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Whipple shield performance in the shatter regime
International Journal of Impact Engineering 38 (2011) 504e510
Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Whipple shield performance in the shatter regime S. Ryan a, *, M. Bjorkman b, E.L. Christiansen c a
USRA Lunar and Planetary Institute, 3600 Bay Area Blvd, Houston, TX 77058, USA Jacobs Engineering, 2224 Bay Area Blvd, Houston, TX 77058, USA c NASA Johnson Space Center, 2101 NASA Pkwy, Houston, TX 77058, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Available online 26 January 2011
A series of hypervelocity impact tests have been performed on aluminum alloy Whipple shields to investigate failure mechanisms and performance limits in the shatter regime. Test results demonstrated a more rapid increase in performance than predicted by the latest iteration of the JSC Whipple shield ballistic limit equation (BLE) following the onset of projectile fragmentation. This increase in performance was found to level out between 4.0 and 5.0 km/s, with a subsequent decrease in performance for velocities up to 5.6 km/s. For a detached spall failure criterion, the failure limit was found to continually decrease up to a velocity of 7.0 km/s, substantially varying from the BLE, while for perforation-based failure an increase in performance was observed. An existing phenomenological ballistic limit curve was found to provide a more accurate reproduction of shield behavior that the BLE, prompting an investigation of appropriate models to replace linear interpolation in shatter regime. A largest fragment relationship was shown to provide accurate predictions up to 4.3 km/s, which was extended to the incipient melt limit (5.6 km/s) based on an assumption of no additional fragmentation. Alternate models, including a shock enhancement approach and debris cloud cratering model are discussed as feasible alternatives to the proposed curve in the shatter regime, due to conflicting assumptions and difficulties in extrapolating the current approach to oblique impact. These alternate models require further investigation. Ó 2010 Published by Elsevier Ltd.
Keywords: Hypervelocity impact Orbital debris Whipple shield Ballistic limit
1. Introduction In 1947 Fred Whipple suggested that a thin “bumper”, when placed in front of the pressure hull of a space vehicle, would substantially increase the vehicle’s level of protection against impacting meteors. From Apollo through to the International Space Station, the Whipple shield concept has provided the baseline for shielding against the impact of micrometeoroids and orbital debris (MMOD). Over the range of impact velocities relevant for Earthorbiting spacecraft, the performance of a Whipple shield is characterized in three parts: low velocity, shatter, and hypervelocity. In the low velocity regime, the projectile perforates the shield bumper plate, and propagates to the rear wall, intact (albeit possibly deformed and eroded). Transition to the shatter regime occurs once the impact shock amplitudes are sufficient to induce fragmentation of the projectile. Within the shatter regime, further increases in projectile velocity result in increased projectile fragmentation, transitioning from a small number of solid fragments to a multitude of small, finely dispersed mixed phase debris cloud (solid and molten fragments). Transition to the hypervelocity regime is defined by the point at which the rear wall failure mechanism * Corresponding author. Tel.: þ61 (0)3 96267706; fax: þ61 (0)3 96268999. E-mail address: [email protected] (S. Ryan). 0734-743X/$ e see front matter Ó 2010 Published by Elsevier Ltd. doi:10.1016/j.ijimpeng.2010.10.022
changes from cratering-based to impulsive, similar to that induced by a blast wave. Increased impact speeds within the hypervelocity regime are expected to increase the debris cloud kinetic energy, resulting in a decrease in shielding performance. Ballistic limit equations (BLEs) are used to design and evaluate the performance of shields for MMOD protection. For a metallic Whipple shield, the new non-optimum equation (NNO) [1], or variations thereof (e.g. [2]) are commonly used. These equations are based on cratering relationships in the low velocity regime, kinetic energy scaling in the hypervelocity regime, and a linear interpolation between the two in the shatter regime. With a debris environment increasingly dominated by manmade debris, vehicles operating in low earth orbit (LEO) are subjected to slower median encounter velocities. As such, the performance of shielding at velocities in the shatter regime is increasingly important to mission risk predictions. In this paper, the results of an experimental impact study to characterize failure limits of an aluminum alloy Whipple shield in the shatter regime are presented. 2. Predicting the failure limit of metallic Whipple shields For this analysis, a recent iteration of the Whipple shield ballistic limit equation, referred to herein as the JSC Whipple shield equation, is used. This equation is based on the NNO approach and
S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
505
Nomenclature d S t V
diameter (cm) shield spacing (cm) thickness (cm) velocity (km/s) impact angle (deg) density (g/cm3) yield strength (MPa)
q r s
Subscripts b bumper c critical f fragment n normal p projectile w rear wall Fig. 1. Phenomenological ballistic limit curve for an Al-alloy Whipple shield (reproduced from [5]) with velocity limits.
incorporates a selection of modifications proposed by Reimerdes et al. [2], namely the de-rating of shield performance (F*2) in the hypervelocity regime for bumpers that are insufficiently thick to effectively fragment the projectile. Proposed modifications to the low velocity regime limit velocity, VLV, based on bumper thickness to projectile diameter ratio from [2] are also incorporated, albeit in the original form proposed in [3]. The JSC Whipple shield ballistic limit equation is defined as: In the low velocity regime, i.e. Vn VLV:
" dc ¼
tw ðs=40Þ1=2 þtb 0:6ðcosqÞ
#18=19 (1)
5=3 1=2 2=3 rp V
where VLV ¼ 2.60 if tb/dp 0.16 and 1.436 (tb/dp) if tb/dp < 0.16 In the hypervelocity regime, i.e.: Vn 7 km/s:
dc ¼ 3:918 F2*
2=3 tw S1=3 ðs=70Þ1=3 1=3 1=9 rp rb ðVcos qÞ2=3
(2)
In the shatter regime, i.e. VLV < Vn < 7 km/s, linear interpolation is applied:
dc ¼ dc ðVLV Þ þ
ðdc ðVHV Þ dc ðVLV ÞÞ ðVn VLV Þ VHV VLV
(3)
The de-rating factor, F*2, is calculated as:
( F2* ¼
1 rS=D 2
where,
tb =dp c ¼
t =d 2 ðtb =dp Þ ð pÞ r rS=D 1 1 þ t b=d ðtb =dp Þc S=D ð b p Þc
0:2 rp =rb for S=dp 30 0:25 rp =rb for S=dp < 30
Although providing a reasonable and conservative simplification of shield performance for risk assessment, the linear interpolation in the shatter regime may not accurately reproduce the actual behavior observed for this shield type. In 1970, Swift et al. [4] reported on a series of hypervelocity impact experiments that were performed on aluminum shields with constant spacing and bumper thickness, while the shield rear wall thickness was varied in order to determine the failure threshold. To effectively describe the types of damage observed in target photographs, and in an effort to better characterize the impact performance of a dual-wall structure, Hopkins et al. [5] defined a phenomenological ballistic limit curve (BLC), shown in Fig. 1. The authors defined the characteristics of each region based on their observation of rear wall damage, summarized as follows. In region I, the typical damage observed was a single crater e indicating an intact projectile. In region II, typical damage graduated from a few fairly large craters to a multitude of small craters as a result of the onset and escalation of projectile fragmentation. In region III, the appearance of damage remained rather constant, with each individual crater increasing in size and depth. In region IV, solid fragment craters similar in appearance to those in region II/III were increasingly interspersed by soft contour craters made by molten fragments. Throughout region V, soft contour craters caused by molten material were the dominant damage observable. Region VI damage was characterized by a mixture of molten and vapor damage, where the
tb =dp tb =dp c tb =dp tb =dp c
for for
(5)
and rS/D is ratio between the required rear wall thickness when the bumper thickness is zero, and when it is equal to the limit (tb/dp)c, i.e.:
t ðt ¼ 0Þ w b tw tb =dp ¼ tb =dp c . 19=18 5=3 1=2 2=3 0:6,dp ðs=40Þ1=2 ðcosqÞ rp VHV tb ¼ 3=2 1=3 1=9 rb ðVHV cosqÞ2=3 S1=3 ðs=70Þ1=3 dp =3:918$rp
rS=D ¼
(6)
(4)
failure transitions from penetration and perforation to rupture and tearing at the upper velocity limits. Damage in region VII was similar to that of pressing by high-pressure gas. 3. Impact testing A total of 82 hypervelocity impact tests were performed with spherical Al2017-T4 projectiles on aluminum alloy Whipple shields nominally identical to those tested by Swift et al. [4]. The thickness of the Al6061-T6 bumper (tb ¼ 0.079 cm), shield spacing (S ¼ 5.08 cm) and projectile diameter (dp ¼ 0.3175 cm) were constant, while the Al6061-T6 rear wall thickness (tw) was varied to determine the shield failure limits. Tests were performed at varying impact angles (0 /45 /60 ) and over a range of velocities
506
S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
Fig. 2. Schematic of the Whipple shield test setup.
from 2.27 to 7.20 km/s. In all the tests, failure was defined as the onset of detached spall (SP), however distinction was made between spalled targets and those clearly perforated (P). It should be noted that the failure criteria is different to that used in the original investigation, which assessed failure through use of dye penetrant or gas leak (i.e. perforation). A schematic of the test configuration is shown in Fig. 2. A summary of the test conditions and results is given in Table 1. Of the 82 tests there were 23 perforated targets, 17 spalled targets, and 42 pass results. Of the 17 detached spall results, only one was the result of an oblique impact. The test results are plotted in Fig. 3 against the JSC Whipple shield BLE (Eqs. (1)e(3)). Following the onset of projectile fragmentation, the shield performance is shown to improve more
rapidly in the shatter regime than predicted by the BLE. For normal impact, the shield performance is shown to decrease with increasing velocity in the upper ranges of the shatter regime (V > 4.7 km/s). It is also apparent that the BLE does a better job of predicting perforation rather than spall above 5.8 km/s. In Figs. 4 and 5 a series of rear wall damage photographs are shown for normal incidence tests with impact velocities between 2.6 and 6.9 km/s. A clear progression from solid projectile impact (HD9920118), to fragmentation initiation (HD9902117), through increasing degrees of fragmentation (HD9920060eHD9920057) and the onset of melting (HD9920159eHD9920116) can be observed on the front side of the targets, which corresponds well with the descriptions from [5]. The rear side view shows that the targets all fractured internally (incipient spall) without any detachment of material. An interesting feature of the targets in Fig. 5 is that the lower speed (2.6/3.0 km/s) and higher-speed (6.0/6.6 km/s) tests all show a single, clearly defined bulge (plugging or scabbing), while the tests at 4.7 and 5.5 km/s demonstrate more complex damage features with rings of fracture zones or bulges.
4. Definition of a new curve The ballistic limit of the Whipple shield configuration tested in this study was shown in Fig. 3 to contain additional features in the
Table 1 Overview of Whipple shield test conditions and results. Test no.
q (deg)
V (km/s)
tw (cm)
Result
Test no.
q (deg)
V (km/s)
tw (cm)
Result
HD9920118 HD9920117 HD9920060 HD9920057 HD9920159 HD9920151 HD9920116 HD9820223 HD9920028 HD9820222 HD9820221 HD9920030 HD9920115 HD9820220 HD9920104 HD9920040 HD9920003 HD9920152 HD9920058 HD9920014 HD9920065 HD9920004 HD9920032 HD9920114 HD9820217 HD9920066 HD9920009 HD9920005 HD9920034 HD0020217 HD0020241 HD0020154 HD0120055 HD0020112 HD0020194 HD0120005 HD0020048 HD0020050 HD0020152 HD0020121 HD0020051
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45 45 45 45 45 45 45 45 45 45 45 45
2.6 3 4.7 5.5 6 6.6 6.9 2.5 2.6 3 4.4 4.5 5.7 6.6 2.7 3.1 5.8 5.9 5.9 6 6.4 6.4 6.4 6.5 6.5 6.7 6.7 6.9 6.9 2.29 2.32 2.42 2.48 2.56 2.63 2.75 3.00 4.60 5.61 5.69 6.00
0.4572 0.4572 0.254 0.3175 0.4064 0.4064 0.4572 0.254 0.3175 0.3175 0.2032 0.2286 0.254 0.1803 0.4064 0.4064 0.254 0.3175 0.2286 0.2032 0.254 0.2032 0.2286 0.3175 0.1803 0.4064 0.1803 0.254 0.3175 0.2286 0.2286 0.4064 0.2032 0.3175 0.254 0.2032 0.4064 0.4064 0.4064 0.3175 0.2286
NP NP NP NP NP NP NP P P P P P P P SP SP SP SP SP SP SP SP SP SP SP SP SP SP SP NP NP NP NP NP NP NP NP NP NP NP NP
HD0020010 HD0020122 HD0020047 HD0020053 HD9920272 HD0020011 HD9920271 HD0020087 HD0020078 HD0020052 HD0020077 HD9920270 HD0020113 HD0020088 HD0020153 HD0020242 HD0020116 HD0020171 HD0020219 HD0120003 HD0020170 HD0120079 HD0020015 HD9920275 HD9920274 HD0020062 HD0020014 HD0020195 HD0020169 HD0020090 HD0020123 HD9920273 HD0020061 HD0120004 HD0020119 HD0020060 HD0020019 HD0020013 HD0020117 HD0020059 HD0020118
45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60
6.50 6.53 6.60 7.20 3.10 4.50 4.50 5.45 5.60 5.70 5.93 6.50 6.59 6.68 6.81 2.27 2.37 2.63 2.67 2.76 2.93 3.03 3.10 3.10 4.30 4.37 4.50 4.56 5.09 5.71 6.00 6.50 6.95 2.66 4.51 5.67 6.00 6.60 5.70 5.94 6.80
0.4064 0.254 0.3175 0.3175 0.3175 0.3175 0.2032 0.254 0.2286 0.2032 0.2032 0.2286 0.254 0.254 0.254 0.1803 0.2286 0.2032 0.1600 0.1803 0.2032 0.1803 0.2286 0.254 0.2286 0.1803 0.2032 0.2286 0.2286 0.254 0.254 0.2032 0.2286 0.127 0.1600 0.1803 0.2032 0.1803 0.2032 0.2286 0.2032
NP NP NP NP P P P P P P P P P P P NP NP NP NP NP NP NP NP NP NP NP NP NP NP NP NP NP NP P P P P P SP NP SP
S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
507
Fig. 3. Test results and ballistic limit curves for 0 (left) and 45 (right).
shatter regime that are not captured in the linear interpolation of current BLEs. These features, however, appear to be well reproduced by the phenomenological curve proposed by Hopkins et al. [5]. The primary deviation from the linear curve is the increased rate of performance enhancement following incipient fragmentation (i.e. region II), followed by a decrease in performance due to increased kinetic energy of fully fragmented particles prior to the onset of insipient melt (i.e. region III). Piekutowski [6] investigated the formation of debris clouds during hypervelocity impact, and characterized the largest fragment, df, generated for a variety of bumper thickness to projectile diameter ratios (tb/dp) across a range of impact velocities, defined as:
df ¼ 204:8 V 2:24 for tb =dp ¼ 0:049;
0:819 2:24 df ¼ 18:072 tb =dp V for dp ¼ 9:53 mm
(9)
An alternate empirical relationship, derived from test data on multiple projectile diameters was found to provide a superior fit (shown in Fig. 6), defined as:
df =dp ¼ 5:62e0:86
V
0:166 tb =dp
Vþ0:134
(10)
(7)
If perforation of a Whipple shield rear wall in the initial stages of the shatter regime is assumed to occur due to cratering by the largest fragment in the debris cloud, the ballistic limit can be determined by an extension of Eq. (1). In terms of required rear wall thickness, the ballistic limit is calculated as:
(8)
tw ¼
and
df ¼ 147:1 V 2:24 for tb =dp ¼ 0:084:
A third power law expression was developed for tests on targets with tb/dp ¼ 0.233, from which a generalized equation, in terms of tb/dp, could be determined, defined as:
19=18
0:733 df
5=3 1=2 rp
ðcosqÞ
V 2=3 tb ð40=sÞ1=2
Fig. 4. Rear wall (front view) damage profile with increasing impact velocity (clockwise from top left).
(11)
508
S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
Fig. 5. Rear wall (rear view) damage profile with increasing impact velocity (clockwise from top left).
It should be noted that Eq. (11) is based on the detached spall failure condition, compared to Eq. (1), for which perforation is used. The failure criterion is embedded in the scaling constant, which changes from 0.6 (Eq. (1)) to 0.733 (Eq. (11)) for detached spall. Hopkins et al. [5] estimated that complete fragmentation for alon-al impact, and therefore the limit of Eq. (10), occurred at 4.3 km/ s (for the specific configuration investigated). In region III, therefore, it is assumed that increased rear wall damage is due to the increased kinetic energy of individual solid fragments (hence, damage “retains a rather constant appearance throughout, with each tiny crater becoming larger and deeper as the velocity increases”). Assuming a constant fragment diameter, the cratering
relationship states that the required rear wall thickness increases at a rate proportional to V2/3. In Fig. 7 test data from Table 1 and [4] are plotted along with the JSC Whipple shield BLE (Eqs. (1)e(3)) and a new curve based on largest fragment cratering in the shatter regime. Between 3.0 and 4.3 km/s, the Piekutowski-based largest fragment diameter relationship is used to calculate rear wall thickness (Eq. (11)). Crateringbased failure by the largest fragment is extended through to 5.6 km/s, based on the assumption of no additional fragmentation above 4.3 km/s (i.e. largest fragment diameter is constant). This approach was found, for this particular configuration, to converge with the JSC Whipple shield BLE at 5.6 km/s. For velocities
Fig. 6. Largest fragment relationships extrapolated from Piekutowski [6] (Eq. (9)) and Eq. (10). Test data from [6].
S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
509
Fig. 7. JSC Whipple shield BLE and a new curve based on largest fragment cratering in the shatter regime.
above 5.6 km/s, the original BLE is used. In order to combine the JSC Whipple shield equation in the low velocity regime (i.e. Vn 2.6 km/s) with the largest fragment analysis above 3.0 km/s, a quadratic Bézier curve is used with the parameters P0 ¼ VLV, P1 ¼ VLV þ 0.1, P2 ¼ (P1 P0) þ P1. The new curve is shown to provide an excellent fit with the pail/fail test data, however it is still unable to accurately predict spall at velocities above 5.8 km/s. 5. Discussion The proposed largest fragment-based cratering modification of the curve in the shatter regime is shown in Fig. 7 to provide a good level of agreement with the test data. However, there remain a number of areas that require further investigation. The relationship for determining the diameter of the largest fragment in the debris cloud is based on test data that extends to 6.7 km/s. Therefore, the assumption of complete fragmentation at 4.3 km/s, based on the description of target damage by Swift et al. [5], is questionable. If the largest fragment diameter is shown to continually decrease through velocities of 5.6 km/s, the increase in required rear wall thickness between 4.3 and 5.6 km/s cannot be predicted with the existing model. Lloyd [7] reports on modifications made to the SCAN endgame simulation code for high density fragment spray impacts
that account for enhanced damage due to sequential and simultaneous, localized impacts. For simultaneous impact of multiple fragments with limited separation, superposition of the induced shock waves within the target plate can lead to premature internal fracture and spallation. It is possible that as the diameter of the largest projectile fragment decreases, the influence of secondary fragments, and the resulting superposition of shock and release waves within the target rear wall, may become more relevant, thus accounting for the decrease in shield performance predicted between 4.3 and 5.6 km/s. This type of approach, i.e. failure due to a cumulative effect from multiple fragments, agrees well with the appearance of rear side target damage in this velocity range (see Fig. 5). An alternative approach to largest fragment-based cratering is proposed by Cohen [8], who calculates cratering depth into a semiinfinite Whipple shield rear wall by a shattered fragment cloud rather than an individual fragment. Assuming a constant evacuated volume, based on average fragment area and penetration depth (i.e. no overlapping of craters), the penetration depth of the debris cloud is dependent on the debris cloud spray angle, and shield spacing, S. From [8]:
Df PN ¼ 0:0085 dp 2Pf
!2
rp rw sw
Fig. 8. A comparison of Cohen’s debris cloud cratering model with test data.
v tanðwÞ
2
S dp
1 t 2 1þ b dp
510
S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
where PN is penetration depth (cm), Df is the average crater diameter (cm), Pf is the average fragment penetration depth (cm), and w is the spray cone half angle (deg). The model is valid for impacts in which solid fragment cratering is the dominant damage mechanism, which is assumed to occur at velocities below the midpoint between incipient and complete melt (i.e. 5.6 and 7.0 km/ s for al-on-al impact). Above the velocity required for incipient melt, the penetration depth is decreased according to the mass fraction of the projectile that is molten. As a result, the Cohen model qualitatively reproduces some key features of Swift et al.’s phenomenological curve (see Fig. 8), namely the trends described in regions IIeIV. The Cohen model is a simple mechanical approach, based on a number of empirical factors that may provide more agreeable results with further investigation. 6. Summary and conclusions In order to investigate the performance of a Whipple shield in the shatter regime, a number of hypervelocity impact tests were performed in which the projectile diameter, bumper thickness, and shield spacing were constant, while rear wall thickness was varied in order to determine failure limits. The projectile, and target configuration were nominally identical to that investigated in [5]. A recent iteration of the new nonoptimum Whipple shield BLE, termed the JSC Whipple shield equation, which includes modifications for including the effect of bumper thickness in the hypervelocity regime, and the effect of tb/dp ratio on projectile fragmentation initiation (i.e. VLV), was shown to vary from the test data, particularly in the shatter regime. Following the onset of projectile fragmentation, a more rapid increase in shield performance was noted between velocities of 3.1 and 4.0 km/s than predicted by the BLE. At velocities between 4.0 and 5.0 km/s, performance was relatively constant, followed by a decrease between 5.0 and 6.0 km/s for perforation-based failure. For detached spall, the required rear wall thickness was found to increase linearly up to 7 km/s. The trends observed in the test data were more accurately described by a phenomenological curve from [5] which divided shield performance into seven velocity regions based on the state of the material within the debris cloud (intact, shattered, molten, vaporized). An effort was made to develop an analytical/empirical equation suitable for modification of the JSC Whipple shield BLE in the shatter regime that would more effectively capture the behavior described in Hopkins’ phenomenological curve. A general equation to determine the diameter of the largest projectile fragment in the
debris cloud was defined, based on data from [6], and found to agree well with test data for velocities between 3.0 and 4.3 km/s. Assuming no additional fragmentation at velocities above 4.3 km/s, the resulting increase in kinetic energy of the largest remnant projectile fragment was shown to effectively predict the increased in required rear wall thickness observed in the experiments for velocities up to 5.6 km/s. The modified curve was found to converge (for this particular configuration) with the baseline BLE at 5.6 km/s, enabling extrapolation to velocities above the incipient melt condition. Contradictions in the assumption of constant fragment diameter between 4.3 and 5.6 km/s were discussed, and alternate models introduced. One technique involves the inclusion of shock superpositioning due to simultaneous impact of closely spaced fragments on the target rear wall, resulting in enhanced penetration and decreased spallation thresholds. An alternate model, proposed by Cohen [8] used for determining cratering depth of predominantly solid fragment debris clouds, was evaluated and found to qualitatively reproduce the trends of the test data. This simplified model is independent of the number of fragments, and based on an assumption of constant evacuated volume, resulting in a penetration depth dependent on spray angle and shield spacing. The model contains a number of empirical constants that may be suitable for adjustment in order to fit the Whipple shield test data in the shatter regime. References [1] Christiansen EL. Design and performance equations for advanced meteoroid and debris shields. International Journal of Impact Engineering 1993;14: 145e56. [2] Reimerdes HG, Noelke D, Schaefer FK. Modified Cour-Palais/Christiansen damage equations for double-wall structures. International Journal of Impact Engineering 2006;33:645e54. [3] Piekutowsi AJ. Fragment-initiation threshold for spheres impacting at hypervelocity. International Journal of Impact Engineering 2003;29:563e74. [4] Swift H, Preonas D, Dueweke P, Bertke R. Response of materials to impulsive loading. AFML-TR-70e135. Dayton: Air Force Materials Laboratory, WrightPatterson Air Force Base; 1970. [5] Hopkins AK, Lee TW, Swift HF. Material phase transformation effects upon performance spaced bumper systems. Journal of Spacecraft 1972;9(5): 342e5. [6] Piekutowski AJ. Formation and description of debris clouds produced by hypervelocity impact. CR-4707. Dayton, NASA: University of Dayton Research Institute; 1996. [7] Lloyd R. Advanced multiple impact endgame model against ballistic missile payloads. In: I.R. Crewther (ed). 19th international symposium of ballistics. Interlaken: IBS 7e11 May 2001. [8] Cohen L. A debris cloud cratering model. International Journal of Impact Engineering 1995;17:229e40.
Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Whipple shield performance in the shatter regime S. Ryan a, *, M. Bjorkman b, E.L. Christiansen c a
USRA Lunar and Planetary Institute, 3600 Bay Area Blvd, Houston, TX 77058, USA Jacobs Engineering, 2224 Bay Area Blvd, Houston, TX 77058, USA c NASA Johnson Space Center, 2101 NASA Pkwy, Houston, TX 77058, USA b
a r t i c l e i n f o
a b s t r a c t
Article history: Available online 26 January 2011
A series of hypervelocity impact tests have been performed on aluminum alloy Whipple shields to investigate failure mechanisms and performance limits in the shatter regime. Test results demonstrated a more rapid increase in performance than predicted by the latest iteration of the JSC Whipple shield ballistic limit equation (BLE) following the onset of projectile fragmentation. This increase in performance was found to level out between 4.0 and 5.0 km/s, with a subsequent decrease in performance for velocities up to 5.6 km/s. For a detached spall failure criterion, the failure limit was found to continually decrease up to a velocity of 7.0 km/s, substantially varying from the BLE, while for perforation-based failure an increase in performance was observed. An existing phenomenological ballistic limit curve was found to provide a more accurate reproduction of shield behavior that the BLE, prompting an investigation of appropriate models to replace linear interpolation in shatter regime. A largest fragment relationship was shown to provide accurate predictions up to 4.3 km/s, which was extended to the incipient melt limit (5.6 km/s) based on an assumption of no additional fragmentation. Alternate models, including a shock enhancement approach and debris cloud cratering model are discussed as feasible alternatives to the proposed curve in the shatter regime, due to conflicting assumptions and difficulties in extrapolating the current approach to oblique impact. These alternate models require further investigation. Ó 2010 Published by Elsevier Ltd.
Keywords: Hypervelocity impact Orbital debris Whipple shield Ballistic limit
1. Introduction In 1947 Fred Whipple suggested that a thin “bumper”, when placed in front of the pressure hull of a space vehicle, would substantially increase the vehicle’s level of protection against impacting meteors. From Apollo through to the International Space Station, the Whipple shield concept has provided the baseline for shielding against the impact of micrometeoroids and orbital debris (MMOD). Over the range of impact velocities relevant for Earthorbiting spacecraft, the performance of a Whipple shield is characterized in three parts: low velocity, shatter, and hypervelocity. In the low velocity regime, the projectile perforates the shield bumper plate, and propagates to the rear wall, intact (albeit possibly deformed and eroded). Transition to the shatter regime occurs once the impact shock amplitudes are sufficient to induce fragmentation of the projectile. Within the shatter regime, further increases in projectile velocity result in increased projectile fragmentation, transitioning from a small number of solid fragments to a multitude of small, finely dispersed mixed phase debris cloud (solid and molten fragments). Transition to the hypervelocity regime is defined by the point at which the rear wall failure mechanism * Corresponding author. Tel.: þ61 (0)3 96267706; fax: þ61 (0)3 96268999. E-mail address: [email protected] (S. Ryan). 0734-743X/$ e see front matter Ó 2010 Published by Elsevier Ltd. doi:10.1016/j.ijimpeng.2010.10.022
changes from cratering-based to impulsive, similar to that induced by a blast wave. Increased impact speeds within the hypervelocity regime are expected to increase the debris cloud kinetic energy, resulting in a decrease in shielding performance. Ballistic limit equations (BLEs) are used to design and evaluate the performance of shields for MMOD protection. For a metallic Whipple shield, the new non-optimum equation (NNO) [1], or variations thereof (e.g. [2]) are commonly used. These equations are based on cratering relationships in the low velocity regime, kinetic energy scaling in the hypervelocity regime, and a linear interpolation between the two in the shatter regime. With a debris environment increasingly dominated by manmade debris, vehicles operating in low earth orbit (LEO) are subjected to slower median encounter velocities. As such, the performance of shielding at velocities in the shatter regime is increasingly important to mission risk predictions. In this paper, the results of an experimental impact study to characterize failure limits of an aluminum alloy Whipple shield in the shatter regime are presented. 2. Predicting the failure limit of metallic Whipple shields For this analysis, a recent iteration of the Whipple shield ballistic limit equation, referred to herein as the JSC Whipple shield equation, is used. This equation is based on the NNO approach and
S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
505
Nomenclature d S t V
diameter (cm) shield spacing (cm) thickness (cm) velocity (km/s) impact angle (deg) density (g/cm3) yield strength (MPa)
q r s
Subscripts b bumper c critical f fragment n normal p projectile w rear wall Fig. 1. Phenomenological ballistic limit curve for an Al-alloy Whipple shield (reproduced from [5]) with velocity limits.
incorporates a selection of modifications proposed by Reimerdes et al. [2], namely the de-rating of shield performance (F*2) in the hypervelocity regime for bumpers that are insufficiently thick to effectively fragment the projectile. Proposed modifications to the low velocity regime limit velocity, VLV, based on bumper thickness to projectile diameter ratio from [2] are also incorporated, albeit in the original form proposed in [3]. The JSC Whipple shield ballistic limit equation is defined as: In the low velocity regime, i.e. Vn VLV:
" dc ¼
tw ðs=40Þ1=2 þtb 0:6ðcosqÞ
#18=19 (1)
5=3 1=2 2=3 rp V
where VLV ¼ 2.60 if tb/dp 0.16 and 1.436 (tb/dp) if tb/dp < 0.16 In the hypervelocity regime, i.e.: Vn 7 km/s:
dc ¼ 3:918 F2*
2=3 tw S1=3 ðs=70Þ1=3 1=3 1=9 rp rb ðVcos qÞ2=3
(2)
In the shatter regime, i.e. VLV < Vn < 7 km/s, linear interpolation is applied:
dc ¼ dc ðVLV Þ þ
ðdc ðVHV Þ dc ðVLV ÞÞ ðVn VLV Þ VHV VLV
(3)
The de-rating factor, F*2, is calculated as:
( F2* ¼
1 rS=D 2
where,
tb =dp c ¼
t =d 2 ðtb =dp Þ ð pÞ r rS=D 1 1 þ t b=d ðtb =dp Þc S=D ð b p Þc
0:2 rp =rb for S=dp 30 0:25 rp =rb for S=dp < 30
Although providing a reasonable and conservative simplification of shield performance for risk assessment, the linear interpolation in the shatter regime may not accurately reproduce the actual behavior observed for this shield type. In 1970, Swift et al. [4] reported on a series of hypervelocity impact experiments that were performed on aluminum shields with constant spacing and bumper thickness, while the shield rear wall thickness was varied in order to determine the failure threshold. To effectively describe the types of damage observed in target photographs, and in an effort to better characterize the impact performance of a dual-wall structure, Hopkins et al. [5] defined a phenomenological ballistic limit curve (BLC), shown in Fig. 1. The authors defined the characteristics of each region based on their observation of rear wall damage, summarized as follows. In region I, the typical damage observed was a single crater e indicating an intact projectile. In region II, typical damage graduated from a few fairly large craters to a multitude of small craters as a result of the onset and escalation of projectile fragmentation. In region III, the appearance of damage remained rather constant, with each individual crater increasing in size and depth. In region IV, solid fragment craters similar in appearance to those in region II/III were increasingly interspersed by soft contour craters made by molten fragments. Throughout region V, soft contour craters caused by molten material were the dominant damage observable. Region VI damage was characterized by a mixture of molten and vapor damage, where the
tb =dp tb =dp c tb =dp tb =dp c
for for
(5)
and rS/D is ratio between the required rear wall thickness when the bumper thickness is zero, and when it is equal to the limit (tb/dp)c, i.e.:
t ðt ¼ 0Þ w b tw tb =dp ¼ tb =dp c . 19=18 5=3 1=2 2=3 0:6,dp ðs=40Þ1=2 ðcosqÞ rp VHV tb ¼ 3=2 1=3 1=9 rb ðVHV cosqÞ2=3 S1=3 ðs=70Þ1=3 dp =3:918$rp
rS=D ¼
(6)
(4)
failure transitions from penetration and perforation to rupture and tearing at the upper velocity limits. Damage in region VII was similar to that of pressing by high-pressure gas. 3. Impact testing A total of 82 hypervelocity impact tests were performed with spherical Al2017-T4 projectiles on aluminum alloy Whipple shields nominally identical to those tested by Swift et al. [4]. The thickness of the Al6061-T6 bumper (tb ¼ 0.079 cm), shield spacing (S ¼ 5.08 cm) and projectile diameter (dp ¼ 0.3175 cm) were constant, while the Al6061-T6 rear wall thickness (tw) was varied to determine the shield failure limits. Tests were performed at varying impact angles (0 /45 /60 ) and over a range of velocities
506
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Fig. 2. Schematic of the Whipple shield test setup.
from 2.27 to 7.20 km/s. In all the tests, failure was defined as the onset of detached spall (SP), however distinction was made between spalled targets and those clearly perforated (P). It should be noted that the failure criteria is different to that used in the original investigation, which assessed failure through use of dye penetrant or gas leak (i.e. perforation). A schematic of the test configuration is shown in Fig. 2. A summary of the test conditions and results is given in Table 1. Of the 82 tests there were 23 perforated targets, 17 spalled targets, and 42 pass results. Of the 17 detached spall results, only one was the result of an oblique impact. The test results are plotted in Fig. 3 against the JSC Whipple shield BLE (Eqs. (1)e(3)). Following the onset of projectile fragmentation, the shield performance is shown to improve more
rapidly in the shatter regime than predicted by the BLE. For normal impact, the shield performance is shown to decrease with increasing velocity in the upper ranges of the shatter regime (V > 4.7 km/s). It is also apparent that the BLE does a better job of predicting perforation rather than spall above 5.8 km/s. In Figs. 4 and 5 a series of rear wall damage photographs are shown for normal incidence tests with impact velocities between 2.6 and 6.9 km/s. A clear progression from solid projectile impact (HD9920118), to fragmentation initiation (HD9902117), through increasing degrees of fragmentation (HD9920060eHD9920057) and the onset of melting (HD9920159eHD9920116) can be observed on the front side of the targets, which corresponds well with the descriptions from [5]. The rear side view shows that the targets all fractured internally (incipient spall) without any detachment of material. An interesting feature of the targets in Fig. 5 is that the lower speed (2.6/3.0 km/s) and higher-speed (6.0/6.6 km/s) tests all show a single, clearly defined bulge (plugging or scabbing), while the tests at 4.7 and 5.5 km/s demonstrate more complex damage features with rings of fracture zones or bulges.
4. Definition of a new curve The ballistic limit of the Whipple shield configuration tested in this study was shown in Fig. 3 to contain additional features in the
Table 1 Overview of Whipple shield test conditions and results. Test no.
q (deg)
V (km/s)
tw (cm)
Result
Test no.
q (deg)
V (km/s)
tw (cm)
Result
HD9920118 HD9920117 HD9920060 HD9920057 HD9920159 HD9920151 HD9920116 HD9820223 HD9920028 HD9820222 HD9820221 HD9920030 HD9920115 HD9820220 HD9920104 HD9920040 HD9920003 HD9920152 HD9920058 HD9920014 HD9920065 HD9920004 HD9920032 HD9920114 HD9820217 HD9920066 HD9920009 HD9920005 HD9920034 HD0020217 HD0020241 HD0020154 HD0120055 HD0020112 HD0020194 HD0120005 HD0020048 HD0020050 HD0020152 HD0020121 HD0020051
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 45 45 45 45 45 45 45 45 45 45 45 45
2.6 3 4.7 5.5 6 6.6 6.9 2.5 2.6 3 4.4 4.5 5.7 6.6 2.7 3.1 5.8 5.9 5.9 6 6.4 6.4 6.4 6.5 6.5 6.7 6.7 6.9 6.9 2.29 2.32 2.42 2.48 2.56 2.63 2.75 3.00 4.60 5.61 5.69 6.00
0.4572 0.4572 0.254 0.3175 0.4064 0.4064 0.4572 0.254 0.3175 0.3175 0.2032 0.2286 0.254 0.1803 0.4064 0.4064 0.254 0.3175 0.2286 0.2032 0.254 0.2032 0.2286 0.3175 0.1803 0.4064 0.1803 0.254 0.3175 0.2286 0.2286 0.4064 0.2032 0.3175 0.254 0.2032 0.4064 0.4064 0.4064 0.3175 0.2286
NP NP NP NP NP NP NP P P P P P P P SP SP SP SP SP SP SP SP SP SP SP SP SP SP SP NP NP NP NP NP NP NP NP NP NP NP NP
HD0020010 HD0020122 HD0020047 HD0020053 HD9920272 HD0020011 HD9920271 HD0020087 HD0020078 HD0020052 HD0020077 HD9920270 HD0020113 HD0020088 HD0020153 HD0020242 HD0020116 HD0020171 HD0020219 HD0120003 HD0020170 HD0120079 HD0020015 HD9920275 HD9920274 HD0020062 HD0020014 HD0020195 HD0020169 HD0020090 HD0020123 HD9920273 HD0020061 HD0120004 HD0020119 HD0020060 HD0020019 HD0020013 HD0020117 HD0020059 HD0020118
45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60
6.50 6.53 6.60 7.20 3.10 4.50 4.50 5.45 5.60 5.70 5.93 6.50 6.59 6.68 6.81 2.27 2.37 2.63 2.67 2.76 2.93 3.03 3.10 3.10 4.30 4.37 4.50 4.56 5.09 5.71 6.00 6.50 6.95 2.66 4.51 5.67 6.00 6.60 5.70 5.94 6.80
0.4064 0.254 0.3175 0.3175 0.3175 0.3175 0.2032 0.254 0.2286 0.2032 0.2032 0.2286 0.254 0.254 0.254 0.1803 0.2286 0.2032 0.1600 0.1803 0.2032 0.1803 0.2286 0.254 0.2286 0.1803 0.2032 0.2286 0.2286 0.254 0.254 0.2032 0.2286 0.127 0.1600 0.1803 0.2032 0.1803 0.2032 0.2286 0.2032
NP NP NP NP P P P P P P P P P P P NP NP NP NP NP NP NP NP NP NP NP NP NP NP NP NP NP NP P P P P P SP NP SP
S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
507
Fig. 3. Test results and ballistic limit curves for 0 (left) and 45 (right).
shatter regime that are not captured in the linear interpolation of current BLEs. These features, however, appear to be well reproduced by the phenomenological curve proposed by Hopkins et al. [5]. The primary deviation from the linear curve is the increased rate of performance enhancement following incipient fragmentation (i.e. region II), followed by a decrease in performance due to increased kinetic energy of fully fragmented particles prior to the onset of insipient melt (i.e. region III). Piekutowski [6] investigated the formation of debris clouds during hypervelocity impact, and characterized the largest fragment, df, generated for a variety of bumper thickness to projectile diameter ratios (tb/dp) across a range of impact velocities, defined as:
df ¼ 204:8 V 2:24 for tb =dp ¼ 0:049;
0:819 2:24 df ¼ 18:072 tb =dp V for dp ¼ 9:53 mm
(9)
An alternate empirical relationship, derived from test data on multiple projectile diameters was found to provide a superior fit (shown in Fig. 6), defined as:
df =dp ¼ 5:62e0:86
V
0:166 tb =dp
Vþ0:134
(10)
(7)
If perforation of a Whipple shield rear wall in the initial stages of the shatter regime is assumed to occur due to cratering by the largest fragment in the debris cloud, the ballistic limit can be determined by an extension of Eq. (1). In terms of required rear wall thickness, the ballistic limit is calculated as:
(8)
tw ¼
and
df ¼ 147:1 V 2:24 for tb =dp ¼ 0:084:
A third power law expression was developed for tests on targets with tb/dp ¼ 0.233, from which a generalized equation, in terms of tb/dp, could be determined, defined as:
19=18
0:733 df
5=3 1=2 rp
ðcosqÞ
V 2=3 tb ð40=sÞ1=2
Fig. 4. Rear wall (front view) damage profile with increasing impact velocity (clockwise from top left).
(11)
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S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
Fig. 5. Rear wall (rear view) damage profile with increasing impact velocity (clockwise from top left).
It should be noted that Eq. (11) is based on the detached spall failure condition, compared to Eq. (1), for which perforation is used. The failure criterion is embedded in the scaling constant, which changes from 0.6 (Eq. (1)) to 0.733 (Eq. (11)) for detached spall. Hopkins et al. [5] estimated that complete fragmentation for alon-al impact, and therefore the limit of Eq. (10), occurred at 4.3 km/ s (for the specific configuration investigated). In region III, therefore, it is assumed that increased rear wall damage is due to the increased kinetic energy of individual solid fragments (hence, damage “retains a rather constant appearance throughout, with each tiny crater becoming larger and deeper as the velocity increases”). Assuming a constant fragment diameter, the cratering
relationship states that the required rear wall thickness increases at a rate proportional to V2/3. In Fig. 7 test data from Table 1 and [4] are plotted along with the JSC Whipple shield BLE (Eqs. (1)e(3)) and a new curve based on largest fragment cratering in the shatter regime. Between 3.0 and 4.3 km/s, the Piekutowski-based largest fragment diameter relationship is used to calculate rear wall thickness (Eq. (11)). Crateringbased failure by the largest fragment is extended through to 5.6 km/s, based on the assumption of no additional fragmentation above 4.3 km/s (i.e. largest fragment diameter is constant). This approach was found, for this particular configuration, to converge with the JSC Whipple shield BLE at 5.6 km/s. For velocities
Fig. 6. Largest fragment relationships extrapolated from Piekutowski [6] (Eq. (9)) and Eq. (10). Test data from [6].
S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
509
Fig. 7. JSC Whipple shield BLE and a new curve based on largest fragment cratering in the shatter regime.
above 5.6 km/s, the original BLE is used. In order to combine the JSC Whipple shield equation in the low velocity regime (i.e. Vn 2.6 km/s) with the largest fragment analysis above 3.0 km/s, a quadratic Bézier curve is used with the parameters P0 ¼ VLV, P1 ¼ VLV þ 0.1, P2 ¼ (P1 P0) þ P1. The new curve is shown to provide an excellent fit with the pail/fail test data, however it is still unable to accurately predict spall at velocities above 5.8 km/s. 5. Discussion The proposed largest fragment-based cratering modification of the curve in the shatter regime is shown in Fig. 7 to provide a good level of agreement with the test data. However, there remain a number of areas that require further investigation. The relationship for determining the diameter of the largest fragment in the debris cloud is based on test data that extends to 6.7 km/s. Therefore, the assumption of complete fragmentation at 4.3 km/s, based on the description of target damage by Swift et al. [5], is questionable. If the largest fragment diameter is shown to continually decrease through velocities of 5.6 km/s, the increase in required rear wall thickness between 4.3 and 5.6 km/s cannot be predicted with the existing model. Lloyd [7] reports on modifications made to the SCAN endgame simulation code for high density fragment spray impacts
that account for enhanced damage due to sequential and simultaneous, localized impacts. For simultaneous impact of multiple fragments with limited separation, superposition of the induced shock waves within the target plate can lead to premature internal fracture and spallation. It is possible that as the diameter of the largest projectile fragment decreases, the influence of secondary fragments, and the resulting superposition of shock and release waves within the target rear wall, may become more relevant, thus accounting for the decrease in shield performance predicted between 4.3 and 5.6 km/s. This type of approach, i.e. failure due to a cumulative effect from multiple fragments, agrees well with the appearance of rear side target damage in this velocity range (see Fig. 5). An alternative approach to largest fragment-based cratering is proposed by Cohen [8], who calculates cratering depth into a semiinfinite Whipple shield rear wall by a shattered fragment cloud rather than an individual fragment. Assuming a constant evacuated volume, based on average fragment area and penetration depth (i.e. no overlapping of craters), the penetration depth of the debris cloud is dependent on the debris cloud spray angle, and shield spacing, S. From [8]:
Df PN ¼ 0:0085 dp 2Pf
!2
rp rw sw
Fig. 8. A comparison of Cohen’s debris cloud cratering model with test data.
v tanðwÞ
2
S dp
1 t 2 1þ b dp
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S. Ryan et al. / International Journal of Impact Engineering 38 (2011) 504e510
where PN is penetration depth (cm), Df is the average crater diameter (cm), Pf is the average fragment penetration depth (cm), and w is the spray cone half angle (deg). The model is valid for impacts in which solid fragment cratering is the dominant damage mechanism, which is assumed to occur at velocities below the midpoint between incipient and complete melt (i.e. 5.6 and 7.0 km/ s for al-on-al impact). Above the velocity required for incipient melt, the penetration depth is decreased according to the mass fraction of the projectile that is molten. As a result, the Cohen model qualitatively reproduces some key features of Swift et al.’s phenomenological curve (see Fig. 8), namely the trends described in regions IIeIV. The Cohen model is a simple mechanical approach, based on a number of empirical factors that may provide more agreeable results with further investigation. 6. Summary and conclusions In order to investigate the performance of a Whipple shield in the shatter regime, a number of hypervelocity impact tests were performed in which the projectile diameter, bumper thickness, and shield spacing were constant, while rear wall thickness was varied in order to determine failure limits. The projectile, and target configuration were nominally identical to that investigated in [5]. A recent iteration of the new nonoptimum Whipple shield BLE, termed the JSC Whipple shield equation, which includes modifications for including the effect of bumper thickness in the hypervelocity regime, and the effect of tb/dp ratio on projectile fragmentation initiation (i.e. VLV), was shown to vary from the test data, particularly in the shatter regime. Following the onset of projectile fragmentation, a more rapid increase in shield performance was noted between velocities of 3.1 and 4.0 km/s than predicted by the BLE. At velocities between 4.0 and 5.0 km/s, performance was relatively constant, followed by a decrease between 5.0 and 6.0 km/s for perforation-based failure. For detached spall, the required rear wall thickness was found to increase linearly up to 7 km/s. The trends observed in the test data were more accurately described by a phenomenological curve from [5] which divided shield performance into seven velocity regions based on the state of the material within the debris cloud (intact, shattered, molten, vaporized). An effort was made to develop an analytical/empirical equation suitable for modification of the JSC Whipple shield BLE in the shatter regime that would more effectively capture the behavior described in Hopkins’ phenomenological curve. A general equation to determine the diameter of the largest projectile fragment in the
debris cloud was defined, based on data from [6], and found to agree well with test data for velocities between 3.0 and 4.3 km/s. Assuming no additional fragmentation at velocities above 4.3 km/s, the resulting increase in kinetic energy of the largest remnant projectile fragment was shown to effectively predict the increased in required rear wall thickness observed in the experiments for velocities up to 5.6 km/s. The modified curve was found to converge (for this particular configuration) with the baseline BLE at 5.6 km/s, enabling extrapolation to velocities above the incipient melt condition. Contradictions in the assumption of constant fragment diameter between 4.3 and 5.6 km/s were discussed, and alternate models introduced. One technique involves the inclusion of shock superpositioning due to simultaneous impact of closely spaced fragments on the target rear wall, resulting in enhanced penetration and decreased spallation thresholds. An alternate model, proposed by Cohen [8] used for determining cratering depth of predominantly solid fragment debris clouds, was evaluated and found to qualitatively reproduce the trends of the test data. This simplified model is independent of the number of fragments, and based on an assumption of constant evacuated volume, resulting in a penetration depth dependent on spray angle and shield spacing. The model contains a number of empirical constants that may be suitable for adjustment in order to fit the Whipple shield test data in the shatter regime. References [1] Christiansen EL. Design and performance equations for advanced meteoroid and debris shields. International Journal of Impact Engineering 1993;14: 145e56. [2] Reimerdes HG, Noelke D, Schaefer FK. Modified Cour-Palais/Christiansen damage equations for double-wall structures. International Journal of Impact Engineering 2006;33:645e54. [3] Piekutowsi AJ. Fragment-initiation threshold for spheres impacting at hypervelocity. International Journal of Impact Engineering 2003;29:563e74. [4] Swift H, Preonas D, Dueweke P, Bertke R. Response of materials to impulsive loading. AFML-TR-70e135. Dayton: Air Force Materials Laboratory, WrightPatterson Air Force Base; 1970. [5] Hopkins AK, Lee TW, Swift HF. Material phase transformation effects upon performance spaced bumper systems. Journal of Spacecraft 1972;9(5): 342e5. [6] Piekutowski AJ. Formation and description of debris clouds produced by hypervelocity impact. CR-4707. Dayton, NASA: University of Dayton Research Institute; 1996. [7] Lloyd R. Advanced multiple impact endgame model against ballistic missile payloads. In: I.R. Crewther (ed). 19th international symposium of ballistics. Interlaken: IBS 7e11 May 2001. [8] Cohen L. A debris cloud cratering model. International Journal of Impact Engineering 1995;17:229e40.