13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- on both the hardening exponent (n) and the ratio of initial yield strength over Young’s modulus ( 0 / E σ ). The studies of Kim et al. (2004) [20] have shown that, for a given material, the q-parameters should vary with stress triaxialities. The nucleation parameters, Nε and 0.1 Ns = , determined by Chu and Needleman (1980) [6], are considered reasonable values for the current application. 0.04 Nf = is also suggested and is widely used by many researchers. However, the API x65 steel is a high-grade pipeline steel which is very clean steel and thus void nucleation is not significant and also delayed until very late in the deformation process. For this reason a much smaller value of Nf , 0.0008 is adopted in this study. Some authors suggests that, as a first approximation, initial void volume fraction 0f could be taken as the volume fraction of MnS inclusions, which is estimated from Franklin’s formula [21] 0.001 0.054 % % v f S Mn ⎛ ⎞ = ⎜ − ⎟ ⎝ ⎠ , (18) where, %S and % Mn are the weight-% of sulfur and manganese, respectively. While voids coalescence is automatically determined by two type’s criteria in the extended damage models, 0f is the only unknown parameter and is to be fitted. The void volume fraction at final fracture Ff is strongly dependent on 0f . Since Ff has been considered as an unimportant parameter, it can be extrapolated from the empirical equation by Zhang (2001) [13]: 0 0.15 2 Ff f = + , (19) In the present study, two sets of damage model parameters are involved. For these two groups, Nε , Ns and Nf are to be take the same values as discussed previously; 0f is determined by fitting to the experiment results for one notched tensile bar and then with this 0f , the void volume fraction at final fracture, Ff is obtained by Eq. 19. The critical coalescence porosity cf is decided by the two coalescence criteria: plastic limit load criterion and equivalent plastic strain criterion. The q-values are also disparate for the two groups. The classic values ( 1 1.5 q = and 2 1.0 q = ) are adopted in the first group. For the second, 1 1.704 q = and 2 0.846 q = are interpolated from the Faleskog’s tabulated results based on the measured values of hardening exponent (n) and the ratio of initial yield strength over Young’s modulus ( 0 / E σ ). Both sets of damage model parameters are illustrated in Table 1. Table 1. Damage models parameters Nε Ns Nf 0f cf Ff 1q 2q Set1 0.3 0.1 0.0008 0.000125Criteria10.15025 1.5 1.0 Set2 0.3 0.1 0.0008 0.0005 Criteria2 0.151 1.704 0.846 3.2. Comparison with experimental results The finite element analyses are applied to predict mechanical behavior for the notched tensile bars that had notch root radii of 6mm, 3mm and 1.5 mm. These specimens have different levels of stress triaxiality. Porous metal material based on Gurson plasticity theory is also provided by ABAQUS in both implicit and explicit code; however, the failure definition is only available in
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