International Journal of Fracture

, Volume 178, Issue 1–2, pp 281–298 | Cite as

A non-iterative approach for the modelling of quasi-brittle materials

  • R. Graça-e-Costa
  • J. Alfaiate
  • D. Dias-da-Costa
  • L. J. Sluys
Original Paper


Due to the softening behaviour of quasi-brittle materials, in particular the localisation of initially diffused cracking, convergence problems are often found using an iterative procedure, such as the Newton–Raphson method. This is why a new non-iterative procedure is adopted in this paper, which is inspired by the sequentially linear approach(SLA) (Rots et al. in Eng Fract Mech 75(3–4):590–614, 2008). However, several important differences between the present approach and the SLA are presented. In the present model, multi-linear material laws are adopted such that non-linearities occur only due to changes in loading/unloading states. An incremental solution is obtained until non-convergence occurs, upon which a secant approach is used in a corresponding step. The update of the stiffness in the secant approach is based on information obtained from the previous incremental solution. This method is applied to: (i) softening materials, within the scope of the discrete crack approach, and to (ii) hardening materials. As a consequence, conversely to the smeared crack approach adopted in the SLA, no mesh size sensitivity problems are obtained and there is no need to adjust material parameters. Several numerical examples are shown in order to illustrate the proposed formulation.


NIEM Fracture Non-iterative procedure Sequentially linear analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alfaiate J, Simone A, Sluys LJ (2003) Non-homogeneous displacement jumps in strong embedded discontinuities. Int J Solids Struct 40: 5799–5817. doi: 10.1016/S0020-7683(03)00372-X CrossRefGoogle Scholar
  2. Alfaiate J, Sluys LJ (2002) Analysis of a compression test on concrete using strong embedded discontinuities. In: Mang HA, Rammerstorfer FG, Eberhardsteiner J (eds) WCCM V, fifth world congress on computational mechanics. Wien, AustriaGoogle Scholar
  3. Alfaiate J, Wells GN, Sluys LJ (2002) On the use of embedded discontinuity elements with crack path continuity for mode I and mixed mode fracture. Eng Fract Mech 69(6): 661–686. doi: 10.1016/S0013-7944(01)00108-4 CrossRefGoogle Scholar
  4. Barpi F, Valente S (2000) Numerical simulation of prenotched gravity dam models. J Eng Mech 126(6): 611–619. doi: 10.1061/(ASCE)0733-93399(2000)126:6(611) CrossRefGoogle Scholar
  5. Billington SL (2009) Nonlinear and sequentially linear analysis of tensile strain hardening cement-based composite beams in flexure. In: Hendriks M, Billington SL (eds) Computational modeling workshop on concrete, masonry and on fiber-reinforced composites, pp 7–10, Delf, The NetherlandsGoogle Scholar
  6. Burns NH, Seiss CP (1962) Load-deformation characteristics of beam-column connections in reinforced concrete. Report technical report, Engineering Studies SRS 234. Department of Civil Engineering, University of California, BerkeleyGoogle Scholar
  7. CEB: (1991) CEB-FIP Model Code 1990. Thomas Telford, LondonGoogle Scholar
  8. Crisfield MA (1984) Difficulties with current numerical models for reinforced-concrete and some tentative solutions. Comput Aided Anal Des Concr Struct (1):331–358Google Scholar
  9. Dias-da-Costa D, Alfaiate J, Sluys LJ, Júlio E (2009) Towards a generalization of a discrete strong discontinuity approach. Comput Methods Appl Mech Eng 198(47–48): 3670–3681. doi: 10.1016/j.cma.2009.07.013 CrossRefGoogle Scholar
  10. Dias-da-Costa D, Alfaiate J, Sluys LJ, 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. doi: 10.1007/s10704-009-9432-6 CrossRefGoogle Scholar
  11. Gago A, Milosevic J, Lopes M, Bento R (2011, submitted) Shear strength of rubble stone masonry walls. Bull Earthq EngGoogle Scholar
  12. Galvez JC, Elices M, Guinea GV, Planas J (1998) Mixed mode fracture of concrete under proportional and nonproportional loading. Int J Fract 94(3): 267–284. doi: 10.1023/A:1007578814070 CrossRefGoogle Scholar
  13. Graça-e-Costa R (2005) Modelação de vigas de betão armado reforçadas com chapas metálicas. MSc thesis, Instituto Superior Técnico, Universidade Técnica de Lisboa, PortugalGoogle Scholar
  14. Graça-e-Costa R, Alfaiate J (2006) The numerical analysis of reinforced concrete beams using embedded discontinuities. SDHM Struct Durab Health Monit 1: 11–17. doi: 10.3970/sdhm.2006.002.011 Google Scholar
  15. Gutiérrez MA (2004) Energy release control for numerical simulations of failure in quasi-brittle solids. Commun Numer Methods Eng 20(1): 19–29. doi: 10.1002/cnm.649 CrossRefGoogle Scholar
  16. Invernizzi S, Trovato D, Hendriks MAN, van Graaf AV (2011) Sequentially linear modelling of local snap-back in extremely brittle structures. Eng Struct 33(5): 1617–1625. doi: 10.1016/j.engstruct.2011.01.031 CrossRefGoogle Scholar
  17. Lowes LN (1999) Finite element modeling of reinforced concrete beam-column bridge connections. PhD thesis, University of California, BerkeleyGoogle Scholar
  18. Milosevic J, Bento R, Gago A, Lopes M (2010) Seismic vulnerability of old masonry buildings—SEVERES project. Report 1. Instituto Superior Técnico, Lisbon.
  19. Oliver J, Huespe AE, Cante JC (2008) An implicit/explicit integration scheme to increase computability of non-linear material and contact/friction problems. Comput Methods Appl Mech Eng 197(21–24): 1865–1889. doi: 10.1016/j.cma.2007.11.027 CrossRefGoogle Scholar
  20. Rots JG (2001) The role of structural modelling in preserving Amsterdam architectural city heritage. In: Lourenço PB, Roca P (eds) Historical constructions. Guimarães, Portugal, pp 685–696Google Scholar
  21. Rots JG, Belletti B, Invernizzi S (2008) Robust modeling of RC structures with an ”event-by-event” strategy. Eng Fract Mech 75(3–4): 590–614. doi: 10.1016/j.engfracmech.2007.03.027 CrossRefGoogle Scholar
  22. Schlangen E (1993) Experimental and numerical analysis of fracture process in concrete. PhD thesis, Delft University of Technology, The NetherlandsGoogle Scholar
  23. Slobbe AT, Hendriks MAN, Rots JG (2012) Sequentially linear analysis of shear critical reinforced concrete beams without shear reinforcement. Finite Elem Anal Des 50(0): 108–124. doi: 10.1016/j.finel.2011.09.002 CrossRefGoogle Scholar
  24. Verhoosel CV., Remmers JJC, Gutiérrez MA (2009) A dissipation-based arc-length method for robust simulation of brittle and ductile failure. Int J Numer Methods Eng 77(9): 1290–1321. doi: 10.1002/nme.2447 CrossRefGoogle Scholar
  25. Xing H, Zhang J (2009) Finite element modelling of non-linear deformation of rate-dependent materials using a R-minimum strategy. Acta Geotechnica 4(2): 139–148. doi: 10.1007/s11440-009-0090-7 CrossRefGoogle Scholar
  26. Yamada Y, Yoshimura N, Sakurai T (1968) Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by the finite element method. Int J Mech Sci 10(5): 343–354. doi: 10.1016/0020-7403(68)90001-5 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • R. Graça-e-Costa
    • 1
    • 3
  • J. Alfaiate
    • 1
    • 4
  • D. Dias-da-Costa
    • 2
    • 5
  • L. J. Sluys
    • 6
  1. 1.ICISTLisbonPortugal
  2. 2.INESC CoimbraCoimbraPortugal
  3. 3.Department of Civil Engineering, Instituto Superior de EngenhariaUniversidade do AlgarveFaroPortugal
  4. 4.Department of Civil Engineering and Architecture, Instituto Superior TécnicoTechnical University of LisbonLisbonPortugal
  5. 5.Department of Civil EngineeringUniversity of CoimbraCoimbraPortugal
  6. 6.Department of Civil Engineering and GeosciencesDelft University of TechnologyDelftThe Netherlands

Personalised recommendations