roman H (2014) Safety analysis of modern heritage masonry buildings: box-buildings in Recife, Brazil. Eng Struct 80:222–240CrossRefGoogle Scholar
  44. 44.
    Mendes N, Lourenço PB (2014) Sensitivity analysis of the seismic performance of existing masonry buildings. Eng Struct 80:137–146CrossRefGoogle Scholar
  45. 45.
    Jäger P, Steinmann P, Kuhl E (2008) On local tracking algorithms for the simulation of three-dimensional discontinuities. Comput Mech 42(3):395–406CrossRefzbMATHGoogle Scholar
  46. 46.
    Roth S-N, Léger P, Soulaïmani A (2015) A combined XFEM-damage mechanics approach for concrete crack propagation. Comput Methods Appl Mech Eng 283:923–955MathSciNetCrossRefGoogle Scholar
  47. 47.
    Zhang Y, Lackner R, Zeiml M, Mang HA (2015) Strong discontinuity embedded approach with standard SOS formulation: element formulation, energy-based crack-tracking strategy, and validations. Comput Methods Appl Mech Eng 287:335–366MathSciNetCrossRefGoogle Scholar
  48. 48.
    Saloustros S, Pelà L, Cervera M (2015) A crack-tracking technique for localized cohesive-frictional damage. Eng Fract Mech 150:96–114CrossRefGoogle Scholar
  49. 49.
    Pelà L, Cervera M, Oller S, Chiumenti M (2014) A localized mapped damage model for orthotropic materials. Eng Fract Mech 124–125:196–216CrossRefGoogle Scholar
  50. 50.
    Linder C, Raina A (2013) A strong discontinuity approach on multiple levels to model solids at failure. Comput Methods Appl Mech Eng 253:558–583CrossRefzbMATHGoogle Scholar
  51. 51.
    Motamedi MH, Weed DA, Foster CD (2016) Numerical simulation of mixed mode (I and II) fracture behavior of pre-cracked rock using the strong discontinuity approach. Int J Solids Struct 85–86:44–56CrossRefGoogle Scholar
  52. 52.
    Li J-B, Fu X-A, Chen B-B, Wu C, Lin G (2016) Modeling crack propagation with the extended scaled boundary finite element method based on the level set method. Comput Struct 167:50–68CrossRefGoogle Scholar
  53. 53.
    Wu JY, Li FB, Xu SL (2015) Extended embedded finite elements with continuous displacement jumps for the modeling of localized failure in solids. Comput Methods Appl Mech Eng 285:346–378MathSciNetCrossRefGoogle Scholar
  54. 54.
    Feld-Payet S, Chiaruttini V, Besson J, Feyel F (2015) A new marching ridges algorithm for crack path tracking in regularized media. Int J Solids Struct 71:57–69CrossRefGoogle Scholar
  55. 55.
    Comi C, Perego U (2001) Fracture energy based bi-dissipative damage model for concrete. Int J Solids Struct 38(36–37):6427–6454CrossRefzbMATHGoogle Scholar
  56. 56.
    Pelà L, Cervera M, Roca P (2011) Continuum damage model for orthotropic materials: application to masonry. Comput Methods Appl Mech Eng 200:917–930MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Voyiadjis GZ, Taqieddin ZN, Kattan PI (2009) Theoretical formulation of a coupled elastic-plastic anisotropic damage model for concrete using the strain energy equivalence concept. Int J Damage Mech 18(7):603–638CrossRefGoogle Scholar
  58. 58.
    Mazars J, Hamon F, Grange S (2014) A new 3D damage model for concrete under monotonic, cyclic and dynamic loadings. Mater Struct 48:3779–3793CrossRefGoogle Scholar
  59. 59.
    He W, Wu YF, Xu Y, Fu TT (2015) A thermodynamically consistent nonlocal damage model for concrete materials with unilateral effects. Comput Methods Appl Mech Eng 297:371–391MathSciNetCrossRefGoogle Scholar
  60. 60.
    Pereira LF, Weerheijm J, Sluys LJ (2016) A new rate-dependent stress-based nonlocal damage model to simulate dynamic tensile failure of quasi-brittle materials. Int J Impact Eng 94:83–95CrossRefGoogle Scholar
  61. 61.
    Lemaitre J, Chaboche JL (1978) Aspect phenomenologique de la rupture par endommagement. J Mec Appl 2(3):317–365Google Scholar
  62. 62.
    Simo JC, Ju JW (1987) Strain- and stress-based continuum damage models-I. Formulation. Int J Solids Struct 23(7):821–840CrossRefzbMATHGoogle Scholar
  63. 63.
    Oliver J, Cervera M, Oller Martinez SH, Lubliner J (1990) Isotropic damage models and smeared crack analysis of concrete. In: Proceedings SCI-C computer aided analysis and design of concrete structures, Feb, pp 945–957Google Scholar
  64. 64.
    Bazant Z, Oh B (1983) Crack band theory for fracture of concrete. Mater Struct 16:155–177Google Scholar
  65. 65.
    Cervera M (2003) Viscoelasticity and rate-dependent continuum damage models, monography N-79, technical report, BarcelonaGoogle Scholar
  66. 66.
    Oliver J (1989) A consistent characteristic length for smeared cracking models. Int J Numer Methods Eng 28(2):461–474CrossRefzbMATHGoogle Scholar
  67. 67.
    Wu J-Y, Cervera M (2015) On the equivalence between traction- and stress-based approaches for the modeling of localized failure in solids. J Mech Phys Solids 82:137–163MathSciNetCrossRefGoogle Scholar
  68. 68.
    Cervera M, Wu J-Y (2015) On the conformity of strong, regularized, embedded and smeared discontinuity approaches for the modeling of localized failure in solids. Int J Solids Struct 71:19–38CrossRefGoogle Scholar
  69. 69.
    ASTM:C496/C496M (2011) Standard test method for splitting tensile strength of cylindrical concrete specimens, vol 336. ASTM International, West Conshohocken, PA, pp 1–5Google Scholar
  70. 70.
    ASTM:D3967-08 (2008) Standard test method for splitting tensile strength of intact rock core specimens. ASTM International, West Conshohocken, PAGoogle Scholar
  71. 71.
    COMET (2013) Coupled mechanical and thermal analysis.
  72. 72.
    GiD (2014) The personal pre and post-processor.
  73. 73.
    EN (Eurocode 2) (1992) Design of concrete structures. Technical report, LondonGoogle Scholar
  74. 74.
    Augenti N, Parisi F, Prota A, Manfredi G (2011) In-plane lateral response of a full-scale masonry subassemblage with and without an inorganic matrix-grid strengthening system. J Compos Constr 15(4):578–590CrossRefGoogle Scholar
  75. 75.
    Parisi F, Lignola GP, Augenti N, Prota A, Manfredi G (2011) Nonlinear behavior of a masonry subassemblage before and after strengthening with inorganic matrix-grid composites. J Compos Constr 15(5):821–832CrossRefGoogle Scholar
  76. 76.
    EN 1998-1 (Eurocode 8) (2003) Design of structures for earthquake resistance, part 1 general rules seismic actions and rules for buildingsGoogle Scholar
  77. 77.
    Fajfar P (1999) Capacity spectrum method based on inelastic demand spectra. Earthq Eng Struct Dyn 28:979–993CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringTechnical University of Catalonia, UPC-BarcelonaTechBarcelonaSpain
  2. 2.International Center for Numerical Methods in Engineering (CIMNE)BarcelonaSpain

Personalised recommendations