## 5 Numerical Results and Experimental Comparisons  
Non-linear static analysis have been performed for the two material configurations under displacement-controlled loading conditions to replicate the experimental tests, using the cohesive parameters obtained in the previous section. The deformed shape with the out-of-plane displacements contour plot is reported in Figure 11 at different values of the applied displacement for the model with the \"Low Bridging (LB)\" material configuration. The delamination initiates from the right edge of the doubler at an applied displacement of 2.31 mm (Figure 11a). As the damage propagates, the displacement distribution loses its symmetry, as shown in Figure 11b where the deformed shape at the maximum load is presented. When the doubler is almost completely separated from the skin (Figure 11c), the specimen fails. An identical sequence of events occurs for the \"High Bridging (HB)\" material configuration, with the delamination propagation delayed due to the higher fracture toughness. The numerical results in terms of load vs. applied displacement for the \"Low Bridging (LB)\" and \"High Bridging (HB)\" material configurations are compared with the data of the experimental tests [23] in Figure 12a,b, respectively. For each material configuration, three specimens were tested, and the displacements were measured using two laser sensors (L1 and L2) positioned close to the center of the specimen. For both the models, the numerical stiffness is slightly overestimated, although it still falls within the range of the experimental data. The reason is probably related to the numerical boundary conditions, where the displacement in the vertical direction of the nodes located on the supports is blocked. During the experimental tests these nodes can slide on the supports, resulting in a higher compliance of the structure. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Low Bridging (LB)** & & **High Bridging (HB)** \\ \hline n & & 0.9899 & n & 0.9479 \\ m & & 0.7419 & m & 0.4305 \\ G\({}_{1}\) & [J/m\({}^{2}\)] & 230 & G\({}_{1}\) & [J/m\({}^{2}\)] & 310 \\ \(\tau_{1c}\) & [MPa] & 39.59 & \(\tau_{1c}\) & [MPa] & 37.91 \\ G\({}_{2}\) & [J/m\({}^{2}\)] & 80 & G\({}_{2}\) & [J/m\({}^{2}\)] & 420 \\ \(\tau_{2c}\) & [MPa] & 0.40 & \(\tau_{2c}\) & [MPa] & 2.08 \\ \hline \end{tabular}
\end{table} 
Table 2: Cohesive laws parameters. Figure 11: Deformed shape with out-of-plane displacements contour plot at an applied displacement of: (a) 2.31 mm; (b) 3.33 mm; (c) 4.68 mm (“Low Bridging”). The effect of the different R-curves of the two material configurations is clearly visible observing the comparisons of the maximum load between the experimental tests and the numerical models, reported in Table 3. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multicolumn{2}{c}{**Low Bridging (LB)**} & \multicolumn{4}{c}{**High Bridging (HB)**} \\ \hline \multirow{3}{*}{EXP} & LB\#1 & 1.58 kN & \multirow{3}{*}{EXP} & HB\#1 & 1.77 kN \\ \cline{2-5}  & LB\#2 & 1.52 kN & & HB\#2 & 2.04 kN \\ \cline{2-5}  & LB\#3 & 1.30 kN & & HB\#3 & 2.08 kN \\ \hline NUM & & 1.25 kN & NUM & & 1.64 kN \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental and Numerical maximum loads. Figure 12: Numerical-experimental Comparisons of the load-displacement curves for: (**a**) “Low Bridging”; (**b**) “High Bridging”. The higher fracture toughness results in an increase in the maximum load in the experimental tests of 34% on average. Although both the numerical models slightly underestimate the maximum load, they exhibit the same trend of the experimental results with a 31% increase in the maximum load for the \"High Bridging (HB)\" material configuration. The lower maximum load displayed by the numerical models can be explained by the fact that the R-curve effect for the opening mode II has been neglected, since no experimental data were available.