Issue 10

M. Paggi, Frattura ed Integrità Strutturale, 10 (2009) 43-55; DOI: 10.3221/IGF-ESIS.10.06 49   2 2 2 2 IC IC , , , , , , ;1 fl u u i u u u d l E N h a R K h K d                      (9) where  is a dimensionless function. Note that Eq. (9) has been derived fro m Eq. (8) by choosing u  , IC K and the plastic zone size as the main variables whose suitable combination provides a dimensionless number. At this point, we want to see if the number of quantities involved in the relationship (9) can be reduced further from height. In close analogy with the procedure carried out for the Paris’ law, we assume incomplete self-similarity in 1  , 3  , 4  , 5  and 8  (the same conditions apply to the dimensionless numbers as in the derivation of Eq. (4)) , obtaining:           1 2 4 3 5 5 3 1 2 4 2 3 4 3 1 2 3 4 2 2 2 2 6 7 2 2 IC IC 2 2 6 7 2 2 2 2 IC 1 , , , , 1 f u u u f u v N h a R K f K h v a R f K                                                                     (10) Eq. (10) represents a generalized Wöhler relationship of fatigue and encompasses the empirical S-N curves as limit cases. For instance, the S-N curve in Fig. 1a can be approximated by the Basquin power law: 1 n n n u fl N N k            (11) where u   is the range of stress at static failure, fl   is the fatigue limit corresponding to a conventional fatigue life of 7 1 10 N    cycles,   is the stress range corresponding to a fatigue life N and k is a constant. Equating the first and the third terms in Eq. (11), we obtain he following power-law equation:   1 n n n u u n R N                 (12) Hence, the Basquin power law is predicted as a limit case o f Eq. (10) when 1 n    , 2 3 4 0       and 5 n   . Figure 3 : the effect of incomplete self-similarity in 3  , 4  , 5  and 8  on the S-N curves. Therefore, in this framework, the incomplete self-similarity exponent 1  simply corresponds to Basquin parameter n  that can be evaluated as the slope of the   vs. N curve in a bilogarithmic diagram (see Fig. 3). For 1 N  , the