Issue 10

M. Paggi, Frattura ed Integrità Strutturale, 10 (2009) 43-55; DOI: 10.3221/IGF-ESIS.10.06 47 dependence of the crack growth rate on the stress-intensity factor range, which is experimentally confirmed by the use of the Paris’ law for concrete [15-17]. Therefore, complete self-similarity in 1  cannot be accepted and the assumption of incomplete self-similarity holds whenever K  is far lower than IC K , i.e., for 1 1   . As far as 3  is concerned, this number is expected to be particularly important in quasi-brittle materials, since it compares the structural size-scale h with the plastic zone size. As regards 4  (or * 4  for FRC), we know that, in metals, incomplete self-similarity is attained for p / 1 d r  , i.e., for 4 1   [25,30]. A similar reasoning applies for 5  , that is incomplete self-similarity in this number is expected when the crack length is comparable with the process zone siz e [17] . As regards 8  , a dependence on R is very often observed and therefore incomplete self-similarity in 8  is expected for 8 0 1    . Hence, assuming incomplete self-similarity in the dimensionless variables 1  , 3  , 4  , 5  and 8  , the following generalized representation of fatigue crack growth is derived starting fro m Eq.(3):     1 2 4 3 5 3 5 1 2 4 3 1 2 3 4 2 3 4 2 2 2 IC 1 2 6 7 2 2 IC IC IC 1 2 6 7 2 2 2 2 2(1 ) IC d (1 ) , , d , , (1 ) f u u u f u v K a K h a R N K K f K K h v a R f K                                                                           (4) where the exponent i  and, consequently, the dimensionless function 1  , cannot be determined from considerations of dimensional analysis alone. Moreover, it is important to remark that the exponents i  may depend on the dimensionless numbers i  . Eq. (4) can be regarded as a generalized Paris’ law (see the classical expression d d m a N C K   superimposed t o Fig. 1b) , in which the main functional dependencies of the parameter C have been now explicitated. Figure 2 : the effect of incomplete self-similarity in 3  , 4  , 5  and 8  on the Paris’ curves. In this mathematical framework, it emerges that the incomplete self-similarity exponent 1  simply corresponds to the Paris’ law parameter m that can be evaluated as the slope of the bilogarithmic d d a N vs. K  curve (se e Fig. 2) . Usually, m varies from 5 up to 30 in quasi-brittle materials [31] , a value particularly high as compared to metals, where m ranges between 2 and 4. For 1 K   , the intercept of the straight line with the vertical axis provides the logarithm of the multiplicative parameter C (see Fig. 2) . Under the assumptions of incomplete self-similarity discussed above, C is no longer a true constant, but it is rather a power-law function of h , f v and a . Regarding the exponents 2  and 3  , it is reasonable to expect them less than zero, with a reduction of C by increasing either the size of the specimen or the amount of fibres. In fact, this would imply a shift of the vertical asymptote corresponding to max IC K K  towards