## 2 Experimental and Numerical Failure Analysis of Adhesive Joints  
### Theoretical Background  
Considering the composite blade is subject to combined loads, it is essential to account for the interactions of opening mode, shearing mode, and mixed mode. This study's analytical approach to adhesives encompasses four stages. Initially, the onset of failure is assessed using the quadratic nominal stress criterion (QNS criterion) proposed by [PERSON] et al. [20], as outlined in Equation (1). This assessment is based on the adhesive's strength. Subsequently, upon the initiation of failure, crack growth occurs. To analyze this, a fracture energy rate-based crack growth criterion, considering mixed mode conditions as per Equation (2) proposed by [PERSON] and [PERSON] [21], is applied for progressive failure analysis. \[\left(\frac{\sigma_{n}}{N_{max}}\right)^{2}+\left(\frac{\sigma_{s}}{T_{max}} \right)^{2}+\left(\frac{\sigma_{t}}{S_{max}}\right)^{2}=1, \tag{1}\]  
\[G_{IC}+(G_{IIC}-G_{IC})\bigg{(}\frac{G_{II}}{G_{I}+G_{II}}\bigg{)}^{\eta}=G_{C},G_{C}=\frac{3p\delta}{2 ba}. \tag{2}\]  
where \(N_{max}\) represents the normal tensile strength, \(T_{max}\) represents the interlaminar shear strength, \(S_{max}\) denotes the transverse interlaminar shear strength, \(\sigma_{n}\) denotes the nominal stress, \(\sigma_{s}\) denotes the shear stress, \(\sigma_{t}\) is the transverse shear stress, \(G_{IC}\) is the critical fracture toughness of mode I, \(G_{IIC}\) is the critical fracture toughness of mode II, \(G_{I}\) is the fracture toughness of mode I, \(G_{II}\) is the fracture toughness of mode II, \(G_{C}\) is the critical fracture toughness, and \(\eta\) represents the material exponent of the mixed mode. As a third step, the VCCT is employed for the analytical interpretation based on the criteria for crack initiation and growing fracture toughness values. VCCT is a widely used and well-developed method for analyzing adhesive joints and has been successfully applied to various issues, including complex delamination, skin tightener detachment, and adhesive joint failure. The VCCT was originally developed by [PERSON] and [PERSON] [22] to calculate the strain energy release rate from crack tips by modifying [PERSON]'s principle of crack closure integration. VCCT operates on the principle that the energy released during crack growth is equivalent to the energy needed to restore the crack to its initial length. Typically, nodes at the crack tips exhibit a bound configuration. As shown in Figure 1 under mode I loading, the rate of energy release is calculated using the opening displacement (vertical displacement) and the vertical crack closure force (connectivity) as follows: \(v_{1,6}\) is displacement 6 to 1 node. \(F_{2,5}\) is load 5 to 2. Conditions for determining fracture toughness according to crack propagation are as follows:  
\[G_{IC}=\frac{1}{2}\frac{F_{2,5}v_{1,6}}{bd}, \tag{3}\]  
Crack propagation is assumed to occur if the following condition is satisfied:  
\[f=\frac{G_{I}}{G_{IC}}\geq 1.0, \tag{4}\]  
where \(G_{I}\) and \(G_{IC}\) are the calculated energy release rate and the critical strain energy release rate of matter, respectively. Then, 1, 2, 5, and 6 denote the specific node locations. The contact between node 2 and node 5 breaks, allowing the crack to propagate by \(d\) and forming a new crack tip. As a fourth step, while VCCT is traditionally applied under a singular load condition, this study necessitates additional analytical criteria to evaluate crack growth life under  
Figure 1: Finite element propagation characteristics of pure mode 1.  
fatigue loads, including cycle numbers. To address this, the NASA-2017 report introduced the VCCT-XFEM method [23] based on [PERSON]'s theory [24]. This approach involves applying a material factor derived from load, displacement, and crack growth length data obtained from fatigue crack growth tests to the analytical technique. The resistance to fatigue crack growth is then determined using Equations (5)-(7).