# Shock Initiation of a Satellite Tank under Debris Hypervelocity Impact
## 4 Initiation of the Satellite Tank under Debris Impact
### Simulation of the Debris Impacting Satellite Tank  
Numerical simulation is conducted by ANSYS/LS-DYNA software to reproduce a hypothetical scenario, in which the spherical aluminum debris with a diameter of 4 cm impacts the satellite tank at the velocity of 10 km/s. The satellite tank is made of titanium and filled with liquid hydrazine. It is 50 cm in diameter, 120 cm in height, and 2 mm in wall thickness. The impact point is located at the middle of the satellite tank. The simulation model consists of the debris, satellite tank and liquid hydrazine, as displayed in Figure 8.  
Lagrange grids are constructed for the debris and tank, whilst Euler grids are constructed for liquid hydrazine. ALE algorithm, initial velocity, contact type, and material parameters of aluminum and liquid hydrazine are the same as those of HYP-2 simulation model. Numerical simulations were carried out by setting the mesh sizes of liquid hydrazine to 0.1 cm, 0.5 cm, 1 cm, 2 cm, 3 cm, and 4 cm. Results show that when the mesh size is smaller than 2 cm, the shock wave pressure along the impact line remains basically unchanged, indicating that the shock wave pressure is independent of the mesh size. Therefore, the mesh size of liquid hydrazine is set to 2 cm. The mesh sizes of the projectile and satellite tank are 0.5 cm and 2 cm, respectively. The computing time step is 0.6 \(\upmu\)s.  
As the parameters describing titanium in Johnson-Cook model are unknown, the titanium tank is modelled using the material model MAT_PLASTIC_KINEMATIC. The material parameters of titanium  
Figure 8: The simulation model of the debris impacting the satellite tank and the right side is the model cut in half. Figure 7: Pressure-time curve of element \(A\) in liquid hydrazine; \(P_{max}\) represents the peak pressure. are listed in Table 6. Where, \(E\) is the Young's modulus, \(\upsilon\) is the [PERSON]'s ratio, \(\sigma_{s}\) is the yield strength, and \(\sigma_{b}\) is the tensile strength. Figure 9 exhibits the pressure nephogram of the shock wave in satellite tank at 6 \(\upmu\)s, 34 \(\upmu\)s, 55 \(\upmu\)s, 98 \(\upmu\)s, 143 \(\upmu\)s, and 192 \(\upmu\)s. When the debris penetrates the satellite tank, a hemispherical shock wave is formed at the impact point. The velocity of the shock wave is higher than that of the debris, so the debris lags behind the shock wave. In the process of shock wave propagation, part of the wave front energy is converted into the internal energy of liquid hydrazine and the deformation energy of tank wall. The irreversible energy loss occurs on the wave front, which results in the continuous decrease in shock wave pressure. When \(t\) = 192 \(\upmu\)s, the shock wave reaches the back wall of the satellite tank. To determine the attenuation law of shock wave in the tank, the peak pressures of the elements on the debris impact line are obtained. These discrete pressures are fitted into a curve by the least square method, as depicted in Figure 10a. The expression of the fitting curve \(P(t)\) is as follows (the pressure unit is kbar):  
\[P(t)=802.10977e^{-\frac{t}{10.00728}}+94.04187e^{-\frac{t}{72.6948}}-1.60227. \tag{48}\]  
The regular residuals \(\varepsilon\) between the pressure calculated by \(P(t)\) and the pressure obtained by simulation are shown in Figure 10b. They are evenly distributed and fall within the range of (\(-5\%\) to \(5\%\)), which demonstrates that the fitting curve \(P(t)\) is reliable. The pressure duration in satellite tank can be calculated as follows:  
\[802.10977e^{-\frac{t}{10.00728}}+94.04187e^{-\frac{t}{72.6948}}-1.60227=138. \tag{49}\]  
The solution to equation (49) is \(t\) = 25.38 \(\upmu\)s, which means that the shock wave pressure attenuates to the C-J pressure at \(t\) = 25.38 \(\upmu\)s. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & \(\rho\) **(kg/m\({}^{3}\))** & \(E\) **(GPa)** & \(\upsilon\) & \(\sigma_{s}\) **(MPa)** & \(\sigma_{b}\) **(MPa)** \\ \hline Titanium & 4500 & 118 & 0.34 & 895 & 824 \\ \hline \hline \end{tabular}
\end{table}