## 3 Fracture Parameters for 3D Crack  
### The Theoretical Derivation of Energy Release Rate for I-II-III Mixed Mode Crack  
Considering the meaning of SIFs, \(K_{I}\), \(K_{II}\), \(K_{III}\) denote the intensity of tensile stress, in-plane stress and out-of-plane stress, as shown in Figure 13. When crack kink occurs, the local SIFs should be recalculated. A local \(x^{\prime}-y^{\prime}-z^{\prime}\) coordinate system at the tip of the kink is defined and assumes that the remote stress fields in polar coordinates defined the local stress field. The transformation of the \(x^{\prime}-y^{\prime}\) plane is shown in Figure 14.  
Figure 13: Stress field at crack tip. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Crack Angle (\({}^{\circ}\))**} & \multirow{2}{*}{**Sample Number**} & **Calculated Extension** & **Experimental** & **Experimental Extension** & \multirow{2}{*}{**Error (\({}^{\circ}\))**} \\  & & **Angle (\({}^{\circ}\))** & **Extension Angle (\({}^{\circ}\))** & **Angle (Average) (\({}^{\circ}\))** & \\ \hline \multirow{3}{*}{30} & 1 & \(-60\) & \(-59\) & & \\  & 2 & \(-60\) & \(-59.5\) & & \\  & 3 & \(-60\) & \(-57.6\) & & \\ \hline \multirow{3}{*}{60} & 1 & \(-43\) & \(-41.5\) & & \\  & 2 & \(-43\) & \(-40.8\) & \(-41.17\) & 4.26 \\ \cline{1-1}  & 3 & \(-43\) & \(-41.2\) & & \\ \hline \multirow{3}{*}{90} & 1 & 0 & 0 & & \\  & 2 & 0 & 0 & 0 & \\ \cline{1-1}  & 3 & 0 & 0 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculated extension angle and experimental extension angle. Stress fields ahead of the crack front for Mode I, Mode II and Mode III at an angle \(\theta\) in polar coordinates are shown in Formula (6). \[ \begin{array}{l}\sigma_{rr}=\frac{K_{I}}{\sqrt{2\pi r}}\Big{[}\frac{5}{4}cos \Big{(}\frac{\theta}{2}\Big{)}-\frac{1}{4}cos\Big{(}\frac{3\theta}{2}\Big{)} \Big{]}+\frac{K_{II}}{\sqrt{2\pi r}}\Big{[}\frac{5}{4}sin\Big{(}\frac{\theta}{ 2}\Big{)}+\frac{3}{4}sin\Big{(}\frac{3\theta}{2}\Big{)}\Big{]}\\ \sigma_{\theta\theta}=\frac{K_{I}}{\sqrt{2\pi r}}\Big{[}\frac{3}{4}cos\Big{(} \frac{\theta}{2}\Big{)}+\frac{1}{4}cos\Big{(}\frac{3\theta}{2}\Big{)}\Big{]}+ \frac{K_{II}}{\sqrt{2\pi r}}\Big{[}-\frac{3}{4}sin\Big{(}\frac{\theta}{2} \Big{)}-\frac{3}{4}sin\Big{(}\frac{3\theta}{2}\Big{)}\Big{]}\\ \tau_{r\theta}=\frac{K_{II}}{\sqrt{2\pi r}}\Big{[}\frac{1}{4}sin\Big{(}\frac {\theta}{2}\Big{)}+\frac{1}{4}sin\Big{(}\frac{3\theta}{2}\Big{)}\Big{]}+\frac {K_{II}}{\sqrt{2\pi r}}\Big{[}\frac{1}{4}cos\Big{(}\frac{\theta}{2}\Big{)}+ \frac{3}{4}cos\Big{(}\frac{3\theta}{2}\Big{)}\Big{]}\\ \sigma_{z}=v(\sigma_{rr}+\sigma_{\theta\theta})\\ \tau_{rz}=\frac{K_{II}}{\sqrt{2\pi r}}sin\Big{(}\frac{\theta}{2}\Big{)}\\ \tau_{\theta z}=\frac{K_{III}}{\sqrt{2\pi r}}cos\Big{(}\frac{\theta}{2}\Big{)} \end{array}  \tag{6}\]  
The directions of the local stress field for the crack tip in a local \(x^{\prime}-y^{\prime}-z^{\prime}\) coordinate system are consistent with those of the polar coordinate, as shown in Figure 14. Thus, the stress at the crack tip between the polar coordinate and local \(x^{\prime}-y^{\prime}-z^{\prime}\) coordinate has the following relationship:  
\[\left. \begin{array}{l}\sigma_{y^{\prime}y^{\prime}}=\sigma_{\theta\theta}\\ \tau_{x^{\prime}y^{\prime}}=\tau_{r\theta}\\ \tau_{y^{\prime}z^{\prime}}=\tau_{\theta z}\end{array} \right\} \tag{7}\]  
The local stress intensity factors at the arbitrary extend angle \(\alpha\) were deduced by summing the normal and shear stresses, respectively:  
\[\left. \begin{array}{l}k^{\prime}_{I}(\alpha)=\sigma_{y^{\prime}y^{\prime}} \sqrt{2\pi r}=C_{11}K_{I}+C_{12}K_{II}\\ k^{\prime}_{II}(\alpha)=\tau_{x^{\prime}y^{\prime}}\sqrt{2\pi r}=C_{21}K_{I}+ C_{22}K_{II}\\ k^{\prime}_{III}(\alpha)=\tau_{y^{\prime}Z^{\prime}}\sqrt{2\pi r}=C_{33}K_{III} \end{array} \right\} \tag{8}\]  
Combining Formula (6)-(8), the coefficients can be calculated using Formula (9):  
\[