## 3 Numerical Applications and Discussion
### Test Case 2: Validation and Applicability of the Mixed Mode I and II Fast Procedure  
The mixed mode I and II _Fast_ procedure has been also implemented in the ABAQUS FEM code [41] by using a Python script interface. The effectiveness of this enhanced version of the fast procedure compared to the mode I fast procedure has been tested on CFRP stiffened panels with a circular delamination with variable size and depth. The structural configuration [40] adopted for this second test-case has been schematized in Figure 13. The mechanical properties of the adopted (T800/924) material system are reported in Table 3. The stacking sequences of the skin and each of the stringers are respectively [(+45\({}^{\circ}\)/-45\({}^{\circ}\)/0\({}^{\circ}\)/90\({}^{\circ}\))\({}_{4}\)]\({}_{5}\) and [(-45\({}^{\circ}\)/+45\({}^{\circ}\)/0\({}^{\circ}\))\({}_{2}\)]\({}_{52}\). In particular, two configurations, named S#1 and S#4, have been analyzed, characterized by bay circular delaminations with a radius of 25 mm and 17.5 mm, respectively. For each configuration, different delamination depths have been considered, and the influence of the size and depth of the delamination on the delamination growth initiation is assessed. The VCCT method has been adopted to prove the effectiveness of the proposed mixed mode I and II fast procedure in terms of critical load and ERR distribution along the delamination front. Figure 12: Application limits of linear procedure. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mathbf{E_{1}}\) & \(\mathbf{E_{2}=E_{3}}\) & \(\mathbf{G_{12}=G_{13}}\) & \(\mathbf{G_{23}}\) & \(\mathbf{\
u_{12}=\
u_{13}}\) & \(\mathbf{\
u_{23}}\) & \(\mathbf{G_{lc}}\) & \(\mathbf{G_{ll}}\) & \(\mathbf{t_{p}}\) \\ (**GPa**) & (**GPa**) & (**GPa**) & (**GPa**) & (**c**) & (**) & (**J/m\({}^{2}\)**) & (**J/m\({}^{2}\)**) & (**mm**) \\ \hline
155.0 & 8.57 & 7.4 & 4.8 & 0.33 & 0.52 & 280 & 519 & 0.125 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: T800/924 material properties. In Figure 14, the global model and the local thicker and thinner sub-laminates are shown. Indeed, the delamination depth is compatible with low velocity impact induced damages. For both configurations, different delamination depths have been considered to assess the effectiveness and the limits of applicability of proposed mixed mode I and II fast procedure with respect to the standard mode I fast procedure. In Tables 4 and 5, the delamination growth initiation applied strains obtained by the standard VCCT approach, by the mode I _Fast_ approach, and by the mixed mode I and II _Fast_ approach for different delamination depths have been reported for the SS#1 and SS#4 configurations, respectively. Moreover, the delamination growth initiation applied strain deviations of both mode I _Fast_ and mixed mode I and II _Fast_ approaches with respect to the VCCT method have been reported as well. Figure 14: Details of global-local approach adopted in FE simulation. Figure 13: Test case 2 geometrical description (dimensions in mm) and boundary conditions. Figures 15 and 16 compare the results of the mode I _Fast_ and mixed mode I and II _Fast_ approaches, in terms of delamination growth initiation load deviation from the VCCT results, of the SS#1 and SS#4 configurations, respectively. Indeed, the mixed mode I and II _Fast_ procedure allows the extension of the range of applicability of the mode I _Fast_ procedure, reducing the error in term of delamination growth initiation load evaluation. Then, the strain energy release rate (SERR) distributions for the analyzed configurations have been investigated in order to assess the effectiveness of the mixed mode I and II _Fast_ procedure in predicting the actual growth location along the delamination front. Again, the effectiveness of the mixed mode I and II _Fast_ approach can be appreciated by comparing the results, in terms of mode I and mode II SERR distributions, to the VCCT ones.