## 2 Materials and Methods
### Numerical Modeling
#### 2.2.2 Material Modeling  
Considering its high strain capability, good manufacturability, and low cost, the ASP was modeled using stainless steel 304 material (SS304). Due to strain rate dependency in the blast simulation, a modified Johnson-Cook (J-C) model was adapted for the simulation. According to this constitutive model, the effective plastic stress \(\sigma_{eff}^{p}\) is given by the following equation:  
\[\sigma_{eff}^{p}=\Big{(}A+B\Big{(}\epsilon_{eff}^{p}\Big{)}^{n}\Big{)}\Big{(}1+ \frac{\dot{\epsilon}}{\dot{\epsilon}_{0}}\Big{)}^{C}\Big{(}1-\bigg{(}\frac{T- T_{r}}{T_{m}-T_{r}}\bigg{)}^{m}\Big{)} \tag{2}\]  
The first term represents material plasticity, where \(\epsilon_{eff}^{p}\) is the effective plastic strain, \(A\) is the yield strength, and \(B\) and \(n\) are strain-hardening parameters. The second term accounts for the strain rate dependency, where \(\dot{\epsilon}\) is the strain rate, \(\dot{\epsilon}_{0}\) is the reference strain rate, and \(C\) is the strain rate sensitivity coefficient. The final term addresses temperature dependency, where \(T\), \(T_{r}\), and \(T_{m}\) are the working temperature, room temperature, and melting temperature, respectively, and \(m\) is the thermal softening coefficient.  
All material parameters can be determined through curve fitting of experimental data. The modified J-C parameters for SS304 were obtained from curve fitting of the stress-strain curve by references [14; 53; 54], as shown in Figure 4. These material parameters are summarized in Table 2.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \(E\) & \(\
u\) & \(\rho\) & \(A\) & \(B\) & \(n\) & \(\dot{\epsilon}_{0}\) & \(C\) & \(T_{r}\) & \(T_{m}\) & \(m\) & \(C_{p}\) & \(\chi\) & \(\alpha\) \\ (**GPa**) & (**-**) & (**kg/m3**) & (**MPa**) & (**MPa**) & (**-**) & (**s\({}^{-1}\)) & (**-**) & (**K**) & (**K**) & (**-**) & (**J/kg\(\cdot\)K**) & (**-**) & (**K\({}^{-1}\)) \\ \hline
200 & 0.3 & 7900 & 310 & 1872 & 0.96 & 0.001 & 0.016 & 293 & 1673 & 1 & 500 & 0.9 & \(1.5\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Modified Johnson-Cook parameters of 304 stainless steel.  
Figure 3: Finite element (FE) model of the ASP subjected to air-blast loading at the center. The ASP is modeled with quarter symmetry due to its symmetric response. The red, green, and blue lines represent the fixed boundary condition and the symmetric boundary conditions in the \(y\)–\(z\) and \(x\)–\(z\) planes, respectively.  
For the material failure criterion, the Cockcroft-Latham damage evolution rule \((D)\) was used to simulate the failure of the modified J-C model. This criterion is based on the concept that ductile fracture occurs when the accumulated plastic strain energy in a material reaches a critical value [55]. Mathematically, the Cockcroft-Latham damage criterion can be written as  
\[D=\frac{D_{\text{C}}}{W_{\text{C}}}\text{max}(\sigma_{1},0)\varepsilon_{eff}^{p} \tag{3}\]  
where \(D_{\text{C}}\) is the critical damage parameter, \(\sigma_{1}\) is the maximum principal stress, and \(W_{\text{C}}=\int_{0}^{\varepsilon_{f}}\text{max}(\sigma_{1},0)de_{eff}^{p}\) is the plastic strain energy, with \(\varepsilon_{f}\) as the strain failure. The value of \(D_{\text{C}}\) is 1 [52], while \(W_{\text{C}}\) is obtained from the stress-strain curve, with a value of 296 MPa. In this study, failure criteria were not applied during the data sampling process but were only considered for the application of the AFV subsystem.