International Journal of Fracture

, Volume 154, Issue 1–2, pp 27–49 | Cite as

Computational modeling of size effects in concrete specimens under uniaxial tension

Original Paper


The paper presents a follow-up study of numerical modeling of complicated interplay of size effects in concrete structures. The major motivation is to identify and study interplay of several scaling lengths stemming from the material, boundary conditions and geometry. Methods of stochastic nonlinear fracture mechanics are used to model the well published results of direct tensile tests of dog-bone specimens with rotating boundary conditions. Firstly, the specimens are modeled using microplane material and also fracture-plastic material laws to show that a portion of the dependence of nominal strength on structural size can be explained deterministically. However, it is clear that more sources of size effect play a part, and we consider two of them. Namely, we model local material strength using an autocorrelated random field attempting to capture a statistical part of the combined size effect, scatter inclusive. In addition, the strength drop noticeable with small specimens which was obtained in the experiments could be explained either by the presence of a weak surface layer of constant thickness (caused e.g. by drying, surface damage, aggregate size limitation at the boundary, or other irregularities) or three dimensional effects incorporated by out-of-plane flexure of specimens. The latter effect is examined by comparison of 2D and 3D models with the same material laws. All three named sources (deterministic-energetic, statistical size effects and the weak layer effect) are believed to be the sources most contributing to the observed strength size effect; the model combining all of them is capable of reproducing the measured data. The computational approach represents a marriage of advanced computational nonlinear fracture mechanics with simulation techniques for random fields representing spatially varying material properties. Using a numerical example, we document how different sources of size effects detrimental to strength can interact and result in relatively complicated quasibrittle failure processes. The presented study documents the well known fact that the experimental determination of material parameters (needed for the rational and safe design of structures) is very complicated for quasibrittle materials such as concrete.


Size effect Scaling Random field Weak boundary Crack band Dog-bone specimens Quasibrittle failure Crack initiation Stochastic simulation Characteristic length Weibull integral Microplane model Fracture-plastic model 


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  1. Barenblatt GI (1996) Scaling, self-similarity, and intermediate asymptotics No. 14 in Cambridge texts in applied mathematics. Cambridge University Press, CambridgeGoogle Scholar
  2. Bažant ZP, Oh BH (1983) Crack band theory for fracture of concrete. Mater Struct 16: 155–177Google Scholar
  3. Bažant ZP, Oh BH (1986) Efficient numerical integration on the surface of a sphere. Zeitschrift für angewandte Mathematik und Mechanik (ZAMM) Berlin 66(1): 37–49MATHCrossRefGoogle Scholar
  4. Bažant ZP, Pang SD (2007) Activation energy based extreme value statistics and size effect in brittle and quasibrittle fracture. J Mech Phys Solids 55(1): 91–131CrossRefADSGoogle Scholar
  5. Bažant ZP, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca Raton and LondonGoogle Scholar
  6. Bažant ZP, Caner FC, Carol I, Adley MD, Akers SA (2000) Microplane model M4 for concrete: I. Formulation with work-conjugate deviatoric stress. J Eng Mech (ASCE) 126(9): 944–961CrossRefGoogle Scholar
  7. Bažant ZP, Pang SD, Vořechovský M, Novák D (2007a) Energetic-statistical size effect simulated by SFEM with stratified sampling and crack band model. Int J Numer Methods Eng 71(11): 1297–1320. doi: 10.1002/nme.1986 CrossRefGoogle Scholar
  8. Bažant ZP, Vořechovský M, Novák D (2007b) Asymptotic prediction of energetic-statistical size effect from deterministic finite element solutions. J Eng Mech (ASCE) 133(2):153–162. doi: 10.1061/(ASCE)0733-9399(2007) 133:2(153) Google Scholar
  9. Buckingham E (1914) On physically similar systems; illustrations of the use of dimensional equations. Phys Rev 4(4): 345–376. doi: 10.1103/PhysRev.4.345 CrossRefADSGoogle Scholar
  10. Caner FC, Bažant ZP (2000) Microplane model M4 for concrete: II. algorithm and calibration. J Eng Mech, ASCE 126(9): 954–961CrossRefGoogle Scholar
  11. Carmeliet J, de Borst R (1995) Stochastic approaches for damage evolution instandard and non-standard continua. Int J Solids Struct 32(8–9): 1149–1160. doi: 10.1016/0020-7683(94)00182-V MATHCrossRefGoogle Scholar
  12. Carmeliet J, Hens H (1994) Probabilistic nonlocal damage model for continua with random field properties. J Eng Mech, ASCE 120(10): 2013–2027CrossRefGoogle Scholar
  13. Červenka V, Pukl R (2005) Atena program documentation. Technical report, Červenka Consulting, Prague, Czech Republic.
  14. Červenka J, Bažant ZP, Wierer M (2005) Equivalent localization element for crack band approach to mesh-sensitivity in microplane model. Int J Numer Methods Eng 62(5): 700–726MATHCrossRefGoogle Scholar
  15. de Borst R, Gutiérrez MA, Wells GN, Remmers JJC, Askes H (2004) Cohesive-zone models, higher-order continuum theories and reliability methods for computational failure analysis. Int J Numer Methods Eng 60(1): 289–315. doi: 10.1002/nme.963 MATHCrossRefGoogle Scholar
  16. Dyskin A, van Vliet M, van Mier J (2001) Size effect in tensile strength caused by stress fluctuations. Int J Fract 108: 43–61CrossRefGoogle Scholar
  17. Fisher RA, Tippett LHC (1928) Limiting forms of the frequency distribution of the largest and smallest member of a sample. Proc Camb Philos Soc 24: 180–190MATHCrossRefGoogle Scholar
  18. Gutiérrez MA, de Borst R (2002) Deterministic and probabilistic material length-scales and their role in size-effect phenomena. In: Corotis R, Schuëller G, Shinozuka M (eds) Structural safety and reliability: proceedings of the eighth International Conference on Structural Safety and Reliability ICoSSaR ’01, A.A. Balkema Publishers, Netherlands; Swets & Zeitinger, Newport Beach, California, USA, pp 129–136, abstract page 114Google Scholar
  19. Hordijk D (1991) Local approach to fatigue of concrete. PhD thesis, Delft University of Technology, Delft, The Netherlands, ISBN 90/9004519-8Google Scholar
  20. Jirásek M (1998) Nonlocal models for damage and fracture: comparison of approaches. Int J Solids Struct 35(31–32): 4133–4145MATHCrossRefGoogle Scholar
  21. Lehký D, Novák D (2002) Nonlinear fracture mechanics modeling of size effect in concrete under uniaxial tension. In: Schießl P (ed) 4th International Ph.D. Symposium in Civil Engineering, volume 2, Millpress, Rotterdam, Munich, Germany, pp 410–417Google Scholar
  22. Liu P, Der Kiureghian A (1986) Multivariate distribution models with prescribed marginals and covariances. Probab Eng Mech 1(2): 105–111CrossRefGoogle Scholar
  23. Mazars J, Pijaudier-Cabot G, Saouridis C (1991) Size effect and continuous damage in cementious materials. Int J Fract 51: 159–173Google Scholar
  24. Novák D, Lawanwisut W, Bucher C (2000) Simulation of random fields based on orthogonal transformation of covariance matrix and latin hypercube sampling. In: Schuëller (ed) International Conference on Monte Carlo Simulation MC 2000, Swets & Zeitlinger, Lisse (2001), Monaco, Monte Carlo, pp 129–136Google Scholar
  25. Novák D, Bažant ZP, Vořechovský M (2003a) Computational modeling of statistical size effect in quasibrittle structures. In: Der Kiureghian A, Madanat S, Pestana JM (eds) ICASP 9, International Conference on Applications of Statistics and Probability in Civil Engineering, held in San Francisco, USA, Millpress, Rotterdam, Netherlands, pp 621–628Google Scholar
  26. Novák D, Vořechovský M, Rusina R (2003b) Small-sample probabilistic assessment—FREET software. In: Der Kiureghian A, Madanat S, Pestana JM (eds) ICASP 9, International Conference on Applications of Statistics and Probability in Civil Engineering, held in San Francisco, USA, Millpress, Rotterdam, Netherlands, pp 91–96Google Scholar
  27. Novák D, Vořechovský M, Rusina R (2006) FReET—Feasible Reliability Engineering Efficient Tool. Technical report, Brno/Červenka Consulting, Czech Republic., program documentation—Part 2—User Manual
  28. Pang SD, Bažant ZP, Le JL (2009) Statistics of strength of ceramics: Finite weakest-link model and necessity of zero threshold. Int J Fract. doi: 10.1007/s10704-009-9317-8
  29. Pietruszczak S, Mróz Z (1981) Finite element analysis of deformation of strain softening materials. Int J Numer Methods Eng 17: 327–334. doi: 10.1002/nme.1620170303 MATHCrossRefGoogle Scholar
  30. Pijaudier-Cabot G, Bažant ZP (1987) Nonlocal damage theory. J Eng Mech, ASCE 113: 1512–1533CrossRefGoogle Scholar
  31. Remmers RJC, de Borst R, Needleman A (2003) A cohesive segments method for the simulation of crack growth. Comput Mech 31(1–2): 69–77. doi: 10.1007/s00466-002-0394-z MATHCrossRefGoogle Scholar
  32. van Mier J, van Vliet M (2003) Influence of microstructure of concrete on size/scale effects in tensile fracture. Eng Fract Mech 70: 2281–2306CrossRefGoogle Scholar
  33. van Vliet M (2000) Size effect in tensile fracture of concrete and rock. PhD thesis, Delft University of Technology, Delft, The NetherlandsGoogle Scholar
  34. van Vliet M, van Mier J (1998) Experimental investigation of size effect in concrete under uniaxial tension. In: Mihashi H, Rokugo K (eds) FRAMCOS-3. Aedificatio Publishers, Japan, pp 1923–1936Google Scholar
  35. van Vliet M, van Mier J (1999) Effect of strain gradients on the size effect of concrete in uniaxial tension. Int J Fract 95: 195–219CrossRefGoogle Scholar
  36. van Vliet M, van Mier J (2000a) Experimental investigation of size effect in concrete and sandstone under uniaxial tension. Eng Fract Mech 65: 165–188CrossRefGoogle Scholar
  37. van Vliet M, van Mier J (2000b) Size effect of concrete and sandstone. Eng Fract Mech 45: 91–108Google Scholar
  38. Vashy A (1892) Sur les lois de similitude en physique. Annales télégraphiques 19: 25–28Google Scholar
  39. Vořechovský M (2004a) Statistical alternatives of combined size effect on nominal strength for structures failing at crack initiation. In: Stibor M (ed) Problémy lomové mechaniky IV (Problems of Fracture Mechanics IV), Brno University of Technology, Academy of Sciences—Institute of physics of materials of the ASCR, pp 99–106, invited lectureGoogle Scholar
  40. Vořechovský M (2004b) Stochastic fracture mechanics and size effect. PhD thesis, Brno University of Technology, Brno, Czech RepublicGoogle Scholar
  41. Vořechovský M (2007) Interplay of size effects in concrete specimens under tension studied via computational stochastic fracture mechanics. Int J Solids Struct 44(9): 2715–2731. doi: 10.1016/j.ijsolstr.2006.08.019 MATHCrossRefGoogle Scholar
  42. Vořechovský M (2008) Simulation of simply cross correlated random fields by series expansion methods. Struct safety 30(4): 337–363. doi: 10.1016/j.strusafe.2007.05.002 CrossRefGoogle Scholar
  43. Vořechovský M, Chudoba R (2006) Stochastic modeling of multi-filament yarns: II. Random properties over the length and size effect. Int J Solids Struct 43(3–4): 435–458. doi: 10.1016/j.ijsolstr.2005.06.062 MATHCrossRefGoogle Scholar
  44. Vořechovský M, Novák D (2004) Modeling statistical size effect in concrete by the extreme value theory. In: Walraven J, Blaauwendaad J, Scarpas T, Snijder B (eds) 5th International Ph.D. Symposium in Civil Engineering, held in Delft, The Netherlands, A.A. Balkema Publishers, London, UK, vol 2, pp 867–875Google Scholar
  45. Vořechovský M, Bažant ZP, Novák D (2005) Procedure of statistical size effect prediction for crack initiation problems. In: Carpinteri A (ed) ICF XI 11th International Conference on Fracture, held in Turin, Italy, Politecnico di Torino, pp CD–ROM proc, abstract page 1166Google Scholar
  46. Vořechovský M, Chudoba R, Jeřábek J (2006) Adaptive probabilistic modeling of localization, failure and size effect of quasi-brittle materials. In: Soares et al (eds) III European Conference on Computational Mechanics (ECCM-2006), held in Lisbon, Portugal, National Laboratory of Civil Engineering, Springer, p 286 (abstract), full papers on CD-ROMGoogle Scholar
  47. Weibull W (1939) The phenomenon of rupture in solids. Roy Swedish Inst Eng Res (Ingenioersvetenskaps Akad Handl) Stockholm 153: 1–55Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Institute of Structural MechanicsBrno University of TechnologyBrnoCzech Republic

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