Skip to main content
SpringerLink
Log in

A homogenization-based quasi-discrete method for the fracture of heterogeneous materials

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

Abstract

The understanding and the prediction of the failure behaviour of materials with pronounced microstructural effects is of crucial importance. This paper presents a novel computational methodology for the handling of fracture on the basis of the microscale behaviour. The basic principles presented here allow the incorporation of an adaptive discretization scheme of the structure as a function of the evolution of strain localization in the underlying microstructure. The proposed quasi-discrete methodology bridges two scales: the scale of the material microstructure, modelled with a continuum type description; and the structural scale, where a discrete description of the material is adopted. The damaging material at the structural scale is divided into unit volumes, called cells, which are represented as a discrete network of points. The scale transition is inspired by computational homogenization techniques; however it does not rely on classical averaging theorems. The structural discrete equilibrium problem is formulated in terms of the underlying fine scale computations. Particular boundary conditions are developed on the scale of the material microstructure to address damage localization problems. The performance of this quasi-discrete method with the enhanced boundary conditions is assessed using different computational test cases. The predictions of the quasi-discrete scheme agree well with reference solutions obtained through direct numerical simulations, both in terms of crack patterns and load versus displacement responses.

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

Similar content being viewed by others

References

  1. Hettich T, Hund A, Ramm E (2008) Modeling of failure in composites by X-FEM and level sets within a multiscale framework. Comput Methods Appl Mech Eng 197:414

    Article  MATH  Google Scholar 

  2. Nguyen V, Stroeven M, Sluys L (2012) Multiscale failure modeling of concrete: micromechanical modeling, discontinuous homogenization and parallel computations. Comput Methods Appl Mech Eng 201–204:139

    Article  MathSciNet  Google Scholar 

  3. van der Pluijm R (1999) Out-of-plane bending of masonry—behaviour and strength. Ph.D. thesis, Technische Universiteit Eindhoven

  4. Smilauer V, Hoover C, Bazant Z, Caner F, Waas A, Shahwan K (2011) Multiscale simulation of fracture of braided composites via repetitive unit cells. Eng Frac Mech 78:901

    Article  Google Scholar 

  5. Kouznetsova V, Geers M, Brekelmans W (2004) Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech Eng 193:5525

    Article  MATH  Google Scholar 

  6. Mercatoris B, Massart T (2009) Assessment of periodic homogenisation-based multiscale computational schemes for quasi-brittle structural failure. Int J Multiscale Comput Eng 7:153–170

    Article  Google Scholar 

  7. Feyel F (2003) A multilevel finite element method (FE\(^2\)) to describe the response of highly non-linear structures using generalized continua. Comput Methods Appl Mech Eng 192:3233

    Article  MATH  Google Scholar 

  8. Inglis H, Geubelle P, Matous K (2008) Boundary condition effects on multiscale analysis of damage localization. Philos Mag 88:2373

    Article  Google Scholar 

  9. Coenen E, Kouznetsova V, Geers M (2010) Computational homogenization for heterogeneous thin sheets. Int J Numer Methods Eng 83:1180

    Article  MATH  Google Scholar 

  10. Mercatoris B, Massart T (2011) A coupled two-scale computational scheme for the failure of periodic quasi-brittle thin planar shells and its application to masonry. Int J Numer Methods Eng 85:1177

    Article  MATH  Google Scholar 

  11. Özdemir I, Brekelmans W, Geers M (2008) Computational homogenization for heat conduction in heterogeneous solids. Int J Numer Methods Eng 73:185

    Google Scholar 

  12. Coenen E, Kouznetsova V, Geers M (2012) Novel boundary conditions for strain localization analyses in microstructural volume elements. Int J Numer Methods Eng 90:1

    Article  MATH  MathSciNet  Google Scholar 

  13. Massart T, Peerlings R, Geers M (2005) A dissipation-based control method for the multi-scale modelling of quasi-brittle materials. C.R. Mécanique 333:521

    Article  Google Scholar 

  14. Massart T, Peerlings R, Geers M (2007) An enhanced multi-scale approach for masonry walls computations with localisation of damage. Int J Numer Methods Eng 69:1022

    Article  MATH  Google Scholar 

  15. Mesarovic S, Padbidri J (2005) Minimal kinematic boundary conditions for simulations of disordered microstructures. Philos Mag 85:65

    Article  Google Scholar 

  16. Lloberas-Valls O, Rixen D, Simone A, Sluys L (2012) Multiscale domain decomposition analysis of quasi-brittle heterogeneous materials. Int J Numer Methods Eng 89:1337

    Article  MATH  MathSciNet  Google Scholar 

  17. Ghosh S, Lee K, Raghavan P (2001) A multi-level computational model for multi-scale damage analysis in composite and porous materials. Int J Solids Struct 38:2335

    Article  MATH  Google Scholar 

  18. Ghosh S, Bai J, Raghavan P (2007) Concurrent multi-level model for damage evolution in microstructurally debonding composites. Mech Mater 39:241

    Article  Google Scholar 

  19. Ladevèze P, Loiseau O, Dureisseix D (2001) A micro-macro and parallel computational strategy for highly heterogeneous structures. Int J Numer Methods Eng 51:121

    Article  Google Scholar 

  20. Ibrahimbegovic A, Markovic D (2003) Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures. Comput Methods Appl Mech Eng 192:3089

    Article  MATH  Google Scholar 

  21. Magoulès F, Roux FX (2006) Lagrangian formulation of domain decomposition methods: a unified theory. Appl Math Model 30:593

    Article  MATH  Google Scholar 

  22. Maday Y, Magoulès F (2006) Absorbing interface conditions for domain decomposition methods: a general presentation. Comput Methods Appl Mech Eng 195:3880

    Article  MATH  Google Scholar 

  23. Farhat C, Roux FX (1991) A method of finite element tearing and interconnecting and its parallel solution algorithm. Int J Numer Methods Eng 32:1205

    Article  MATH  Google Scholar 

  24. Garikipati K, Hughes T (1998) A study of strain localization in a multiple scale framework—the one-dimensional problem. Comput Methods Appl Mech Eng 159:193

    Article  MATH  MathSciNet  Google Scholar 

  25. Garikipati K, Hughes T (2000) A variational multiscale approach to strain localization—formulation for multidimensional problems. Comput Methods Appl Mech Eng 188:39

    Article  MATH  MathSciNet  Google Scholar 

  26. Kouznetsova V (2002) Computational homogenization for the multi-scale analysis of multi-phase materials, Ph.D. thesis, Technische Universiteit Eindhoven

  27. Belytschko T, Loehnert S, Song JH (2008) Multiscale aggregating discontinuities: a method for circumventing loss of material stability. Int J Numer Methods Eng 73:869

    Article  MATH  MathSciNet  Google Scholar 

  28. Belytschko T, Song JH (2010) Coarse-graining of multiscale crack propagation. Int J Numer Methods Eng 81:537

    MATH  MathSciNet  Google Scholar 

  29. Nguyen V, Lloberas-Valls O, Stroeven M, Sluys L (2011) Homogenization-based multiscale crack modelling: from micro-diffusive damage to macro-cracks. Comput Methods Appl Mech Eng 200:1220

    Article  MATH  Google Scholar 

  30. Nguyen V, Stroeven M, Sluys L (2012) An enhanced continuous-discontinuous multiscale method for modeling mode-I cohesive failure in random heterogeneous quasi-brittle materials. Eng Frac Mech 79:78

    Article  Google Scholar 

  31. Anthoine A (1995) Derivation of the in-plane elastic characteristics of masonry through homogenization theory. Int J Solids Struct 32:137

    Article  MATH  Google Scholar 

  32. Michel J, Moulinec H, Suquet P (1999) Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Eng 172:109

    Article  MATH  MathSciNet  Google Scholar 

  33. Lourenço P, Rots J (1997) Multisurface interface model for analysis of masonry structures. J Eng Mech 123:660

    Article  Google Scholar 

  34. Page A (1978) Finite element model for masonry. J Struct Div ASCE 104:1267

    Google Scholar 

  35. Haach V, Vasconcelos G, Lourenço P (2011) Numerical analysis of concrete block masonry beams under three point bending. Eng Struct 33:3226

    Article  Google Scholar 

Download references

Acknowledgments

The first author was sponsored by the Fonds de la Recherche Scientifique F.R.S.-FNRS of Belgium (post-doctoral research grant ‘Chargè de Recherches’ No. 1.2.093.10.F). The authors also acknowledge the support of F.R.S.-FNRS Belgium (Grant No. 1.5.032.09.F) for the intensive computational facilities used for this work.

Author information

Authors and Affiliations

  1. BATir Department, Université Libre de Bruxelles (ULB), CP194/2, 50 av. F. D. Roosevelt, 1050 , Brussels, Belgium

    P. Z. Berke & T. J. Massart

  2. Mechanics of Materials, Depatment of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB , Eindhoven, The Netherlands

    R. H. J. Peerlings & M. G. D. Geers

Authors
  1. P. Z. Berke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. H. J. Peerlings
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. J. Massart
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. M. G. D. Geers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Z. Berke.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berke, P.Z., Peerlings, R.H.J., Massart, T.J. et al. A homogenization-based quasi-discrete method for the fracture of heterogeneous materials. Comput Mech 53, 909–923 (2014). https://doi.org/10.1007/s00466-013-0939-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-013-0939-3

Keywords

Search

Navigation