## 8 Multiple Impacts of Space Debris Particles  
Since the service life of spacecraft is assumed to be 10-20 years, it is natural that protective screens should withstand multiple impacts from SD particles. Let us determine the mass of the incoming SD particles that the armored screen #5 can protect, avoiding penetration. To reduce the number of calculations, it is necessary to predict the maximum  
Figure 12: Cross-sections of heterogeneous screens with a steel concentration of 38%: (**a**) Screen #6 with a uniform distribution of 316L steel within the A356 aluminum alloy; (**b**) Screen #7 with a direct gradient distribution of steel; (**c**) Screen #8 with a reverse gradient distribution of steel. Figure 13: Calculation results of the interaction between an aluminum sphere and heterogeneous screens. Cross-sections of screens with a thickness equal to the grid step: (**a**) screen #6; (**b**) screen #7; (**c**) screen #8.  
ballistic velocity of the screen from the mass of the incoming particle. [PERSON]'s work [47] proposes a formula for the engineering estimation of the ballistic velocity:  
\[\frac{1}{2}m_{p}V_{bl}^{2}=\pi r^{2}hY, \tag{13}\]  
where \(m_{p}\) is the particle mass, \(V_{bl}\) is the ballistic velocity, \(r\) is the hole radius, \(h\) is the screen thickness, \(Y\) is the yield strength of the screen material. As the ballistic velocity needs to be evaluated, let us assume that the combination of screen parameters is some constant value. Then, the expression for ballistic velocity is as follows:  
\[V_{bl}=\frac{C}{\sqrt{m_{p}}}, \tag{14}\]  
To calibrate the constant \(C\), two or three calculations need to be carried out to determine the ballistic velocity at a given particle mass. In our case, \(C\sim 1.01\). Figure 14 illustrates the applicability of this formula and the results of numerical calculations. This Approximation (14) provides a 10% error margin for the calculation. Let us perform the calculations for the sequential collision of a group of eight particles with the protective screens, as shown in Figure 5. Random values close to 4 \(\upmu\)s were taken as time intervals, and the particles were spatially located in a circle of 0.5 cm radius with the first particle at the center, as is indicated in Figure 12. The most resistant screens were selected for testing, namely, screen #5, which is a composite with volumetric reinforcement; screen #4, which is a layered screen of A356/316L; and screen #7, which is a heterogeneous screen with a direct gradient of 316L steel on an A356 matrix. The calculation results are presented in Figure 15, which shows cross-sections of protective screens with a thickness of approximately one step of the computational grid cut out from the central region. The sequential multiple impact loading by SD particles leads to the propagation of shock waves through the deformed and partially damaged materials of the screens, resulting in further damage and fracture of the materials. The common feature of all screens is the process of damage and fracture of the A356 aluminum alloy, which is quite brittle. For the volumetrically reinforced composite, delamination between the steel elements and the aluminum matrix is characteristic, as can be seen in Figure 15a, and individual grains appear on the back side of the screen due to spallation processes. The mass of such grains is 0.1 mg, and their velocity is 50-70 m/s, so they do not pose a significant danger to the protected devices. Figure 14: Dependence of the maximum ballistic velocity on the mass of the incoming particle according to the Formula (14). Crosses represent the results of numerical calculations. The two-layer A356/316L screen withstands a similar impact load because the back side of the screen is made of 316L steel, which has sufficient strength against the spallation. In this case, the size of the crater volume is the largest among the compared protective screens since the A356 aluminum alloy quickly becomes damaged and starts to break down, as shown in Figure 15b.