Issue 38

R. Fincato et alii, Frattura ed Integrità Strutturale, 38 (2016) 231-236; DOI: 10.3221/IGF-ESIS.38.31 234 where  is the plastic multiplier and superscript + indicates that just the tensile contributions are considered. s 1 , s 2 , and s 3 are material parameters; s 1 and s 2 affect the energy release rate, Y [2, 5], and s 3 is a threshold for the cumulative plastic strain after which damage begins (the term in the Macaulay brackets is null until H = s 3 ). m  is the mean stress and G and K are the shear and bulk moduli, respectively. N UMERICAL TESTS he constitutive equations were implemented in commercial finite element code, Abaqus 6.14, via a user subroutine, and they were used to simulate a monotonic extension of an A533B steel bar. A similar numerical and experimental test was conducted by Bonora et al. [1], and it was used as a reference for our simulation. The sample geometry and boundary conditions were taken from the literature (Fig. 2). For simplicity, one eighth of the sample was modelled, applying symmetric constraints on the cut sections. For the mesh discretization, 7870 linear brick elements were used for 9548 nodes. Figure 2 : Sketch of the steel bar (grey area indicates the modelled portion). The material parameters were obtained from the calibration uniaxial extension test in Fig. 7 and they are reported in Tab. 1. E 200 GPa ν 0.3 F 0 345 MPa h 1 , h 2 1.0; 11 c, χ 200; 0.9 s 1 , s 2 , s 3 2.5; 1.0; 0.65 Table 1 : A533B steel material parameters for the subloading surface model. The kinematic hardening contribution was neglected and the following isotropic hardening law was used.       h H h H dF F F h e F h h e dH 2 2 0 1 0 1 2 1 1 ,             (8) The s3 damage parameter in the first of Eqs. (7) is set to activate the damage after the cumulative isotropic hardening variable, H, reaches a threshold of 65%, and an element deletion occurs whenever the damage variable is 0.70 at the Gauss point, which is assumed as a critical value for void coalescence and crack formation. Fig. 4–6 show the damage contour fields before and after the first crack formation, and near the end of the analysis. The crack is initiated corresponding to the notch, but not at the surface, and it rapidly propagates in the centre of the cross section in accordance with the results obtained by Bonora et al. [1]. This can be explained by the competition between the stress triaxiality and the plastic strains, which both affect the damage variable as shown in Eq. (7). The evolution of these two element entities in the cross section of Fig. 3 shows that the stress triaxiality is higher at the bar core (point B) than on the surface (point A), whereas the opposite is observed for the cumulative plastic strain (Fig. 9 and 10). T