## 2 Dimensional Analysis  
When an explosion occurs inside a box structure, the degree of damage can generally be assessed and measured by the maximum deflection of the wall plate \(\delta\). The degree of structural damage is influenced by many factors, such as total explosive mass \(W\), explosive blast energy \(E\), structural material yield strength \(\sigma_{y}\), material density and sound velocity \(\rho_{\mathrm{s}}\) and \(C_{\mathrm{S}}\), air density and sound velocity \(\rho_{0}\) and \(C_{0}\), structural characteristic dimension \(L\), and wall plate thickness \(h\). The main influencing factors and the relationship between them were identified through dimensional analysis. The damage to the box-shaped structure is expressed as  
\[\delta=f\big{(}W,E,\sigma_{y},\rho_{\mathrm{s}},C_{\mathrm{s}},\rho_{0},C_{0}, L,H\big{)} \tag{1}\]  
The explosion is assumed to occur in ideal air, where the speed of sound _late deformation energy ratio_. \(C_{0}\) and density \(\rho_{0}\) are considered to be constant. The problem has eight physical quantities, including seven independent variables and one dependent variable, that involve three basic measures of mass \(M\), length \(L\), and time \(T\), as shown in Table 2.  
Using the Vaschy-Buckingham theorem, five dimensionless quantities can be obtained, including four dimensionless independent variables and one dimensionless dependent variable  
\[\Pi_{0}=F(\Pi_{1},\Pi_{2},\Pi_{3},\Pi_{4}) \tag{2}\]  
where  
\[\Pi_{0}=\frac{\delta}{h},\Pi_{1}=\frac{E}{\sigma_{y}L^{2}h},\,\Pi_{2}=\frac{W }{\rho_{\mathrm{s}}L^{2}h},\Pi_{3}=\frac{\rho_{\mathrm{s}}C_{\mathrm{s}}^{2} }{\sigma},\Pi_{4}=\frac{L}{h} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{3}{c}{**Dependent**} & \multicolumn{8}{c}{**Independent Variables**} \\ \multicolumn{3}{c}{**Variable**} & \(\delta\) & \(W\) & \(E\) & \(\sigma_{y}\) & \(\rho_{\mathrm{s}}\) & \(C_{\mathrm{S}}\) & \(L\) & \(H\) \\ \hline Quantitative & \(L\) & \(M\) & \(ML^{2}T^{-2}\) & \(ML^{-1}T^{-2}\) & \(ML^{-3}\) & \(LT^{-1}\) & \(L\) & \(L\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Variables and quantitative. to obtain the dimensionless function  
\[\frac{\delta}{h}=F\bigg{(}\frac{E}{\sigma_{y}L^{2}h},\frac{W}{\rho_{\rm s}L^{2}h}, \frac{\rho_{\rm s}C_{\rm s}^{2}}{\sigma},\frac{L}{h}\bigg{)} \tag{4}\]  
The above equation shows that the structural deformation deflection is a function of the four dimensionless expressions above, and each expression has a clear physical meaning: \(\frac{E}{\sigma_{y}L^{2}h}\) is the ratio of blast energy to structural deformation energy, which characterizes the transformation and consumption between blast energy and plastic deformation energy of the wall plate; \(\frac{W}{\rho_{\rm s}L^{2}h}\) is the ratio of explosive mass to the mass of the member, which characterizes the transformation and consumption between the explosive energy and the kinetic energy of the wall plate; \(\frac{\rho_{\rm s}C_{\rm s}^{2}}{\sigma}\) for the ratio of material strength and stiffness, which characterizes the ratio of the energy generated by rupture or tearing to the energy deformed, and \(\frac{L}{h}\) for the geometric scale of the structural elements. The energy generated by the explosion is mainly divided into wall kinetic energy, plastic deformation energy, and rupture fracture energy, and a portion of the energy is transferred to the adjacent box through the breakage [19]. Compared to other energy, the fracture energy consumed by the wall panel rupture is very small.