## 3 Numerical Applications and Discussion
### Test Case 2: Validation and Applicability of the Mixed Mode I and II Fast Procedure
In particular, in Figures 17 and 18, the comparisons between SERR considering a delamination between the fifth and sixth ply (+45\({}^{\circ}\)/-45\({}^{\circ}\) interface), for SS#1 and SS#4 configurations respectively, are shown. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Delamination** & **Delamination** & \(\epsilon_{\sigma}\) & **Error** & **Analysis Method** \\
**Interface** & **Depth** & \(\epsilon_{\sigma}\) & **Error** & **Analysis Method** \\ \hline
4 th–5 th (90\({}^{\circ}\)/+45\({}^{\circ}\)) & 12.5\% & 2415.33 & & VCCT \\
4 th–5 th (90\({}^{\circ}\)/+45\({}^{\circ}\)) & 12.5\% & 2887.44 & 19.5\% & Mode I Fast \\  & & 3104.18 & 22.2\% & Mixed mode I and II Fast \\ \hline
5 th–6 th (+45\({}^{\circ}\)/-45\({}^{\circ}\)) & 15.62\% & 2678.00 & & VCCT \\
2788.06 & 3.9\% & Mixed mode I and II Fast \\ \hline
6 th–7 th (-45\({}^{\circ}\)/0\({}^{\circ}\)) & 18.7\% & 3607.00 & & VCCT \\
3284.56 & 9.8\% & Mode I Fast \\
3260.72 & 0.4\% & Mixed mode I and II Fast \\ \hline
7 th–8 th (0\({}^{\circ}\)/90\({}^{\circ}\)) & 21.8\% & 3938.00 & & VCCT \\
3266.97 & 20.5\% & Mode I Fast \\
3268.79 & 11.6\% & Mixed mode I and II Fast \\ \hline
8 th–9 th (90\({}^{\circ}\)/+45\({}^{\circ}\)) & 25\% & 3752.00 & & VCCT \\
3267.52 & 8.0\% & Mode I Fast \\
3260.72 & 4.1\% & Mixed mode I and II Fast \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results summary for SS#1 damage configuration (25 mm radius delamination). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Delamination** & **Delamination** & \(\epsilon_{\sigma}\) & **Error** & **Analysis Method** \\
**Interface** & **Depth** & \(\epsilon_{\sigma}\) & **Error** & **Analysis Method** \\ \hline
4 th–5 th (90\({}^{\circ}\)/+45\({}^{\circ}\)) & 12.5\% & 2890.67 & & VCCT \\
32843.92 & 1.6\% & Mode I Fast \\
3288.00 & 3007.27 & 3.9\% & Mixed mode I and II Fast \\ \hline
5 th–6 th (+45\({}^{\circ}\)/-45\({}^{\circ}\)) & 15.62\% & 3065.76 & 7.2\% & Mode I Fast \\
3265.35 & 3.9\% & Mixed mode I and II Fast \\ \hline
6 th–7 th (-45\({}^{\circ}\)/0\({}^{\circ}\)) & 18.7\% & 4804.00 & & VCCT \\
3261.94 & 10.1\% & Mode I Fast \\
3265.97 & 6.5\% & Mixed mode I and II Fast \\ \hline
7 th–8 th (0\({}^{\circ}\)/90\({}^{\circ}\)) & 21.8\% & 4734.76 & 16.2\% & Mode I Fast \\
3265.97 & 4954.13 & 11.0\% & Mixed mode I and II Fast \\ \hline
8 th–9 th (90\({}^{\circ}\)/+45\({}^{\circ}\)) & 25\% & 5926.00 & & VCCT \\
3268.88 & 1.5\% & Mode I Fast \\
3269.88 & 0.9\% & Mixed mode I and II Fast \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results summary for SS#4 damage configuration (17.5 mm radius delamination). Figure 16: Error of critical load evaluation versus depth for SS#4 damage configuration. Figure 17: Comparison of strain energy release rate (SERR) distribution for SS#1 damage configuration. Figure 15: Error of critical load evaluation versus depth for SS#1 damage configuration. In both cases (of Figures 17 and 18), the implemented mixed mode I and II _Fast_ approach provides good results, showing an excellent agreement with SERR distribution predicted with the non-linear technique. The distribution of mode II SERR is related to the radial component of the displacement field on the delamination front. Hence, it is strongly influenced by depth of delamination and ply interface. Indeed, nonlinear analysis performed on SS#1 damage configuration with six plies for delaminated sub-laminate, at \(-45^{\circ}/0^{\circ}\) interface, shows (Figure 19) that radial component distribution on delamination front is far from \(G_{II}\) distribution probably because the strain energy release rate distribution undergoes significant variations from the delamination buckling condition to delamination growth condition, eventually, passing through the global buckling of the skin. Figure 20 shows the predicted SERR distribution compared to VCCT. As expected, the radial displacement for the SS#1 damage configuration is not suitable for the evaluation of the \(G_{II}\) distribution. Figure 19: Radial displacement field from non-linear analysis.