Scholarly article on topic 'HVI Ballistic Performance Characterization of Non-parallel Walls'

HVI Ballistic Performance Characterization of Non-parallel Walls Academic research paper on "Chemical engineering"

CC BY-NC-ND
0
0
Share paper
Academic journal
Procedia Engineering
OECD Field of science
Keywords
{Hypervelocity / "Orbital Debris" / Non-Parallel / Hydrocode / Hydrodynamics / Ballistic / Hypervelocity / Impact / Micrometeoroid / MMOD / HVI}

Abstract of research paper on Chemical engineering, author of scientific article — W.E. Bohl, J.E. Miller, E.L. Christiansen, B.A. Davis

Abstract The parallel, Double Wall, hypervelocity impact shield configuration has been heavily tested and characterized for ballistic performance for normal and oblique impacts for the ISS and other programs. However, in many locations with spacecraft designs, the rear wall cannot be modeled as being parallel or concentric with the outer bumper wall and in cases with a cylindrical outer wall, the effective non-parallel angle commonly varies as the outer wall impact location changes. This complicates micrometeoroid and orbital debris assessment of critical spacecraft components located within outer spacecraft walls. Based on a study combining hypervelocity impact testing and hydrodynamic impact simulations on multiple shield configurations including non-parallel first and rear walls, this paper provides equation adjustments for use with the double wall Ballistic Limit Equation (BLE) for a variety of impact speeds, non-parallel wall angles and impact obliquities.

Academic research paper on topic "HVI Ballistic Performance Characterization of Non-parallel Walls"

Available online at www.sciencedirect.com

SciVerse ScienceDírect Procedía

Engineering

ELSEVIER Procedía Engineering 58 (2013) 21 -30 ;

www.elsevier.com/locate/procedia

The 12th Hypervelocity Impact Symposium

HVI Ballistic Performance Characterization of Non-Parallel Walls

W.E. Bohla*, J.E. Miller3, E.L. Christiansenb, and B.A. Davisb

aLockheed Martin SSC, Denver, CO bNASA Johnson Spacecraft Center, Houston,

Abstract

The parallel, Double Wall, hypervelocity impact shield configuration has been heavily tested and characterized for ballistic performance for normal and oblique impacts for the ISS and other programs. However, in many locations with spacecraft designs, the rear wall cannot be modeled as being parallel or concentric with the outer bumper wall and in cases with a cylindrical outer wall, the effective non-parallel angle commonly varies as the outer wall impact location changes. This complicates micrometeoroid and orbital debris assessment of critical spacecraft components located within outer spacecraft walls. Based on a study combining hypervelocity impact testing and hydrodynamic impact simulations on multiple shield configurations including non-parallel first and rear walls, this paper provides equation adjustments for use with the double wall Ballistic Limit Equation (BLE) for a variety of impact speeds, non-parallel wall angles and impact obliquities.

© 2013 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the Hypervelocity Impact Society

Keywords: Hypervelocity; Orbital Debris; Non-Parallel; Hydrocode; Hydrodynamics; Ballistic; Hypervelocity; Impact; Micrometeoroid; MMOD; HVI.

Nomenclature

d diameter (cm)

k constant from Ref. 1

S separation (cm)

t thickness (cm)

V velocity (km/s) Greek symbols

a angle between walls (°)

0 obliquity (°)

p density (g/cm3)

er normalized yield strength from Ref. 1

Subscripts

B first wall

FP flight path

h high velocity

1 impact

l low velocity

p projectile

RW rear wall

JJ_parallel_

* Corresponding author. Tel.: +0-303-977-9740; E-mail address: william.e.bohl@lmco.com.

1877-7058 © 2013 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the Hypervelocity Impact Society

doi: 10.1016/j.proeng.2013.05.005

1. Introduction

The Double-Wall, "Whipple" Shield [1] has been the subject of many hypervelocity impact studies and has proven to be an effective shield system for Micro-Meteoroid and Orbital Debris (MMOD) impacts for spacecraft. The US modules of the International Space Station (ISS), with bumper shields offset from the pressure holding rear walls, provide examples of effective on-orbit use of the double wall shield. The concentric cylinder shield configuration which commonly has a large radius of curvature relative to separation distance is effectively represented for testing and analysis as a system of two parallel plates. As such, the parallel plate double wall configuration has been heavily tested and characterized for shield performance for normal and oblique impacts for the ISS and other programs.

While the parallel wall approximations have proven effective for many cases, the rear wall cannot always be modeled as being parallel or concentric with the outer bumper wall. As represented in Fig. 1, often there is an included angle between the two walls. For these impact geometries the core of the concept of double wall shield performance is applicable as the threat particle hits the outer bumper wall (first wall in a two wall stack) and then diffuses as it proceeds to the rear wall and is ultimately defeated prior to impacting the critical component; however, there is the additional obliquity effect that is due to the walls being non-parallel. Furthermore, with a cylindrical outer wall, the included angle can vary as the impact location on the outer wall changes. This non-parallel included angle complicates assessment of critical spacecraft components located within outer spacecraft walls.

Fig. 1. Representative non-parallel plate double wall MMOD shielding with the projectile flight path at oblique angle to first (outer) wall

To address the multiple obliquity effects presented by this configuration, a study including hypervelocity impact tests and hydrodynamic simulations has been undertaken by the NASA Johnson Space Center (JSC) Hypervelocity Impact Technology (HVIT) Group and the Multipurpose Crew Vehicle program. In this paper the results of the test series and hydrodynamic simulations are discussed along with the adaptations made to the current parallel wall ballistic limit equation that account for the independent influences of the impact obliquity as measured to the first and rear wall.

2. Hypervelocity impact test series

Hypervelocity impact tests have been performed by the JSC / HVIT Group at the NASA JSC White Sands Test Facility (WSTF) Hypervelocity Impact Range to develop reference points for hydrodynamic simulations and ballistic performance model development of a dual-wall, non-parallel Al6061-T6 shield system for ballistic limit equation refinements that account for the independent influences of the impact obliquity as measured to the first wall and the rear wall [2]. Fig. 2a is an image of a representative test article used in the test with the projectile flight path drawn in as a red arrow. The 0.0635 cm (0.025") first wall and 0.16 cm (0.063") rear wall test article was selected as being representative of the parallel plate 0.46-scale ISS test article of Piekutowski, et al. [3] which was used with approximately 7 and 9 km/s, 0° impact obliquity testing. The angles of concern to this study are drawn in Fig. 2b. The impact obliquity, 9j is the angle between the projectile velocity vector and the surface normal vector of the first wall and is positive when the projectile velocity vector is in the direction of the joint. The rear wall obliquity, 0RW, is the angle between the projectile velocity vector and the surface normal vector of the rear wall. The angle between the plates, a, is related to the impact and rear wall obliquities by

a - 0j + Orw.

The impact point for the test series is selected such that the distance between the plates, SFP, is 4.7 cm (1.85") corresponding to the distance between the parallel plates of Ref. 3.

Within the test series 45° and 90° angles between the plates are considered. For the targets with 45° between the plates, impact obliquities of 0°, 22.5° and 45° are considered. For the targets with 90° between the plates, only a 45° impact obliquity is considered. In addition to the impact obliquities and target types, the impact velocity is varied between two nominal impact velocities of 3.6 and 7.2 km/s. The projectile material is Al2017-T4 for all tests. Adjacent to the rear wall is an Al2024-T3 witness plate separated by 2" from the rear wall. A test matrix of the nine tests performed in this campaign is given in Table 1. Included in this table are the facility test number, target plate angle, the projectile diameter, impact velocity, and obliquity, along with, rear wall damage level. This study defines detached spall or perforation of the rear wall as failure.

Fig. 2. a) Representative non-parallel plate impact test article and, b) impact obliquity and non-parallel plate angle definitions (red arrow indicates flight path of the projectile

In Fig. 3 the images of the rear walls for the tests of Table 1 are given. For cases where the rear wall damage consists of only craters, the image of the forward surface of the rear wall is given, and for cases of attached/detached spall or perforation the rear surface image of the rear wall is given. In all of the images, the top of the image is in the direction of the joint between the first and rear wall. For the case where the angle between the plates, a, and the impact obliquity, 0, are both 45° the rear wall is normal to the flight path; consequently, the combination of 9t at 0° and 45° with a non parallel wall angle of 45° are the limits of normal impact on the first and rear wall, respectively. The combination of a at 45° and 0i at 22.5° is the bisector of these two limits, and the combination of a at 90° and 0/ at 45° is double this bisector.

Table 1. Nonparallel plate test matrix

HITF # a dp Vi Oi Rear Wall Rear Side Damage

(°) (cm) (km/s) (°)

12008 90 0.20 7.23 45 Craters/Max Deflection 0.34 mm

12009 45 0.26 7.10 0 Craters/Max Deflection 0.19 mm

12010 45 0.26 7.27 22.5 Detached Spall/9.5 mm x 9.9 mm

12011 45 0.22 6.74 45 Perforation/2.3 mm x 9.4 mm

12012 90 0.22 3.60 45 Perforation/3.6 mm x 3.2 mm

12013 45 0.20 3.44 0 Craters/Max Deflection 0.47 mm

12014 45 0.30 3.83 0 Attached Spall/Max Deflection 2.4 mm

12015 45 0.20 3.51 45 Perforation/2.5 mm x 3.1 mm

12024 45 0.30 7.15 0 Attached Spall/Max Deflection 2.0 mm

3. Hydrodynamic Modeling

Hydrodynamic modeling has been performed to assess the interdependency of the impact obliquity angle and the nonparallel plate angle on the ballistic performance of the double wall shield system. The ANSYS® AUTODYN® 3D [4]

modeling was performed using wall thicknesses and materials that are consistent with the hypervelocity test series. The standoff distance between the walls is measured along the projectile flight path and is consistent with the testing.

HITF12008 (Front Surface)

HITF 12009 ; Surface)

HITF 12010

HITF 12011

HITF 12012 (Rear Surface)

0 10mm -

■ ■ ' & I *

|iiii|ini|

HITF 12013 (Front Surface)

; n ' v

" ' . ;

I I I • I • I Otts«

HITF 12014 (Rear Surface

HITF 12015 (Rear Surface)

HITF 12024 (Rear Surface)

0 10mm

Fig. 3. Rear wall images of the non-parallel plate tests with image scale (top of the image is in the direction of the plate joint)Hydrodynamic impact modeling

3.1. Model calibration and material properties

The material parameters used with the equation of state, constitutive equation and failure model have been calibrated through an intensive study of parallel wall hypervelocity test results against AUTODYN® 2D modeling of the same. The material properties used with the modeling for the first wall, the rear wall and the projectile are provided in Table 2, Table 3 and Table 4 respectively. Descriptions of all of the properties can be found in the materials model section of Ref. 5, and the units for the value are provided with the property in the table. Based on the calibration modeling, the AUTODYN® supplied "Shock' equation of state performed well without modification for the aluminum 6061-T6 walls. A general "Johnson Cook" aluminum strength model is used for strength [6]. The Al2017-T4 material is not available in the AUTODYN library; however, the properties of Al2024-T351 are similar to Al2017-T4 and provided in the AUTODYN library. For this study, the Al2024-T351 models are used as a surrogate for the projectile material. The Grady Spall failure model with a critical strain value of 0.15 is used to model fracture of the first wall and proj ectile. The Hydro (Pmin) failure model, with the tensile failure parameter set to 900 MPa, provided the best correlation with the rear wall damage levels seen with the parallel wall hypervelocity impact test data and is used to model fracture of the rear wall.

Table 2. Aluminum 6061-T6 first wall material properties

Equation of State: Shock

Property Value (units) Property Value (units)

Reference density Gruneisen coefficient Parameter C1 Parameter S1 Parameter Quadratic S2 Relative volume, VE/V0 2.70300E+00 (g/cm3 ) 1.97000E+00 (none ) 5.24000E+03 (m/s ) 1.40000E+00 (none ) 0.00000E+00 (s/m ) 0.00000E+00 (none ) Relative volume, VB/V0 Parameter C2 Parameter S2 Reference Temperature Specific Heat Thermal Conductivity 0.00000E+00 (none ) 0.00000E+00 (m/s ) 0.00000E+00 (none ) 3.00000E+02 (K ) 8.85000E+02 (J/kgK ) 0.00000E+00 (J/mKs )

Strength Model: Johnson Cook

Property Value (units) Property Value (units)

Shear Modulus Yield Stress Hardening Constant Hardening Exponent Strain Rate Constant 2.76000E+07 (kPa ) 2.65000E+05 (kPa ) 4.26000E+05 (kPa ) 3.40000E-01 (none ) 1.50000E-02 (none ) Thermal Softening Exp Melting Temperature Ref. Strain Rate (/s) Strain Rate Correction 1.00000E+00 (none ) 7.75000E+02 (K ) 1.00000E+00 (none ) 1st Order

Failure Model: Grady Spall

Property Value (units) Property Value (units)

Critical Strain Value Crack Softening 1.50000E-01 (none ) No Stochastic failure No

Table 3. Aluminum 6061-T6 rear wall material properties

Equation of State: Shock

Property Value (units) Property Value (units)

Reference density Gruneisen coefficient Parameter C1 Parameter S1 Parameter Quadratic S2 Relative volume, VE/V0 2.70300E+00 (g/cm3 ) 1.97000E+00 (none ) 5.24000E+03 (m/s ) 1.40000E+00 (none ) 0.00000E+00 (s/m ) 0.00000E+00 (none ) Relative volume, VB/V0 Parameter C2 Parameter S2 Reference Temperature Specific Heat Thermal Conductivity 0.00000E+00 (none ) 0.00000E+00 (m/s ) 0.00000E+00 (none ) 3.00000E+02 (K ) 8.85000E+02 (J/kgK ) 0.00000E+00 (J/mKs )

Strength Model: Johnson Cook

Property Value (units) Property Value (units)

Shear Modulus Yield Stress Hardening Constant Hardening Exponent Strain Rate Constant 2.76000E+07 (kPa ) 2.65000E+05 (kPa ) 4.26000E+05 (kPa ) 3.40000E-01 (none ) 1.50000E-02 (none ) Thermal Softening Exp Melting Temperature Ref. Strain Rate (/s) Strain Rate Correction 1.00000E+00 (none ) 7.75000E+02 (K ) 1.00000E+00 (none ) 1st Order

Failure Model: Hydro (Pmin)

Property Value (units) Property Value (units)

Hydro Tensile Limit Reheal -9.00000E+05 (kPa ) Yes Crack Softening Stochastic failure No No

Table 4. Aluminum 2024-T351 projectile properties

Equation of State: Shock

Property Value (units) Property Value (units)

Reference density 2.78500E+00 (g/cm3 ) Relative volume, VB/V0 0.00000E+00 (none )

Gruneisen coefficient 2.00000E+00 (none ) Parameter C2 0.00000E+00 (m/s )

Parameter C1 5.32800E+03 (m/s ) Parameter S2 0.00000E+00 (none )

Parameter S1 1.33800E+00 (none ) Reference Temperature 3.00000E+02 (K )

Parameter Quadratic S2 0.00000E+00 (s/m ) Specific Heat 8.63000E+02 (J/kgK )

Relative volume, VE/V0 0.00000E+00 (none ) Thermal Conductivity 0.00000E+00 (J/mKs )

Strength Model: Johnson Cook

Property Value (units) Property Value (units)

Shear Modulus 2.76000E+07 (kPa ) Thermal Softening Exp 1.00000E+00 (none )

Yield Stress 2.65000E+05 (kPa ) Melting Temperature 7.75000E+02 (K )

Hardening Constant 4.26000E+05 (kPa ) Ref. Strain Rate (/s) 1.00000E+00 (none )

Hardening Exponent 3.40000E-01 (none ) Strain Rate Correction 1st Order

Strain Rate Constant 1.50000E-02 (none )

Failure Model: Grady Spall

Property Value (units) Property Value (units)

Critical Strain Value 1.50000E-01 (none ) Stochastic failure No

Crack Softening No

3.2. Non-parallel wall modeling

3D hydrodynamic modeling has been used to determine the ballistic performance of the shield system by finding the projectile sizes that are just defeated and those that just perforate or produce simulated release of detached spall from the rear wall for a range of non-parallel wall angles and the two nominal impact velocities (3.6 km/s and 7.2 km/s). The 3D models are constructed using Z plane symmetry with normal impact projectile velocities in the -Y direction and the first wall modeled in the X-Z plane. The rear walls are modeled using the Lagrange solver and a mesh having 8 elements (9 nodes) through its 0.16 cm (0.063") thickness. The 3D elements are approximately even sided, and with the 0 through 45 degree non-parallel cases, the rear walls are modeled with 125 elements along their 2.5 cm width and 225 elements along their 4.5 cm length. Thus for these angles a total of 225,000 elements are used to model the rear wall. With nonparallel angles above 45°, the rear wall is lengthened to 5.5 cm with 281 elements along the length (281,000 total elements). The rear walls are constrained with a velocity fixed boundary condition applied to nodes across the width at its two ends to simulate the boundary conditions of the restrained targets.

Due to the large deformation of the initial impact, the first walls and the projectiles are modeled using SPH (smooth particle hydrodynamics). A 0.0053 cm SPH element size is used with the majority of the cases and it provided 11 SPH elements through the 0.0635 cm (0.025") thickness of the first wall. For simulations of impacts normal to the first wall surface, the first wall was modeled as a disk having a radius approximately 10% - 20% larger than the projectile diameter. With oblique flight path simulations, ellipses are used to represent the first wall, with the semi-major axis sized to provide similar edge distances when accounting for the obliquity angle effects. The semi-minor axis is set in the same manner as the radius of the disks for normal first wall impact. The 0.0053 cm SPH nominal element size is reduced for very small projectiles for increased elements through its diameter, and the nominal element size is increased for very large projectiles to keep the total SPS count below 500,000. In keeping the first wall SPH element size equal to that of the projectile this resulted in greater or lesser elements through the first wall thickness. A 0.004 cm SPH size is used with the smallest projectile considered (1.6 mm) which resulted in approximately 15 SPH elements though the first wall thickness, and a 0.008 cm SPH size is used with the largest projectile considered (6.2 mm) which resulted in approximately 7 SPH elements through the first wall thickness.

With all of the 3D simulations the standoff distance between the two walls is modeled as 4.7cm (1.85") measured along the flight path. The left side of Fig. 4 provides the model layout for a representative simulation having a normal flight path into the first wall and a 45° non-parallel angle. The right side of Fig. 4 shows the developed debris cloud shortly before impact with the rear wall. The state of the rear wall after debris cloud impact is determined by density contours with a lower bound of 2.35 g/cm3 to indicate material separation due to fracture.

Impact simulations have been performed to model the tests performed in the previous section. These simulations test the simulation model against the test shots and provide additional data to characterize the interdependence between the impact obliquity and the non-parallel angle on overall ballistic performance. Table 5 provides the size of the projectiles simulated at impact velocities of 3.6 km/s, and Table 6 provides the same for impact velocities of 7.2 km/s. The table is organized by impact configuration including the first wall impact obliquity and the non-parallel plate angle. Impact simulations using the

given projectile size that showed perforation or detached spall are shown here in red and above the shield failed line. The simulations that show that the shield integrity is maintained are in black and below the shield failed line. When actual test velocities deviated significantly from the nominal values, those test velocities are substituted to facilitate a direct comparison.

Fig. 4. Left; representative non-parallel plate model setup showing the first wall (bumper), rear wall and the projectile, and Right; debris cloud prior to rear wall impact

The closest simulation matches to the test cases include the corresponding HITF numbers (from Table 1) on Tables 5 and 6. In general the simulations achieved good agreement with the tests. This was especially the case with the higher velocity simulations, Table 6, where the Failed/Not Failed results of the simulations matched the tests. However, there did appear to be a systematic bias with the simulations for increased projectile breakup at the lower test velocities (Table 5) where three Failed/No Failed deviations from the tests occurred; however, the deviation in diameter from the test results never exceeded 0.6 mm. Clearly there is limited test data and a plot that is provided on page 49 of Ref 3 illustrates the degree of data scatter that is encountered with Whipple Shield testing. The simulated damage achieved good agreement with observed damage at the higher test velocities which is more typical of the MMOD threat for most missions.

Table 5. Simulation results with approximately 3.6 km/s impact velocities based on the test configurations

Test Configuration (0„ a)

Low T1 (45°, 90°) T2 (0°, 45°) T3 (22.5°, 45°) T4 (45°, 45°)

Velocity d (mm) V (km/s) HITF # d (mm) V (km/s) HITF # d (mm) V HITF (km/s) # d (mm) V (km/s) HITF #

Failed 2.8 3.6 3.0 3.83 12014 2.0 3.6 2.2 3.6

Not Failed 2.6 3.6 2.8 3.6 1.8 3.6 2.0 3.6 12015

2.2 3.6 12012 2.6 3.6 1.5 3.6

Table 6. Simulation results with approximately 7.2 km/s impact velocities based on the test configurations

Test Configuration (0„ a)

High T1 (45°, 90°) T2 (0°, 45°) T3 (22.5°, 45°) T4 (45°, 45°)

Velocity d (mm) V (km/s) HITF # d (mm) V (km/s) HITF # d (mm) V HITF (km/s) # d (mm) V (km/s) HITF #

2.4 7.2 3.4 7.2 2.6 7.27 12010 2.2 7.2

Failed 2.2 7.2 3.2 7.2 2.2 7.27 2.2 6.74 12011

Not Failed 2.0 7.2 12008 3.0 7.2 12024 2.0 7.27 2.0 7.2

Given the performance of the developed simulation model with respect to the tests, simulations have been performed to bound the ballistic limit projectile diameter with non-parallel angles of 0°, 30°, 45°, 60° and 75°, and the results are summarized in Tables 7 & 8 in a similar format to those in Tables 5 & 6. In all cases the projectile impacts the first wall normal to the surface, which means the rear wall obliquity is the same as the non-parallel angle that is shown. The 3.0 mm simulation of the 45° non-parallel wall configuration has been performed at 3.83 km/s to match the test condition. As in Tables 5 & 6 projectile diameters corresponding to shield failure are shown in red, and projectile diameters that are defeated by the shield are shown in black. The two tables clearly show that the critical projectile size for rear wall failure has strong dependence on the non-parallel angle at both 3.6 and 7.2 km/s. The 2.6 mm, 7.2 km/s, 0° simulation closely matched one of the tests performed by Piekutowski, et al [3]. The test, No. 4-2010 with a 2.6 mm projectile fired at 7.25 km/s at 0° impact obliquity, was judged a pass (not failed) but the rear wall's rear surface exhibited a blister, an open crack and a spall ring. The simulation was classified as failed based on indication of very small detached spall from the rear wall (no perforation).

Table 7. Simulation results with 3.6 km/s impact velocity, normal obliquity and various non-parallel angles

3.6 km/s Non-Parallel Angle

0° 30° 45° 60° 75°

d (mm) d (mm) d (mm) d (mm) d (mm)

2.2 2.2 4.4

Failed 2.0 2.0 3.0* 4.2 5.8

Not Failed 1.8 1.8 2.8 4.0 5.6

1.6 2.6 3.6 5.4

* V = 3.83Km/s

Table 8. Simulation results with 7.2 km/s impact velocity, normal obliquity and various non-parallel angles

7.2 km/s Non-Parallel Angle

0° 30° 45° 60° 75°

d (mm) d (mm) d (mm) d (mm) d (mm)

2.8 3.4 4.4 6.2

Failed 2.6 3.0 3.2 4.2 6.0

Not Failed 2.4 2.8 3.0 4.0 5.8

2.6 3.6 5.6

4. Ballistic performance model development

With the non-parallel plate tests along with the hydrodynamic simulations to extend the available states, extensions to the ballistic performance model developed by Christiansen, E. L. [1] have been made to provide a smooth extension of the parallel wall model developed in that work. The parallel wall model as developed in Ref. 1 is a piecewise model on impact velocity and impact obliquity with a high velocity critical particle size given by

where onset is defined as VHi = l/CoslBj] km/s, and ftp , pg, pnW, and tmv are the projectile, first wall, rear wall densities in gjcm1, parallel wall separation and the thickness of the rear wall in cm, respectively. The terms k-r and ap, are

fit factors associated with aluminum for which this equation is developed. For the low velocity portion of the ballistic performance curve, the critical diameter is given by

where onset is defined as Vt = 3/Cvsfy]^1 km/s, and t3 is the thickness of the first wall in cm, fc;, c-;, and 0; are also fit

factors associated with aluminum. The intermediate combinations of impact velocities and obliquities are interpolated linearly with impact velocity while impact obliquity is held constant.

To accommodate non-parallel wall effects an attempt has been made to separate the obliquity effects to the controlling variable. In this separation it is assumed that the relevant thickness in the model is the thickness along the flight path; therefore, the oblique thickness is then given by the ratio of the normal thickness to the cosine of the relevant angle. The obliquity effect on separation distance is to reduce the debris cloud expansion as the debris has less length to expand towards the joint of the non-parallel walls over an impact on parallel plates; to this end, flight path distance is reduced by the cosine of the average obliquity, which is equal to half of the angle between the front and rear walls. In addition to these flight path effects, considering the low velocity impacts as momentum conserving interactions the velocity component normal to the wall is the damaging component; consequently, the normal component of the velocity needs to be considered. Using these assumptions, the modified form of the ballistic performance model from Ref. 1 becomes

An additional sixth root of impact obliquity is needed in Eqn. 5 to match Eqn. 3. As the high velocity onset represents the onset of fragmentation/melt of the projectile, its dependence on impact obliquity remains the same as that in Ref. 1; however, as the low velocity onset represents the retention of plastic response it becomes a function of both the impact and rear wall obliquity and is given by

_ a STT/acnsIgprtf]

" dw^itf^™.!«/»]' (6)

where the argument of the cosine in the denominator is the square root of the absolute value of the product of the impact and rear wall obliquity. As the angle between the walls corresponds to zero in a parallel wall impact, this onset equation reduces to that of Ref. 1 for parallel walls.

The critical diameter relationships to impact characteristics described in Eqn. 4, 5 and 6 collectively go to that described in Ref. 1 for the parallel wall configuration as shown in Fig. 5a. In this figure, the refined model is shown as solid curves while the original model is shown as dashed curves of a slightly darker shade for impact obliquities of 0°, 30°, 45° and 60° to first wall normal. The figure uses the material configuration of the test cases, and the parallel separation distance used is 4.7 cm (1.85"). The dynamic yield strength of Al6061-T6 used in the figure is 344.7 MPa (50 ksi) to better match the data from Ref. 3, and this dynamic yield strength is held constant among the two models for a direct comparison.

With the capability to reproduce the parallel wall model, the limit of first wall normal impact for non-parallel walls is shown in Fig. 5b. In this figure the revised ballistic performance model (solid curves) is compared to simulation points (circle data points) and impact tests (diamond data points). In the case of data points, open points are impact conditions that exceeded the shield's performance limit, and closed points are impact conditions where shield integrity is maintained. As can be seen in the figure, the revised model performs well relative to the impact tests and simulations with the exception at very shallow interaction angles with the rear wall. This is likely due to better projectile breakup as seen in the simulations than what is assumed in the impact models. This increased breakup moves the first interaction between the projectile and the rear wall closer to the first wall and puts a steeper component of projectile debris into the rear wall; however, as the projected area on the first wall that achieves a given separation is proportional to the cosine of the rear wall angle this difference between model and simulation results in a small non-conservativeness in risk calculations. Moreover, as the critical projectile size grows rapidly for these shallow interactions, a greater contribution to risk could be from other critical components normal to the first wall.

The comparison of the revised model with the impact tests (diamonds) and hydrodynamic simulations (circles) of 45° and 90° angles between the first and rear wall are shown in Fig. 6a and Fig. 6b, respectively. In this figure, the impact obliquity is varied between 0°, 22.5° and 45° relative to the first wall normal for the target with 45° between first and rear wall, and is limited to 45° for the target with 90° between the first and rear wall. The comparison is again favorable with the obtained data where open points are shield failures and closed points are shield passes.

0 5 10 15

Impact Velocity (km/s)

0 -30 45 -60 -75

Fig. 5. a) Comparison of refined aluminum double-wall ballistic performance model (solid curves) with Ref. 1 (dashed curves) for impact obliquities of 0°, 30°, 45° and 60° to target normal in parallel wall configuration, and b) comparison of refined aluminum double-wall ballistic performance model (solid curves) with simulations (circles) and tests (diamonds) for normal first wall impacts in 0°, 30°, 45°, 60° and 75° to rear wall normal configurations (open and closed data points are shield fail and pass, respectively).

-0 22.5 45

5 10 15

Impact Velocity (km/s)

Fig. 6. Comparison of refined aluminum double-wall ballistic performance model (solid curves) with simulations (circles) and tests (diamonds) for various impact obliquities in a) 45° and b) 90° non-parallel wall configurations (open and closed data points are shield fail and pass, respectively).

5. Conclusions

The high and low velocity regime Double Wall (Whipple Shield) ballistic limit equations are modified to account for the interdependence of the flight path obliquity angle to the first wall and rear wall. When used with a linear interpolation between the two velocity regimes, the modified ballistic limit equations are shown to perform well in predicting shield performance with the variety of angle combinations tested. Hydrodynamic simulations extended the available test data and aided the establishment of the sensitivity of the non-parallel wall angle on overall shield system performance. Future work is needed to characterize the performance of non-parallel walls at higher test velocities and additional hydrodynamic modeling refinement could be performed at impact velocities in excess of 7 km/s. Additionally, further work on the failure model, which is critical for characterizing projectile breakup and rear wall failure, would be beneficial.

References

[1] Christiansen, E. L., 2003. Meteoroid/Debris Shielding, NASA/TP-2003-210788, pp. 45-49.

[2] Davis, B. A., 2012. Non-Parallel Plate Hypervelocity Impact (HVI) Study, NASA, JSC-66364.

[3] Piekutowski, A. J., Poormon, K. L., Christiansen, E. L. and Davis, B. A., 2010. Performance of Whipple shields at impact velocities above 9 km/s, Proceedings of the 11th Hypervelocity Impact Symposium, International Hypervelocity Impact Society, pp. 495-503.

[4] AUTODYN User Manual Version 12.1, ANSYS, Inc., Cannonsburg, PA, November 2009.

[5] AUTODYN Theory Manual, ANSYS - Century Dynamics, Cannonsburg, PA, 2005, pp. 144-204.

[6] Johnson, G. R. and Cook, W. H., 1983. A constitutive model and data for metals subjected to large strains, high strain rates, and high temperatures, Seventh International Symposium on Ballistics, The Hague, The Netherlands.