Skip to main content
SpringerLink
Log in

Modelling fatigue in quasi-brittle materials with incomplete self-similarity concepts

  • Original Article
  • Published:
Materials and Structures Aims and scope Submit manuscript

Abstract

In this study, a generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth is proposed in order to highlight and explain the deviations from the classical power–law equations used to characterize the fatigue behaviour of quasi-brittle materials. According to this theoretical approach, the microstructural-size (related to the volumetric content of fibers in fiber-reinforced concrete), the crack-size, and the size-scale effects on the Paris’ law and on the Wöhler equation are presented within a unified mathematical framework. Relevant experimental results taken from the literature are used to confirm the theoretical trends and to determine the values of the incomplete self-similarity exponents. All this information is expected to be useful for the design of experiments, since the role of the different dimensionless numbers governing the phenomenon of fatigue is herein elucidated. Finally, a numerical model based on damage mechanics and nonlinear fracture mechanics is proposed for the prediction of uniaxial S–N curves, showing how to efficiently use the information gained from dimensional analysis and how the shape of the S–N curves is influenced by the parameters of the damage model.

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

Similar content being viewed by others

References

  1. Hordijk DA (1992) Tensile and tensile fatigue behaviour of concrete experiments, modelling and analyses. Heron 37:1–77

    Google Scholar 

  2. Wöhler A (1860) Versuche über die festiykeit eisenbahnwagenuchsen. Z. Bauwesen 10

  3. Basquin OH (1910) The exponential law of endurance tests. Proc ASTM 10:625–630

    Google Scholar 

  4. Murdock JW, Kesler CE (1959) Effect of range of stress on fatigue strength of plain concrete beams. ACI J 55:221–232

    Google Scholar 

  5. Tepfers R (1979) Tensile fatigue strength of plain concrete. ACI J 76:919–993

    Google Scholar 

  6. Butler JE (2008) The flexural fatigue performance of plain and fibrous concretes. Strain 26:135–139

    Article  Google Scholar 

  7. Ramakrishnan V, Oberling G, Tatnall P (1987) Flexural fatigue strength of steel fiber reinforced concrete, Fibre Reinforced Concrete—Properties and Applications, SP-105-13, ACI Special Publication. ACI, Detroit, pp 225–245

  8. Johnston CD, Zemp RW (1991) Flexural fatigue performance of steel fiber reinforced concrete—influence of fiber content aspect ratio and type. ACI Mater J 88:374–383

    Google Scholar 

  9. Zhang J, Stang H (1998) Fatigue performance in flexure of fiber reinforced concrete. ACI Mater J 95:58–67

    Google Scholar 

  10. Zhang J, Stang H, Li VC (2000) Experimental study on crack bridging in FRC under uniaxial fatigue tension. ASCE J Mater Civ Eng 12:66–73

    Article  Google Scholar 

  11. Zhang J, Li VC, Stang H (2001) Size effect on fatigue in bending of concrete. ASCE J Mater Civ Eng 13:446–453

    Article  Google Scholar 

  12. Singh SP, Mohammadi Y, Madan SK (2006) Flexural fatigue strength of steel fibrous concrete containing mixed steel fibres. J Zhejiang Univ Sci A 7:1329–1335

    Article  Google Scholar 

  13. Paris P, Gomez M, Anderson W (1961) A rational analytic theory of fatigue. Trend Eng 13:9–14

    Google Scholar 

  14. Paris P, Erdogan F (1963) A critical analysis of crack propagation laws. J Basic Eng Trans ASME 58D:528–534

    Google Scholar 

  15. Bažant ZP, Xu K (1991) Size effect in fatigue fracture of concrete. ACI Mater J 88:390–399

    Google Scholar 

  16. Bažant ZP, Shell WF (1993) Fatigue fracture of high strength concrete and size effect. ACI Mater J 90:472–478

    Google Scholar 

  17. Kolloru SV, O’Neil EF, Popovics JS, Shah SP (2000) Crack propagation in flexural fatigue of concrete. ASCE J Eng Mech 126:891–898

    Article  Google Scholar 

  18. Carpinteri Al, Paggi M (2007) Self-similarity and crack growth instability in the correlation between the Paris’ constants. Eng Fract Mech 74:1041–1053

    Article  Google Scholar 

  19. Paggi M, Carpinteri Al (2009) Fractal and multifractal approaches for the analysis of crack-size dependent scaling laws in fatigue. Chaos Solitons Fractals 40:1136–1145

    Article  MATH  Google Scholar 

  20. Matsumoto T, Li VC (1999) Fatigue life analysis of fiber reinforced concrete with a fracture mechanics based model. Cem Concr Compos 21:249–261

    Article  Google Scholar 

  21. Roe KL, Siegmund T (2003) An irreversible cohesive zone model for interface fatigue crack growth simulation. Eng Fract Mech 70:209–232

    Article  Google Scholar 

  22. Barenblatt GI, Botvina LR (1980) Incomplete self-similarity of fatigue in the linear range of fatigue crack growth. Fatigue Fract Eng Mater Struct 3:193–202

    Article  Google Scholar 

  23. Barenblatt GI (1996) Scaling, self-similarity and intermediate asymptotics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  24. Ciavarella M, Paggi M, Carpinteri Al (2008) One, no one, and one hundred thousand crack propagation laws: a generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth. J Mech Phys Solids 56:3416–3432

    Article  MATH  Google Scholar 

  25. Carpinteri Al, Paggi M (2009) A unified interpretation of the power laws in fatigue and the analytical correlations between cyclic properties of engineering materials. Int J Fatigue 31:1524−1531

  26. Buckingham E (1915) Model experiments and the form of empirical equations. ASME Trans 37:263–296

    Google Scholar 

  27. Carpinteri A (1981) Static and energetic fracture parameters for rocks and concretes. RILEM Mater Struct 14:151–162

    Google Scholar 

  28. Carpinteri A (1982) Notch sensitivity in fracture testing of aggregative materials. Eng Fract Mech 16:467–481

    Article  Google Scholar 

  29. Carpinteri A (1983) Plastic flow collapse vs. separation collapse in elastic-plastic strain-hardening structures. RILEM Mater Struct 16:85–96

    Google Scholar 

  30. Spagnoli A (2005) Self-similarity and fractals in the Paris range of fatigue crack growth. Mech Mater 37:519–529

    Article  Google Scholar 

  31. Chan K (1995) A scaling law for fatigue crack initiation in steels. Scr Metal Mater 32:235–240

    Article  Google Scholar 

  32. Carpinteri Al, Paggi M (2010) A unified fractal approach for the interpretation of the anomalous scaling laws in fatigue and comparison with existing models. Int J Fract. 161:41–52

    Article  Google Scholar 

  33. Fleck NA, Kang KJ, Ashby MF (1994) The cyclic properties of engineering materials. Acta Metall Mater 42:365–381

    Article  Google Scholar 

  34. Carpinteri Al (1994) Fractal nature of material microstructure and size effects on apparent mechanical properties. Mech Mater 18:89–101

    Article  Google Scholar 

  35. Carpinteri An, Spagnoli A, Vantadori S (2009) A fractal approach to finite-life fatigue strength. Int J Fatigue 31:927–933

    Article  Google Scholar 

  36. Newman JC Jr, Phillips EP, Swain MH (1999) Fatigue-life prediction methodology using small-crack theory. Int J Fatigue 21:109–119

    Article  Google Scholar 

  37. Carpinteri Al, Chiaia B, Cornetti P (2002) A scale-invariant cohesive crack model for quasi-brittle materials. Eng Fract Mech 69:207–217

    Article  Google Scholar 

  38. Li VC (1992) Post-crack scaling relations for fiber-reinforced cementitious composites. ASCE J Mater Civ Eng 4:41–57

    Article  Google Scholar 

  39. Visalvanich K, Naaman AE (1982) Fracture modelling for fiber reinforced cementitious materials. Program Report for NSF Grant ENG 77-23534. Department of Materials Engineering, University of Illinois at Chicago Circle, Chicago, IL

Download references

Author information

Authors and Affiliations

  1. Politecnico di Torino, Department of Structural and Geotechnical Engineering, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    Marco Paggi

Authors
  1. Marco Paggi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Paggi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Paggi, M. Modelling fatigue in quasi-brittle materials with incomplete self-similarity concepts. Mater Struct 44, 659–670 (2011). https://doi.org/10.1617/s11527-010-9656-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1617/s11527-010-9656-y

Keywords

Search

Navigation