## 2 Finite Element Model  
### Damage Tolerance and Stress Intensity Factor  
Damage tolerance refers to the ability of a structure to resist damage caused by defects, cracks or other damages during specified unrepaired service life. In short, it refers to the initial defect in aircraft structure and the degree of defect development in use [16]. Therefore, the damage tolerance design concept involves acknowledging whether structures have initial defects before use. However, it must consider the defect or damage according to the method of design of the prescribed maintenance use during the period of growth control in a certain range. In the meantime, the structure should satisfy the requirement of the provisions of residual strength (including defect or crack structure bearing capacity) to ensure the safety and reliability of aircraft structure [17].  
In order to determine a safe service life of components, it is necessary to be able to calculate the time at which the crack develops and breaks. Therefore, for damage tolerance analysis, it is necessary to provide information about the structural strength changes with the crack size and determine the crack propagation life [18]. The SIF describes the severity of the stress distribution around the crack tip. It is used as a fracture parameter to describe the crack growth in linear elastic fracture mechanics (LEFM), which is related to the size of the characteristic crack and load of the structure [19]. Only by determining the SIF correctly can the critical length and fatigue life of structural elements before failure be calculated accurately. According to the direction of load, the crack can be divided into type I crack, type II crack and type III crack. Among them, the load direction of type I crack is perpendicular to the crack surface. The loading direction of type II crack is parallel to the crack surface and perpendicular to the crack line. The load direction of the type III crack is parallel to the crack surface and crack line. M-Integral is used to calculate the stress intensity factor, and the expression of M-Integral energy is [20; 21]  
\[\overline{M}^{(1,2)}=\int_{\Gamma}\!\!\left(\sigma_{ij}^{(1)}\frac{\partial u _{i}^{(2)}}{\partial x_{1}}+\sigma_{ij}^{(2)}\frac{\partial u_{i}^{(1)}}{ \partial x_{1}}-W^{(1,2)}\delta_{1j}\right)\!\frac{\partial q}{\partial x_{j}} ds,\ \ \ \ \ i=1,2,3;j=1,2 \tag{1}\]  
\[W^{(1,2)}=\sigma_{ij}^{(1)}\varepsilon_{ij}^{(2)}=\sigma_{ij}^{(2)}\varepsilon _{ij}^{(1)}, \tag{2}\]  
where \(\Gamma\) is the integral loop around the crack tip; \(u_{i}^{(1)},u_{i}^{(2)}\) is the component of the displacement vector; \(\delta_{1j}\) is the Kronecker delta and q is a function that is 1 at the stress intensity evaluation point and zero on the outer boundary of the domain of integration; \(W^{(1,2)}\) is the strain energy density.  
The form of M-integral is same as the form of interaction integral. The definition of M in terms of the crack tip field variables that can be obtained from an FEM analysis (superscript (1)) and from the theoretical expressions of \(K_{I}\), \(K_{II}\) and \(K_{III}\) (superscript (2)).  
\[K_{I}=K_{I}^{(1)}+K_{I}^{(2)},\ \ \ K_{II}=K_{II}^{(1)}+K_{II}^{(2)},\ \ \ K_{III}=K_{III}^{(1)}+K_{III}^{(2)},\]  
The definition of M in terms of Ks and material properties is defined as [22]  
\[M^{(1,2)}=\frac{2\!\left(1\!-\!v^{2}\right)}{E}K_{I}^{(1)}K_{I}^{(2)}+\frac{2 \!\left(1\!-\!