Computational Mechanics

, Volume 58, Issue 4, pp 549–585 | Cite as

Cohesive surface model for fracture based on a two-scale formulation: computational implementation aspects

  • S. Toro
  • P. J. Sánchez
  • J. M. Podestá
  • P. J. Blanco
  • A. E. Huespe
  • R. A. Feijóo
Original Paper


The paper describes the computational aspects and numerical implementation of a two-scale cohesive surface methodology developed for analyzing fracture in heterogeneous materials with complex micro-structures. This approach can be categorized as a semi-concurrent model using the representative volume element concept. A variational multi-scale formulation of the methodology has been previously presented by the authors. Subsequently, the formulation has been generalized and improved in two aspects: (i) cohesive surfaces have been introduced at both scales of analysis, they are modeled with a strong discontinuity kinematics (new equations describing the insertion of the macro-scale strains, into the micro-scale and the posterior homogenization procedure have been considered); (ii) the computational procedure and numerical implementation have been adapted for this formulation. The first point has been presented elsewhere, and it is summarized here. Instead, the main objective of this paper is to address a rather detailed presentation of the second point. Finite element techniques for modeling cohesive surfaces at both scales of analysis (FE\(^2\) approach) are described: (i) finite elements with embedded strong discontinuities are used for the macro-scale simulation, and (ii) continuum-type finite elements with high aspect ratios, mimicking cohesive surfaces, are adopted for simulating the failure mechanisms at the micro-scale. The methodology is validated through numerical simulation of a quasi-brittle concrete fracture problem. The proposed multi-scale model is capable of unveiling the mechanisms that lead from the material degradation phenomenon at the meso-structural level to the activation and propagation of cohesive surfaces at the structural scale.


Multi-scale cohesive models Computational homogenization Heterogeneous material failure Embedded Finite Elements (EFEM) 



S.Toro, P. J. Sánchez and A. E. Huespe acknowledge the financial support from CONICET (grant PIP 2013-2015 631) and from the European Research Council under the European Unions Seventh Framework Programme (FP/2007–2013) / ERC Grant Agreement N. 320815 (ERC Advanced Grant Project Advanced tools for computational design of engineering materials COMP-DES-MAT). P. J. Blanco and R. A. Feijóo acknowledge the financial support provided by the Brazilian agencies CNPq and FAPERJ.


  1. 1.
    Dugdale D (1960) Yielding of steel sheets containing slits. J Mech Phys Solids 8:100–108CrossRefGoogle Scholar
  2. 2.
    Barenblatt G (1962) The mathematical theory of equilibrium of cracks in brittle fracture. Adv Appl Mech 7:55–129MathSciNetCrossRefGoogle Scholar
  3. 3.
    Needleman A (1987) A continuum model for void nucleation by inclusion debonding. J Appl Mech 54(3):525–531CrossRefzbMATHGoogle Scholar
  4. 4.
    Xu XP, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42:1397–1434CrossRefzbMATHGoogle Scholar
  5. 5.
    Needleman A (2014) Some issues in cohesive surface modeling. Procedia IUTAM 10:221–246CrossRefGoogle Scholar
  6. 6.
    Falk, ML, Needlemann A, Rice JR (2001) A critical evaluation of dynamic fracture simulation using cohesive surfaces. J Phys IV Pr-5-43–Pr-5-50Google Scholar
  7. 7.
    Xu XP, Needleman A (1995) Numerical simulations of dynamic crack growth along an interface. Int J Fract 74(4):289–324CrossRefGoogle Scholar
  8. 8.
    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):279–297CrossRefGoogle Scholar
  9. 9.
    Hutchinson JW, Evans AG (2000) Mechanics of materials: top-down approaches to fracture. Acta Mater 48(1):125–135CrossRefGoogle Scholar
  10. 10.
    Tvergaard V (2001) Crack growth predictions by cohesive zone model for ductile fracture. J Mech Phys Solids 49(9):2191–2207CrossRefzbMATHGoogle Scholar
  11. 11.
    Siegmund T, Brocks W (2000) A numerical study on the correlation between the work of separation and the dissipation rate in ductile fracture. Eng Fract Mech 67(2):139–154CrossRefGoogle Scholar
  12. 12.
    Huespe AE, Needleman A, Oliver J, Sánchez PJ (2009) A finite thickness band method for ductile fracture analysis. Int J Plast 25(12):2349–2365CrossRefGoogle Scholar
  13. 13.
    Huespe AE, Needleman A, Oliver J, Sánchez PJ (2012) A finite strain, finite band method for modeling ductile fracture. Int J Plast 28(1):53–69CrossRefGoogle Scholar
  14. 14.
    Hillerborg A, Modéer M, Petersson PE (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement Concr Res 6(6):773–781CrossRefGoogle Scholar
  15. 15.
    Bažant ZP (2002) Concrete fracture models: testing and practice. Eng Fract Mech 69(2):165–205CrossRefGoogle Scholar
  16. 16.
    Elices M, Guinea GV, Gomez J, Planas J (2002) The cohesive zone model: advantages, limitations and challenges. Eng Fract Mech 69(2):137–163CrossRefGoogle Scholar
  17. 17.
    Oliver J, Huespe AE, Pulido MDG, Chaves E (2002) From continuum mechanics to fracture mechanics: the strong discontinuity approach. Eng Fract Mech 69:113–136CrossRefGoogle Scholar
  18. 18.
    Oliver J (2000) On the discrete constitutive models induced by strong discontinuity kinematics and continuum constitutive equations. Int J Solids Struct 37:7207–7229MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tvergaard V, Hutchinson JW (1992) The relation between crack growth resistance and fracture process parameters in elasto-plastic solids. J Mech Phys Solids 40:1377–1397CrossRefzbMATHGoogle Scholar
  20. 20.
    Xia L, Shih CF (1995) Ductile crack growth I. A numerical study using computational cells with micrstructurally based length scales. J Mech Phys Solids 43:233–259CrossRefzbMATHGoogle Scholar
  21. 21.
    Vernerey FranckJ, Liu Wing Kam, Moran Brian, Olson Gregory (2008) A micromorphic model for the multiple scale failure of heterogeneous materials. J Mech Phys Solids 56(4):1320–1347MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sánchez PJ, Blanco PJ, Huespe AE, Feijóo RA (2013) Failure-oriented multi-scale variational formulation: micro-structures with nucleation and evolution of softening bands. Comput Methods Appl Mech Eng 257:221–247MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gitman IM, Askes H, Sluys LJ (2007) Representative volume: Existence and size determination. Eng Fract Mech 74:2518–2534CrossRefGoogle Scholar
  24. 24.
    Nguyen VP, Lloberas-Valls O, Stroeven M, Sluys LJ (2010a) On the existence of representative volumes for softening quasi-brittle materials - a failure zone averaging scheme. Comput Methods Appl Mech Eng 199:3028–3038CrossRefzbMATHGoogle Scholar
  25. 25.
    Belytschko T, Loehnert S, Song JH (2008) Multiscale aggregating discontinuities: a method for circumventing loss of material stability. Int J Numer Methods Eng 73(6):869–894MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Belytschko T, Song JH (2010) Coarse-graining of multiscale crack propagation. Int J Numer Methods Eng 81(5):537–563MathSciNetzbMATHGoogle Scholar
  27. 27.
    Geers MGD, Kouznetsova VG, Brekelmans WAM (2010) Multi-scale computational homogenization: Trends and challenges. J Comput Appl Math 234:2175–2182CrossRefzbMATHGoogle Scholar
  28. 28.
    Bosco E, Kouznetsova VG, Geers MGD (2015) Multi-scale computational homogenization–localization for propagating discontinuities using x-fem. Int J Numer Methods Eng 102(3–4):496–527MathSciNetCrossRefGoogle Scholar
  29. 29.
    Nguyen VP, Lloberas-Valls O, Sluys LJ, Stroeven M (2010b) Homogenization-based multiscale crack modelling. Comput Methods Appl Mech Eng 200:1220–1236CrossRefzbMATHGoogle Scholar
  30. 30.
    Verhoosel CV, Remmers JJC, Gutiérrez MA, de Borst R (2010) Computational homogenization for adhesive and cohesive failure in quasi-brittle solids. Int J Numer Methods Eng 83:1155–1179CrossRefzbMATHGoogle Scholar
  31. 31.
    Oliver J, Caicedo M, Roubin E, Huespe AE, Hernández JA (2015) Continuum approach to computational multiscale modeling of propagating fracture. Comput Methods Appl Mech Eng 294:384–427MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kulkarni MG, Geubelle PH, Matous K (2009) Multi-scale modeling of heterogeneous adhesives: Effect of particle decohesion. Mech Mater 41:573–583CrossRefGoogle Scholar
  33. 33.
    Blanco PJ, Sánchez PJ, de Souza Neto EA, Feijóo RA (2016a) Variational foundations and generalized unified theory of RVE-based multiscale models. Arch Comput Methods Eng 23:191–253. doi: 10.1007/s11831-014-9137-5 MathSciNetCrossRefGoogle Scholar
  34. 34.
    Hill R (1972) On constitutive macro-variables for heterogeneous solids at finite strain. Proc R Soc Lond 326:131–147CrossRefzbMATHGoogle Scholar
  35. 35.
    Mandel J (1971) Plasticit classique at viscoplasticit., CISIM Lecture NotesSpringer, BerlinGoogle Scholar
  36. 36.
    de Souza EA (2006) Neto and R.A. Feijóo. Variational foundation on multi-scale constitutive models of solids: small and large strain kinematical formulation. LNCC Research & Development. Report 16Google Scholar
  37. 37.
    de Souza Neto EA, Feijóo RA (2008) On the equivalence between spatial and material volume averaging of stress in large strain multi-scale solid constitutive models. Mech Mater 40:803–811CrossRefGoogle Scholar
  38. 38.
    Perić D, de Souza Neto RA, Feijóo M Partovi, Carneiro Molina AJ (2011) On micro-to-macro transitions for multi-scale analysis of non-linear heterogeneous materials. Int J Numer Methods Eng 87:149–170CrossRefzbMATHGoogle Scholar
  39. 39.
    de Souza Neto EA, Feijóo RA (2011) Variational foundations of large strain multiscale solid constitutive models: Kinematical formualtion. In: Vaz M, de Souza Neto EA, Muoz-Rojas PA (eds) Advanced computational materials modeling. From classical to multi-scale techniques. Wiley, Weinheim, pp 341–378Google Scholar
  40. 40.
    Blanco PJ, Sánchez PJ, de Souza Neto EA, Feijóo RA (2016b) The method of multiscale virtual power for the derivation of a second-order mechanical model. Mech Mater 99:53–67CrossRefGoogle Scholar
  41. 41.
    Blanco PJ, Giusti SM (2014) Thermomechanical multiscale constitutive modeling: accounting for microstructural thermal effects. J Elast 115:27–46MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    de Souza Neto EA, Blanco PJ, Sánchez PJ, Feijóo RA (2015) An rve-based multiscale theory of solids with micro-scale inertia and body force effects. Mech Mater 80:136–144CrossRefGoogle Scholar
  43. 43.
    Toro S, Sánchez PJ, Huespe AE, Giusti SM, Blanco PJ, Feijóo RA (2014) A two-scale failure model for heterogeneous materials: numerical implementation based on the finite element method. Int J Numer Methods Eng 97(5):313–351MathSciNetCrossRefGoogle Scholar
  44. 44.
    Toro S, Sánchez PJ, Blanco PJ, de Souza Neto EA, Huespe AE, Feijóo RA (2016) Multiscale formulation for material failure accounting for cohesive cracks at the macro and micro scales. Int J Plast 76:75–110CrossRefGoogle Scholar
  45. 45.
    Simo J, Oliver J, Armero F (1993) An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput Mech 12:277–296MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Oliver J, Dias IF, Huespe AE (2014) Crack-path field and strain-injection techniques in computational modeling of propagating material failure. Comput Methods Appl Mech Eng 274:289–348MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Miehe C, Koch A (2002) Computational micro-to-macro transition of discretized microstructures undergoing small strain. Arch Appl Mech 72:300–317CrossRefzbMATHGoogle Scholar
  48. 48.
    Oliver J, Huespe AE, Blanco S, Linero DL (2005) Stability and robustness issues in numerical modeling of material failure with the strong discontinuity approach. Comput Methods Appl Mech Eng 195(52):7093–7114CrossRefzbMATHGoogle Scholar
  49. 49.
    Manzoli OL, Gamino AL, Rodrigues EA, Claro GKS (2012) Modeling of interfaces in two-dimensional problems using solid finite elements with high aspect ratio. Comput Struct 94:70–82CrossRefGoogle Scholar
  50. 50.
    Carol I, López CM, Roa O (2001) Micromechanical analysis of quasi-brittle materials using fracture-based interface elements. Int J Numer Methods Eng 52(1–2):193–215CrossRefGoogle Scholar
  51. 51.
    Unger JF, Eckardt S (2011) Multiscale modeling of concrete. Arch Comput Methods Eng 18(3):341–393CrossRefzbMATHGoogle Scholar
  52. 52.
    Bocca P, Carpinteri A, Valente S (1990) Size effects in the mixed mode crack propagation: softening and snap-back analysis. Eng Fract Mech 35(1):159–170CrossRefGoogle Scholar
  53. 53.
    Feyel F, Chaboche JL (2000) FE\(^2\) multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Comput Methods Appl Mech Eng 183:309–330CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • S. Toro
    • 1
    • 2
  • P. J. Sánchez
    • 1
    • 2
  • J. M. Podestá
    • 1
  • P. J. Blanco
    • 3
    • 4
  • A. E. Huespe
    • 1
    • 5
  • R. A. Feijóo
    • 3
    • 4
  1. 1.CIMEC-UNL-CONICETSanta FeArgentina
  2. 2.GIMNI-UTN-FRSFSanta FeArgentina
  3. 3.LNCC/MCTI, Laboratório Nacional de Computação CientíficaPetrópolisBrazil
  4. 4.INCT-MACC, Instituto Nacional de Ciência e Tecnologia em Medicina Assistida por Computação CientíficaPetrópolisBrazil
  5. 5.Centre Internacional de Métodes Numérics en Enginyeria (CIMNE)BarcelonaSpain

Personalised recommendations