# Ductile Fracture Behavior of Mild and High-Tensile Strength Shipbuilding Steeds
## Appendix A  
### Definition of the Stress State  
For an isotropic solid material, the stress tensor can be represented by the ordered principal stresses (\(\sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\)) or using its invariants  
\[I_{1}=\text{tr}\left[\sigma\right]=\sigma_{1}+\sigma_{2}+\sigma_{3} \tag{1}\]  
\[J_{2}=\frac{1}{2}\mathbf{s}:\mathbf{s}=\frac{1}{6}\{(\sigma_{1}-\sigma_{2})^{ 2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\} \tag{2}\]  
\[J_{3}=\det\left[\mathbf{s}\right]=(\sigma_{1}-\sigma_{m})(\sigma_{2}-\sigma_{ m})(\sigma_{3}-\sigma_{m}) \tag{3}\]  
Here, \(I_{1}\) represents the first invariant of the Cauchy stress tensor \(\sigma\). The invariants of the deviatoric stress tensor, \(\mathbf{s}\), are denoted as \(J_{2}\) and \(J_{3}\). The stress tensor may be decomposed into hydrostatic and deviatoric parts  
\[\sigma=\frac{1}{3}I_{1}\mathbf{I}+\mathbf{s}=\sigma_{m}\mathbf{I}+\mathbf{s} \tag{4}\]  
where \(\mathbf{I}\) is the identity matrix and \(\sigma_{m}=(\sigma_{1}+\sigma_{2}+\sigma_{3})/3\) is the hydrostatic stress.  
The stress state is characterized by the two non-dimensional ratios of the invariants or the principal stresses. In this paper, the stress triaxiality, \(\eta\), and Lode angle parameter, \(\bar{\theta}\), were used as the stress state parameters. The stress triaxiality (\(-\infty<\eta<\infty\)), which represents the first invariant, is obtained by normalizing hydrostatic stress, \(\sigma_{m}\), with the von Mises equivalent stress, \(\bar{\sigma}=\sqrt{3J_{2}}\)  
\[\eta=\frac{\sigma_{m}}{\sigma}=\frac{I_{1}}{3\sqrt{3J_{2}}} \tag{10}\]  
The third invariant can be normalized as follows  
\[\xi=\frac{27}{2}\frac{J_{3}}{\sigma^{3}}=\cos\left[3\theta\right] \tag{11}\]  
where \(\theta\) is the Lode angle (\(0\leq\theta\leq\pi/3\)). The normalized Lode angle, which is called the Lode angle parameter (\(-1\leq\bar{\theta}\leq 1\)), is expressed as follows  
\[\bar{\theta}=1-\frac{6\theta}{\pi}=1-\frac{2}{\pi}\arccos\left[\xi\right] \tag{12}\]  
\(\bar{\theta}=0\) designates generalized shear (e.g., pure shear or plane strain tension), whereas \(\bar{\theta}=1\) corresponds to axisymmetric tension and \(\bar{\theta}=-1\) represents axisymmetric compression. An alternative measure of the third invariant is the so-called Lode parameter, \(L\)  
\[L=\frac{\sigma_{2}-\sigma_{n}}{\tau_{max}}=\frac{2\sigma_{2}-\sigma_{1}-\sigma _{3}}{\sigma_{1}-\sigma_{3}} \tag{13}\]  
where maximum shear stress, \(\tau_{max}\), and normal stress, which acts on the same plane as maximum shear stress, \(\sigma_{n}\), are defined as follows  
\[\tau_{max}=\frac{\sigma_{1}-\sigma_{3}}{2} \tag{14}\]  
\[\sigma_{n}=\frac{\sigma_{1}+\sigma_{3}}{2} \tag{15}\]  
The stress vector can be defined in the Cartesian coordinate system using the principal stresses \((\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3})\). Alternatively, a cylindrical coordinate system may be used based on \((\bar{\sigma},\eta,\theta)\). The transformation from \((\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3})\) to \((\bar{\sigma},\eta,\bar{\theta})\) is obtained as follows  
\[\sigma_{1} =\bar{\sigma}\left(\eta+\frac{2}{3}\cos\left[\frac{\pi}{6}\left(1 -\bar{\theta}\right)\right]\right) \tag{16a}\] \[\sigma_{2} =\bar{\sigma}\left(\eta+\frac{2}{3}\cos\left[\frac{\pi}{6}\left( 3+\bar{\theta}\right)\right]\right)\] (16b) \[\sigma_{3} =\bar{\sigma}\left(\eta-\frac{2}{3}\cos\left[\frac{\pi}{6}\left( 1+\bar{\theta}\right)\right]\right) \tag{16c}\]  
In the cylindrical coordinate system, the direction of the stress vector can be defined using stress triaxiality and Lode angle parameter. If these two parameters, which define the stress state, does not vary until fracture, a proportional loading path is experienced.  
Under plane stress condition (\(\sigma_{3}=0\)), the stress state can be represented by \(\sigma_{2}/\sigma_{1}\). In that case, the following relation holds  
\[\bar{\theta}=1-\frac{2}{\pi}\arccos\left[-\frac{27}{2}\eta\left(\eta^{2}- \frac{1}{3}\right)\right] \tag{17}\]