Issue 29

R. Serpieri et alii, Frattura ed Integrità Strutturale, 29 (2014) 284-292; DOI: 10.3221/IGF-ESIS.29.24 288 recognize that 0 0 /   I I I K s a nd 0 0 /   II II II K s . Finally, the areas under the bilinear curves in each mode represent the fracture energies in pure modes I and II and are denoted by cI G and cII G , respectively. Figure 3 : Decomposition of each infinitesimal area of microplanes into damaged and undamaged parts. The assumption is then made that 0 0 / /  I cI II cII s s s s a nd the following ‘ductility’ parameter  i s introduced: 0 0 1 1      I II cI cII s s s s (6) According to [6], the evolution of the damage variable k D is given by: 1 max 0, min 1, 1                            k k k D (7) where     2 2 0 0 max 1 kI kII k history I II s s s s                    s s (8) The above evolution law for the damage variable k D c ould allow to consider different values of the fracture energies [11] in modes I and II, cI G and cII G . However, it is shown in [10] that thermodynamically consistency is more ‘neatly’ achieved if these two values are taken to be equal. In fact, there is an ongoing debate in the scientific community concerning the underlying physical justification for a mode-mixity dependent fracture energy, that for zero or positive mode-I opening typically increases from the lowest value in pure mode I to the highest value in pure mode II, to the point that many authors have questioned whether the mode-II fracture energy can be considered as a true material property at all [20]. In this aspect lies one of the primary strengths of the proposed approach, because the different contributions to the energy dissipation, i.e. to the measured fracture energy, are accounted for separately in the model and with a simple, yet physically sound, approach. In particular, the fracture energies cI G and cII G only represent here the energy dissipation due to the rupture of bonds; regardless of any theoretical reason on how thermodynamic consistency can be ensured, physical arguments suggest that such energy should be equal. N UMERICAL SIMULATIONS OF A PULL - OUT TEST Experimental set up and finite element model he model briefly described in the previous section has been implemented in a finite-step algorithm, providing the solution of the interface law at the integration points of interface elements, and coded in a user-subroutine (UMAT) for the finite-element code ABAQUS. To validate the efficiency of the model in capturing the concrete- steel interface interaction against experimental results published in the literature, the pull-out test of s SD30 steel bar from a cylindrical block of concrete, studied by Shima et al. [18] has been numerically simulated. The experimental apparatus devised by Shima et al. is reported in Fig. 4(a). The diameter D of the steel bar is 19.5mm, and the set-up and specimen geometry are such that an initial 10 195  D mm long unbonded region in the vicinity of the loaded end of the bar is created. These features have been purposefully designed by Shima and coworkers in order to move the stress concentration well inside the concrete block and therefore avoid concrete damage and splitting at the top end of the concrete block. To this end, an initial 195mm long clay sleeve surrounding the bar in this region was inserted. T