Issue 10

M. Paggi, Frattura ed Integrità Strutturale, 10 (2009) 43-55; DOI: 10.3221/IGF-ESIS.10.06 46 Considering a state with no explicit time dependence, it is possible to apply the Buckingham’s  Theorem [26] t o reduce the number of parameters involved in the problem (see also [27-29] for the application to the quasi-static case). As a result, we have:   2 2 2 2 IC th 2 2 2 IC IC IC IC IC 2 IC d , , , , , ;1 d u u u u u i u K K a K E h d a R N K K K K K K                                  (2) where i  ( 1, , 7) i   are dimensionless numbers. Note that 3  corresponds to the square of the dimensionless number Z introduced by Barenblatt and Botvina [13] and to the inverse of the square of the brittleness number s introduced by Carpinteri [17-19]. The number 5  was firstly considered by Spagnoli [8] f or the analysis of the crack-size dependence of the Paris’ law parameters. Here, it has to be noted that, according to Irwin, the ratio   2 IC / u K  , which is used to made dimensionless the variables h , d and a , is proportional to the plastic zone size, p r . The number 4  is related to the microstructural length scale and has been introduced in [24] in order to interpret the effect of the grain size in metals on the coefficient C of the Paris’ law according to the experimental findings by Chan [30]. As far as plain concrete is concerned, this microstructural length scale should correspond to the average size of the aggregates. For FRC, the variable d should correspond to the fibre diameter and Eq. (2) c an be rewritten in order to put into evidence the fibre volumetric content. This parameter is computed as the ratio between the total area of fibres contained in a transversal cross-section of the specimen and the cross-section area of the specimen itself,     2 / 100 / f f v A A f d h    . Here, the function f represents a shape function which depends on the transversal cross-section of the fibres and on their amount. For a square specimen cross-section of side h and for circular fibres we have 100 / 4 f f n    , where f n is the total number of crossed fibres. In this case, instead of considering the dimensionless representation (2), where all the variables with a physical dimension of a length are made dimensionless using the plastic zone size, it is more convenient to use the following alternative form:   2 2 2 IC th 2 2 IC IC IC IC 2 IC d , , , , , , ;1 d u u u u i u K K a K d l E h a R N K K K h K d K                                 (3) where the dimensionless number 4  has been replaced by * 4 4 3 / / / f d h v f       . Moreover, the number 6 / l d   (fibre length/fibre diameter) has also been suitably introduced in Eq. (3) and it corresponds to the fibre aspect ratio. At this point, we want to see if the number of quantities involved in the relationships (2) or (3) can be reduced further from seven (or height). This can occur either in the case of complete or incomplete self-similarities in the corresponding dimensionless numbers. In the former situation, the dependence of the mechanical response on a given dimensionless number, say i  , disappears and we can say that i  is non essential for the representation of the physical phenomenon. In the latter situation, a power-law dependence on i  can be proposed, which usually characterizes a physical situation intermediate between two asymptotic behaviours. Considering the dimensionless number 1 IC / K K    , it has to be noticed that it rules the transition from the asymptotic behaviours characterized by the condition of nonpropagating cracks, when th K K    , to the pure Griffith- Irwin instability, when cr K K    . Moreover, incomplete self-similarity in 1  would correspond to a power-law