# Ductile Fracture Behavior of Mild and High-Tensile Strength Shipbuilding Steeds
## 2 Ductile Fracture Model
### Parameter Sensitivity  
Before discussing the sensitivity of the fracture strain to model parameters, the dependencies of the model to the stress triaxiality and the Lode angle parameter need to be introduced. Figure 1 shows the equivalent plastic strain to fracture obtained using the proposed model with the parameters \(C_{1}=0.85,C_{2}=2.0,C_{3}=0.2\). Geometrically, the fracture locus is represented as a surface in the space of \((\varepsilon_{p},\eta,\bar{\theta})\). The fracture locus is asymmetric with respect to the Lode angle parameter. However, its dependence on stress triaxiality is monotonic. Figure 2 gives an example 2D plot of the fracture locus in the plane of \((\tilde{\epsilon}_{p},\eta)\) for several fixed values of the Lode angle parameter. The stress triaxiality dependence is a monotonically decreasing one, which is in agreement with the previous stress-triaxiality-dependent models [4, 5, 11]. At high stress triaxialities, the effects of the Lode angle vanish and all three curves converge to a single one. Figure 3 shows a 2D plot of the fracture locus in the plane \((\tilde{\epsilon}_{p},\tilde{\theta})\) for several fixed values of the stress triaxiality. The asymmetric nature of the Lode angle dependency is more evident in this figure. The Lode angle dependency explains the difference in ductility observed for axisymmetric tension-compression and generalized shear state states. The parameter \(C_{1}\) regulates the effects of the maximum shear stress and maximum principal tensile stress. For the limiting case of \(C_{1}=1\), the proposed model reduces to the normalized maximum shear-stress criterion, which is not stress-triaxiality-dependent. The other limiting case, \(C_{1}=0\), reduces the proposed model to the normalized CL criterion (the modified CL criterion proposed by [13]), in which case both the Lode angle- and stress triaxiality-dependencies are retained. Figure 4 presents several plots of the proposed fracture model under plane stress conditions considering several \(C_{1}\) values. The parameter \(C_{1}\) has no effect on the strain to fracture for in-plane shear \((\eta=0)\). Large  
Figure 1: 3D fracture locus obtained using the proposed model with the parameters \(C_{1}=0.85,C_{2}=2.0,C_{3}=0.2\). Figure 2: Stress triaxiality dependence of the proposed fracture model for \(C_{1}=0.85,C_{2}=2.0,C_{3}=0.2\). values yield a fracture locus with a noticeable hump at uniaxial tension \((\eta=1/3)\) and a plane strain tension \((\eta=1/\sqrt{3})\) \"valley\", i.e., a trough between uniaxial tension and equi-biaxial tension \((\eta=2/3)\). This effect is due mainly to the Lode angle dominance on the stress-state dependency of ductility. In addition, stress triviality sensitivity decreases with increasing \(C_{1}\), as the maximum shear stress criterion dominates. Hence, \(C_{1}\) plays an important in regulating the stress triaxiality and Lode angle competition. Moreover, a stress-triaxiality-dependent cut-off region value for each \(C_{1}\) value is noticeable in Figure 4.  
Ductile fracture will not occur below the cut-off value. Figure 4: Effect of \(C_{1}\) on the proposed fracture model for \(C_{2}=2.0\), \(C_{3}=0.2\). Figure 3: Lode angle parameter dependence of the proposed fracture model for \(C_{1}=0.85,C_{2}=2.0,C_{3}=0.2\). In the general case, the border of the cut-off region can be derived from Equation (5). The strain to fracture approaches infinity when the following condition is met  
\[C_{1}\left(\frac{1}{\sqrt{3}}\cos\left[\frac{\pi}{6}\bar{\theta}\right]\right)+ \left(1-C_{1}\right)\left(\frac{\left\langle 3\eta+2\cos\left[\frac{\pi}{6}\left(1- \bar{\theta}\right)\right]\right\rangle}{3}\right)=0 \tag{7}\]  
This equation defines a plane in the space of \((\varepsilon_{p},\eta,\bar{\theta})\). Figure 7 shows the projection of the cut-off plane to the plane of \((\eta,\bar{\theta})\) for \(C_{1}=0.6\). As mentioned earlier, the parameter \(C_{1}\) controls the shape of the cut-off plane.