## 4 Numerical Results and Discussion
### Comparison of Load-Displacement Curve and Delamination Growth Process  
The DCB load-displacement curve of the PD and experimental results are compared as shown in Figure 12. The load-displacement curve of PD delamination growth is shown in Figure 13. The results in Figure 12 show that the PD-based analytical result agrees well with experimental results. It is also noted that the modeling load-displacement curve of PD delamination growth shows a zigzag-shaped curve in Figure 13. This is due to the fact that when the delamination front grows for a grid spacing, the displacement-controlled load will drop. Before the next delamination front growth happens, the load will slightly increase with the increase of displacement. A similar phenomenon is also observed in the load-displacement curves obtained from experiments, as shown in Figure 12. However, due to the complex specimen manufacturing process, preset delamination condition, and experimental setup condition, the load-displacement curve obtained from experiments show significant discrepancy between different specimens.  
Figure 11: Influence of different grid spacing, PD modeling load–displacement.  
Figure 12: Comparison of load–displacement curve between DCB test and PD (dx = 0.25 mm, dL = 0.008 mm).  
Quantitative delamination growth process is also compared between PD and experiment as presented in Table 3, and the delamination growth process damage contours are illustrated in Figure 14. In Table 3, the average delamination growth process of the five DCB test samples from 50 (initial delamination) to 65 mm (shown in Table 1) is compared with the modeling results. From the results, it noted that the present PD simulation results for the composite DCB delamination growth process agree well with the experimental results. The maximum difference between PD and experimental average results in displacement is \(-14.10\%\), and is \(11.78\%\) in load. In Figure 14, the delamination damage coefficient \(\varphi_{\text{delamination}}^{(16)(17)}\) is calculated using Equation (24). The delamination length growing from 50 (initial delamination) to 65 mm is vividly shown. It is worth noting that the PD delamination growth process shown in Figure 14 is \"spontaneous\". It does not need presetting of delamination path, nor non-physical stabilization parameters which are required for FEM modeling [26]. Also, it does not require the refinement of delamination front grid spacing.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**DELL (mm)**} & \multicolumn{2}{c}{**Experimental Avg.**} & \multicolumn{2}{c}{**PD**} & \multicolumn{2}{c}{**Difference**} \\  & **Disp.** & **Load** & **Disp.** & **Load** & **Disp.** & **Load** \\  & **(mm)** & **(N)** & **(mm)** & **(N)** & **(mm)** & **(N)** \\ \hline
50 & 5.09 & 91.35 & 4.39 & 88.66 & \(-13.78\%\) & \(-2.95\%\) \\
51 & 5.30 & 87.66 & 4.55 & 87.16 & \(-14.10\%\) & \(-0.57\%\) \\
52 & 5.43 & 85.07 & 4.72 & 85.64 & \(-13.00\%\) & \(0.67\%\) \\
53 & 5.49 & 85.23 & 4.89 & 84.13 & \(-11.00\%\) & \(-1.30\%\) \\
54 & 5.52 & 84.14 & 5.06 & 82.62 & \(-8.29\%\) & \(-1.81\%\) \\
55 & 5.70 & 81.99 & 5.23 & 81.24 & \(-8.26\%\) & \(-0.91\%\) \\
60 & 6.09 & 67.14 & 6.16 & 75.05 & \(1.16\%\) & \(11.78\%\) \\
65 & 6.84 & 63.64 & 7.16 & 69.72 & \(4.66\%\) & \(9.55\%\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison between PD delamination growth with experimental average results (DELL: delamination length).  
Figure 13: Zigzag-shaped load–displacement curve of DCB test delamination growth process.