Issue 10

M. Paggi, Frattura ed Integrità Strutturale, 10 (2009) 43-55; DOI: 10.3221/IGF-ESIS.10.06 48 higher values of K  . This trend is consistent with the common observation that the fracture toughness of concrete is an increasing function of the size of the tested sample [32] and of the fibre volumetric content [10]. On the contrary, short cracks are expected to grow faster than long cracks [19] and therefore the exponent 4  is expected to be positive valued. The same reasoning holds for the loading ratio. It is important to note that the above scaling law has been derived by considering the stress-intensity factor range as the driving parameter for crack growth. However, in quasi-brittle materials, where the cohesive crack model is routinely applied to describe crack propagation, it is also possible to find numerical results relating the crack growth rate to the range of energy release rate, G  . Power-laws of the Paris form relating the crack growth rate to G  still hold [33] and Eq. (1) could be replaced by the following expression:   F th d , , ; , , , , , ;1 d u a F G G G h d a E R N       (5) where the fracture energy, F G , has been suitably introduced instead of the fracture toughness IC K and the existence of a threshold range of energy release rate, th G  , has been postulated. Hence, Eq. (3) would become: th F F F F F d , , , , , , ;1 d u u u u G G a G d l E h a R N G G G h G d                (6) where we recognize that the dimensionless number governing the size-scale effects is no longer equal to 2 2 IC 1/ / u s h K   as in Eq. (3), but it is now represented by the inverse of the energy brittleness number, E F 1/ / u s h G   . However, it is possible to demonstrate that Eq. (6) implies Eq. (3) and viceversa. Therefore, the dimensionless representation in Eq. (3) is equivalent to that in Eq. (6) and no additional dimensionless numbers have to be introduced. Recalling the Irwin’s relationship between the stress-intensity factor and the energy release rate, it is possible to recast Eq. (6) in terms of stress-intensity factors. In doing that, we note that the energy brittleness number E s is equal to the stress brittleness number s times the inverse of the dimensionless number / u E  [34], i.e.,   2 E / / u s s E   . Therefore, E s is nothing but a linear combination of two dimensionless numbers already defined in Eq. (3) . Hence, considering plane stress conditions, Eq. (6) can be rewritten as follows: 2 IC th 2 2 IC IC IC IC d , , , , , , ;1 d u u u u K K E E a K d l E h a R N E K K K h K d                (7) which can be further manipulated by replacing the Young’s modulus with the tensile strength, since the value of the ratio between these two variables is already taken into consideration by the dimensionless number / u E  . Therefore, after such a substitution, Eq. (7) becomes identical to Eq. (3). The inverse implication is straightforward and can be gained by applying the inverse of the Irwin’s relationship. T HE GENERALIZED W ÖHLER REPRESENTATION o far, the crack growth rate has been chosen as the main output parameter characterizing the phenomenon of fatigue crack growth. However, it is also possible to consider the number of cycles, N , as the parameter representative of fatigue. Following this alternative route, we postulate again the following functional dependence:   IC , , ; , , , , , ;1 u fl N F K h d a E R         (8) where the definition of the governing variables is provided in Tab. 1. Considering a state with no explicit time dependence, it is possible to apply the Buckingham’s  Theorem [26] to reduce the number of parameters involved in the problem: S