## 3 Results
### FE Analysis for Mode I, Mode II, and Mixed-Mode Tests  
To analytically evaluate the crack growth behavior according to each test mode, the fracture toughness was calculated under the same test conditions, as shown in Table 6. Through NL analysis for each test mode, we selected values of 1.12 kJ/m\({}^{2}\) and 2.02 kJ/m\({}^{2}\) for Modes I and II, respectively. For Mode III, we used the same value as Mode II [27]. In addition, for the Mixed-mode, we selected a material factor of 1.85. First, in Figure 12a,b, which show the displacement-load curve as the analysis result under Mode I conditions and the crack growth behavior and fracture toughness, respectively, it can be observed that the test and analysis exhibited similar trends. A pre-crack test was performed at 3 mm to ensure the homogeneity of the initial crack shape. Additionally, the analysis considered that crack growth behavior was confirmed from 50 mm (including pre-crack length) because crack propagation was generally simulated as a cohesive failure. \begin{table}
\begin{tabular}{c c c} \hline \hline  & & 2 axis\({}_{3}\)/FM/3/3 axis\({}_{2}\) \\ \hline \multirow{2}{*}{**Fracture toughness [kJ/m\({}^{2}\)]**} & **Mode I, \(G_{I}\)** & 1.12 \\  & **Mode II, \(G_{I}\)** & 2.02 \\  & **Mode III, \(G_{III}\)** & 2.02 \\ \hline \multirow{2}{*}{**Material constant**} & **B–K criterion** & 1.85 \\  & **[Mixed-mode], \(\eta\)** & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Fracture toughness test results of hybrid composites. Figure 11: Fracture cross-section of Mixed-mode test results: (**a**) c = 82.9 mm, mode mixture ratio = 0.23; and (**b**) c = 31.2 mm, mode mixture ratio = 0.64. From Figure 13, it can be deduced that the crack grew as the stress propagated from the center of the specimen to the edge. Figure 13a shows the initial crack selection, the shape of the final crack, and the in-plane stress as the principal stress. Figure 13b shows the crack growth behavior with the applied displacement and the average stress at the critical point. Particularly, Figure 13b shows a similar trend as the fracture surface shown in Figure 11a based on the locations of the initial crack and crack growth. In other words, the reliability of the crack growth model in Mode I was established by confirming the similarity between the progressive fracture analysis method implemented using finite element analyses and the test. To analytically evaluate the crack growth behavior of Mode II, a finite element analysis was performed under the same test conditions. Figure 14a shows the resulting displacement-load curve, which indicates that the deviation between the test and analysis result varied from that of Mode I. It was found that the softening effect led to greater non-linearity in the analysis compared to that in the test [28]. In addition, as seen in Figure 14b, which illustrates the fracture toughness with respect to the crack growth length, similar trends were observed at the location of the initial crack, whereas an error began to manifest during the crack growth due to the softening effect. Furthermore, it is believed that the rapid change in fracture toughness at a crack growth length of 45 mm was due to the  
Figure 12: Comparative analysis of specimen-level FE analysis results and Mode I test results: (**a**) \(P-\delta\) curve and (**b**) \(G_{I}-a\) curve. Figure 13: FE analysis result shape of Mode I: (**a**) in-plane stress contour and (**b**) crack growth length at applied displacement. increase in the load relative to the crack growth rate as the location of the crack growth approached the loading point. To closely analyze this crack behavior, the crack growth length and stress propagation with respect to the load-displacement were examined, as shown in Figure 15. Figure 15a shows the final crack growth length and the principal stress shape in the composite material. In contrast to Mode I, the crack appears to propagate from the edge to the center of the specimen.