Issue 10

M. Paggi, Frattura ed Integrità Strutturale, 10 (2009) 43-55; DOI: 10.3221/IGF-ESIS.10.06 44 behaviour of metals [2,3]. Hence, the empirical S-N curve, relating the fatigue life or cycles to failure, N , to the applied stress range,   or S , is often used both for plain [4-6] a nd fibre-reinforced concretes [7-12]. A schematic representation of the Wöhler’s diagram is shown in Fig. 1a, where the cyclic stress range, max min       , is plotted as a function of the number of cycles to failure, N . In this diagram, we introduce the range of stress at static failure, max min min (1 ) u u u R              , where u  is the material tensile strength , and we define the endurance or fatigue limit , fl   , as the stress range that a sample will sustain without fracture for 7 1 10 N    cycles, which is a conventional value that can be thought of as “infinite” life. With the advent of fracture mechanics, a more ambitious task was undertaken, i.e., to predict, or at least understand, the propagation of cracks. Plotting the crack growth rate, d a /d N , as a function of the stress-intensity factor range, max min K K K    , most of the experimental data can be interpreted in terms of a power-law relationship, i.e., according to the so-called Paris’ law [13,14]. In spite of the fact that this approach requires a more elaborate testing procedure than for the S-N curves, it has also been applied to plain concrete [15-17] (see a schematic representation of the Paris’ curve in Fig. 1b). Note that the power-law representation presents deviations for very high values of K  approaching cr IC (1 ) K R K    [18], where IC K is the material fracture toughness , or for very low values of K  approaching the threshold stress-intensity factor range , th K  [19]. Again, in close analogy with the concept of fatigue limit, the fatigue threshold is defined in a conventional way as the value of K  below which the crack grows at a rate of less than 9 1 10   m/cycle. (a) (b) Figure 1 : schemes of the (a) Wöhler and (b) Paris’ curves with the related fatigue parameters. Yet, the fatigue behaviour of quasi-brittle materials in bending or even in compression is not completely understood in terms of all the influential variables, such as the type of the loading cycle, cycling rate, structural size, crack length and, perhaps most important of all for FRC, fibres parameters. For instance, it has been experimentally shown in [15-17] that the coefficients entering the Paris’ law depend on the material composition, on the structural size, and on the crack length, potentially explaining the large scatter in the reported values available in the literature. This result has also the fundamental implication that the Paris coefficients cannot be treated as true material constants. Similarly, the S-N curves are found to be dependent on the size of the tested specimen [11], on the loading conditions [1], as well as on the material composition [8]. From the modelling point of view, an important research effort towards simulating S-N curves of plain concrete and FRC in cyclic bending was put forward in [11,20] . This required the use of nonlinear fracture mechanics theories coupled with a damage law for modelling the degradation of the concrete cohesive crack stresses and of the fibres bridging in the process zone due to cyclic loading. However, in spite of the useful predictive capabilities of these models, their complex mathematical formulation limits their applicability to design purposes and confines them to the research environment. Moreover, although the damage model can be tuned on the basis of experimental data as proposed in [10], other choices for the expression of the damage variable are also possible (see e.g. [21]) and the connection existing between the evolution of damage and the outcome of the fatigue simulation has not yet been completely clarified.