Issue 10

M. Paggi, Frattura ed Integrità Strutturale, 10 (2009) 43-55 ; DOI: 10.3221/IGF-ESIS.10.06 50 intercept of the straight line with the vertical axis provides the logarithm of the multiplicative parameter u   (see Fig. 3). Under the assumptions of incomplete self-similarity discussed above, we expect 2  less than zero, with a reduction of u   by increasing the size of the specimen. Note that this trend is fully consistent with the well-known size-scale effects on the tensile strength of quasi-brittle materials [32] a nd with the recent findings in [35]. Similarly, we expect 4 0   as for metals [36]. As regards the fibre volumetric content and the loading ratio, an opposite trend is expected, with 3  and 5  positive valued (see Fig. 3) . A PPLICATION TO QUASI - BRITTLE MATERIALS n this section, the generalized Paris and Wöhler representations of fatigue are applied to plain concrete and FRC, providing a dimensional analysis interpretation to some of the most relevant experimental trends found in the literature. More specifically, the effect of the microstructural size (fibre diameter), of the crack size and of the structural size are carefully discussed. The effect of the microstructural size As shown in Section Generalized mathematical representations of fatigue , the use of fibres is beneficial from the fatigue standpoint, since an increase in fibre diameter (or, equivalently, in volumetric content) corresponds to a decrease in crack growth rate for a given value of K  and to an increase in the cycles to failure for a given value of   . The quantify such an effect, the experimental flexural fatigue data by Johnston and Zemp [8] can be profitably used. They tested square concrete beams in cyclic bending without fibres or with smooth wire fibres with different volumetric contents ( f v  0.5, 1.0 and 1.5%). All the fibres have an aspect ratio / 75 l d  . The original S-N results are reported in the bilogarithmic diagram of Fig. 4, along with the best-fitting power-law regression curves. According to Eq. (10), the exponent of N corresponds to 1 1/  and we note that it is almost independent of the fibre volumetric content, ranging from 0.0218  to 0.0249  . For this type of fibres, the average value of 1  is then equal to 43  . On the other hand, the multiplicative coefficient of the variable N corresponds to u   and this parameter is significantly affected by the presence of fibres. Plotting the values of u   vs. the volumetric content i n Fig. 5 and determining the best-fitting power-law equation, we find 0.26 6.37 u f v    . According to Eq. (10) evaluated in correspondence of 1 N  , the exponent of f v is equal to 3 1 /    , leading 3 1 0.26 11.3      . Figure 4 : The effect of the fibre volumetric content, f v , on the S-N curves as a result of incomplete self-similarity in 4  (experimental data from [8], aspect ratio / 75 l d  ). I