Issue 18

G. Del Piero et alii, Frattura ed Integrità Strutturale, 18 (2011) 5-13; DOI: 10.3221/IGF-ESIS.18.01 10 if kl > 2  .  The latter are concentrated on an interval ( a, a + l y ) of length l y = 2  / k < l . The initial slope of the inelastic response curve is positive for kl <  , and negative for kl >  . It is usual to consider the material constant l i =  / k (26) as an internal length of the material. From the preceding analysis it follows that the inelastic regime starts with  a full-size solution and a work-hardening response if l < l i ,  a full-size solution and a strain-softening response if l i < l < 2 l i ,  a localized solution and a strain-softening response if l > 2 l i . Catastrophic failure again occurs when the slope of the response curve becomes  . This never happens for l < l i . For l i < l < 2 l i , this happens when 0 ) ('' 2/ 2/ tan 1 )0(''          c w kl kl   (27) where  c is the elongation at the onset of the inelastic regime. For l > 2 l i , this happens when 0 ) ('' )0(''   c y w l l   (28) The occurrence of catastrophic rupture at the onset is the response predicted by Griffth’s theory of brittle fracture. For all remaining situations, the subsequent response depends on the behavior of  away from the origin. Depending on the form assumed for   as a function of  , it may happen that an initially full-size solution localizes for some  >  c , and in some cases it may also happen that a localized solution becomes full-size. Catastrophic fracture may take place in both localized and full-size solutions.  N UMERICAL SIMULATIONS ith an appropriate choice of the expression of  , the non-local model gives the possibility of describing many experimental situations in which rupture is preceded by more or less extended regimes of inelastic deformation, as well as many intermediate situations between the extreme cases of a totally brittle and a totally ductile response. However, the non-homogeneous character of the solutions in the non-local model makes hopeless any attempt of describing the evolution of the inelastic deformation by a simple differential equation, like Eq. (15) in the case of the local model. For this reason, we made a series of numerical simulations. Here we present the current status of our study, which is still in progress. The purpose of our simulations was to determine the shape of the function  giving the best reproducion of the response curves of two experimental tests, one on a steel bar and one on a concrete specimen. For  we chose a piecewise polynomial C 2 representation. This was obtained by subdividing the domain {   0 } into N intervals, in each of which  was taken to be a third-order polynomial, and by imposing the continuity of the function and of its first and second derivatives at the nodal points of the subdivision. The number and position of the nodal points determine the accuracy of the approximation. The advantage of a better approximation obtained with large N is paid with the disadvantage of working with a larger number of material constants. For the steel bar we made two series of simulations, one with N = 3 and one with N = 6. The results of the first series are shown in Fig. 2. In it, we see the response curves obtained numerically for a bar length l equal to 40, 80, 160 mm. They show that a short bar is more ductile than a long bar, since for short bars catastrophic failure takes place for larger  . This is a manifestation of the size effect . In the same figure, the curve for l = 80 mm is compared with the experimental curve, represented by the dotted line, for a bar of the same length. We see a very good agreement, except for the initial horizontal plateau exhibited by the experimental curve and not reproduced in the simulation. The second series of simulations was done with the purpose of eliminating this discrepancy, by increasing the number N of parameters. The result is shown in Fig. 3, where we see an almost perfect agreement between experiment and simulation.  W