Issue 47

R. Fincato et alii, Frattura ed Integrità Strutturale, 47 (2019) 231-246; DOI: 10.3221/IGF-ESIS.47.18 243 Moreover, the effect of the tangential plasticity is again quite relevant, increasing the values of the P-D damage in B’ and C’ of 91% and 147%, respectively. This phenomenon is due to the fact that in correspondence of the points along the loading direction (i.e. the points A, B, C, A’, B’ and C’) the stress state keeps alternating between two extremes of tension and compression, constantly triggering a non-negligible component of the tangential stress rate, see Fig. 8. The magnitude of the cumulative tangential plastic strain rate is higher around 105 mm from the column base due to the effect of the compressive load P and of the boundary conditions which causes the typical ‘elephant foot’ buckling mode. The magnitude of the lateral displacements for the SIDE A and SIDE B (see Fig. 3a) is reported in the following Fig. 9. In detail, the two graphs in Fig. 9 display the computed radial displacements at the unloading points where the displacement amplitudes are ±3δ y , ±6δ y and ±8δ y . A comparison with the experimental data reported in [36] is also offered for the conditions ±3δ y and ±6δ y . As mentioned before, the buckling takes place at the base of the column between 0 and 200 mm. The DSS model is capable to catch quite well the magnitude of the lateral displacements in both sides. There is a slight underestimation of the heights where the maximum lateral displacements are located, however, this phenomenon is probably due to the welding conditions at the base of the real specimens which influence the deformed shape of the pier. The same mismatch can be observed in the numerical analyses of Van Do et al. [36] and Gao et al. [34]. C ONCLUSIONS he present paper introduced a coupled elastoplastic and damage model for the evaluation of a thin wall steel bridge subjected to unidirectional and bidirectional non-proportional loading. The ductile damage was modeled with a modified version of the Mohr-Coulomb failure criteria that proved to give reliable results not only for granular materials (i.e. soil, rock, concrete) but also for metallic materials [1,9,10,12]. Moreover, the ductile damage evolution law was modified by the authors to take into account a different accumulation during non-proportional loading paths. Experimental works [8–10,38] pointed out that the total deformation at fracture is higher during proportional loading, suggesting that, in case of non-proportional loading, a different mechanism for the damage accumulation has to be considered. In the present paper, the authors took into account a previous idea developed in Hashiguchi and Tsutsumi [16] and in Tsutsumi and Kaneko [25] for explaining the acceleration of the damage during non-proportional loading. The damage rate is a function not only of the plastic strain rate, generated along the normal to the plastic potential, but also of a tangential inelastic stretch generated by a deviatoric component of the stress rate, tangential to the yield surface. It is worth mentioning that this additional contribution arises only during the non-coaxiality of the directions of the principal stresses and strains and it is negligible whenever the coaxiality is re-established. This allows considering the different mechanisms of the ductile damage accumulation during proportional and non-proportional loading conditions. The main results of the paper can be summarized in the following points: • The implementation of the constitutive equations of the DSS model, with a damage law accounting for the tangential plastic contributions, are shown in a simple two steps numerical example. A damage acceleration, function of material parameters, is triggered whenever non-proportional loading condition are realized. • The calibration of the model parameters was done reproducing a uniaxial tensile test data for the SS400 steel. The elastoplastic constants, chosen in this work, approximate well the uniaxial behavior of the SS400 steel reported in Goto et al. [37]. • A single experimental test is not enough for the calibration of the MC constants, since the definition of the failure envelope is a non-linear function in the   , , f    space. Therefore, an adjustment of the MC constants was done during the calibration of the steel column subjected to unidirectional loading. • The investigations on the pier under unidirectional loading revealed that a better approximation of the real structure behavior can be achieved considering the contribution to the damage of the inelastic tangential stretch. • The analyses on the local behavior of the structure pointed out that the damage accumulation is accelerated considerably by the NP-D law. However, a future campaign of investigation on the crack formation at the base of the structure is needed to characterize the ductile damage behavior of the pier. Future works will be focused to design experimental and numerical analyses for a better understanding of the ductile damage evolution in non-proportional loading since it still represents an open challenge in the CDM community. R EFERENCES [1] Algarni, M., Bai, Y., Choi, Y. (2015). A study of Inconel 718 dependency on stress triaxiality and Lode angle in plastic T