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.
Similar content being viewed by others
Abbreviations
- a :
-
crack length [L]
- a in :
-
initial crack length [L]
- a 0 :
-
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 [−]
- N cr :
-
minimum number of cycles for the validity of the Wöhler’s regime [−]
- N th :
-
maximum number of cycles for the validity of the Wöhler’s regime [−]
- R :
-
loading ratio [−]
- v cr :
-
critical crack growth rate for unstable crack growth [L]
- v th :
-
crack growth rate for infinite life (threshold value) [L]
- Δ K :
-
stress-intensity factor range [FL−3/2]
- Δ K th :
-
fatigue threshold [FL−3/2]
- Δ σ :
-
stress range [FL−2]
- Δ σ fl :
-
fatigue limit [FL−2]
- K IC :
-
fracture toughness [FL−3/2]
- σ y :
-
yield strength [FL−2]
- σ 0 :
-
coefficient of the Wöhler’s power-law [FL−2]
References
Lazzeri, L., Salvetti, A., 1996. An experimental evaluation of fatigue crack growth prediction models. Proc. 1996 ASIP Conference, Vol. I, 477–508
Paris, P.C., 1962. The growth of cracks due to variations in load. Doctoral Dissertation, Lehigh University, Lehigh
Paris PC, Erdogan F (1963) A critical analysis of crack propagation laws. ASME J Basic Eng 85D:528–534
Paris PC, Gomez MP, Anderson WP (1961) A rational analytic theory of fatigue. Trend Eng 13:9–14
Jones R, Molent L, Pitt S (2008) Similitude and the Paris crack growth law. Int J Fatigue 30:1873–1880
Weibull W (1951) A statistical distribution function of wide applicability. ASME J Appl Mech A6:293–297
Beretta S, Zerbst U (2011) Damage tolerance of railway axles. Eng Fract Mech 78:713–862
Luke M, Varfolomeev I, Lütkepohl K, Esderts A (2011) Fatigue crack growth in railway axles: assessment concept and validation tests. Engng Fract Mech 78:714–730
Makino T, Kato T, Hirakawa K (2011) Review of the fatigue damage tolerance of high-speed railway axles in Japan. Engng Fract Mech 78:810–825
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
Barenblatt GI (1980) Scaling, self-similarity and intermediate asymptotics. Cambridge University Press, Cambridge, p 1996
Ritchie RO (2005) Incomplete self-similarity and fatigue-crack growth. Int J Fract 132:197–203
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 56:3416–3432
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 31:1524–1531
Carpinteri A, Paggi M (2011) Dimensional analysis and fractal modelling of fatigue crack growth. J ASTM Int 8:1–13
Paggi M, Carpinteri A (2009) Fractal and multifractal approaches for the analysis of crack-size dependent scaling laws in fatigue. Chaos, Solitons Fractals 40:1136–1145
Wöhler, A., 1860. Versuche über die Festigkeit Eisenbahnwagenachsen. Z. Bauwesen 10
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 161:41–52
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 33:477–483
Pugno N, Ciavarella M, Cornetti P, Carpinteri A (2006) A generalized Paris’ law for fatigue crack growth. J Mech Phys Solids 54:1333–1349
Acknowledgments
The support of the Italian Ministry of Education, University and Research to the Project FIRB 2010 Future in Research “Structural mechanics models for renewable energy applications” (RBFR107AKG) is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Carpinteri, A., Paggi, M. The effect of crack size and specimen size on the relation between the Paris and Wöhler curves. Meccanica 49, 765–773 (2014). https://doi.org/10.1007/s11012-014-9908-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11012-014-9908-y