, 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

  • Alberto Carpinteri
  • Marco Paggi


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.


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

List of symbols


crack length [L]


initial crack length [L]


El Haddad characteristic short-crack length [L]


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


grain size [L]


crack growth rate [L]


elastic modulus [FL−2]


specimen size [L]


Paris’ power-law exponent


Wöhler’s power-law exponent


number of cycles [−]


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


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


loading ratio [−]


critical crack growth rate for unstable crack growth [L]


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

Greek symbols


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


fatigue threshold [FL−3/2]


stress range [FL−2]


fatigue limit [FL−2]


fracture toughness [FL−3/2]


yield strength [FL−2]


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



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.


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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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