Issue 10

M. Paggi, Frattura ed Integrità Strutturale, 10 (2009) 43-55 ; DOI: 10.3221/IGF-ESIS.10.06 52 “two different sized cracks embedded into two different sized bodies subjected to the same stress-intensity factor range should grow at the same rate”. As a support to the theories against the similitude hypothesis, we mention the experimental results by Newman et al. [37], who observed that “in the threshold regime there is something missing in the (closure) model”, and those by Forth et al. [36], revealing that similitude does not hold in Region I (the near-threshold region) and also in the lower portion of Region II. To overcome this problem, Molent et al. [39] and Jones et al. [40] have recently proposed a generalized Frost and Dugdale crack growth law, assuming that the crack growth rate is proportional to the accumulated plastic strain, averaged over a characteristic length ahead of the crack tip, ' (1 '/ 2) ' d / d m m a N C a K    , where C ’ and m ’ are regarded as material constants. This equation states that d / d a N is not only a function of the stress- intensity factor range, but also of the crack length. Such a generalized Frost and Dugdale crack growth equation was successfully used to predict the growth of near micron sized cracks in both coupon and full scale aircraft fatigue tests and interpret a large amount of experimental data that could not be modelled using the Paris’ law. For concrete, a detailed experimental examination of crack propagation in flexural fatigue [17] has shown that the crack growth rate is not a monotonic increasing function of the crack length. For cracks shorter than the crack length at peak load in quasi-static monotonic loading, a deceleration stage was found, where d a /d N is a decreasing function of a . Afterwards, an acceleration stage takes place and d a /d N can be well approximated according to the classical Paris’ law [17]. To model the deceleration stage, Kolloru et al. [17] proposed an empirical relationship between d a /d N and the crack length, apparently independent of the Paris’ law. Actually, it can be interpreted as a particular case of our proposed generalized Paris’ law, simply allowing a crack-size dependence of the coefficient C , i.e., setting 1 2 ( / 2) n n C a   4 1 2 ( / 2) n n    , where 1 n and 2 n are the power-law exponents for the two regimes found in [17]. The effect of the structural size The effect of the structural size can be highlighted by considering the experimental data by Bažant and Xu [15] for normal strength concrete and by Bažant and Shell [16] for high strength concrete, subsequently re-examined by Spagnoli [41]. From their experimental results on self-similar beams tested in cyclic bending, the computed Paris’ law exponent 1 m   was found to be dependent on the structural size. Plotting m vs. 3  i n Fig. 7, we recognize that m is a linear decreasing function of 3  for each type of concrete. The effect of the ultimate tensile strength is also important, since it affects the dimensionless number 7 / u E    . For normal strength concrete we have 7 9686   , whereas for high strength concrete we have 7 4303   and the slope of the linear relationship between m and 3  turns out to be an increasing function of 7  (see Fig. 7) . Figure 7 : Size-scale effects on the Paris’ law exponent 1 m   for normal and high strength concrete (experimental data from [15,16] reinterpreted in [41]) .