## 3 Finite Element (FE) Investigation
### Calibration of Fracture Parameters  
To determine the fracture locus of structural steel SS275, the equivalent plastic strain and stress triaxiality at the onset of fracture were measured by using a hybrid experimental finite element method, which was found to give a good approximation of such fracture parameters in the literature [14; 24]. The finite element model, which incorporates the plastic material model shown in Figure 5, was verified against DIC analysis results concerning the distribution of the equivalent plastic strain, as shown in Figure 6, for tension and in-plane shear specimens. It is noted that the strain at the fracture displacement could not be calculated successfully through DIC analysis due to the large deformation, which leads to the loss of local speckle pattern. For this reason, the comparison in the strain distribution was made near the fracture displacement. It can be seen from Figure 6 that the strain distribution of DIC analysis for tension specimen is rather unsymmetric. This is attributed to the misalignment of the specimen. However, the FE models were able to replicate the plastic deformation of the tested specimens with reasonable accuracy.  
The existing studies [23; 25; 26] showed that ductile fracture in tension coupon generally initiates at the center of the necking region, while the fracture for in-plane shear specimen occurs at the edge. Thus, elements at the corresponding locations were selected for calibrating the fracture parameters, as marked in Figure 6. To illustrate the relationships between fracture parameters, the corresponding parameter paths of the critical element over the loading history are presented in Figure 7. It shows that the tension specimens exhibited larger variation in stress triaxiality than the shear specimen. This can be explained by the fact that the tension specimen experienced large plastic deformation from the ultimate strength point to the completed fracture point, in contrast to the shear specimen in which the fracture occurred just after the ultimate strength point. In the present study, the average value of stress triaxiality, given by Equation (6) was used to account for such characteristic.  
\[\eta_{avg}=\frac{1}{\bar{\varepsilon}_{f}}\int_{0}^{\bar{\varepsilon}_{f}} \eta(\bar{\varepsilon})d\bar{\varepsilon} \tag{6}\]Table 2 lists the calibrated fracture parameters used in determining the material parameters \(C_{1}\) and \(C_{2}\) in Equation (5). Through the optimization procedure with the aid of FE analysis, it was found that \(C_{1}\) of 0.84 and \(C_{2}\) of 1.60 were appropriate in predicting the fracture initiation of both tension and in-plane shear specimens composed of SS275 structural steel. Figure 8 shows the constructed fracture locus with these fracture parameters. For a nonlinear strain path, the fracture initiation criterion can be expressed as the integral form as below:  
\[D(\overline{\epsilon})=\int_{0}^{\overline{\epsilon}_{f}}\frac{d\overline{ \epsilon}}{\overline{\epsilon}_{f}(\eta)}=1 \tag{7}\]  
where \(D(\overline{\epsilon})\) is the accumulated damage index, and \(\overline{\epsilon}_{f}(\eta)\) is the fracture strain given by Equation (5). When this index is equal to unity, the fracture will occur. In the present study, the Damage evolution law for use in ABAQUS material model was set with the equivalent plastic displacement at failure to be 0.01. In this way, the FE models can simulate a sudden loss in the load capacity when the damage index reaches a critical value.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Specimen**} & **Fracture Strain** & **Averaged Stress Triaxiality** \\  & \(\epsilon_{f}\) & \(\eta_{arg}\) \\ \hline Tension & 1.1739 & 0.4588 \\ Shear & 0.8709 & 0.0532 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calibrated fracture parameters.  
Figure 6: Strain contours obtained from DIC calculation and FE analysis: (**a**) Uniaxial tension specimen; (**b**) In-plane shear specimen.