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Hypervelocity impact performance of aluminum egg-box panel enhanced Whipple shield
Acta Astronautica 119 (2016) 48–59
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Hypervelocity impact performance of aluminum egg-box panel enhanced Whipple shield$ Xiaotian Zhang n, Tao Liu, Xiaogang Li, Guanghui Jia School of Astronautics, Beihang University, Beijing 100191, China
a r t i c l e in f o
abstract
Article history: Received 10 March 2015 Received in revised form 9 October 2015 Accepted 22 October 2015 Available online 10 November 2015
The enhanced Whipple shield is widely used on the space stations against space debris hypervelocity impact. Numerical investigation is conducted in this paper to test the concept of using the egg-box panel as the enhanced layer. Taking the flat panel as the benchmark and then the enhanced flat panel is substituted with the egg-box panel. For objective comparison all the egg-box panels have the same areal density as the benchmark flat panel. The performance is tested from the aspects of impact area, cell size and the axial offset, etc. The egg-box panel absorbs more incoming debris energy and generates less large fragments from itself. It can be fabricated by the stamping and forming of the aluminum flat sheet, the cost of manufacture would not be significantly increased. The egg-box panel has better performance than the flat panel and is appropriate to be the enhanced layer of the Whipple shield. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Space debris Spacecraft shielding Hypervelocity impact Finite element method Numerical simulation
1. Introduction Space debris is the increasing threatens to manned spacecraft. Protective shield must be designed and installed over the key parts of the spacecraft to withstand the hypervelocity impact of space debris. Whipple shield [1–6] is widely used on spacecraft since 1960s. An aluminum bumper is placed over a distance of the wall of the spacecraft. The thickness of the bumper and the distance between the bumper and the rear wall are carefully designed so that the debris can be fully fragmented after perforating the bumper. The damage to the rear wall can be therefore reduced. Typical Whipple shield can withstand 1–5 mm space debris. To further enhance ☆ Supported by the National Natural Science Foundation of China (11502010) and the Fundamental Research Funds for the Central Universities of China (YWF-14-YHXY-022). n Corresponding author. E-mail addresses: [email protected] (X. Zhang), [email protected] (T. Liu), [email protected] (X. Li), [email protected] (G. Jia).
http://dx.doi.org/10.1016/j.actaastro.2015.10.013 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.
the performance of the shield, stuffed Whipple shield is developed. A stuffed layer is inserted between the front bumper and the rear wall. The stuffed Whipple shield is applied on the International Space Station [7]. The stuffed layer is made of Nextel/Kevlar material. The protective ability is raised upto 10 mm with stuffed Whipple shield. The stuffed layer is widely investigated in recent years. Various materials are tested under hypervelocity impact for performance comparison. For example: Aluminum foam/Aluminum foam sandwich panel [8,9], aluminum net [10,11], honeycomb sandwich panel [12,13], fiber laminates [14,15], amorphous alloy [16,17], etc. The new types of “stuffed” Whipple shield are general called enhanced Whipple shield. These panels can be basically categorized into two classes: the new topology of aluminum (aluminum foam, aluminum net) and the new materials (fiber laminates, amorphous alloy). The honeycomb sandwich panel, which takes the carbon fiber reinforced plastic as the face sheet, can be seen as the combination of the concepts of new topology (honeycomb) and new material (face sheet).
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Fig. 1. Geometries of the single and double layer egg-box panels.
Davidson [17] tested the idea of fabricating the amorphous alloy sheet into the “egg-box” panel with a semisolid forging technique (Fig. 1). The individual single-layer and multi-layer egg-box panels are used in the hypervelocity tests and the results are compared with the honeycomb sandwich panel. Using the egg-box geometry the normal impact of the fragments on the flat panel can be turned into the inclined impact. According to the JSC single wall ballistic limit equation [18] (BLE): 0
0:5 118 19
0:25 ρb
Bt BH dc ¼ @ b k K v
ρp
cos θ C
C 23 A
ð1Þ
where dc is the critical perforation diameter; t b is the bumper thickness; BH is the material Brinell hardness; ρb and ρp are the material density of the bumper and the projectile respectively; v is the impact velocity; θ is the incidence; C is the sound speed; k and K are constants. The critical perforation diameter of the projectile dc is proportional to the ð cos θÞ 2=3 . Therefore with the same sheet thickness, the larger inclined angle θ results in the larger critical diameter, i.e. the better protective performance. Since the egg-box geometry can change the local impact incidence to about 45°, the ballistic limit can be promoted. When refer to using the egg-box panel as the enhanced layer of the Whipple shield, the non-parallel BLE [19] is a good support to our hypothesis. Considering the obliquity of the walls, the Whipple shield BLE is as follows: 2 1 ρRW t RW 3 13 2 1 1 dc;h ¼ kh ρp 3 ρb 9 S cos ðα=2Þ 2 σhv 3 ð2Þ cos ðθRW Þ 0 dc;l ¼ kl @
1
1
1 σ 2l t RW ct ρ 1 þ l b b Aρp 2 v cos ðθi Þ 3 v cos ðθRW Þ cos ðθRW Þ cos ðθi Þ
1 1 cos ðθi Þ 6 3
ð3Þ
The subscript “h” is for high speed; “l” is for low speed; “RW” is for rear wall; “i” is for impact. S is the distance between the front bumper and the rear wall. α is the angle between the front bumper and the rear wall. θ is the obliquity. σ is the strength of the material. “kh” and “kl” are constants. If the front bumper is normal to the projectile
velocity vector and the intermediate bumper is inclined, θi ¼ 0 and α ¼ θRW . Eqs. (2) and (3) become: 1 2 1 2 cos ðθRW =2Þ 2 1 1 1 dc;h ¼ kh ρp 3 ρB 9 S2 ρRW t RW 3 σ 3h V i 3 23 cos ðθRW Þ 12 cos ðθRW =2Þ ¼ A1 ð4Þ 2 cos ðθRW Þ 3 1 4 2 1 dc;l ¼ k1 ρp 2 ðV i Þ 3 σ 2l t RW cos ðθRW Þ 3 1 þ cl t B ρB cos ðθRW Þ 3 4 1 ¼ A2 B1 cos ðθRW Þ 3 þB2 cos ðθRW Þ 3
ð5Þ
As the configuration and the impact condition are all fixed except θRW , the parameters A1, A2, B1, B2 are all positive constants. Therefore both dc;h and dc;l increases with θRW , so does the ballistic limit. In this paper the aluminum egg-box panel is used as the enhanced layer of the Whipple shield. The effectiveness and efficiency of the egg-box panel enhanced Whipple shield are evaluated by numerical simulations.
2. Numerical methods The simulation approach used in this paper is the finite element method (FEM) with the node-separation fracture technique [20–23]. In the standard FE mesh, the adjacent elements share their nodes. For example, in “standard mesh”, the elements E1–E4 share the node N5 (Fig. 2). In the node separation FE mesh the shared node is copied, and one copy of the node is assigned to each element which shares the original node. Define Ni as the ith node in the standard FE mesh. Define EH as the Hth element. Define the attach-to-element set of the Ni as: n o ð6Þ AEi ¼ EHj Ni A EHj One copy of Ni is assigned to each element in AEi. N ji is the jth copied node of Ni (Fig. 3 “node-coincident mesh”). Define the node set: n o ð7Þ Si ¼ Nji N1i ; N2i ⋯; NijAEi j All the Nji in Si have the same spatial position and velocity vector. At the beginning of the explicit integration
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Fig. 2. Stand FE mesh and node-coincident mesh.
Fig. 3. The FEM simulation output of triple plate shield.
iteration, a constraint is added to each Si. All the nodes in Si share the same degree of freedom. The node acceleration update implementation contains two steps: 1) calculating the acceleration of each node copy in the element it attaches to; 2) calculating the average acceleration as the shared value for the set: X 1 P ai ¼ ð8Þ U mHj aji jSi j U mH j j EHj A AEi
Ni A Si
where aji is the acceleration vector of Nji , which is calculate through the constitute equation and momentum equation of EHj . mHj is the mass of EHj . ai is the shared acceleration vector for the set Si . In the explicit iteration the constraint of the node set, the stress or strain of which meets the fracture criterion, is released and then a crack is generated in the mesh. Some of the finite elements would encounter severe distortion in the impact process. The distortion is inherently caused by material phase change [20]. These distorted elements are identified and eliminated from the simulation. Although the energy contained in the distorted elements is deleted at the same time and system energy loss occurs, from “pure solid” perspective the energy in hypervelocity impact process is not conservative because some of the materials would encounter phase change and
become gas or liquid and take away some of the energy. The phase changed material, especially the gas, does not move according to the solid mechanics/dynamics governing equations. Considering the gas material impacting with other solid material based on the solid momentum equation would lead to overestimation of the wall damage. Therefore the elimination of the phase changed elements and energy from the simulation is generally reasonable. In the typical fracture erosion technique of FE method the crack is generated by deleting a series of elements. The large deformed elements, that would encounter distortion, are also deleted. In contrast, in the node-separation technique the crack is generated by releasing node set constraint and only the distort elements are deleted. In this way much more elements are preserved. More discussion about the node-separation technique and element distortion treatment please see the reference [20]. Though compared to the SPH method, the node separation FE technique is not very widely used in high velocity impact simulation, our technique is built by secondary development in LS-dyna. This means that the basic FE algorithms, material constitutive model, and the node constraint technique are all from LS-dyna. We only extend the detailed implementation and performance of the node separation technique.
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Table 1 Johnson-Cook material model parameters. Parameters
Al 2024
Shear modulus (kPa) Yield stress (kPa) Harding constant (kPa) Harding exponent Strain rate constant Thermal softening exponent melting temperature (K)
2.76E þ07 2.65E þ05 4.26E þ05 0.34 0.015 1 775
This approach can simulate the debris cloud and the structure damage under hypervelocity impact with good accuracy and efficiency. A brief example of multi-shock simulation is given for verification of the numerical method and as the benchmark for the comparisons in the next chapters. The spherical projectile, the front bumper, the second bumper and the rear wall are all aluminum alloys. The diameter of the sphere is 7.9 mm; the impact velocity is 5.6 km/s. The thickness of both the front and second bumper is 1 mm. The spacing between the two bumpers and that between the second bumper and the rear wall is 50 mm. The thickness of the rear wall is 4.8 mm. The constitutive model we used in the simulations is Johnson-Cook model [24], and the equation of state is Gruneisen equation [25]. The material parameters used in the simulations are listed in Tables 1 and 2. The fracture criterion for controlling the separation of the node set is the “pressure threshold”. This criterion is found to be simple and effective in hypervelocity impact simulation [20]. The threshold of pressure is 0.012 MBar. Fig. 3 shows the node-separation FEM output of the Whipple shield enhanced with a flat panel. The materials of the projectile, the front bumper, the rear wall and the enhanced panel are all aluminum alloys. The diameter of the projectile is 7.9 mm. The impact velocity is 5.6 km/s. The thicknesses of the front bumper and the enhanced layer are both 1 mm, and that of the rear wall is 4.8 mm. The simulated diameter of the hole in the front bumper is 11.7 mm; the test result is 12.6 mm; the error is 7%. The simulated diameter of the enhanced panel is 32.8 mm; the test result is 37 mm, the error is 11%. The rear wall is not perforated in the simulation, so is the test result. The error is acceptable. More details of the node separation FEM technique and the comparison with SPH method can be found in the reference [20]. The node separation FEM technique results in higher accuracy and efficiency than the SPH method under close model resolution in the triple wall simulation. Based on the case above, we rotate the enhanced panel by 45° in order to evaluate the idea of inclined impact. Other configuration of this case is exactly the same as the first case. The simulation output is shown in Fig. 4a. The maximum crater depth in the rear wall is decreased by 80% (from 5.2 mm to 1.04 mm). However to cover the same projection area on the rear wall, the inclined panel needs more mass. For objective comparison of the horizontal panel and the inclined panel,
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the sheet thickness of the enhanced panel is decreased so that they cover the same area of the rear wall with the same mass, i.e. equal areal density. Other configuration of this case is exactly the same as the first case. The sheet thickness of the inclined panel is modified to
δ45 ¼ ð cos 451Þ2 δ0
ð9Þ
Fig. 4b shows the simulation result with the new sheet thickness. The current crater depth in the rear wall is 2.42 mm. The damage is larger than Fig. 4a, but is still 53% smaller than the benchmark case. We can infer the basic idea of changing the normal impact into the inclined impact is effective. Fig. 4c shows the velocity vectors of the fragments. After the debris cloud hit the enhanced panel, some of the debris cut through the panel, while the others change their flowing direction and move along the panel surface due to the resistance of the panel. The “drainage” effect can turn part of the vertical momentum to the lateral direction. The further penetration ability of the debris cloud is therefore reduced. Actually the thickness along the impact direction in the third case is the same as the first case. However the damage of the rear wall is still reduced by 53% relative to the first case. Therefore the “drainage” effect, i.e. the movement of the debris, which cannot be implied by the BLE formula, is also very important. However considering the space occupation, the way of rotating the whole enhanced panel will lead to large spatial penalty, and cannot be used in practice. The egg-box panel can change the local normal impact into the inclined impact and has a good control of the space occupation.
3. Egg-box panel modeling For objective comparison of the protective performance, the areal density of the egg-box panel should be set the same as the flat panel. Fig. 5 provides the geometrical relationship between all sides in the 1/4 “cell” of the egg-box panel. The “pyramid” in the egg-box panel is consisted of four equilateral triangles. The lengths in Fig. 5 satisfy: c ¼ 1=2 U a pffiffiffi d ¼ 3=2 U a pffiffiffi h ¼ 2=2 Ua e ¼ 1=2 U b pffiffiffi f ¼ 3=2 U b pffiffiffi g ¼ 2=2 Ub sin θ ¼ g=f ¼
pffiffiffiffiffiffiffiffi 2=3
ða bÞ=2 ¼ δ= sin θ
ð10Þ
We can infer: pffiffiffi b ¼ a 6δ
ð11Þ
Therefore given the cell side length a and the wall thickness of the panel δ as the inputs, all the other edges can be determined. The areal density of the 1/4 cell, i.e. the areal
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4. Hypervelocity impact simulation of egg-box panel enhanced Whipple shield
density of the egg-box panel, is: pffiffiffi pffiffiffi ρ ρ 3δ pffiffiffi 2 a 6 δ þ a2 þ a a 6 δ ρA ¼ V ¼ A A 12 þ
δ2 pffiffiffi
6a 3 δ
2
! ð12Þ
The areal density for the flat panel on the same projection area is: ρ ρ 1 2 a δ ρA ¼ V ¼ ð13Þ A A 4 where δ is the thickness of the horizontal flat panel. In order to make the areal density of both the egg-box panel and the flat panel the same, the wall thicknesses should satisfy: pffiffiffi pffiffiffi 2 pffiffiffi 3δ a 6 δ þ a2 þa a 6 δ 12 þ
δ2 pffiffiffi 2
1 6a 3 δ ¼ a2 δ 4
The egg-box panel enhanced Whipple shield with different parameters and configurations are simulated. The comparison and analysis will be made in the next chapters. Here the general descriptions of all the main cases are summarized. The basic cell size of egg-box panel is 10 mm; 3/5/20 mm cases are also simulated for comparison. Table 3 lists the configurations and parameters of the typical simulation cases. The three cases in Chapter 2 are also included in the table (No. 1–3). Case no. 2 and 3 use case no. 1 as the benchmark (as the # marked in the table); case no. 5–14 use case no. 4 as the benchmark (as the n marked in the table). All the information (impact condition and structure configuration) that is not specified is the same as the corresponding benchmark case. In case no. 4, the second panel in case no. 1 is replaced with the egg-box panel that has the equal areal density; other configuration
ð14Þ
Fig. 6 illustrates the constitution of one “unit” of the egg-box panel. The 1/4 cell is rotationally arrayed by 90° to form the whole pyramid, i.e. one cell. One unit is formed with two upward and two downward cells. The full eggbox panel can be formed by translationally arraying the unit along two mutually perpendicular directions (as shown in Fig. 1).
Table 2 Gruneisen equation of state parameters. Parameters
Al
Gruneisen coefficient 1.97 C1 (m/s) 5386 S1 1.339 Reference temperature (K) 300 Specific Heat (J/kg/K) 884
Fig. 5. Geometrical relationship between the edges in 1/4 cell.
Fig. 4. Simulation output of the 45° inclined stuffed panel cases. (a) The benchmark case with the enhanced panel inclined by 45°; (b) the thickness of the enhanced plate is decreased so that it has the same areal density as the benchmark; (c) velocity vectors of the case in (b).
X. Zhang et al. / Acta Astronautica 119 (2016) 48–59
is exactly the same as case no. 1. All the enhanced layers except case no. 2 have the same areal density as case no. 1. The edges of the three panels are set as the nonereflection boundary. We have tested the boundary conditions. Aiming at the hypervelocity impact ballistic limit simulation of the shield we found that the boundary condition does not influence the damage much. We think the reason is that the boundary condition mainly affects the elastic wave propagation but the perforation and damage of the panel is mainly affected by the plastic deformation and material failure. Fig. 7 shows the simulation images of the cases. The left image in each block presents the side view at 22 μs when the debris cloud is just touching the rear wall. The right image presents the residual of the enhanced panel (bottom view) at 60 μs when the whole impact, deformation and fragmentation process has been stable. The block of case no. 14 is an exception. The right image is the front bumper, because in that case the egg-box panel is used as the front bumper. Fig. 8 shows the crater in the rear wall of all the cases. The data of the crater depth can be found in Table 3. In cases no. 4–14 (egg-box cases), to keep the areal density constant the sheet thickness is calculated for each case. The “seen thickness” effect and the debris movement effect both exist in the cases. But generally the sheet thickness, and then the seen thickness, does not vary much from each other. Therefore the main effect that
Fig. 6. One unit of the egg-box panel (including 4 cells).
53
causes the different damages in the rear wall would be the debris movement restricted by the egg-box geometry. 4.1. Impact area The geometry of the egg-box panel is not isotropic, so the different area would have different protective performances. Three typical areas are chosen for comparison: the saddle (case no. 4), the peak (case no. 5) and the valley (case no. 6). The crater depth data are listed in Table 3. The smaller crater depth indicates that the rear wall gets less damaged and the shield has better performance. If the cell edge length is 10 mm, the case with the projectile aiming at the peak leads to the smallest damage in the rear wall (0.83 mm), while the saddle is the medium (0.50 mm), the valley is the largest (1.53 mm). All the performances are better than the enhanced layer of the flat panel (5.2 mm). The egg-box panel is generally better than the flat panel under the specified impact conditions and the structural configurations. Moreover the comparison indicates that the peak area can absorb most power of the debris cloud behind the front bumper and best protects the rear wall from the penetration. Fig. 9 illustrates the egg-box panel and the debris cloud behind that panel at 22 μs. The debris cloud is consisted of many projectile fragments and a few large egg-box panel fragments. The front bumper fragments can hardly be seen. The velocities of the egg-box panel fragments are not very large. The further damage to the rear wall is mainly determined by the front bumper fragments. Comparing to Fig. 3, we can find that the egg-box panel can absorb more incoming debris energy and generates less debris from itself than the flat panel. The different impact areas would lead to different movement tendencies when the debris cloud behind the front bumper contacts the egg-box panel. The moving direction of the debris is marked with the blue arrows in Fig. 9a/b/c. Since the cell of the egg-box panel has two types of upward and downward, the debris would flow along the downward pyramids and would be resisted by the upward pyramids. Therefore for the peak case, the debris diverge along the four directions; for the valley case,
Table 3 Summary list of the egg-box panel stuffed Whipple shield simulation cases. Case no.
Location of egg-cell plate
Cell size (mm)
Thickness (mm)
Additional configuration description
Crater depth in rear wall (mm)
1 2 3 4* 5 6 7 8 9 10 11 12 13 14
None None None Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Front bumper
None None None 10 10 10 3 5 20 20 10 10 10 10
1 1 0.5 0.568 0.568 0.568 0.566 0.564 0.572 0.572 0.568 0.568 0.286 0.568
Triple-plate Enhanced plate 45° inclined Enhanced plate 45° inclined Impact at saddle Impact at peak Impact at valley – – – Impact at peak Offset upward by 20 mm Offset downward by 20 mm Double layer egg-box plate
5.20 1.04 2.42 0.83 0.50 1.53 1.48 1.13 1.03 2.42 3.46 0.90 1.90 5.00
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Fig. 7. Simulated debris cloud at 22 μs and the residual of the egg-box panel (the mark on the top left of each block is the case number).
Fig. 8. Crater in the rear wall of all the cases.
the debris converge to the center; and for the saddle case, the debris diverge along two opposite directions and converge along the other two. From the side views we can see that because of the divergence of the debris in front of the egg-box panel in the peak case, fewer and more dispersed debris are generated behind the panel; and the specific impulse of the debris cloud is therefore smaller. The specific impulse is the local debris momentum divided by the projection area on the target plate. As a result the damage, which is mainly determined by the specific impulse, in the rear wall is smaller. Oppositely for the valley case the convergence
leads to more residual large fragments near the central axis, and that causes larger damage in the rear wall. Fig. 9g and h shows the 45° and 45° side views. From the perspective of Fig. 9g, the impact point is the peak and the profile of the debris cloud is similar to Fig. 9e; while from the perspective of Fig. 9h, the impact area is the valley and the profile of the debris cloud is similar to Fig. 9f. Therefore the performance of the saddle area is between the peak and the valley, so is the damage in the rear wall. Looking at the detail of Fig. 9e, a needle like large fragment that is consisted of the material of the egg-box
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Fig. 9. Debris cloud comparison of different impact area at 22 μs. (For interpretation of the reference to color in this figure, the reader is reffered to the web version of this article.)
panel (green) can be seen among the debris cloud. This needle like large fragment, which is found in all the peak cases, easily causes a severe damage in the rear wall. In case no. 5 the needle like fragment dose not play the decisive role, but this situation gets changed with the cell size. This issue will be further discussed in the Section 4.2. 4.2. Cell size The cell size is studied with the simulations and comparisons of case no. 4, 7, 8, 9, and 10. Fig. 10 shows the deformations of the egg-box panel with different cell size. Since the projectile and the front bumper is the same in the four cases, the debris clouds are exactly the same before they contact the egg-box panel. The different cell size leads to the different cut effect to the debris cloud. When the cell size is equal to 10 mm, the main contact area is one unit (four cells); when the cell size is equal to 20 mm, the main contact area is smaller than one unit; when the cell size is equal to 5 mm and 3 mm, the debris
clouds are spread into several units. In the bigger cell size cases the process that the debris cloud hit the inclined surface of the pyramid, is similar to the process in case no. 3 (Fig. 4b). The normal impact is turned into the inclined impact and the protective performance is promoted by the egg-box geometry. However when the cell size is small, especially is similar to that of the individual fragments, the effect of the egg-box geometry is not significant. If the cell size gets further smaller, the egg-box panel gets close to the flat panel. As the limit case if the cell size is equal to 0, the egg-box panel is exactly the same as the flat panel. Fig. 11 shows the curve of the crater depth in the rear wall versus the cell size. The crater depth decreases by 84% as the cell size increases from 0 to 10 mm. The crater depth does not vary much when the cell size is larger than 10 mm (slightly increase). With the former analysis of Fig. 10 we can see that when the cell size is large than 10 mm the main part of the debris cloud is restricted in one unit. So we can infer that for the better performance of
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Fig. 10. Deformation comparison of the egg-box panel of different cell size at 12 μs.
the egg-box panel, the cell size should be set large enough so that the main part of the debris cloud is restricted in one unit. The projectile diameter in all the cases is 7.9 mm (cell size/projectile diameter is equal to 10/7.9 is equal to 1.3), the impact velocity is 5.6 km/s. Since the expansion of the debris cloud behind the front bumper increases with the impact velocity, a larger cell size will be needed in higher velocity conditions. Therefore under the specified impact conditions and the structural configurations the cell size is suggested to be set to 41.3 times of the largest intended projectile diameter that the shield can withstand. Moreover when the cell size is equal to 20 mm, if impact at the peak area, the crater depth in the rear wall is 2.42 mm, that is larger than the depth of the saddle case.
This situation is different for cell size is equal to 10 mm. Fig. 12 illustrates the deformation sequence of the peak case (overlapped). The peak cell is squashed by the debris cloud with a high velocity and forms a sharp penetrator. The penetrator is made of the egg-box panel material. This process is very similar to the mechanism of explosively formed penetrator (EFP). The difference only lies in the source that generates the high pressure. Here the high pressure to the peak cell (called “liner” in the EFP issue) is formed by the impact of the debris cloud rather than the explosion. Therefore when the cell size is over large and the impact area is the peak, the penetrator would be generated. And this sharp penetrator would aggravate the damage to the rear wall. This effect is not significant in the small cell size cases (e.g. 3 mm and 5 mm), and is
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significant but not decisive in the medium cell size cases (e.g. 10 mm), but is decisive in the large cell size cases (e.g. 20 mm). Therefore in the design of egg-box panel, the cell size should not be over large. Along with the former analysis, under the specified impact conditions and the structural configurations the cell size is suggested to be 1.2–1.5 times of the maximum diameter of the intended projectile that the shield can withstand. 4.3. Egg-box panel axial offset The egg-box panel is translated upward/downward by 20 mm for comparison to find the best offset distance. From Table 3, the crater depth significantly increases when moving upward, while slightly increases when moving downward. For better understanding of this result, two aspects need to be analyzed: the specific impulse and the relative cell size of the egg-box panel. First, when the stable debris cloud is formed behind the front bumper the fragments approximately keep uniform linear motion. The debris cloud expands with a constant speed when moving toward the enhanced layer and the total momentum of the debris cloud is also constant. The smaller the distance between the front bumper and the
57
egg-box panel is, the smaller expansion of the debris cloud is, so is the smaller contact area of the debris cloud on the egg-box panel (Fig. 13). Since the total momentum of the debris cloud is nearly constant, the specific impulse that is the decisive factor to the damage increases with the decreasing of the contact area. Therefore translating the egg-box upward will increase the power of the debris cloud and will aggravate the damage. More analysis about the motion of the debris cloud and the specific impulse, please see the reference [26,27]. Second, according to the analysis in the former section, the protective performance of the egg-box panel depends on the relative cell size to the fragment size. Since the debris cloud behind the front bumper expands with moving downward, translating the egg-box panel upward results in the same effect of increasing the relative cell size to the incoming debris cloud size. According to Fig. 11, further increasing the cell size when cell size 410 mm does not significantly affect the damage in the rear wall, i.e. the protective performance of the egg-box panel does not change much. To sum up the two aspects, moving the egg-box panel upward will enhance the power of the debris cloud and the protective performance of the egg-box does not change much, therefore the damage in the rear wall will increase. Oppositely moving downward the egg-box panel will result in the decreasing of the specific impulse of the debris cloud. But at the same time the relative cell size to the contacted debris cloud also decreases and therefore the protective ability of the egg-box panel decreases (Fig. 11). The two aspects cancel each other to some extent. As a result the damage in the rear wall does not change much. The result shows that the egg-box panel is better to be set in the middle of the front bumper and the rear wall, or a little closer to the rear wall. 4.4. Extra discussion
Fig. 11. Curve of the crater depth in the rear wall versus the cell size.
Two extra configurations are also tested: the double layer egg-box panel and substituting the front bumper with the egg-box panel. Keeping the total areal density
Fig. 12. Formation of the sharp penetrator in the peak impact case with large cell size.
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5. Conclusion
Fig. 13. Expansion of the debris cloud and the contact area in the eggbox panel.
invariant, the egg-box is divided into two layers, each of which has half of the original areal density. Since the peak area and the valley area is the strongest and the weakest part respectively, the second layer egg-box panel is therefore turned upside down (Fig. 1). From the block no. 13 in Fig. 7, the central large fragment, which is consisted of the projectile material, still exists. The power of debris cloud can be measured with the specific impulse alone the initial impact velocity direction. The specific impulse depends on two main aspects: the mass and the velocity. The effect of the bumper is to decrease the specific impulse of the debris cloud. With the constant areal density, the larger body density results in the more concentrated mass distribution along the thickness direction. The bumper with larger body density can better break the large fragments and disperse the mass of the debris cloud. Oppositely the smaller body density results in the more dispersed mass distribution along the thickness direct. The bumper with smaller body density can better reduce the axial velocity of the debris cloud. The good design of the shield is the optimized combination of dispersing mass and reducing velocity. From the simulation output of case no. 13, the double layer egg-box panel does not effectively break the central large fragment. Then the mass becomes the decisive factor of the specific impulse of the debris cloud. Finally the damage in the rear wall is aggravated. To sum up, the single layer egg-box panel is better than the double layer one with the unified total areal density. Moreover, in the case no. 14, the flat enhanced panel is reserved and the front bumper is substituted with the eggbox panel. From the block no. 14 of Fig. 7, the distribution of the debris cloud is also very concentrated. This will cause the large damage in the rear wall. Comparing to the flat panel, the egg-box panel has smaller wall thickness but larger spatial expansion (total thickness) along the impact direction. The egg-box panel can be seen as decreasing the body density while keeping the areal density constant. Therefore the eggbox panel is better at reducing velocity rather than dispersing mass. The simulation output indicates that in this configuration the mass is the decisive factor of the debris cloud power, and therefore the damage of the rear wall is aggravated. To sum up, the egg-box panel may be not appropriate to be used as the front bumper and is better to be used as the enhanced layer.
The hypervelocity impact performance of the aluminum egg-box panel enhanced Whipple shield are investigated in this paper. The finite element models of the egg-box panel enhanced Whipple shield are built. Combined with the node separation technique, the hypervelocity impact process is simulated. Using the flat panel enhanced Whipple shield as the benchmark and keeping the areal density constant, the enhanced flat panel is substituted with the egg-box panel. Many configurations are simulated and compared to find the better parameters for the egg-box panel. Generally the eggbox panel has better performance than the flat panel. It can absorb more incoming debris energy and generates less debris from itself than the flat panel. The impact area analysis indicates that the peak area is strongest, the saddle area is the next and the valley area is the weakest. The cell size analysis shows that the performance of the egg-box panel is not linearly related to the cell size. Both the over large and over small cell size leads to the reduction of the protective performance. Moreover if the impact point is the peak and the cell size is too large, a sharp penetrator made of the egg-box panel material would be generated. The sharp penetrator significantly increases the damage in the rear wall. Secondly, the distance between the front bumper and the egg-box panel should be large enough, so that the debris cloud behind the front bumper can fully disperse. However if the egg-box panel is too close to the rear wall, the relative cell size to the debris cloud is decreased. As a result the performance of the egg-box panel would also decrease. Therefore the egg-box panel is suggested to be set in the middle of the front bumper and the rear wall, or a little closer to the rear wall. Finally it is not appropriate to use the egg-box panel as the front bumper, using it as the enhanced layer is better. With the specified total areal density, the single layer egg-box panel is better than the double layer egg-box panel. In this paper we present a primary analysis of the egg-box panel performance with numerical simulations. The egg-box panel can be fabricated by the stamping and forming to the aluminum flat panel, the cost of manufacture would not be significantly increased. The egg-box panel is appropriate to be the enhanced layer of the Whipple shield. In the future work we will do further simulations to verify our inferences and conclusions at more extensive impact conditions and structure configurations. Other simulation method, like SPH, will be introduced for comparisons. The experimental test is also planned for verification.
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[5] S. Ryan, E.L. Christiansen, A ballistic limit analysis programme for shielding against micrometeoroids and orbital debris, Acta Astronaut 69 (5–6) (2011) 245–257. [6] W.P. Schonberg, The development of ballistic limit equations for dual-wall spacecraft shielding: a concise history and suggestions for future development, Proceedings of the 49th AIAA/ASME/ASCE/AHS/ ASC Structures Structural Dynamics and Materials Conference, Schaumburg, 2008. [7] Committee on International Space Station Meteoroid/Debris Risk Management, Protecting the Space Station from Meteoroids and Orbital Debris, National Academies Press, 1997, ISBN-10: 0309056306 ISBN-13: 978-0309056304, http://www.amazon.com/ Protecting-Station-Meteoroids-Orbital-compass/dp/0309056306. [8] S. Ryan, E.L. Christiansen, Hypervelocity impact testing of advanced materials and structures for micrometeoroid and orbital debris shielding, Acta Astronaut 83 (2013) 216–231. [9] S. Ryan, T. Hedman, E.L. Christiansen, Honeycomb vs. foam: evaluating potential upgrades to ISS module shielding, Acta Astronaut 67 (7–8) (2010) 818–825. [10] N.N. Myagkov, T.A. Shumikhin, L.N. Bezrukov, Experimental and numerical study of peculiarities at high-velocity interaction between a projectile and discrete bumpers, Int. J. Impact Eng. 37 (9) (2010) 980–994. [11] M. Higashide, M. Tanaka, Y. Akahoshi, et al., Hypervelocity impact tests against metallic meshes, Int. J. Impact Eng. 33 (1–12) (2006) 335–342. [12] W.P. Schonberg, F. Schäfer, R. Putzar, Hypervelocity impact response of honeycomb sandwich panels, Acta Astronaut 66 (3–4) (2010) 455–466. [13] M. Lambert, F.K. Schäfer, T. Geyer, Impact damage on sandwich panels and multi-layer insulation, Int. J. Impact Eng. 26 (1–10) (2001) 369–380. [14] M. Rudolph, F. Schafer, R. Destefanis, et al., Fragmentation of hypervelocity aluminum projectiles on fabrics, Acta Astronaut 76 (2012) 42–50. [15] A. Francesconi, C. Giacomuzzo, S. Kibe, et al., Effects of high-speed impacts on CFRP plates for space applications, Adv. Space Res. 50 (5) (2012) 539–548.
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[16] L. Hamill, S. Roberts, M. Davidson, et al., Hypervelocity impact phenomenon in bulk metallic glasses and composites**, Adv. Eng. Mater 16 (1) (2014) 85–93. [17] M. Davidson, S. Roberts, G. Castro, et al., Investigating amorphous metal composite architectures as spacecraft shielding, Adv. Eng. Mater 15 (1–2) (2013) 27–33. [18] K.B. Hayashida, J.H. Robinson, Single Wall Penetration Equations, , 1991 NASA TM-103565 (NASA Marshall Space Flight Center). [19] W.E. Bohl, J.E. Miller, E.L. Christiansen, et al., HVI ballistic performance characterization of non-parallel Walls, Procedia Eng. 58 (2013) 21–30. [20] X.T. Zhang, X.G. Li, T. Liu, et al., Element fracture technique of FEM for hypervelocity impact simulation, Adv. Space Res. 55 (2015) 2293–2304. [21] M. Lee, E.Y. Kim, Y.H. Yoo, Simulation of high speed impact into ceramic composite systems using cohesive-law fracture model, Int. J. Impact Eng. 35 (12) (2008) 1636–1641. [22] D.R. Ambur, N. Jaunky, R.E. Lawson, et al., Numerical simulations for high-energy impact of thin plates, Int. J. Impact Eng. 25 (7) (2001) 683–702. [23] N.F. Knight, N. Jaunky, R.E. Lawson, et al., Penetration simulation for uncontained engine debris impact on fuselage-like panels using LSDYNA, Finite Elem. Anal. Des. 36 (2) (2000) 99–133. [24] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, Proceedings of the 7th International Symposium on Ballistics, The Hague, 1983. [25] Selected Hugoniots, LA-4167-MS, Los Alamos Report, 1969. [26] X. Zhang, G. Jia, H. Huang, A fast numerical approach for Whipple shield ballistic limit analysis, Acta Astronaut 93 (2014) (2014) 112–120. [27] X.T. Zhang, G.H. Jia, H. Huang, An approach for constituting double/ multi wall BLE by single wall BLE of spacecraft shield, Int. J. Impact Eng. 69 (2014) (2014) 114–121.
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Hypervelocity impact performance of aluminum egg-box panel enhanced Whipple shield$ Xiaotian Zhang n, Tao Liu, Xiaogang Li, Guanghui Jia School of Astronautics, Beihang University, Beijing 100191, China
a r t i c l e in f o
abstract
Article history: Received 10 March 2015 Received in revised form 9 October 2015 Accepted 22 October 2015 Available online 10 November 2015
The enhanced Whipple shield is widely used on the space stations against space debris hypervelocity impact. Numerical investigation is conducted in this paper to test the concept of using the egg-box panel as the enhanced layer. Taking the flat panel as the benchmark and then the enhanced flat panel is substituted with the egg-box panel. For objective comparison all the egg-box panels have the same areal density as the benchmark flat panel. The performance is tested from the aspects of impact area, cell size and the axial offset, etc. The egg-box panel absorbs more incoming debris energy and generates less large fragments from itself. It can be fabricated by the stamping and forming of the aluminum flat sheet, the cost of manufacture would not be significantly increased. The egg-box panel has better performance than the flat panel and is appropriate to be the enhanced layer of the Whipple shield. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Space debris Spacecraft shielding Hypervelocity impact Finite element method Numerical simulation
1. Introduction Space debris is the increasing threatens to manned spacecraft. Protective shield must be designed and installed over the key parts of the spacecraft to withstand the hypervelocity impact of space debris. Whipple shield [1–6] is widely used on spacecraft since 1960s. An aluminum bumper is placed over a distance of the wall of the spacecraft. The thickness of the bumper and the distance between the bumper and the rear wall are carefully designed so that the debris can be fully fragmented after perforating the bumper. The damage to the rear wall can be therefore reduced. Typical Whipple shield can withstand 1–5 mm space debris. To further enhance ☆ Supported by the National Natural Science Foundation of China (11502010) and the Fundamental Research Funds for the Central Universities of China (YWF-14-YHXY-022). n Corresponding author. E-mail addresses: [email protected] (X. Zhang), [email protected] (T. Liu), [email protected] (X. Li), [email protected] (G. Jia).
http://dx.doi.org/10.1016/j.actaastro.2015.10.013 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.
the performance of the shield, stuffed Whipple shield is developed. A stuffed layer is inserted between the front bumper and the rear wall. The stuffed Whipple shield is applied on the International Space Station [7]. The stuffed layer is made of Nextel/Kevlar material. The protective ability is raised upto 10 mm with stuffed Whipple shield. The stuffed layer is widely investigated in recent years. Various materials are tested under hypervelocity impact for performance comparison. For example: Aluminum foam/Aluminum foam sandwich panel [8,9], aluminum net [10,11], honeycomb sandwich panel [12,13], fiber laminates [14,15], amorphous alloy [16,17], etc. The new types of “stuffed” Whipple shield are general called enhanced Whipple shield. These panels can be basically categorized into two classes: the new topology of aluminum (aluminum foam, aluminum net) and the new materials (fiber laminates, amorphous alloy). The honeycomb sandwich panel, which takes the carbon fiber reinforced plastic as the face sheet, can be seen as the combination of the concepts of new topology (honeycomb) and new material (face sheet).
X. Zhang et al. / Acta Astronautica 119 (2016) 48–59
49
Fig. 1. Geometries of the single and double layer egg-box panels.
Davidson [17] tested the idea of fabricating the amorphous alloy sheet into the “egg-box” panel with a semisolid forging technique (Fig. 1). The individual single-layer and multi-layer egg-box panels are used in the hypervelocity tests and the results are compared with the honeycomb sandwich panel. Using the egg-box geometry the normal impact of the fragments on the flat panel can be turned into the inclined impact. According to the JSC single wall ballistic limit equation [18] (BLE): 0
0:5 118 19
0:25 ρb
Bt BH dc ¼ @ b k K v
ρp
cos θ C
C 23 A
ð1Þ
where dc is the critical perforation diameter; t b is the bumper thickness; BH is the material Brinell hardness; ρb and ρp are the material density of the bumper and the projectile respectively; v is the impact velocity; θ is the incidence; C is the sound speed; k and K are constants. The critical perforation diameter of the projectile dc is proportional to the ð cos θÞ 2=3 . Therefore with the same sheet thickness, the larger inclined angle θ results in the larger critical diameter, i.e. the better protective performance. Since the egg-box geometry can change the local impact incidence to about 45°, the ballistic limit can be promoted. When refer to using the egg-box panel as the enhanced layer of the Whipple shield, the non-parallel BLE [19] is a good support to our hypothesis. Considering the obliquity of the walls, the Whipple shield BLE is as follows: 2 1 ρRW t RW 3 13 2 1 1 dc;h ¼ kh ρp 3 ρb 9 S cos ðα=2Þ 2 σhv 3 ð2Þ cos ðθRW Þ 0 dc;l ¼ kl @
1
1
1 σ 2l t RW ct ρ 1 þ l b b Aρp 2 v cos ðθi Þ 3 v cos ðθRW Þ cos ðθRW Þ cos ðθi Þ
1 1 cos ðθi Þ 6 3
ð3Þ
The subscript “h” is for high speed; “l” is for low speed; “RW” is for rear wall; “i” is for impact. S is the distance between the front bumper and the rear wall. α is the angle between the front bumper and the rear wall. θ is the obliquity. σ is the strength of the material. “kh” and “kl” are constants. If the front bumper is normal to the projectile
velocity vector and the intermediate bumper is inclined, θi ¼ 0 and α ¼ θRW . Eqs. (2) and (3) become: 1 2 1 2 cos ðθRW =2Þ 2 1 1 1 dc;h ¼ kh ρp 3 ρB 9 S2 ρRW t RW 3 σ 3h V i 3 23 cos ðθRW Þ 12 cos ðθRW =2Þ ¼ A1 ð4Þ 2 cos ðθRW Þ 3 1 4 2 1 dc;l ¼ k1 ρp 2 ðV i Þ 3 σ 2l t RW cos ðθRW Þ 3 1 þ cl t B ρB cos ðθRW Þ 3 4 1 ¼ A2 B1 cos ðθRW Þ 3 þB2 cos ðθRW Þ 3
ð5Þ
As the configuration and the impact condition are all fixed except θRW , the parameters A1, A2, B1, B2 are all positive constants. Therefore both dc;h and dc;l increases with θRW , so does the ballistic limit. In this paper the aluminum egg-box panel is used as the enhanced layer of the Whipple shield. The effectiveness and efficiency of the egg-box panel enhanced Whipple shield are evaluated by numerical simulations.
2. Numerical methods The simulation approach used in this paper is the finite element method (FEM) with the node-separation fracture technique [20–23]. In the standard FE mesh, the adjacent elements share their nodes. For example, in “standard mesh”, the elements E1–E4 share the node N5 (Fig. 2). In the node separation FE mesh the shared node is copied, and one copy of the node is assigned to each element which shares the original node. Define Ni as the ith node in the standard FE mesh. Define EH as the Hth element. Define the attach-to-element set of the Ni as: n o ð6Þ AEi ¼ EHj Ni A EHj One copy of Ni is assigned to each element in AEi. N ji is the jth copied node of Ni (Fig. 3 “node-coincident mesh”). Define the node set: n o ð7Þ Si ¼ Nji N1i ; N2i ⋯; NijAEi j All the Nji in Si have the same spatial position and velocity vector. At the beginning of the explicit integration
50
X. Zhang et al. / Acta Astronautica 119 (2016) 48–59
Fig. 2. Stand FE mesh and node-coincident mesh.
Fig. 3. The FEM simulation output of triple plate shield.
iteration, a constraint is added to each Si. All the nodes in Si share the same degree of freedom. The node acceleration update implementation contains two steps: 1) calculating the acceleration of each node copy in the element it attaches to; 2) calculating the average acceleration as the shared value for the set: X 1 P ai ¼ ð8Þ U mHj aji jSi j U mH j j EHj A AEi
Ni A Si
where aji is the acceleration vector of Nji , which is calculate through the constitute equation and momentum equation of EHj . mHj is the mass of EHj . ai is the shared acceleration vector for the set Si . In the explicit iteration the constraint of the node set, the stress or strain of which meets the fracture criterion, is released and then a crack is generated in the mesh. Some of the finite elements would encounter severe distortion in the impact process. The distortion is inherently caused by material phase change [20]. These distorted elements are identified and eliminated from the simulation. Although the energy contained in the distorted elements is deleted at the same time and system energy loss occurs, from “pure solid” perspective the energy in hypervelocity impact process is not conservative because some of the materials would encounter phase change and
become gas or liquid and take away some of the energy. The phase changed material, especially the gas, does not move according to the solid mechanics/dynamics governing equations. Considering the gas material impacting with other solid material based on the solid momentum equation would lead to overestimation of the wall damage. Therefore the elimination of the phase changed elements and energy from the simulation is generally reasonable. In the typical fracture erosion technique of FE method the crack is generated by deleting a series of elements. The large deformed elements, that would encounter distortion, are also deleted. In contrast, in the node-separation technique the crack is generated by releasing node set constraint and only the distort elements are deleted. In this way much more elements are preserved. More discussion about the node-separation technique and element distortion treatment please see the reference [20]. Though compared to the SPH method, the node separation FE technique is not very widely used in high velocity impact simulation, our technique is built by secondary development in LS-dyna. This means that the basic FE algorithms, material constitutive model, and the node constraint technique are all from LS-dyna. We only extend the detailed implementation and performance of the node separation technique.
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Table 1 Johnson-Cook material model parameters. Parameters
Al 2024
Shear modulus (kPa) Yield stress (kPa) Harding constant (kPa) Harding exponent Strain rate constant Thermal softening exponent melting temperature (K)
2.76E þ07 2.65E þ05 4.26E þ05 0.34 0.015 1 775
This approach can simulate the debris cloud and the structure damage under hypervelocity impact with good accuracy and efficiency. A brief example of multi-shock simulation is given for verification of the numerical method and as the benchmark for the comparisons in the next chapters. The spherical projectile, the front bumper, the second bumper and the rear wall are all aluminum alloys. The diameter of the sphere is 7.9 mm; the impact velocity is 5.6 km/s. The thickness of both the front and second bumper is 1 mm. The spacing between the two bumpers and that between the second bumper and the rear wall is 50 mm. The thickness of the rear wall is 4.8 mm. The constitutive model we used in the simulations is Johnson-Cook model [24], and the equation of state is Gruneisen equation [25]. The material parameters used in the simulations are listed in Tables 1 and 2. The fracture criterion for controlling the separation of the node set is the “pressure threshold”. This criterion is found to be simple and effective in hypervelocity impact simulation [20]. The threshold of pressure is 0.012 MBar. Fig. 3 shows the node-separation FEM output of the Whipple shield enhanced with a flat panel. The materials of the projectile, the front bumper, the rear wall and the enhanced panel are all aluminum alloys. The diameter of the projectile is 7.9 mm. The impact velocity is 5.6 km/s. The thicknesses of the front bumper and the enhanced layer are both 1 mm, and that of the rear wall is 4.8 mm. The simulated diameter of the hole in the front bumper is 11.7 mm; the test result is 12.6 mm; the error is 7%. The simulated diameter of the enhanced panel is 32.8 mm; the test result is 37 mm, the error is 11%. The rear wall is not perforated in the simulation, so is the test result. The error is acceptable. More details of the node separation FEM technique and the comparison with SPH method can be found in the reference [20]. The node separation FEM technique results in higher accuracy and efficiency than the SPH method under close model resolution in the triple wall simulation. Based on the case above, we rotate the enhanced panel by 45° in order to evaluate the idea of inclined impact. Other configuration of this case is exactly the same as the first case. The simulation output is shown in Fig. 4a. The maximum crater depth in the rear wall is decreased by 80% (from 5.2 mm to 1.04 mm). However to cover the same projection area on the rear wall, the inclined panel needs more mass. For objective comparison of the horizontal panel and the inclined panel,
51
the sheet thickness of the enhanced panel is decreased so that they cover the same area of the rear wall with the same mass, i.e. equal areal density. Other configuration of this case is exactly the same as the first case. The sheet thickness of the inclined panel is modified to
δ45 ¼ ð cos 451Þ2 δ0
ð9Þ
Fig. 4b shows the simulation result with the new sheet thickness. The current crater depth in the rear wall is 2.42 mm. The damage is larger than Fig. 4a, but is still 53% smaller than the benchmark case. We can infer the basic idea of changing the normal impact into the inclined impact is effective. Fig. 4c shows the velocity vectors of the fragments. After the debris cloud hit the enhanced panel, some of the debris cut through the panel, while the others change their flowing direction and move along the panel surface due to the resistance of the panel. The “drainage” effect can turn part of the vertical momentum to the lateral direction. The further penetration ability of the debris cloud is therefore reduced. Actually the thickness along the impact direction in the third case is the same as the first case. However the damage of the rear wall is still reduced by 53% relative to the first case. Therefore the “drainage” effect, i.e. the movement of the debris, which cannot be implied by the BLE formula, is also very important. However considering the space occupation, the way of rotating the whole enhanced panel will lead to large spatial penalty, and cannot be used in practice. The egg-box panel can change the local normal impact into the inclined impact and has a good control of the space occupation.
3. Egg-box panel modeling For objective comparison of the protective performance, the areal density of the egg-box panel should be set the same as the flat panel. Fig. 5 provides the geometrical relationship between all sides in the 1/4 “cell” of the egg-box panel. The “pyramid” in the egg-box panel is consisted of four equilateral triangles. The lengths in Fig. 5 satisfy: c ¼ 1=2 U a pffiffiffi d ¼ 3=2 U a pffiffiffi h ¼ 2=2 Ua e ¼ 1=2 U b pffiffiffi f ¼ 3=2 U b pffiffiffi g ¼ 2=2 Ub sin θ ¼ g=f ¼
pffiffiffiffiffiffiffiffi 2=3
ða bÞ=2 ¼ δ= sin θ
ð10Þ
We can infer: pffiffiffi b ¼ a 6δ
ð11Þ
Therefore given the cell side length a and the wall thickness of the panel δ as the inputs, all the other edges can be determined. The areal density of the 1/4 cell, i.e. the areal
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X. Zhang et al. / Acta Astronautica 119 (2016) 48–59
4. Hypervelocity impact simulation of egg-box panel enhanced Whipple shield
density of the egg-box panel, is: pffiffiffi pffiffiffi ρ ρ 3δ pffiffiffi 2 a 6 δ þ a2 þ a a 6 δ ρA ¼ V ¼ A A 12 þ
δ2 pffiffiffi
6a 3 δ
2
! ð12Þ
The areal density for the flat panel on the same projection area is: ρ ρ 1 2 a δ ρA ¼ V ¼ ð13Þ A A 4 where δ is the thickness of the horizontal flat panel. In order to make the areal density of both the egg-box panel and the flat panel the same, the wall thicknesses should satisfy: pffiffiffi pffiffiffi 2 pffiffiffi 3δ a 6 δ þ a2 þa a 6 δ 12 þ
δ2 pffiffiffi 2
1 6a 3 δ ¼ a2 δ 4
The egg-box panel enhanced Whipple shield with different parameters and configurations are simulated. The comparison and analysis will be made in the next chapters. Here the general descriptions of all the main cases are summarized. The basic cell size of egg-box panel is 10 mm; 3/5/20 mm cases are also simulated for comparison. Table 3 lists the configurations and parameters of the typical simulation cases. The three cases in Chapter 2 are also included in the table (No. 1–3). Case no. 2 and 3 use case no. 1 as the benchmark (as the # marked in the table); case no. 5–14 use case no. 4 as the benchmark (as the n marked in the table). All the information (impact condition and structure configuration) that is not specified is the same as the corresponding benchmark case. In case no. 4, the second panel in case no. 1 is replaced with the egg-box panel that has the equal areal density; other configuration
ð14Þ
Fig. 6 illustrates the constitution of one “unit” of the egg-box panel. The 1/4 cell is rotationally arrayed by 90° to form the whole pyramid, i.e. one cell. One unit is formed with two upward and two downward cells. The full eggbox panel can be formed by translationally arraying the unit along two mutually perpendicular directions (as shown in Fig. 1).
Table 2 Gruneisen equation of state parameters. Parameters
Al
Gruneisen coefficient 1.97 C1 (m/s) 5386 S1 1.339 Reference temperature (K) 300 Specific Heat (J/kg/K) 884
Fig. 5. Geometrical relationship between the edges in 1/4 cell.
Fig. 4. Simulation output of the 45° inclined stuffed panel cases. (a) The benchmark case with the enhanced panel inclined by 45°; (b) the thickness of the enhanced plate is decreased so that it has the same areal density as the benchmark; (c) velocity vectors of the case in (b).
X. Zhang et al. / Acta Astronautica 119 (2016) 48–59
is exactly the same as case no. 1. All the enhanced layers except case no. 2 have the same areal density as case no. 1. The edges of the three panels are set as the nonereflection boundary. We have tested the boundary conditions. Aiming at the hypervelocity impact ballistic limit simulation of the shield we found that the boundary condition does not influence the damage much. We think the reason is that the boundary condition mainly affects the elastic wave propagation but the perforation and damage of the panel is mainly affected by the plastic deformation and material failure. Fig. 7 shows the simulation images of the cases. The left image in each block presents the side view at 22 μs when the debris cloud is just touching the rear wall. The right image presents the residual of the enhanced panel (bottom view) at 60 μs when the whole impact, deformation and fragmentation process has been stable. The block of case no. 14 is an exception. The right image is the front bumper, because in that case the egg-box panel is used as the front bumper. Fig. 8 shows the crater in the rear wall of all the cases. The data of the crater depth can be found in Table 3. In cases no. 4–14 (egg-box cases), to keep the areal density constant the sheet thickness is calculated for each case. The “seen thickness” effect and the debris movement effect both exist in the cases. But generally the sheet thickness, and then the seen thickness, does not vary much from each other. Therefore the main effect that
Fig. 6. One unit of the egg-box panel (including 4 cells).
53
causes the different damages in the rear wall would be the debris movement restricted by the egg-box geometry. 4.1. Impact area The geometry of the egg-box panel is not isotropic, so the different area would have different protective performances. Three typical areas are chosen for comparison: the saddle (case no. 4), the peak (case no. 5) and the valley (case no. 6). The crater depth data are listed in Table 3. The smaller crater depth indicates that the rear wall gets less damaged and the shield has better performance. If the cell edge length is 10 mm, the case with the projectile aiming at the peak leads to the smallest damage in the rear wall (0.83 mm), while the saddle is the medium (0.50 mm), the valley is the largest (1.53 mm). All the performances are better than the enhanced layer of the flat panel (5.2 mm). The egg-box panel is generally better than the flat panel under the specified impact conditions and the structural configurations. Moreover the comparison indicates that the peak area can absorb most power of the debris cloud behind the front bumper and best protects the rear wall from the penetration. Fig. 9 illustrates the egg-box panel and the debris cloud behind that panel at 22 μs. The debris cloud is consisted of many projectile fragments and a few large egg-box panel fragments. The front bumper fragments can hardly be seen. The velocities of the egg-box panel fragments are not very large. The further damage to the rear wall is mainly determined by the front bumper fragments. Comparing to Fig. 3, we can find that the egg-box panel can absorb more incoming debris energy and generates less debris from itself than the flat panel. The different impact areas would lead to different movement tendencies when the debris cloud behind the front bumper contacts the egg-box panel. The moving direction of the debris is marked with the blue arrows in Fig. 9a/b/c. Since the cell of the egg-box panel has two types of upward and downward, the debris would flow along the downward pyramids and would be resisted by the upward pyramids. Therefore for the peak case, the debris diverge along the four directions; for the valley case,
Table 3 Summary list of the egg-box panel stuffed Whipple shield simulation cases. Case no.
Location of egg-cell plate
Cell size (mm)
Thickness (mm)
Additional configuration description
Crater depth in rear wall (mm)
1 2 3 4* 5 6 7 8 9 10 11 12 13 14
None None None Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Enhanced layer Front bumper
None None None 10 10 10 3 5 20 20 10 10 10 10
1 1 0.5 0.568 0.568 0.568 0.566 0.564 0.572 0.572 0.568 0.568 0.286 0.568
Triple-plate Enhanced plate 45° inclined Enhanced plate 45° inclined Impact at saddle Impact at peak Impact at valley – – – Impact at peak Offset upward by 20 mm Offset downward by 20 mm Double layer egg-box plate
5.20 1.04 2.42 0.83 0.50 1.53 1.48 1.13 1.03 2.42 3.46 0.90 1.90 5.00
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Fig. 7. Simulated debris cloud at 22 μs and the residual of the egg-box panel (the mark on the top left of each block is the case number).
Fig. 8. Crater in the rear wall of all the cases.
the debris converge to the center; and for the saddle case, the debris diverge along two opposite directions and converge along the other two. From the side views we can see that because of the divergence of the debris in front of the egg-box panel in the peak case, fewer and more dispersed debris are generated behind the panel; and the specific impulse of the debris cloud is therefore smaller. The specific impulse is the local debris momentum divided by the projection area on the target plate. As a result the damage, which is mainly determined by the specific impulse, in the rear wall is smaller. Oppositely for the valley case the convergence
leads to more residual large fragments near the central axis, and that causes larger damage in the rear wall. Fig. 9g and h shows the 45° and 45° side views. From the perspective of Fig. 9g, the impact point is the peak and the profile of the debris cloud is similar to Fig. 9e; while from the perspective of Fig. 9h, the impact area is the valley and the profile of the debris cloud is similar to Fig. 9f. Therefore the performance of the saddle area is between the peak and the valley, so is the damage in the rear wall. Looking at the detail of Fig. 9e, a needle like large fragment that is consisted of the material of the egg-box
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Fig. 9. Debris cloud comparison of different impact area at 22 μs. (For interpretation of the reference to color in this figure, the reader is reffered to the web version of this article.)
panel (green) can be seen among the debris cloud. This needle like large fragment, which is found in all the peak cases, easily causes a severe damage in the rear wall. In case no. 5 the needle like fragment dose not play the decisive role, but this situation gets changed with the cell size. This issue will be further discussed in the Section 4.2. 4.2. Cell size The cell size is studied with the simulations and comparisons of case no. 4, 7, 8, 9, and 10. Fig. 10 shows the deformations of the egg-box panel with different cell size. Since the projectile and the front bumper is the same in the four cases, the debris clouds are exactly the same before they contact the egg-box panel. The different cell size leads to the different cut effect to the debris cloud. When the cell size is equal to 10 mm, the main contact area is one unit (four cells); when the cell size is equal to 20 mm, the main contact area is smaller than one unit; when the cell size is equal to 5 mm and 3 mm, the debris
clouds are spread into several units. In the bigger cell size cases the process that the debris cloud hit the inclined surface of the pyramid, is similar to the process in case no. 3 (Fig. 4b). The normal impact is turned into the inclined impact and the protective performance is promoted by the egg-box geometry. However when the cell size is small, especially is similar to that of the individual fragments, the effect of the egg-box geometry is not significant. If the cell size gets further smaller, the egg-box panel gets close to the flat panel. As the limit case if the cell size is equal to 0, the egg-box panel is exactly the same as the flat panel. Fig. 11 shows the curve of the crater depth in the rear wall versus the cell size. The crater depth decreases by 84% as the cell size increases from 0 to 10 mm. The crater depth does not vary much when the cell size is larger than 10 mm (slightly increase). With the former analysis of Fig. 10 we can see that when the cell size is large than 10 mm the main part of the debris cloud is restricted in one unit. So we can infer that for the better performance of
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Fig. 10. Deformation comparison of the egg-box panel of different cell size at 12 μs.
the egg-box panel, the cell size should be set large enough so that the main part of the debris cloud is restricted in one unit. The projectile diameter in all the cases is 7.9 mm (cell size/projectile diameter is equal to 10/7.9 is equal to 1.3), the impact velocity is 5.6 km/s. Since the expansion of the debris cloud behind the front bumper increases with the impact velocity, a larger cell size will be needed in higher velocity conditions. Therefore under the specified impact conditions and the structural configurations the cell size is suggested to be set to 41.3 times of the largest intended projectile diameter that the shield can withstand. Moreover when the cell size is equal to 20 mm, if impact at the peak area, the crater depth in the rear wall is 2.42 mm, that is larger than the depth of the saddle case.
This situation is different for cell size is equal to 10 mm. Fig. 12 illustrates the deformation sequence of the peak case (overlapped). The peak cell is squashed by the debris cloud with a high velocity and forms a sharp penetrator. The penetrator is made of the egg-box panel material. This process is very similar to the mechanism of explosively formed penetrator (EFP). The difference only lies in the source that generates the high pressure. Here the high pressure to the peak cell (called “liner” in the EFP issue) is formed by the impact of the debris cloud rather than the explosion. Therefore when the cell size is over large and the impact area is the peak, the penetrator would be generated. And this sharp penetrator would aggravate the damage to the rear wall. This effect is not significant in the small cell size cases (e.g. 3 mm and 5 mm), and is
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significant but not decisive in the medium cell size cases (e.g. 10 mm), but is decisive in the large cell size cases (e.g. 20 mm). Therefore in the design of egg-box panel, the cell size should not be over large. Along with the former analysis, under the specified impact conditions and the structural configurations the cell size is suggested to be 1.2–1.5 times of the maximum diameter of the intended projectile that the shield can withstand. 4.3. Egg-box panel axial offset The egg-box panel is translated upward/downward by 20 mm for comparison to find the best offset distance. From Table 3, the crater depth significantly increases when moving upward, while slightly increases when moving downward. For better understanding of this result, two aspects need to be analyzed: the specific impulse and the relative cell size of the egg-box panel. First, when the stable debris cloud is formed behind the front bumper the fragments approximately keep uniform linear motion. The debris cloud expands with a constant speed when moving toward the enhanced layer and the total momentum of the debris cloud is also constant. The smaller the distance between the front bumper and the
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egg-box panel is, the smaller expansion of the debris cloud is, so is the smaller contact area of the debris cloud on the egg-box panel (Fig. 13). Since the total momentum of the debris cloud is nearly constant, the specific impulse that is the decisive factor to the damage increases with the decreasing of the contact area. Therefore translating the egg-box upward will increase the power of the debris cloud and will aggravate the damage. More analysis about the motion of the debris cloud and the specific impulse, please see the reference [26,27]. Second, according to the analysis in the former section, the protective performance of the egg-box panel depends on the relative cell size to the fragment size. Since the debris cloud behind the front bumper expands with moving downward, translating the egg-box panel upward results in the same effect of increasing the relative cell size to the incoming debris cloud size. According to Fig. 11, further increasing the cell size when cell size 410 mm does not significantly affect the damage in the rear wall, i.e. the protective performance of the egg-box panel does not change much. To sum up the two aspects, moving the egg-box panel upward will enhance the power of the debris cloud and the protective performance of the egg-box does not change much, therefore the damage in the rear wall will increase. Oppositely moving downward the egg-box panel will result in the decreasing of the specific impulse of the debris cloud. But at the same time the relative cell size to the contacted debris cloud also decreases and therefore the protective ability of the egg-box panel decreases (Fig. 11). The two aspects cancel each other to some extent. As a result the damage in the rear wall does not change much. The result shows that the egg-box panel is better to be set in the middle of the front bumper and the rear wall, or a little closer to the rear wall. 4.4. Extra discussion
Fig. 11. Curve of the crater depth in the rear wall versus the cell size.
Two extra configurations are also tested: the double layer egg-box panel and substituting the front bumper with the egg-box panel. Keeping the total areal density
Fig. 12. Formation of the sharp penetrator in the peak impact case with large cell size.
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5. Conclusion
Fig. 13. Expansion of the debris cloud and the contact area in the eggbox panel.
invariant, the egg-box is divided into two layers, each of which has half of the original areal density. Since the peak area and the valley area is the strongest and the weakest part respectively, the second layer egg-box panel is therefore turned upside down (Fig. 1). From the block no. 13 in Fig. 7, the central large fragment, which is consisted of the projectile material, still exists. The power of debris cloud can be measured with the specific impulse alone the initial impact velocity direction. The specific impulse depends on two main aspects: the mass and the velocity. The effect of the bumper is to decrease the specific impulse of the debris cloud. With the constant areal density, the larger body density results in the more concentrated mass distribution along the thickness direction. The bumper with larger body density can better break the large fragments and disperse the mass of the debris cloud. Oppositely the smaller body density results in the more dispersed mass distribution along the thickness direct. The bumper with smaller body density can better reduce the axial velocity of the debris cloud. The good design of the shield is the optimized combination of dispersing mass and reducing velocity. From the simulation output of case no. 13, the double layer egg-box panel does not effectively break the central large fragment. Then the mass becomes the decisive factor of the specific impulse of the debris cloud. Finally the damage in the rear wall is aggravated. To sum up, the single layer egg-box panel is better than the double layer one with the unified total areal density. Moreover, in the case no. 14, the flat enhanced panel is reserved and the front bumper is substituted with the eggbox panel. From the block no. 14 of Fig. 7, the distribution of the debris cloud is also very concentrated. This will cause the large damage in the rear wall. Comparing to the flat panel, the egg-box panel has smaller wall thickness but larger spatial expansion (total thickness) along the impact direction. The egg-box panel can be seen as decreasing the body density while keeping the areal density constant. Therefore the eggbox panel is better at reducing velocity rather than dispersing mass. The simulation output indicates that in this configuration the mass is the decisive factor of the debris cloud power, and therefore the damage of the rear wall is aggravated. To sum up, the egg-box panel may be not appropriate to be used as the front bumper and is better to be used as the enhanced layer.
The hypervelocity impact performance of the aluminum egg-box panel enhanced Whipple shield are investigated in this paper. The finite element models of the egg-box panel enhanced Whipple shield are built. Combined with the node separation technique, the hypervelocity impact process is simulated. Using the flat panel enhanced Whipple shield as the benchmark and keeping the areal density constant, the enhanced flat panel is substituted with the egg-box panel. Many configurations are simulated and compared to find the better parameters for the egg-box panel. Generally the eggbox panel has better performance than the flat panel. It can absorb more incoming debris energy and generates less debris from itself than the flat panel. The impact area analysis indicates that the peak area is strongest, the saddle area is the next and the valley area is the weakest. The cell size analysis shows that the performance of the egg-box panel is not linearly related to the cell size. Both the over large and over small cell size leads to the reduction of the protective performance. Moreover if the impact point is the peak and the cell size is too large, a sharp penetrator made of the egg-box panel material would be generated. The sharp penetrator significantly increases the damage in the rear wall. Secondly, the distance between the front bumper and the egg-box panel should be large enough, so that the debris cloud behind the front bumper can fully disperse. However if the egg-box panel is too close to the rear wall, the relative cell size to the debris cloud is decreased. As a result the performance of the egg-box panel would also decrease. Therefore the egg-box panel is suggested to be set in the middle of the front bumper and the rear wall, or a little closer to the rear wall. Finally it is not appropriate to use the egg-box panel as the front bumper, using it as the enhanced layer is better. With the specified total areal density, the single layer egg-box panel is better than the double layer egg-box panel. In this paper we present a primary analysis of the egg-box panel performance with numerical simulations. The egg-box panel can be fabricated by the stamping and forming to the aluminum flat panel, the cost of manufacture would not be significantly increased. The egg-box panel is appropriate to be the enhanced layer of the Whipple shield. In the future work we will do further simulations to verify our inferences and conclusions at more extensive impact conditions and structure configurations. Other simulation method, like SPH, will be introduced for comparisons. The experimental test is also planned for verification.
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