# Prediction Method of Blast Wave Impact on Crew Module for Liquid Rocket Explosion on Launch Pad
## 4 Finite ElementModel of Rocket Explosion
### Material Constitutive Models and State Equations  
Three kinds of materials are involved in this finite element model: air, TNT, aluminum alloy. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scenarios**} & **Stages Involved in Explosion** & **Equivalent(kg)** & **Radius(m)** & **Height (m)** & **Detonation Coordinate (x\({}_{0}\), y\({}_{0}\), z\({}_{0}\))/m** \\ \hline \multirow{2}{*}{1} & stage-3 & \(1.11\times 10^{4}\) & 0.81 & 3.26 & (0, 0, 32.57) \\  & stage-1, 2 and boosters & \(4.01\times 10^{4}\) & 1.25 & 5.00 & (0, 0, 10.07) \\  & stage-2 & \(4.98\times 10^{3}\) & 0.62 & 2.49 & (0, 0, 27.57) \\  & stage-1, 3 and boosters & \(4.62\times 10^{4}\) & 1.31 & 5.24 & (0, 0, 27.57) \\  & boosters & \(5.12\times 10^{4}\) & 1.36 & 5.43 & (0, 0, 47.57) \\  & stage-1 & \(1.87\times 10^{4}\) & 0.97 & 3.88 & (0, 0, 11.57) \\  & stage-2 and boosters & \(2.14\times 10^{4}\) & 1.02 & 4.06 & (0, 0, 10.57) \\  & stage-3 & \(1.11\times 10^{4}\) & 0.81 & 3.26 & (0, 0, 32.57) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters setting of TNT model in numerical simulation. Figure 2: Rocket 3D finite element model and gauges distribution. The air is commonly modeled by null material model with a linear polynomial equation of state, EOS_LNIEAR_POLYNOMIAL, which defines the pressure by the following equation:  
\[P=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+\Big{(}C_{4}+C_{5}\mu+C_{6}\mu^{2} \Big{)}E_{0} \tag{6}\]  
where the parameter \(\mu\) is defined as \(\mu=\rho/\rho_{0}\), \(\rho\) is the current density, and \(\rho_{0}\) is a nominal orreference density; \(C_{0}\)-\(C_{6}\) are the equation coefficients; and the parameter \(E_{0}\) is the initial internal energy ofreference specific volume per unit. Table 7 gives the parameters used in the air model by [PERSON], [PERSON] and [PERSON]. [27; 28]. The TNT charge is modeled by the high explosive material model, MAT-HIGH-EXPLOSLVE-BURN, and the Jones-Wilkins-Lee (JWL) equation of state defines the pressure by the following equation:  
\[P=A\bigg{(}1-\frac{\omega}{R_{1}V}\bigg{)}e^{-R_{1}V}+B\bigg{(}1-\frac{\omega} {R_{2}V}\bigg{)}e^{-R_{2}V}+\frac{\omega E}{V} \tag{7}\]  
where \(A\), \(B\), \(R_{1}\), \(R_{2}\), \(\omega\) are the equation coefficients and they all should be tested in an accurate blast analysis; and \(V\) is the initial relative volume. Here we adopt the parameters selected by [PERSON]. [29], as shown in Table 8.  
Various mathematical models have been adopted to represent the flow stress data over a range of temperatures and strain rates. Among all, the Johnson-Cook constitutive model is chosen for aluminum alloy [30] in the research, as it provides a good description of the metal material behavior, subjected to large strain, strain rates and high temperatures. The JC equation is given by  
\[\sigma=\Big{(}A+B\varepsilon_{p}^{n}\Big{)}\Big{(}1+C\ln\dot{\varepsilon}^{*} \Big{)}\bigg{[}1-\Big{(}\frac{T-T_{room}}{T_{m}-T_{room}}\Big{)}^{m}\bigg{]} \tag{8}\]  
where \(\sigma\) is the flow stress, \(\dot{\varepsilon}\) is the equivalent plastic strain, \(A\) is the yield stress of the material at the condition of reference temperature and reference strain, \(B\) is the strain hardening coefficient, \(n\) is the strain hardening exponent, \(C\) and \(m\) are the material constants representing the coefficient of strain rate hardening and thermal softening exponent, respectively. \(\dot{\varepsilon}^{*}\) is the dimensionless strain rate and can be expressed as \(\dot{\varepsilon}^{*}=\dot{\varepsilon}/\dot{\varepsilon}_{r}\), where \(\dot{\varepsilon}\) is the strain rate and \(\dot{\varepsilon}_{r}\) is the reference strain rate. \(T\) is the deformation temperature, \(T_{m}\) is the melting temperature of the investigated material and \(T_{room}\) is the reference temperature, all of which should be absolute temperature (K) [31]. The three items on the right side of Equation (8) are used to describe the phenomena of the work-hardening effect, strain-rate effect and temperature effect, respectively.