## 3 Numerical Applications and Discussion
### Test Case 1: Validation and Applicability of the Mode I Fast Procedure
Moreover, the influence of the mesh discretization in the tangential direction on the critical propagation strain has been assessed in Figure 10. Indeed, the number of elements in the radial direction has been progressively increased, as shown in Figure 10 for the configurations characterized by 56, 80, and 96 elements. According to Figures 9 and 10, the discretization of the elements in the delaminated region has a negligible influence on the critical propagation strain. Finally, the effectiveness of the mode I _Fast_ approach has been demonstrated in Figure 11, where the ERRs predicted by both the standard VCCT and the proposed methods on the delamination front at growth initiation have been compared. Good agreements on the compressive buckling strains \(\varepsilon_{buckling}\) and the compressive delamination initiation strains \(\varepsilon_{del}\) and loads \(F_{del}\) between the proposed linear approach and VCCT can be found as well, as reported in Table 2. Moreover, the advantage of the _Fast_ procedure in terms of computational efforts, respect to the standard VCCT approach, can be appreciated by comparing the computational times reported in Table 2.  
Figure 9: Influence of the element discretization in the radial direction on the critical propagation strain. The mode I _Fast_ procedure is based on the hypothesis that the delamination mainly propagates due to the effect of the fracture mode I. However, this hypothesis limits the field of applicability of the approach to very thin panels with very superficial delaminations. Hence, an investigation of the influence of the delaminated sub-laminate relative thickness on the delamination growth initiation load has been performed and compared to VCCT results. Figure 12 reports the error, with respect to the VCCT approach, on the delamination growth initiation load as a function of the delamination depth. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Approach** & \(\epsilon_{\text{buckling (\mu\varepsilon)}}\) & \(\epsilon_{\text{del (\mu\varepsilon)}}\) & \(\mathbf{F_{del}}\) (kN) & **Computational Time (s)** \\ \hline Fast Procedure & 980.3 & 1700.7 & 366.6 & 295 \\ VCCT & 1024.8 & 1780.0 & 382.9 & 4349 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Critical propagation loads and computational time comparison. Figure 11: Comparison of proposed approach and virtual crack closure technique (VCCT) in terms of normalized energy release rate (ERR) distribution along delamination front. Figure 10: Influence of the element discretization in the tangential direction on the critical propagation strain. According to Figure 12, a limited range of applicability between 10-20% of delamination depth is observed for the mode I _Fast_ approach. Indeed, very thin delaminations lead to a significant delay in delamination growth with respect to the local delamination buckling. This delay involves significant geometrical non-linear effects in contrast with the main hypothesis of linear structural behavior for the applicability of the mode I fast procedure. Moreover, very thick delaminations, involving mixed mode I and mode II growth, do not satisfy the main assumption on the predominance of mode I for the applicability of mode I fast procedure. In order to increase this limited range of applicability and to obtain more accurate results, a modified version of the mode I fast procedure has been investigated which considers the influence of fracture mode II in the evaluation of energy released during the propagation.