## 3 Numerical Simulation of the Debris Impact on Satellite Tank  
### Simulation Model of the Debris Impacting Satellite Tank  
Space debris comes from an invalid spacecraft, the final stages of a rocket, and disintegration debris, which are mainly made of aluminum. Satellite tanks are made of titanium and filled with liquid hydrazine with dimensions of 50 cm in diameter, 120 cm in height, and 2 mm in wall thickness. A numerical simulation was conducted with ANSYS/LS-DYNA software to reproduce a hypothetical scenario where spherical aluminum debris with a radius of 2 cm impacted the satellite tank at the velocity of 7000 m/s in the \(x\) direction. The impact point was located at the middle of the satellite tank. To study the most serious damage caused by the debris impact on satellite tank, the deformation of debris was neglected and the shape of debris was assumed to have remained unchanged.  
Figure 3: (**a**) Pressure nephogram of the shock wave in water. (**b**) Sectional view on the _xoz_ plane. (**c**) Picture captured by a high-speed camera in Varas test [9].  
\begin{table}
\begin{tabular}{c c c c c} \hline \(t\) (\(\upmu\)s) & \(x\) (cm) & \(d_{s}\) (mm) & \(d_{e}\) (mm) & \(\epsilon\)\% \\ \hline \multirow{2}{*}{84} & 1.5 & 35.1 & 33.12 & 5.9 \\  & 5.0 & 20.3 & 19.4 & 4.6 \\ \multirow{2}{*}{140} & 2.0 & 43.5 & 41.9 & 3.8 \\  & 7.5 & 28.9 & 28.2 & 2.5 \\ \hline \end{tabular}
\end{table}
Table 3: Cavity diameter in the simulation results and test results.  
The simulation model consisted of the debris, satellite tank, and liquid hydrazine, as displayed in Figure 4. Lagrange grids were constructed for the debris and tank, whilst Euler grids were constructed for the liquid hydrazine. The coupling between the Lagrange and Euler grids was realized by the key word CONSTRAINED_LAGRANGE_IN_SOLID. The interaction between the debris and tank wall was achieved through the eroding surface-to-surface contact, and the interaction between the debris and liquid hydrazine was achieved through the automatic surface-to-surface contact. The mesh sizes of the debris, satellite tank, and liquid hydrazine were 0.4 cm, 2 cm, and 2 cm, respectively. The whole model consisted of 96,848 elements. The computing time step was 0.6 \(\upmu\)s.  
Since the density, boiling point, and critical temperature of liquid hydrazine are within 2% of the values for water [20], the constitutive equation MAT_NULL and the state equation EOS_GRUNEISEN of water were used to describe the liquid hydrazine. The material parameters of liquid hydrazine are listed in Table 4.  
As the parameters describing titanium in the Johnson Cook model are unknown, the titanium tank was modeled using the material model MAT_PLASTIC_KINEMATIC. The material model uses equivalent stress as the failure criterion. If the equivalent stress of Lagrange grids exceeds the yield strength of titanium, the failure elements will be deleted from the calculation program. This model was used to characterize the perforation on tank walls. The material parameters of titanium are listed in Table 5[21] where \(E\) is the Young's modulus; \(v\) is the Poisson's ratio; \(\sigma_{s}\) is the yield strength; and \(\sigma_{b}\) is the tensile strength.