# Ductile Fracture Behavior of Mild and High-Tensile Strength Shipbuilding Steeds
## 4 Fracture Model Calibration
### Determination of Fracture Model Parameters  
To determine the optimal set fracture model parameters \(\mathbf{\chi}=\{C_{1},C_{2},C_{3}\}\), an optimization problem is set. The loading paths obtained from numerical simulations for each test \(\left(\eta_{i}\left[\bar{\varepsilon}_{p}\right],\bar{\theta}_{i}\left[\bar{ \varepsilon}_{p}\right]\right)\) and the corresponding fracture indicator \(D_{i}=\int_{0}^{\bar{\varepsilon}_{f,i}}\frac{d\bar{\varepsilon}_{p}}{\bar{ \varepsilon}_{f}^{pr}\left[\mathbf{\chi}.\eta_{i}.\bar{\theta}_{i}\right]}\) are used. All four specimens are considered. The cost function is defined as follows  
\[f\left[\mathbf{\chi}\right]=\sum_{i=1}^{4}\left(\int_{0}^{\bar{\varepsilon}_{f,i} }\frac{d\bar{\varepsilon}_{p}}{\bar{\varepsilon}_{f}^{pr}\left[\mathbf{\chi},\eta_ {i}.\bar{\theta}_{i}\right]}-1\right)^{2} \tag{11}\]  
Hence, the optimization problem can be written as  
\[\mathbf{\chi}=\operatorname*{arg\,min}_{\mathbf{\chi}}\left\{f\left[\mathbf{\chi}\right]\right\} \tag{12}\]  
The optimization problem is solved with the aid of the Matlab toolbox. In addition, the constraints on \(C_{1}\), \(C_{2}\) and \(C_{3}\) are constrained as positive values. Table 3 presents the optimal set of the model parameters. Figure 15 presents the resulting 3D fracture loci. The black curves plotted over the 3D fracture loci denote the plane stress condition. Figure 16 shows the plane stress fracture loci.  
Several conclusions may be drawn from the calibrated fracture loci. Among the three steel grades, the ductility limits of normal-strength steel, grade A, showed a relatively low Lode angle dependence which could be best described by a fracture locus similar to the Cockcroft-Latham model. On the other hand, the fracture locus of AH36 elucidates the Lode angle effect. Interestingly, AH36 shows higher ductility for certain stress states, such as uniaxial tension and equi-biaxial tension, compared to A grade steel. DH36 showed the lowest ductility among the three steel grades. The fracture locus of DH36 exhibited a relatively high stress-triaxiality-dependence but moderate Lode angle effect. A similar conclusion is drawn by [32]. For negative stress triaxialities and plane strain tension, AH36 and DH36 have similar ductility levels.  
Figure 17 shows the predicted fracture indicator using the calibrated fracture model at the instant of experimental fracture initiation. Here, the prediction error should be read as deviation from unity. It is apparent that the prediction errors are very low, particularly for the case of the SH specimen. It can concluded that the proposed model could describe the experimental data accurately for all steel grades and test specimens.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Grade** & \(\mathbf{C_{1}}\) & \(\mathbf{C_{2}}\) & \(\mathbf{C_{3}}\) \\ \hline A & 0.069 & 0.648 & 1.154 \\ AH36 & 0.882 & 3.860 & 0.148 \\ DH36 & 0.529 & 1.330 & 0.633 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Identified fracture model parameters.  
Figure 15: Calibrated 3D fracture loci of the steel grades considered.