International Journal of Fracture

, Volume 205, Issue 2, pp 127–138 | Cite as

Numerical and analytical investigation of the size-dependency of the FPZ length in concrete

  • Nassima Aissaoui
  • Mohammed Matallah
Original Paper


The main objective of this paper is to study the size effect on the fracture characteristics in concrete structures. The numerical investigation is based on a mesoscale modeling approach. Analytically, two size effect laws are investigated: the classical Bažant SEL and a new size effect law based on the enrichment of the stress field on the crack tip. The mesoscopic approach is used to study the evolution of the tangential stress along the crack path in order to investigate the fracture process zone variation during the cracking process. In addition, different analytical governing equations are used to evaluate the size-dependency of the FPZ length.


Size effect Fracture FPZ length Mesoscopic modeling 



The authors thank Dr Syed Yasir Alam and Pr Ahmed Loukili from the GEM laboratory (Ecole Centrale de Nantes, France) for providing experimental results.


  1. Abdalla HM, Karihaloo BL (2003) Determination of size-independent specific fracture energy of concrete from threepoint bend and wedge splitting tests. Mag Concr Res 55(2):133–141CrossRefGoogle Scholar
  2. Alam SY (2011) Experimental study and numerical analysis of crack opening in concrete. Dissertation, Ecole Centrale de Nantes, FranceGoogle Scholar
  3. Alam SY, Loukili A, Grondin F (2012) Monitoring size effect on crack opening in concrete by digital image correlation. Eur J Environ Civ Eng 16(7):818–836CrossRefGoogle Scholar
  4. Ayatollahi MR, Akbardoost J (2012) Size effects on fracture toughness of quasibrittle materials—a new approach. Eng Fract Mech 92:89–100CrossRefGoogle Scholar
  5. Bažant ZP, Gettu R, Kazemi MT (1991) Identification of nonlinear fracture properties from size-effect tests and structural analysis based on geometrydependent R-curves. Int J Rock Mech Min Sci 28(1):43–51CrossRefGoogle Scholar
  6. Bažant ZP (2005) Scaling of structural strength, 2nd edn. Elsevier, LondonGoogle Scholar
  7. Bažant ZP, Kazemi MT (1990) Determination of fracture energy, process zone length and brittleness number from size effect, with application to rock and concrete. Int J Fract 44(2):111–131CrossRefGoogle Scholar
  8. Bažant ZP, Pfeiffer PA (1987) Determination of fracture energy from size effect and brittleness number. ACI Mater J 84(6):463–480Google Scholar
  9. Cedolin L, Dei Poli S, Iori I (1983) Experimental determination of the fracture process zone in concrete. Cem Concr Res 13(4):557–567CrossRefGoogle Scholar
  10. Cusatis G, Schauffert EA (2009) Cohesive crack analysis of size effect. Eng Fract Mech 76(14):2163–2173CrossRefGoogle Scholar
  11. Dung N, Lawrence C, La Borderie C, Matallah M, Nahas G (2010) A mesoscopic model for a better understanding of the transition from diffuse damage to localized damage. Eur J Environ Civ Eng 14(6–7):751–776CrossRefGoogle Scholar
  12. Erdogan F, Sih GC (1963) On the crack extension in plates under plane loading and transverse shear. J Basic Eng Trans ASME 85:519–525CrossRefGoogle Scholar
  13. Fichant S, La Borderie C, Pijaudier- Cabot G (1999) Isotropic and anisotropic description of damage in concrete structures. Mech Cohes Frict Mater 4(4):339–359CrossRefGoogle Scholar
  14. Grassl P, Grégoire D, Rojas Solano L, Pijaudier-Cabot G (2012) Mesoscale modelling of the size effect on the fracture process zone of concrete. Int J Solids Struct 49(13):1818–1827CrossRefGoogle Scholar
  15. Grondin F, Matallah M (2014) How to consider the interfacial transition zones in the finite element modelling of concrete? Cem Concr Res 58:67–75CrossRefGoogle Scholar
  16. Guinea GV, Pastor JY, Plans J, Elices M (1998) Stress intensity factor, compliance and CMOD for a general three-point-bend beam. Int J Fract 89(2):103–116CrossRefGoogle Scholar
  17. Hillerborg A, Modéer M, Petersson P-E (1976) Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem Concr Res 6(6):773–781CrossRefGoogle Scholar
  18. Hu XZ, Duan K (2008) Size effect and quasi-brittle fracture: the role of FPZ. Int J Fract 154(1):3–14CrossRefGoogle Scholar
  19. Karihaloo BL (1999) Size effect in shallow and deep-notched quasi-brittle structures. Int J Fract 95(1):379–390CrossRefGoogle Scholar
  20. Karihaloo BL, Abdalla HM, Xiao QZ (2003) Size effect in concrete beams. Eng Fract Mech 70(7–8):979–993CrossRefGoogle Scholar
  21. Karihaloo BL, Abdalla HM, Xiao QZ (2006) Deterministic size effect in the strength of cracked concrete structures. Cem Concr Res 36(1):171–188CrossRefGoogle Scholar
  22. Karihaloo BL, Xiao QZ (2001) Higher order terms of the crack tip asymptotic field for a notched three-point bend beam. Int J Fract 112(2):111–128CrossRefGoogle Scholar
  23. Matallah M, La Borderie C, Maurel O (2010) A practical method to estimate crack openings in concrete structures. Int J Numer Anal Methods Geomech 34(15):1615–1633Google Scholar
  24. Matallah M, Farah M, Grondin F, Loukili A, Rozière E (2013) Sizeindependent fracture energy of concrete at very early ages by inverse analysis. Eng Fract Mech 109:1–16CrossRefGoogle Scholar
  25. Mindess S (1984) The effect of specimen size on the fracture energy of concrete. Cem Concr Res 14(3):431–436CrossRefGoogle Scholar
  26. Morel S (2008) Size effect in quasibrittle fracture: derivation of the energetic size effect law from equivalent LEFM and asymptotic analysis. Int J Fract 154(1):15–26CrossRefGoogle Scholar
  27. Morel S, Dourado N (2011) Size effect in quasibrittle failure: analytical model and numerical simulations using cohesive zone model. Int J Solids Struct 48(10):1403–1412CrossRefGoogle Scholar
  28. Nallathambi P, Karihaloo BL, Heaton BS (1984) Effect of specimen and crack sizes, water/cement ration and coarse aggregate texture upon fracture toughness of concrete. Mag Concr Res 36(129):227–236CrossRefGoogle Scholar
  29. Owen DRJ, Fawkes AJ (1983) Engineering fracture mechanics: numerical methods and applications. Pineridge Press Ltd, SwanseaGoogle Scholar
  30. Petersson PE (1980) Fracture energy of concrete: practical performance and experimental results. Cem Concr Res 10(1):91–101CrossRefGoogle Scholar
  31. Saliba J, Grondin F, Matallah M, Loukili A, Boussa H (2010) Relevance of a mesoscopic modeling for the coupling between creep and damage in concrete. Mech Time-Depend Mater 17(3):481–499CrossRefGoogle Scholar
  32. Shah SP (1990) Experimental methods for determining fracture process zone and fracture parameters. Eng Fract Mech 35(1–3):3–14CrossRefGoogle Scholar
  33. Skarzynski L, Tejchman J (2010) Calculations of fracture process zones on mesoscale in notched concrete beams subjected to three-point bending. Eur J Mech A/Solids 29(4):746–760CrossRefGoogle Scholar
  34. Timoshenko S, Goodier JN (1951) Theory of elasticity. McGraw-Hill, New YorkGoogle Scholar
  35. Weilbull W (1939) The phenomenon of rupture in solids. In: Proceedings of the Royal Swedish Institute of Engineering Research, Ingenioers Vetenskaps Akad. Handl., Stockholm, vol 153, pp 1–55Google Scholar
  36. Williams ML (1957) On the stress distribution at the base of a stationary crack. J Appl Mech 24:109–114Google Scholar
  37. Wu Z, Rong H, Zheng J, Xu F, Dong W (2011) An experimental investigation on the FPZ properties in concrete using digital image correlation technique. Eng Fract Mech 78(17):2978–2990CrossRefGoogle Scholar
  38. Zhang D, Wu K (1999) Fracture process zone of notched three-point-bending concrete beams. Cem Concr Res 29(12):1887–1892CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.RiSAMUniversité de TlemcenTlemcenAlgeria

Personalised recommendations