## 3 Damage Analysis
### Numerical Modeling
Each parameter is multiplied to characterize the cumulative effect of each effect. \[\sigma_{y}=\Big{[}A+B\Big{(}\epsilon_{eff}^{p}\Big{)}^{n}\Big{]}(1+C\ln\varepsilon )\big{[}1-(T_{H})^{m}\big{]} \tag{7}\]  
In Equation (7), \(\epsilon_{eff}^{p}\) is the effective plastic strain; \(\varepsilon=\frac{\epsilon_{eff}^{p}}{\epsilon_{0}}\), where \(\epsilon_{0}\) is the strain rate used to determine \(A\), \(B\), and \(n\); \(T_{H}=\frac{T-T_{R}}{T_{M}-T_{R}}\) is the homologous temperature; \(T_{M}\) is the melting temperature; \(T_{R}\) is the reference temperature; \(\Delta T=\frac{1}{\rho\epsilon_{0}}\int\sigma d\epsilon_{eff}^{p}\), where \(\rho\) is the density, and \(C_{P}\) is the specific heat. The five parameters \(A\), \(B\), \(n\), \(m\) and \(C\) in the model are basic parameters for characterizing the yield strength, where \(A\) is the initial yield strength of the material under the quasi-static strain rate, \(B\) and \(n\) are the flow stress of the strain-hardening behavior under the quasi-static strain rate, \(C\) is the strain rate effect, and \(m\) is the thermal softening effect. In addition to the material properties \(\rho\), \(C_{P}\), and \(T_{M}\), there are also elastic parameters. Usually, the pressure is defined as a function of the volume strain response, and the shear modulus is integrated along the equation of state [11]. The cumulative damage of the material is used to characterize the failure of the material in the J-C constitutive, as shown in Equation:  
\[\varepsilon^{F}=\Bigg{(}D_{1}+D_{2}\exp\Bigg{[}D_{3}\frac{P}{\sigma_{eff}} \Bigg{]}\Bigg{)}(1+D_{4}\ln\varepsilon)(1+D_{5}T_{H}) \tag{8}\]  
where \(D=\sum\frac{\Delta\epsilon_{eff}^{p}}{\epsilon^{p}}\) the material failure occurs when \(D=1\) where \(\epsilon_{eff}\) is the effective stress, \(P\) is the average stress. The parameters of the Johnson-Cook model for the Al7075-T6 Al alloy and the parameters of the Mie-Gruneisen equation of state are shown in Table 1[12]. T300/QY8911 related material parameters are shown in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **7075-T6** \\ \hline Johnson–Cook model parameters & & \\ Density (kg/m\({}^{3}\)) & R0 & 2.81 \\ Poisson’s ratio & PR & 0.33 \\ Shear modulus (GPa) & E & 0.717 \\ Static yield limit (MPa) & A & 0.00546 \\ Strain hardening modulus [13] & B & 0.00678 \\ Strain hardening exponent & n & 0.71 \\ Strain rate coefficient & C & 0.35 \\ Spall type & SPALL & 3 \\ Failure parameters D\({}_{1}\) & D1 & \(-\)0.068 \\ Failure parameters D\({}_{2}\) & D2 & 0.451 \\ Failure parameters D\({}_{3}\) & D3 & \(-\)0.952 \\ Failure parameters D\({}_{4}\) & D4 & 0.036 \\ Failure parameters D\({}_{5}\) & D5 & 0.697 \\ \multicolumn{2}{c}{Mie–Gruneisen EOS parameters} \\ Constants C & C & 0.535 \\ Constants S\({}_{1}\) & S1 & 1.34 \\ Constants \(\gamma\) & GAMAO & 2.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Al7075-T6 Johnson–Cook model and Mie–Gruneisen EOS parameters. The simulated spherical fragments are divided by a uniform mesh with a mesh size of about 0.3 mm, using hexahedral eight-node units with a total number of 56,000 units, as shown in Figure 19.  
The structural metal part of the model mesh uses a hexahedral deca-node unit, and the composite part of the model consists of a 2D shell unit, with a single sub-layer containing three layers of actual layup information. * CONTACT_SURFACE_TO_SURFACE_TIEBREAK [15,16] is used between layers. The total number of model units for the air inlet Al alloy I-beam riveted structure is 241,437, and the total number of model units for the wing composite/Al alloy spacer structure is 1,099,060.