Meccanica

, Volume 49, Issue 4, pp 765–773 | Cite as

The effect of crack size and specimen size on the relation between the Paris and Wöhler curves

Article

Abstract

Paris and Wöhler’s fatigue curves are intimately connected by the physics of the process of fatigue crack growth. However, their connections are not obvious due to the appearance of anomalous specimen-size and crack-size effects. In this study, considering the equations for a notched specimen (or for a specimen where failure is the result of the propagation of a main crack) and the assumption of incomplete self-similarity on the specimen size, the relations between the size-scale effects observed in the Paris and Wöhler’s diagrams are explained. In the second part of the work, the behaviour of physically short cracks is addressed and, considering a fractal model for fatigue crack growth, the crack-size effects on the Paris and Wöhler’s curves are discussed.

Keywords

Fatigue Paris’ curve Wöhler’s curve Incomplete self-similarity Size-scale effects Crack-size effects 

List of symbols

a

crack length [L]

ain

initial crack length [L]

a0

El Haddad characteristic short-crack length [L]

C

coefficient of the Paris’ law [physical dimensions dependent on m, see Eq. (1a)]

d

grain size [L]

da/dN

crack growth rate [L]

E

elastic modulus [FL−2]

h

specimen size [L]

m

Paris’ power-law exponent

n

Wöhler’s power-law exponent

N

number of cycles [−]

Ncr

minimum number of cycles for the validity of the Wöhler’s regime [−]

Nth

maximum number of cycles for the validity of the Wöhler’s regime [−]

R

loading ratio [−]

vcr

critical crack growth rate for unstable crack growth [L]

vth

crack growth rate for infinite life (threshold value) [L]

Greek symbols

ΔK

stress-intensity factor range [FL−3/2]

ΔKth

fatigue threshold [FL−3/2]

Δσ

stress range [FL−2]

Δσfl

fatigue limit [FL−2]

KIC

fracture toughness [FL−3/2]

σy

yield strength [FL−2]

σ0

coefficient of the Wöhler’s power-law [FL−2]

References

  1. 1.
    Lazzeri, L., Salvetti, A., 1996. An experimental evaluation of fatigue crack growth prediction models. Proc. 1996 ASIP Conference, Vol. I, 477–508Google Scholar
  2. 2.
    Paris, P.C., 1962. The growth of cracks due to variations in load. Doctoral Dissertation, Lehigh University, LehighGoogle Scholar
  3. 3.
    Paris PC, Erdogan F (1963) A critical analysis of crack propagation laws. ASME J Basic Eng 85D:528–534CrossRefGoogle Scholar
  4. 4.
    Paris PC, Gomez MP, Anderson WP (1961) A rational analytic theory of fatigue. Trend Eng 13:9–14Google Scholar
  5. 5.
    Jones R, Molent L, Pitt S (2008) Similitude and the Paris crack growth law. Int J Fatigue 30:1873–1880CrossRefMATHGoogle Scholar
  6. 6.
    Weibull W (1951) A statistical distribution function of wide applicability. ASME J Appl Mech A6:293–297Google Scholar
  7. 7.
    Beretta S, Zerbst U (2011) Damage tolerance of railway axles. Eng Fract Mech 78:713–862CrossRefGoogle Scholar
  8. 8.
    Luke M, Varfolomeev I, Lütkepohl K, Esderts A (2011) Fatigue crack growth in railway axles: assessment concept and validation tests. Engng Fract Mech 78:714–730CrossRefGoogle Scholar
  9. 9.
    Makino T, Kato T, Hirakawa K (2011) Review of the fatigue damage tolerance of high-speed railway axles in Japan. Engng Fract Mech 78:810–825CrossRefGoogle Scholar
  10. 10.
    Barenblatt GI, Botvina LR (1980) Incomplete self-similarity of fatigue in the linear range of fatigue crack growth. Fatigue Fract Eng Mater Struct 3:193–202CrossRefGoogle Scholar
  11. 11.
    Barenblatt GI (1980) Scaling, self-similarity and intermediate asymptotics. Cambridge University Press, Cambridge, p 1996Google Scholar
  12. 12.
    Ritchie RO (2005) Incomplete self-similarity and fatigue-crack growth. Int J Fract 132:197–203CrossRefMATHGoogle Scholar
  13. 13.
    Ciavarella M, Paggi M, Carpinteri A (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 56:3416–3432ADSCrossRefMATHGoogle Scholar
  14. 14.
    Carpinteri A, 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 31:1524–1531CrossRefMATHGoogle Scholar
  15. 15.
    Carpinteri A, Paggi M (2011) Dimensional analysis and fractal modelling of fatigue crack growth. J ASTM Int 8:1–13CrossRefGoogle Scholar
  16. 16.
    Paggi M, Carpinteri A (2009) Fractal and multifractal approaches for the analysis of crack-size dependent scaling laws in fatigue. Chaos, Solitons Fractals 40:1136–1145ADSCrossRefMATHGoogle Scholar
  17. 17.
    Wöhler, A., 1860. Versuche über die Festigkeit Eisenbahnwagenachsen. Z. Bauwesen 10Google Scholar
  18. 18.
    Carpinteri A, 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 161:41–52CrossRefMATHGoogle Scholar
  19. 19.
    Plekhov O, Paggi M, Naimark O, Carpinteri A (2011) A dimensional analysis interpretation to grain size and loading frequency dependencies of the Paris and Wöhler curves. Int J Fatigue 33:477–483CrossRefGoogle Scholar
  20. 20.
    Pugno N, Ciavarella M, Cornetti P, Carpinteri A (2006) A generalized Paris’ law for fatigue crack growth. J Mech Phys Solids 54:1333–1349ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Structural, Geotechnical and Building EngineeringPolitecnico di TorinoTurinItaly
  2. 2.IMT Institute for Advanced Studies LuccaLuccaItaly

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