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Ballistic Response of Pyramidal Lattice Truss Structures
A Thesis Presented to
the faculty of the School of Engineering and Applied Science
University of Virginia
In Partial Fulfillment of the requirements for the Degree
Master of Science (Engineering Physics)
By
Christian Joseph Yungwirth
May 2006
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APPROVAL SHEET
The thesis is submitted in partial fulfillment of the
Requirements for the degree of
Master of Science (Engineering Physics)
_______________________________ Author, Christian J. Yungwirth
This thesis has been read and approved by the examining committee:
_______________________________ Thesis advisor, Haydn N.G. Wadley
_______________________________ Committee Chairperson, Dana M.
Elzey _______________________________ Stuart A. Wolf Accepted for
the School of Engineering and Applied Science:
_______________________________ Dean, School of Engineering Applied
Science May 2006
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Abstract Cellular metal structures with periodic “lattice truss”
topologies are being utilized for an
expanding variety of multifunctional applications including
mitigation of the high
intensity dynamic loads created by nearby explosions. In these
situations, the panels are
also exposed to high velocity projectiles and their ballistic
response is then pertinent.
This thesis explores the ballistic resistance of a cellular
pyramidal lattice truss structure
fabricated from both a high ductility, high work hardening rate
304 stainless steel and an
age hardened 6061 aluminum alloy with similar yield strength,
but lower ductility and
significantly smaller work hardening rate. Projectiles made of
1020 carbon steel,
measuring 12.5 mm in diameter, made normal impact with these
sandwich structures.
The pyramidal lattice truss core sandwich panels had a core
relative density of
approximately 3% with cell sizes of approximately 2.54 cm x 2.54
cm x 2.54 cm and 1.5
mm thick faces that were 25.4 mm apart. The stainless steel
structures were first
penetrated at an impact velocity of approximately 450 m/s. Above
this critical velocity,
the exit velocity of the projectile was between 55 and 70% of
the impact velocity. The
sandwich structure outperformed a solid plate of similar
composition, with an equivalent
areal density of 28 kg/m2, exhibiting an exit velocity of the
projectile that was between 67
and 70%. The aluminum alloy structures were penetrated at the
lowest test velocity of
approximately 200 m/s. The exit velocity of the projectile was
between 60 and 92% of
the impact velocity. The stainless steel lattice structures were
then infiltrated with a
polyurethane that had a Tg of -56 °C, a low tensile modulus of
2.76 MPa and a high
elongation to yield of approximately 700%. Infiltration of the
stainless steel lattice with
this low Tg polyurethane exhibited a similar critical velocity
of approximately 450 m/s,
similar to the empty structure. Above the critical velocity, the
exit velocity of the
projectile was between 50 to 55% of the impact velocity at the
expense of doubling the
mass per unit area. Energy was mainly dissipated from the
associated strain fields as the
polymer was transiently displaced outwardly from the projectile.
Methods were
developed to fabricate other “hybrid” lattice truss structures
with various materials
infiltrated into the sandwich panel. These systems contained
ballistic fabrics, a different
polymer system and metal encased ceramics. Two of the systems
prevented penetration
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by projectiles with velocities in the 600 m/s range. The first
system was a polyurethane
infiltrated lattice that had a Tg of 49 °C. The significantly
higher Tg material had a higher
tensile modulus of 1120 MPa and a lower elongation to fracture
of 16%. A second
system containing 304 stainless steel encased alumina prisms,
with the surrounding space
infiltrated with a high Tg polyurethane, also resisted
penetration but at the expense of a
four fold increase in mass per unit area. The success of the
system can be attributed to
the degree of energy absorption of the alumina prisms and the
confinement of the
fragments. After fracturing, the ceramic fragments were
contained in steel tubes and
frictionally dissipated a majority of the remaining kinetic
energy while fracturing the
projectile. The remainder of the kinetic energy appeared to be
dissipated by the
polyurethane and plastic dishing of the rear metal facesheet.
These “hybrid” lattice
systems show significant promise as multifunctional load-bearing
structures that also
possess high ballistic performance.
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Acknowledgements I want to express my gratitude to my advisor
Professor Haydn N.G. Wadley. He has
been instrumental in sharpening my analytical tools and
assisting me in accomplishing
my goals. Allowing a large degree of latitude, he gave me the
opportunity to explore my
ideas and provided the resources to see them until their
conclusion. Along the journey, I
gained an immense professional respect for him and forged a
personal friendship that will
continue after my graduate education.
I want to extend my appreciation to the members of the IPM
Laboratory, in particular
Mrs. Sherri S. Thompson, Dr. Doug T. Queheillalt, Dr. Kumar P.
Dharmasena and Mr.
Rich T. Gregory. Without Sherri’s connection to the group
members or her “greasing the
wheels”, the IPM Laboratory would cease to function. I am
indebted to Doug and Kumar
for their tolerance of my innumerable questions and curiosities.
Their breadth of
knowledge was an invaluable resource that was not taken for
granted. I owe Rich thanks
for keeping the computers operating smoothly and maintaining my
lifeline to the group
over distances despite my occasional indoor headwear.
Additionally, I would like to express my sincere gratitude to
Dr. Mark T. Aronson and his
group at the University of Virginia for conducting chemical
characterizations, Dr. Alan
M. Zakraysek at the Naval Surface Warfare Center for conducting
ballistic testing, Dr.
Steve G. Fishman at the Office of Naval Research for providing
funding for my research
and my other committee members Dr. Dana M. Elzey and Dr. Stuart
A. Wolf.
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Dedication I would like to dedicate my efforts to my grandmother
Agnes V. Sands, my deceased
grandfather Joseph E. Sands, my mother Katherine M. Yungwirth,
my aunt Joann M.
Sands and my aunt Anne M. Sands. My accomplishments are a
testimonial to the abyss
of love and affection they have provided me from the day I was
brought into this world.
Every one of them has provided a safe haven where I could
venture into the farthest
expanses of my imagination and explore each crevice thoroughly.
These explorations
and their support have forged the man that stands today. Through
the trials and
tribulations, they have remained steadfast in their support even
with the occasional mild
opposition. Therefore, I extend my deepest, sincerest gratitude
to each of them and wish
that happiness and prosperity finds them on their continued
journey through life.
Additionally, I would like to make a dedication to all of my
friends and the beloved
people for who I care deeply, particularly Janet and the
Conterelli’s. Janet has been a
pillar of support that I have come to depend and I look forward
to a future rich with
joyous memories shared with her. She is an amazing woman that
epitomizes beauty,
intelligence and loyalty. The Conterelli’s have lovingly
embraced me into their lives and
their hearth, a deed that has earned my eternal gratitude and
appreciation.
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Quotations Ad astra per aspera (A rough road leads to the
stars)
- Plaque dedicated to the crew of Apollo 1 at Launch Complex 34,
Kennedy Space Center
Γνώθι Σεαυτόν (Gnothi Seauton): “know thyself” Μηδέν Άγαν (Meden
Agan): "nothing in excess"
- Inscribed in golden letters at the lintel of the entrance to
the Temple of Apollo at
Delphi He who fights with monsters might take care lest he
thereby become a monster. And if you gaze for long into an abyss,
the abyss gazes also into you.
- Friedrich Nietzsche, Beyond Good and Evil If you aspire to the
highest place, it is no disgrace to stop at the second, or even the
third, place.
- Cicero
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i
Table of Contents Table of Contents
...............................................................................................................
i List of
Figures...................................................................................................................
iii List of Tables
..................................................................................................................
viii List of Symbols
.................................................................................................................
ix Chapter 1.
Introduction........................................................................................................1
1.1 Multifunctional Cellular Materials
........................................................1 1.2
Ballistic Properties of Cellular
Metals...................................................4 1.3
Goals of this
Thesis................................................................................6
1.4 Thesis Outline
........................................................................................6
2. Impact and Plate Penetration Mechanics
........................................................7 2.1
Impact Mechanics
..................................................................................7
2.2 Plate Impact Mechanics
.......................................................................11
3. Materials and
Structures.................................................................................19
3.1 Sandwich Panel
Fabrication.................................................................19
3.2 Relative Density
Relations...................................................................22
3.3 Alloy Mechanical
Properties................................................................23
3.3.1 304 Stainless Steel
.......................................................23 3.3.2 Age
Hardened 6061-T6 Aluminum Alloy ...................24 3.4 Polymer
Infiltrated
...............................................................................25
3.4.1 Hybrid Lattice
Fabrication...........................................25 3.4.2
Polymer
........................................................................26
3.5 Polymer
Characterization.....................................................................28
3.5.1 DSC
Analysis...............................................................28
3.5.2 DMA Analysis
.............................................................29 4.
Ballistic Testing
................................................................................................33
4.1 Stage One Powder
Gun........................................................................33
4.2 Sabot and Projectile
.............................................................................34
4.3 Test Fixture
..........................................................................................35
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5. Empty Lattice Resistance
................................................................................38
5.1 304 Stainless Steel Panel Response
.....................................................38 5.2 304
Stainless Steel Plate Response
......................................................44 5.3 AA6061
Panel
Response......................................................................49
5.4
Discussion............................................................................................55
6. Polymer Infiltration Study
..............................................................................57
6.1 Ballistic
Response................................................................................57
6.2
Discussion............................................................................................63
7. Enhanced Ballistic Lattice Fabrication
...........................................................65 7.1
Concept
Systems..................................................................................65
7.2 Lattice Structure
Fabrication................................................................66
7.3 Double Layer Lattice Relative
Density................................................67 7.4
Infiltration Materials and Methods
......................................................69 7.4.1
Polymers
......................................................................69
7.4.2 Fabric
...........................................................................70
7.4.3 Metal Encased Ceramic Prisms
...................................70 7.5 Hybrid Lattice Relative
Density
..........................................................71 7.6
Material
Properties...............................................................................71
7.6.1 Brazed 304 Stainless Steel
...........................................73 7.6.2 PU 2
.............................................................................74
7.6.2.1 DSC Analysis....................................74 7.6.2.2
DMA Analysis ..................................75 8. Ballistic
Testing
................................................................................................78
8.1 Test
Setup.............................................................................................78
8.2
Results..................................................................................................80
8.2.1 Single Layer Empty
System............................................80 8.2.2 Soft
Polymer Filled System
............................................82 8.2.3 Double Layer
Filled with PU 1.......................................84 8.2.4
Hard Polymer Filled
System...........................................85 8.2.5 Single
layer filled with PU 1 plus Fabric........................87 8.2.6
Ceramic plus PU 2 Filled System
...................................88 8.3
Discussion............................................................................................89
9.
Discussion..........................................................................................................91
10. Conclusions
.....................................................................................................96
References
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iii
List of Figures
Figure 1. a) Photograph and illustration of Alporas®, a
stochastic closed cell aluminum foam manufactured by Foam Tech Co.
Ltd. via titanium hydride particle decomposition. b) Photograph and
illustration of Duocel®, a stochastic open cell aluminum foam
manufactured by ERG Materials and Aerospace Corp. via pressure
casting. Figure 2. Isometric view of a) Hexagonal honeycomb b)
Square honeycomb c) Triangular honeycomb d) Triangular corrugation
e) Diamond corrugation f) navtruss corrugation g) Tetrahedral
lattice truss h) Pyramidal lattice truss and i) 3-D Kagomé lattice
truss structures between solid face sheets. The tetrahedral lattice
truss has three sets of triangular, prismatic voids (0°/60°/120°).
The pyramidal lattice truss possesses similar geometrical voids
running orthogonally (0°/90°) through the lattice. The 3-D Kagomé
lattice truss possesses two sets of similar geometrical void
orientations (0°/60°/120° and 30°/90°/150°). Figure 3.
Bonded-interface test result showing section view with subsurface
accumulated damage beneath the indentation (Left). Finite-element
result showing extent of the plastic zone in terms of contours of
maximum shear stress at 5.0/ =YMaxτ for indenter load NP 1000=
(Right). Distances are expressed in terms of the contact
radius,
mma 326.00 = , for the elastic case of NP 1000= . The bold black
line indicates the
radius of the circle of contact, mma 437.00 = , as determined
from the finite-element
calculation [393H57]. Figure 4. Illustration of a thin target
showing a) bulging b) dishing and c) cratering [407H69]. Figure 5.
Perforation mechanisms [421H69]. Figure 6. Manufacturing process
for making pyramidal lattice truss cored sandwich panels [465H3].
Figure 7. Illustration of the laser welding process for bonding the
truss lattice to proximal and distal facesheet. Figure 8. Cross
section of the single layer empty pyramidal truss lattice along a
nodal line. Figure 9. Unit cell geometry used to derive the
relative density for single layer pyramidal topology. Figure 10.
Uniaxial tension data for as-received 304 stainless steel. Figure
11. Uniaxial tension data for the age hardened 6061-T6 aluminum
alloy. Figure 12. Heat capacity (Rev Cp) of PU 1 as a function of
temperature (°C).
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iv
Figure 13. Storage modulus at a frequency of 1 Hz for PU 1 as a
function of temperature. Figure 14. Predicted values for the
storage and loss modulus of PU 1 at a reference temperature of 25
°C as a function of frequency. Figure 15. Tan δ ( EE ′′′ / ) of PU
1 as a function of frequency. Figure 16. Illustration of the single
stage powder gun used for ballistic studies. The sabot carried a
12.5 mm spherical projectile. Figure 17. Illustration of the sabot
used to carry the projectile. Figure 18. Illustration of setup used
in the blast chamber to mount the samples, measure velocities and
record via high-speed photography. Figure 19. Plot of exit velocity
of the projectile (m/s) as a function of the impact velocity (m/s)
for the 304 stainless steel pyramidal truss lattice. Figure 20.
Plot of energy absorbed (J) as a function of the impact velocity
(m/s) for the 304 stainless steel pyramidal truss lattice system.
Figure 21. a) Cross section of the 304 stainless steel sample that
was impacted at 339.2 m/s and was not fully penetrated, shot 1 b)
Cross section of the entry hole of shot 1 c) Cross section of the
exit hole of shot 1. Figure 22. a) Cross section of the 304
stainless steel sample that was impacted at 810.8 m/s, shot 3 b)
Cross section of the entry hole of shot 3 c) Cross section of the
exit hole of shot 3. Figure 23. a) Cross section of the 304
stainless steel sample that was impacted at 1206.1 m/s, shot 54 b)
Cross section of the entry hole of shot 54 c) Cross section of the
exit hole of shot 54. Figure 24. Plot of exit velocity of the
projectile (m/s) as a function of the impact velocity (m/s) for the
304 stainless steel monolithic plate compared to the 304 stainless
steel pyramidal truss lattice. Figure 25. Plot of energy absorbed
(J) as a function of the impact velocity (m/s) for the 304
stainless steel pyramidal truss lattice system and solid plate.
Figure 26. Cross section of the monolithic 304 stainless steel
plate that was shot at 341.7 m/s, shot 58. Figure 27. Cross section
of the monolithic 304 stainless steel plate that was shot at 509.6
m/s, shot 105.
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v
Figure 28. Cross section of the monolithic 304 stainless steel
plate that was shot at 1226.5 m/s, shot 108. Figure 29. Plot of
exit velocity of the projectile (m/s) as a function of the impact
velocity (m/s) for the age hardened AA6061 aluminum alloy pyramidal
truss lattice compared to the 304 stainless steel pyramidal truss
lattice system. Figure 30. Plot of the energy absorbed (J) as a
function of the impact velocity (m/s) for the 304 stainless steel
sandwich panel and AA6061 aluminum alloy sandwich panel. Figure 31.
a) Cross section of the AA6061 sample that was impacted at 280.1
m/s, shot 114 b) Cross section of the entry hole of shot 114 c)
Cross section of the exit hole of shot 114. Figure 32. a) Cross
section of the AA6061 sample that was impacted at 493.3 m/s, shot
81 b) Cross section of the entry hole of shot 81 c) Cross section
of the exit hole of shot 81. Figure 33. a) Cross section of the
AA6061 sample that was impacted at 1222.2 m/, shot 55 b) b) Cross
section of the entry hole of shot 55 c) Cross section of the exit
hole of shot 55. Figure 34. Plot of exit velocity of the projectile
(m/s) as a function of the impact velocity (m/s) for the 304 hybrid
stainless steel pyramidal truss and the empty 304 stainless
pyramidal truss lattice. Figure 35. Plot of projectile the energy
absorbed (J) as a function of the impact velocity (m/s) for the 304
stainless steel pyramidal truss lattice infiltrated with PU 1 and
the empty 304 stainless pyramidal truss lattice. Figure 36. a)
Cross section of the 304 stainless steel sample infiltrated with PU
1 that was impacted at 370.9 m/s, shot 70 b) Cross section of the
entry hole of shot 70 c) Cross section of the exit hole of shot 70.
Figure 37. a) Cross section of the 304 stainless steel pyramidal
truss lattice filled with PU 1 that was impacted at 515.7 m/s, shot
110 b) Cross section of the entry hole of shot 110 c) Cross section
of the exit hole of shot 110. Figure 38. a) Cross section of the
304 stainless steel pyramidal truss lattice filled with PU 1 that
was impacted at 984.5 m/s, shot 112 b) Cross section of the entry
hole of shot 112 c) Cross section of the exit hole of shot 112.
Figure 39. Schematic illustrations of pyramidal lattice truss
concepts evaluated in the study. a) Empty pyramidal truss lattice
b) Polymer filled in truss lattice c) Ballistic fabric interwoven
between trusses with polymer filling remaining air space d) 304 SS
encased alumina inserted in triangular prismatic voids and
remaining air space filled with polymer
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vi
Figure 40. Unit cell geometry used to derive the relative
density for a double layer pyramidal topology. Figure 41. Uniaxial
tension data for brazed 304 stainless steel. Figure 42. Heat
capacity (Rev Cp) PU 2 as a function of temperature (°C). Figure
43. Storage modulus at a frequency of 1 Hz for PU 2 as a function
of temperature. Figure 44. Predicted values for the storage and
loss modulus of PU 2 at a reference temperature of 25 °C as a
function of frequency. Figure 45. Tan δ ( EE ′′′ / ) of PU 2 as a
function of frequency. Figure 46. Ballistic testing configuration.
Ball bearing projectiles with a radius of 6 mm and weight of 6.9 g
were used. An impact velocity of approximately 600 m/s was used for
all the tests. Figure 47. Projectile impact location for a) Single
and b) Double layer pyramidal lattice truss sandwich panels. Figure
48. Test 1: a) Cross section of the single layer empty pyramidal
truss lattice along a nodal line b) Cross section of the single
layer empty pyramidal truss lattice after a projectile impact of
598 m/s. Figure 49. High-speed photography of a projectile impact
with the empty single layer pyramidal lattice (system 1). Each
figure (a)-(h) depicts a frame of the high-speed photography. The
time in microseconds (μs) is labeled from the initial impact of the
projectile with the proximal facesheet. Figure 50. Position of a
spherical projectile from the proximal facesheet of the empty
single layer pyramidal lattice truss as a function of time. To the
left of the time of impact is before the impact of the projectile
and the right of the time of impact is after impact of the
projectile. Figure 51. Test 2: a) Cross section of the single layer
pyramidal truss lattice filled with PU 1 b) Cross section of the
single layer pyramidal truss lattice filled with PU 1 after a
projectile impact of 616 m/s. Notice the brass breech rupture disk
(b) remaining in the polymer while the projectiles path resealed.
Figure 52. Test 7: a) Cross section of the double layer pyramidal
truss lattice filled with PU 1 b) Cross section of the double layer
pyramidal truss lattice filled with PU 1 after an impact of 613
m/s.
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Figure 53. Test 4: a) Cross section of the single layer
pyramidal truss lattice filled with PU 2 b) Cross section of the
single layer pyramidal truss lattice filled with PU 2 after a
projectile impact of 632 m/s showing approximately 8.5 mm
deflection of the rear face panel. Note that the projectile is
visibly arrested in (b). Figure 54. X-ray tomography of specimen 4.
Side profile (Left). Elevated view above proximal face sheet
(Right). Figure 55. Test 5: a) Cross section of the single layer
pyramidal truss lattice filled with the interwoven fabric and the
PU 1 b) Cross section of the single layer pyramidal truss lattice
filled with interwoven fabric and the PU 1 after a projectile
impact of 613 m/s. Figure 56. Test 6: a) Cross section of the
single layer pyramidal truss lattice filled with 304 stainless
steel prisms and PU 2 b) Cross section of the single layer
pyramidal truss lattice filled with 304 stainless steel prisms and
PU 2 after an impact of 613 m/s.
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viii
List of Tables Table 1. Manufacturer reported properties for the
polyurethane system. Table 2. The impact and exit velocities of the
projectile, nodal disbonding of the distal facesheet and whether
the projectile penetrated the distal facesheet for the 304
stainless steel pyramidal truss lattice sandwich structure. Table
3. The impact and exit velocities of the projectile for a 3 mm
thick 304 stainless steel monolithic plate. Table 4. The impact and
exit velocities of the projectile, nodal disbonding of the distal
facesheet and whether the projectile penetrated the distal
facesheet for the age hardened AA6061-T6 aluminum alloy pyramidal
truss lattice sandwich structure. Table 5. The impact and exit
velocities of the projectile, nodal disbonding of the distal
facesheet and whether the projectile penetrated the distal
facesheet for the 304 stainless steel pyramidal truss lattice
sandwich structure with polyurethane. Table 6. Physical
descriptions of composite lattice truss systems fabricated. Table
7. Manufacturer reported properties for the polyurethane system.
Table 8. Physical properties of AD-94 Al2O3 triangular prisms.
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ix
List of Symbols Δ distance of mutual approach between indenter
and specimen δ parameter used to assess dissipative energy
efficiency ν Poisson’s ratio π pi ρ mass density ρ relative density
θ petal rotation angle at the end of stages σ stress τ shear stress
υ0 initial velocity of projectile υr residual velocity of
projectile ω angle between truss and facesheet a indenter contact
area radius b triangle base height cp heat capacity cpr wave
velocity of projectile ct dilatational wave velocity of target d
diameter h height h0 plate thickness k mass ratio l length m mass
pm mean contact pressure p0 maximum contact pressure (Hertz stress)
r radial distance t thickness w width
⎪⎭
⎪⎬
⎫
zyx
Cartesian coordinates
E Young’s modulus Ec perforation energy of plate Ed energy
absorbed through plate dishing E* contact modulus E ′ storage
modulus E ′′ loss modulus P indenter load force R (reduced) radius
of sphere Tg glass transition temperature V volume W work
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x
Subscripts θ angular cylindrical coordinate a aluminum oxide c
unit cell cr crack f fabric i intermediate plate m base metal p
projectile pu polyurethane r radial cylindrical coordinate t target
tr truss u ultimate y yield z height cylindrical coordinate
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1
Chapter 1 Introduction Cellular metals are a relatively new
class of materials [1-2]. Using foaming or foam derived methods,
various groups developed stochastic topology structures in the
1980’s [1]. Examples of closed and open cell systems are shown in
Figure 1. More recently, methods have begun to be developed to
create open cell topology structures with periodic, or lattice
cells [3] and compliment closed cell periodic systems (e.g.
honeycombs) that have been developed for weight sensitive
structural applications [3-4].
1.1 Multifunctional Cellular Materials
Cellular metal structures with both stochastic (metal foams)
[2-6], Figure 1, and periodic
topologies [5,6], Figure 2, are being utilized for an expanding
variety of structural [3-12],
thermal [13-15], and acoustic damping [2] applications.
a) Closed-cell Metal Foam
b) Open-cell Metal Foam
Figure 1. a) Photograph and illustration of Alporas®, a
stochastic closed cell aluminum foam manufactured by Foam Tech Co.
Ltd. via titanium hydride particle decomposition. b) Photograph and
illustration of Duocel®, a stochastic open cell aluminum foam
manufactured by ERG Materials and Aerospace Corp. via pressure
casting.
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2
Figure 2. Isometric view of a) Hexagonal honeycomb b) Square
honeycomb c) Triangular honeycomb d) Triangular corrugation e)
Diamond corrugation f) navtruss corrugation g) Tetrahedral lattice
truss h) Pyramidal lattice truss and i) 3-D Kagomé lattice truss
structures between solid face sheets. The tetrahedral lattice truss
has three sets of triangular, prismatic voids (0°/60°/120°). The
pyramidal lattice truss possesses similar geometrical voids running
orthogonally (0°/90°) through the lattice. The 3-D Kagomé lattice
truss possesses two sets of similar geometrical void orientations
(0°/60°/120° and 30°/90°/150°). The periodic structures show
significant promise as multifunctional structures when
configured as the cores of sandwich panel structures. In these
scenarios, functions such
as structural load support and thermal management can be
simultaneously exploited
[11,13,15].
Periodic structures consisting of 3-D space filling unit cells
with honeycomb [3,16-17],
corrugation [18] or lattice truss topologies [3,7] are
significantly more structurally
efficient than equivalent relative density metal foams. The
fabrication routes developed
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3
for these periodic cellular systems [19] also enable much higher
strength alloys to be
used. As a result, periodic topology structures can be an order
of magnitude, or more,
stronger than metal foams of the same mass [12].
As the relative density decreases, lattice topologies have been
shown to have higher
strengths than honeycombs and simple corrugations [20]. The
first proposed lattice
structure was lattice block material [21-23]. More recently,
structures based on the octet
truss (i.e. a tetrahedral structure) [24], a pyramidal truss
[25-26], the 3-D Kagomé [27-28]
and various lattices created by weaving or laying up metal wires
and tubes have all been
developed [7]. Figure 2 showed examples. The cell size of these
structures can be varied
from several hundreds of micrometers to several centimeters
using metal folding and
either brazing or spot welding fabrication methods [29,16].
All cellular metals have been shown to possess excellent impact
energy absorption
characteristics [11,30-33]. Typically, these materials exhibit
three regions of deformation
[1]. The first region is an elastic region followed by a plateau
stress region persisting to
plastic strains of around 60-70%. It corresponds to a region
where buckling and plastic
collapse of the cell walls occurs. Finally, after the collapse
of the cells, sufficient
densification of the structure has occurred that cell wall/truss
impingement causes a sharp
rise in stress. This arises because of their very extensive
crush strains at near constant
flow stress. The mechanics of foam deformation and associated
energy absorption have
been reviewed by M. Ashby et al. [2], and includes expressions
for foam elastic modulus,
elastic collapse stress, plastic collapse, strength and
densification strain etc.
Recent experimental and numerical modeling studies indicate that
periodic lattice truss
and honeycomb core sandwich panels enable significant mitigation
of explosion created
shock waves [31-34]. These studies indicate that sandwich panels
fabricated from high
ductility metals (e.g. stainless steels and some aluminum
alloys) with honeycomb, lattice
truss or corrugated cores could provide multifunctional static
load support and blast
protection in air and underwater [36]. If cellular metal
structures of this type were used
for air blast mitigation applications, they would also be
exposed to impact by high
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4
velocity projectiles. Very little is known about the penetration
resistance of these
structures or ways to enhance it.
1.2 Ballistic Properties of Cellular Metals
A study conducted by B. Gama et al. [37] has explored the
ballistic characteristics of a
cellular metal. It investigated metal foams made from low
strength aluminum alloys in
the context of integral armor concepts and reported only modest
system performance
enhancements. In this application, closed-cell aluminum foam
delayed and attenuated
stress wave propagation throughout the composite integral armor
system. The cellular
structure of the metal foam acted as small waveguides and a
geometric dispersion of the
stress waves occurred leading to propagation delays. Damping in
these systems has been
studied by D. Radford et al. at the University of Cambridge
[31-34] and is associated
with thermo-elastic effects. These studies provided little
illumination of the performance
of periodic lattice truss topologies, or sandwich panels
constructed from them when
exposed to high velocity projectiles.
It is to be expected that the two solid faces of a sandwich
panel will each individually
provide some level of projectile propagation resistance. The
penetration of a metal sheet
such as rolled homogenous armor (RHA) [35] by a normal incidence
projectile has been
widely studied [38]. The critical velocity (i.e. the velocity at
which the projectile
penetrates the target) increases linearly with target thickness
[32,35]. The depth of
penetration (DOP) also increases linearly as the projectile
velocity is increased [32, 35].
Experimental studies by A. Almohandes et al. [39] indicated that
distributing the mass of
a plate amongst a pair of plates of equivalent areal density
resulted in a slight lowering of
the ballistic resistance. Theoretical studies by G. Ben-Dor et
al. [40] and experimental
studies by J. Radin and W. Goldsmith [41] indicate that the
distance between such a pair
of plates has little or no effect upon the ballistic resistance
of such systems. Other work
conducted by R. Corran et al. [42] found that two plates in
tight contact had a slightly
higher ballistic limit than an identical pair that was not in
contact. They tentatively
attribute this small effect to a frictional interaction between
layers.
-
5
The lattice truss structure itself might be anticipated to have
some effect upon the
propagation of a projectile provided the projectile impacts the
lattice during penetration
(i.e. the cell spacing is small compared to the projectile
diameter). For example, it might
increase the ballistic performance by deflecting (tipping) the
projectile or causing some
of its energy to be dissipated by plastic deformation/fracture
of the trusses.
Projectile kinetic energy losses during penetration of the face
sheets and the truss
structures are likely to be increased by utilizing metals with
high strength, high fracture
toughness (ductility) and high strain and strain rate hardening
coefficients. Many
austenitic and super austenitic stainless steels [43] have
medium strength levels but high
toughness and strain rate hardening coefficients. Analytical and
experimental results
from a study conducted by S. Shun-cheng et al. [44] showed that
for 304 stainless steel,
the yield stress increased with increasing strain rate until an
upper limit of approximately
2500 s-1. Other materials, such as AA6061-T6 aluminum alloy,
exhibit a decrease in
strength as the strain rate is increased [45]. Recent
developments in the fabrication of
lattice structures from such alloys using perforated metal
folding and brazing techniques
[3,29] now enable an experimental assessment of the ballistic
behavior of sandwich
panels with lattice truss cores to be investigated.
The voids in lattice truss structures provide easy access to the
interior of the sandwich
panel and enable materials to be added that might improve
ballistic resistance. For
example, the voids could be infiltrated with polymers to
dissipate a projectiles kinetic
energy [46], or with ballistic fabrics to arrest fragments
[47,48] or with hard ceramics that
fragment projectiles and impede their penetration [49,50]. The
merits of these are also
presently unclear and no experimental assessments of the
ballistic properties and
deformation mechanisms of these “hybrid” lattice truss
structures have ever been
reported.
-
6
1.3 Goals of this Thesis
This thesis experimentally investigates the ballistic response
of stainless steel and 6061
aluminum alloy pyramidal lattice truss core sandwich structures
using spherical
projectiles with impact velocities up to approximately 1200 m/s.
The stainless steel
sandwich panel structures response is compared to that of a
monolithic plate of
equivalent areal density (mass per unit area). The effects of
filling the lattice void space
with an elastomer are then investigated and the feasibility of
fabricating more
sophisticated “hybrid” sandwich structures containing ceramics
and ballistic fabrics is
added. The study finds significantly enhanced ballistic
resistance can be achieved by this
approach.
1.4 Thesis Outline
The thesis is organized as follows: Chapter 2 presents the
mechanisms of impact and
plate penetration mechanics. Chapter 3 presents the materials
and the fabrication
methodology for the lattice truss sandwich structure. Chapter 4
describes the ballistic
facility used to conduct the experiments and the
sabot-projectile system. Chapter 5
presents the initial impact study of the 304 stainless steel and
the age hardened AA6061-
T6 aluminum alloy mono-layer pyramidal lattice truss sandwich
structures. Chapter 6
presents a study infiltrating the 304 stainless steel pyramidal
lattice truss sandwich
structure with an elastomer. Chapter 7 presents the fabrication
of hybrid systems where
various materials were infiltrated into the structure and
Chapter 8 presents the results of
the study. Chapter 9 summarizes the findings from the studies
while Chapter 10 briefly
lists the conclusions obtained.
-
7
Chapter 2 Impact and Plate Penetration Mechanics
2.1 Impact Mechanics
The impact of a hard projectile with a softer target causes
local deformation (i.e. indent of
both objects). The first attempt to develop a theory of the
local indentation at the contact
between two solid bodies was by Hertz [51], who likened the
problem to an equivalent
one in electrostatics. Hertzian contact mechanics is based on
three key assumptions:
i. The surfaces of the contacting bodies are both continuous,
smooth, nonconforming and form a frictionless contact.
ii. The strains associated with the deformations are small. iii.
Each solid behaves as an elastic half-space in the vicinity of the
contact
zone. The size of the contact area (extent of the deformation
field) is therefore small compared to the size of the bodies.
According to Hertz, if two elastic spheres with radii R1 and R2
are pressed into contact
with a force P, the resultant circular contact area has a
radius, a, such that:
31
*43
⎟⎠⎞
⎜⎝⎛=
EPRa (1),
where E* is the contact modulus defined by:
2
22
1
21* 11
EEE νν −+−= (2).
In equation (2), E and ν are the Young’s modulus and elastic
Poisson’s ratio of each
sphere, respectively. In equation (1), R is the reduced radius
of curvature and is related to
those of the individual components by the relation:
21
11RR
R += (3).
Convex surfaces are taken as positive radii of curvature
(concave surfaces are therefore
taken as negative radii of curvature). If one of the solids is a
plane surface then its
effective radius is infinite so that the reduced radius of the
contact is numerically equal to
-
8
that of the opposing sphere. This is then reduced to the
half-space problem [52]. If we
place a cylindrical coordinate system at the initial point of
contact, the resulting radial
pressure distribution, p(r), is axisymmetric and dependent only
upon the radial distance
from the initial point of contact.
The pressure distribution is semi-elliptical, and of the
form
21
2
2
0 1)( ⎟⎟⎠
⎞⎜⎜⎝
⎛−=
arprp (4),
where 222 yxr += is the radial distance from the initial point
of contact. The maximum
pressure, p0, occurs on the axis of symmetry. This and the mean
pressure, pm, are related:
31
23
2*
206
23
23
⎟⎟⎠
⎞⎜⎜⎝
⎛===
RPE
aPpp m ππ
(5).
The maximum pressure, p0, is also sometimes known as the Hertz
contact stress.
Under this loading, the two spheres move together by a small
displacement, Δ, given by:
31
2*
2
*0
2
169
2 ⎟⎟⎠
⎞⎜⎜⎝
⎛===Δ
REP
Epa
Ra π (6).
Equation (63) is a quasi-static derivation of a sphere making
contact with a sphere or plane
with a load placed on the axis of symmetry to cause a
displacement in the direction of
mutual approach.
In a dynamical derivation of a sphere impacting a flat plane
specimen in the elastic region
[53], the second derivative of the displacement of the plane is
related to the mass of the
projectile, mp, and the force of the projectile impact, P:
Pdt
tdmp −=Δ
2
2 )( (7),
where Δ is the displacement from the flat plane specimen.
-
9
By rearranging equation (6) to give an expression for P(Δ) and
equating it to equation (7),
we obtain the indentation velocity:
*23
21
34
ER
dtdmp
Δ−=
υ (8),
where dtdΔ
=υ . Multiplying both sides of equation (8) by velocity and
integrating from
the impact velocity of the projectile, 0υ , to the final
velocity, 0=fυ , we obtain:
25
*21
20 15
821
Δ= ERmpυ (9).
The left hand side of equation (9) equates the kinetic energy of
the projectile to the strain
energy stored in the specimen.
Equation (9) can be arranged to give an expression for the depth
of penetration, Δ, as a
function of the mass, mp, and the impact velocity, 0υ , of the
projectile:
52
*21
20
16
15⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=Δ
ER
mpυ (10).
This relationship is limited to elastic impacts (i.e. when the
impact velocity is low) and
both objects are made of materials of high strength. It does not
address the plasticity and
fracture that can accompany projectile penetration [52, 53].
An elastic-plastic material will reach the limit of its elastic
behavior at the point beneath
the surface where the maximum contact pressure p0 at the instant
of maximum
compression has reached the von Mises flow criterion. The von
Mises yield criterion for
ductile materials can be written [53]:
( ) ( ) ( )[ ]36
1 22213
232
221
ykσ
σσσσσσ ==−+−+− (11),
where yσ is the yield stress of the impacted (usually softer)
material and iσ are the
principal stress components (i.e. the stress components along
the principal axes) [52].
For the axisymmetrical problem of a sphere impacting a flat
plane, the principal axes are
-
10
with the cylindrical coordinate axes, and thus the principal
stresses are zσ , rσ and θσ
with θσσ =r . Given the relation between the maximum contact
pressure, p0, and the
principal stress components [53], and assuming an elastic
Poisson’s ratio, 3.0=ν , the
maximum value of stress in a thick plate, 062.0 p , and occurs
at a depth (z-direction)
below the surface of a48.0 . Thus by the von Mises yield
criterion the value of 0p for
the onset of plastic yield is given by
ykp σ6.18.20 == (12).
Now by equating equation (6) and (10), we can obtain an
expression for the maximum
contact stress of an elastic impact:
51
20
54
43
*
0 45
3
423
⎟⎠⎞
⎜⎝⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛= υ
π inm
R
Ep (13).
By equating (13) to the von Mises critical contact pressure,
equation (12), it is possible to
obtain an expression relating the kinetic energy of the
projectile to target materials
mechanical properties [53]:
4*53
20
5321
ER
m yinσ
υ ≈ (14).
In the case of a rigid sphere impacting the planar surface of a
large softer body, equation
(14) reduces to
4*
5
0
26
Ey
ρ
συ = (15),
where ρ is the density of the softer (target) material [53].
Analytical treatments of the stress indentation field for
elastic-plastic contact are made
complex by the plasticity zone underneath the impact. The
analysis of the elastic-plastic
stress field of a spherical impact with the surface of a
half-space therefore requires the
use of finite element analysis [54-56]. The actual size and
shape of the plasticity zone
depend on the mechanical properties of the target material,
particularly the ratio of its
Young’s modulus to yield strength, E/σy [57]. A section view of
the subsurface damage
for the Macor® glass-cermamic material is shown in Figure 3
together with the
-
11
corresponding finite-element solution. The residual impression
in the surface made by
the indenter is clearly visible as is the shear-driven
accumulated subsurface damage
resulting from the indentation.
Figure 3. Bonded-interface test result showing section view with
subsurface accumulated damage beneath the indentation (Left).
Finite-element result showing extent of the plastic zone in terms
of contours of maximum shear stress at 5.0/ =YMaxτ for indenter
load NP 1000= (Right). Distances are expressed in terms of the
contact radius,
mma 326.00 = , for the elastic case of NP 1000= . The bold black
line indicates the radius of the circle of contact, mma 437.00 = ,
as determined from the finite-element calculation [57].
2.2 Plate Impact Mechanics
In the 1960’s and early 1970’s, H. Hopkins and H. Kolsky [59],
W. Goldsmith [60-63],
M. Cook [64], A. Olshaker and R. Bjork [65], J. Rinehart and J.
Pearson [66], L. Fugelso
and F. Bloedow [67] and R. Sedgwick [68] conducted experimental
studies to explore the
impact processes and penetration mechanisms in plates. A
compendium on the study of
the mechanics of projectile penetration was published in 1978 by
M. Backman and W.
-
12
Goldsmith [69]. A more recent review by G. Corbett et al. in
1996 [70] has incorporated
copious amounts of experimental data and analytical
interpretations that enable important
penetration mechanisms to be identified.
The analysis of failure mechanisms in finite thickness plates
can be found in the
aforementioned studies of M. Backman and W. Goldsmith and G.
Corbett et al. [69,70].
Permanent deformations, possibly a convolution of two or more
mechanisms, occur for
both the non-penetrated and the penetrated cases. In the
non-penetrated case, there are
two failure modes that can be attributed to the transverse
displacement of a thin1 target
due to plastic deformation, Figure 4 (a) and (b).
Figure 4. Illustration of a thin target showing a) bulging b)
dishing and c) cratering [69].
1 A plate is defined as ‘thin’ if stress and deformation
gradients throughout its thickness do not exist [69]
-
13
The first mode is known as bulging in which the plate deforms to
conform to the nose of
the projectile. Bulging may be considered by the static and
quasi-static methods of
analysis used in metal processing problems [80]. The second
failure mode is induced by
bending, called dishing, and can extend far from the contact
zone. Dishing, unlike
bulging, requires a dynamical explanation of plastic bending,
plastic hinge propagation
and shear banding and/or other fracture modes [70,81-82]. As the
target thickness and
impact velocity increases, these two modes decrease and the
deformation involves
displacement that tends to involve the proximal and distal side
of the target so as to
thicken it with little or no deflection. This process is called
cratering, Figure 4 (c),
common in thick plates, and appropriately describes the effects
of highly local
deformations in targets of any thickness.
As the velocity of the projectile increases, the ductile limit
of target is approached, and
penetration can begin to occur. In the penetrated regime,
failure involving fracture
occurs in plates of thin or intermediate2 thickness. The
fracture occurs from a
combination of mechanisms with one often dominating the others
depending on
projectile/target material characteristics, geometry, velocity
and angle of impact, etc.
[69]. Figure 5 depicts the most common types of failure modes
including that due to the
initial compression wave, fracture in the radial direction,
spalling, scabbing, plugging,
front/rear petaling or fragmentation in the case of brittle
targets and ductile hole
enlargement [63-66, 69-70, 83-93].
2 A plate is defined as ‘intermediate’ if the rear surface
exerts considerable influence on the deformation process during all
(or nearly all) of the penetrator motion [69].
-
14
Figure 5. Perforation mechanisms [69].
-
15
Fracture due to the initial stress wave can be caused by two
different mechanisms
depending on whether the tensile strength or compressive
strength of the target is greater
than the other. If the tensile strength of the target is greater
than its compressive strength
then failure occurs on the distal side, back side, of the plate
from the dilatational wave,
Figure 5 (a). Spalling, similar to the fracture on the distal
side from the initial stress
wave in Figure 5 (a), is a tensile material failure resulting
from the reflection of the initial
compressive transient off the distal side of the target, Figure
5 (c). Reflection of the wave
changes the sign of the pulse thereby placing the target in
tension from compression. The
dilatational wave, produced by the impact, creates a fracture
when the maximum shear
stress of the reflected wave begins to exceed the materials
yield stress [83]. A rough
approximation for the velocity limit, the limit at which the
projectile penetrates the distal
side of the target, of fracture from compressive failure of the
distal side due to impact is
given by [69]:
⎟⎟⎠
⎞⎜⎜⎝
⎛ +
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛
−−
=prpdt
prpdt
pYLim cc
ccdh
ρρρρ
ννσυ
21
2
02121
1 (16).
where cd is the dilatational wave velocity of the target, cpr is
the extensional wave
velocity in the projectile, ρt is the target density, ρp is the
projectile density, dp is the
diameter of the projectile, h0 is the target thickness, σy is
the yield stress of the target and
ν is the Poisson’s ratio of the target.
If the tensile strength of the target is lower than its
compressive strength, then a radial
fracture behind the initial stress wave will result, Figure 5
(b), based on the assumption
that radial stress has exceeded the yield value in tension. A
rough approximation for the
velocity limit of this type of fracture is given by [69]:
( )
( )
⎟⎟⎠
⎞⎜⎜⎝
⎛ +
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛++−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛+−
=prpdt
prpdt
p
py
Lim cccc
dh
dh
ρρρρ
νν
νσ
υ21
2
0
2
0
2121
2112
(17).
-
16
In the ductile separation, voids nucleate through
particle-matrix debonding or through
particle cracking, then they grow by local plastic deformation,
and finally coalesce by the
onset of local instabilities or inhomogeneities [84,85]. A rough
approximation for the
velocity limit of this type of fracture is given by [69]
⎟⎟⎠
⎞⎜⎜⎝
⎛ +
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
−⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=prpdt
prpdt
p
py
Lim cccc
dh
dh
ρρρρ
σ
υ
12
1
21
21
2
0
2
0
(18).
Plugging results as a cylindrical slug, nearly the size of the
projectile is sheared from the
target, Figure 5 (d). The failure occurs due to large shears
around the moving slug.
Generated heat is restricted to an annulus surrounding the slug
and causes a reduction in
material strength, resulting in instability; this is called an
adiabatic shearing process [69].
This catastrophic shear results from interplay between thermal
softening and the low
work, and strain hardening rate of the plate material within the
shear bands [86,87].
Plugging is most common for blunt projectiles impacting thin or
intermediate, hard plates
due to material being geometrically constrained to move ahead of
the projectile.
Analytical models describing the failure mechanism have been
difficult to develop and
tend to be complex, reaching five stages to adequately model the
event [88]. Again,
observed empirical relations have given a rough approximation
for the velocity limit of
this type of fracture [69] given by:
21
221
0 12 0
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛lh
pt
yLim
P
t
edh ρ
ρ
ρσ
υ (19).
where l is the length of the projectile.
Petaling, both frontal and rear, is produced by high radial and
circumferential tensile
stresses after passage of the initial wave near the lip of the
penetration [69,89], Figure 5
(e) and (f). This deformation is the result of bending moments
created by the forward
motion of the plate material being pushed ahead of the
projectile and by inhomogeneities
or weaknesses in the target. Petaling is usually accompanied by
large plastic flows
-
17
and/or permanent flexure. As the material on the distal side of
the plate is further
deformed, the tensile stresses are exceeded and a star-shaped
crack is initiated by the tip
of the projectile [70]. Finally, the sectors are rotated back by
the ensuing motion of the
projectile, forming, often symmetric, petals. Petaling commonly
occurs from ogival or
conical shaped noses on projectiles penetrating thin ductile
plates (h0 / dp < 1). B.
Landkof and W. Goldsmith [91] expanding upon a study conducted
by C. Calder and W.
Goldsmith [93], carried out an experimental and theoretical
investigation of petaling. In
the study, they used an energy balance through multiple stages
of impact to establish an
expression for the final velocity of the projectile given by
( )
( ) 21
22
02
21202
02
6cos1
2122
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
+
−−
−++
=
p
cr
p
d
p
cry
Lim
mhl
mE
mhl
kk
θρπ
θθπσυ
υ (20),
where Ed is the energy absorbed through plate dishing [70], k is
a mass ratio parameter, θ1
and θ2 are the petal rotation angles at the ends of the stages
and lcr is the crack length.
Fragmentation of the projectile and target occur in situations
similar to radial fracture
where the stress wave of the impact creates tensile and
compressive stresses which
exceed those of the projectile and target, Figure 5 (b). A study
conducted by M. Kipp et
al. [92] explores the effect of high-velocity impact
fragmentation, both numerically and
experimentally.
Ductile hole enlargement seems to be a common failure of thick3
plates of medium to
low hardness common from ogival or small-angle conical shaped
projectiles [69], Figure
5 (h). At the beginning of contact, the tip of the projectile
begins displacing material
radially and continues so that a hole in the target is enlarged
along the trajectory of the
projectile. Heavily dependent on projectile shape and projectile
diameter to target
thickness ratio, ductile hole enlargement is favorable instead
of plugging if the following
condition is satisfied with a ogival or small-angle conical
shaped projectile [87]
3 A plate is defined as ‘thick’ if there is an influence of the
distal boundary on the penetration process only after substantial
travel into the target element [69].
-
18
pdh 23
0 > (21).
A quasi-static analysis of the completely symmetrical
enlargement of the hole that
develops at the moving point of the sharp projectile was given
in a classical paper for a
thin infinite elastic-perfectly plastic sheet [69, 94]. This
description was improved by G.
Taylor [95] providing a more precise stress analysis in the
region of significant target
thickening. The work required to expand such a hole to a given
radius R1 is
yhRW σπ 02133.1= (22).
A complex analytical solution to the radial stresses at the hole
and the total resistance to
penetration were formulated by W. Herrmann and A. Jones [96] and
H. Bethe [94]. A
rough approximation for the velocity limit can be found by
equation (23).
Several models describing impact upon plates with a finite
thickness have been proposed
[71-76] but the complexity of the impact event has limited
general closed-form analytical
solutions [77]. To supplement the lack of analytical solutions,
empirical relations,
neglecting plate bending, stretching or dynamic effects beyond
the impact zone, have
been proposed but are of limited utility. These relations are
only applicable in a narrow
set of velocity ranges for a particular type of projectile
geometry. For example, the
Standard Research Institute Formula (SRI) [70] proposes that for
a cylindrical geometry
the critical projectile impact energy, Ec, to penetrate a sample
is given by:
( )0203
7.4213
hlhd
E ppu
c +=σ
(23),
where σu is the ultimate stress, dp is the diameter of the
projectile and h0 is the thickness
of the target. This empirical expression is valid only for 0.1
< h0/dp < 0.6; 0.002 < h0/lp <
0.005; 10 < lp /dp < 50; 5 < lp /dp < 8; lp /h0 <
100 and 21 < v0 < 122 m/s. Other empirical
formulas only make accurate predictions significantly greater
than the target’s ballistic
limit. For example the study conducted by W. Thomson [78,79]
found
⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
3216 200
220
2 ρυσπυυ yp
pr m
hd (24),
where and υr is the residual velocity of the projectile.
-
19
Chapter 3 Materials and Structures
3.1 Sandwich Panel Fabrication
A perforated sheet folding process [29] was used to create
pyramidal truss sandwich
panel structures with a core relative density ( ρ ) between 5
and 6%. A diamond
perforation pattern was die stamped into a 1.9 mm thick (14
gauge) 304 stainless steel
sheet, Figure 6. A similar thickness, 6061-T6 aluminum alloy was
annealed to the O-
condition also die stamped in a similar manner to the 304
stainless steel. The O-
condition annealing was achieved by placing the assembly in a
furnace at 500 °C for 30
minutes and allowed to furnace cool. Afterwards, the sheets were
perforated to create a
2-D array of diamond perforations that were each 5.46 cm in
length and 3.15 cm wide.
Adjacent perforations were separated by 4.0 mm of metal. The
patterned sheets were
then bent as schematically illustrated in Figure 6, to create a
single layer pyramidal truss
lattice with trusses that were 31.75 mm in length and 1.9 x 4.0
mm2 in cross section.
After bending, the annealed O-condition AA6061 trusses were
artificially aged at 165 °C
for 19 hours and then water quenched from the solutionizing
temperature to return them
to their peak strength condition (T6).
-
20
Figure 6. Manufacturing process for making pyramidal lattice
truss cored sandwich panels [3]. The lattice truss panels were
trimmed to form 3x3 pyramidal cell arrays. The 304
stainless steel structures were placed between a pair of 1.5 mm
thick (16 gauge) 304
stainless steel (12.07 cm x 12.70 cm) facesheets and laser
welded at the nodes, Figure 7.
The AA6061 lattices were sandwich between 1.5 mm thick (14
gauge) AA6061 face
sheets with similar dimensions 12.07 cm x 12.70 cm and laser
welded, Figure 7. The 7-
axis CO2 laser was manufactured by LaserDyne (Champlin, MN), and
used 600-1300 W
to control the depth and size of the welds which were conducted
on both alloys.
-
21
Figure 7. Illustration of the laser welding process for bonding
the truss lattice to proximal and distal facesheet. Figure 8 shows
a cross section of the single layer empty pyramidal truss lattice
along a
nodal line.
Figure 8. Cross section of the single layer empty pyramidal
truss lattice along a nodal line.
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22
3.2 Relative Density Relations
The relative density, ρ , is non-dimensional ratio defined as
the volume fraction of truss
members occupying a prescribed unit cell. Ignoring the detailed
geometry located at the
nodes, the relative density of the pyramidal lattice truss core
can be calculated from a unit
cell analysis of the single layer pyramidal unit cell, Figure
9.
Figure 9. Unit cell geometry used to derive the relative density
for single layer pyramidal topology. where ω = 54.7° is the
included angle (the angle between the truss members and the
base
of the pyramid), w is the truss width, t is the truss thickness
and l is the truss length.
Based upon these considerations, the volume, trV , of the truss
members occupying the
single layer pyramidal unit cell shown in Figure 9 is:
lwtVtr 4= (25).
The volume, cV , of the single layer pyramidal unit cell is:
( )( )( ) ωωωωω sincos2sincos2cos2 23llllVc == (26),
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23
Taking the ratio between the volume of the trusses, equation
(25), and volume of the unit
cell, equation (26), we obtain the single layer pyramidal
relative density ( ρ ) expression:
ωω
ρsincos
222lwt
VV
c
tr == (27).
14 gauge thick 304 stainless steel and 12 gauge thick age
hardened AA6061-T6
aluminum alloy panels are used here, with 9.1=t mm, w = 4.0 mm,
l = 31.75 mm and °= 7.54ω . Substituting these values into equation
(27) yields a %3.05.5 ±=ρ . The
304 stainless steel sandwich panel had an areal density of
approximately 28 kg/m2 while
that of the age hardened 6061-T6 aluminum alloy was
approximately 10 kg/m2.
3.3 Alloy Mechanical Properties
3.3.1 304 Stainless Steel
Uniaxial tension specimens were machined from 304 stainless
steel with a 0.61 mm plate
thickness, according to ASTM E-8 guidelines [97]. A
servo-electric universal testing
machine (Model 4208, Instron Corp., Canton, MA) with
self-aligning grips was used to
test each specimen at ambient temperature, approximately 25 °C.
The applied nominal
strain rate for the stainless steel was 0.3 mm/min (10-3 s-1),
and the strain measurements
were made using a linear variable differential transformer
(LVDT) clip-on extensometer
with an accuracy of ±0.5% of the gage length of 50 mm. The
stress as a function of
strain for the as received alloy is plotted in Figure 10. The
elastic modulus measured
approximately 200 GPa, the yield strength measured approximately
255 MPa, the
ultimate yield strength measured approximately 1000 MPa and the
strain to fracture
measured approximately 0.39. The test results approximately
agree with referenced
values [98].
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24
Figure 10. Uniaxial tension data for as-received 304 stainless
steel.
3.3.2 Age Hardened 6061-T6 Aluminum Alloy
Uniaxial tension specimens were machined age hardened 6061-T6
aluminum alloy with a
6.35 mm plate thickness, according to ASTM E-8 guidelines [97].
The equivalent servo-
electric universal testing machine as described in Chapter 3.4.1
was used for testing the
mechanical properties of the alloy. The applied nominal strain
rate for age hardened
6061-T6 aluminum alloy was 0.2 mm/min (10-3 s-1). The stress as
a function strain
response is plotted in Figure 11. The elastic modulus measured
approximately 68 GPa,
the yield strength measured approximately 268 MPa respectively,
the ultimate yield
strength measured approximately 310 MPa and the strain to
fracture measured
approximately 0.15. The test results approximately agree with
referenced values [98].
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25
Figure 11. Uniaxial tension data for the age hardened 6061-T6
aluminum alloy.
3.4 Polymer Infiltrated Sandwich Panels
In an attempt to add ballistic resistance to the sandwich
panels, a low Tg polyurethane, designated PU 1, was infiltrated
into the structure to create a hybrid lattice. This polyurethane
was chosen due to its wide availability and customizable mechanical
properties allowing a polymer with a high elongation to yield to be
chosen easily.
3.4.1 Hybrid Lattice Fabrication
Twenty five 304 stainless steel mono-layer pyramidal lattice
truss structures, with
equivalent dimensions as described in Chapter 3.1, were
fabricated and assembled. The
samples were taped on three sides. PU 1 was poured into the
sandwich structure and the
samples were allowed to cure for twenty four hours at ambient
temperature,
approximately 25 °C.
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26
The relative density of the system can be calculated using a
similar unit cell analysis,
described in Chapter 3.2. Equation (28) shows the relative
density for the 304 stainless
steel pyramidal truss lattice with infiltrated PU 1,
incorporating the different densities of
the metal and the PU 1,
( ) ( )
( )ωωρρωωρ
ρsincos2
44sincos223
23
llwtlwtl
m
trpu +−= (28),
where puρ is the density of the polyurethane, trρ is the density
of the trusses and mρ is
the density of the base metal system. With polyurethane density
of 3.1=pρ g/cc, a truss
density of 97.7=trρ g/cc and a base metal density of 97.7=mρ .
Compared to an all
steel plate, the relative density of the system is approximately
=ρ 27%. The areal
density of the 304 stainless steel pyramidal truss lattice
infiltrated with PU 1 is 55 kg/m2.
3.4.2 Polymer
The PU 1 polymer chosen for the study was a type of
polyurethane, designated PMC-780
Dry [100], formulated by Smooth-On (Easton, PA). PU 1 is a two
component, pliable,
castable elastomer with an approximate twenty-four hour cure
time at room temperature.
Part A is composed mostly of polyurethane prepolymer and a trace
amount of toluene
diisocyanate. Part B is composed of polyol, a proprietary
chemical (NJ Trade Secret
#221290880-5020P), di(methylthio)toluene diamine and
phenylmercuric neodecanoate.
This polyurethane has a low elastic modulus and tensile strength
but a very high
elongation to failure. The term elastomer is loosely applied to
polymers that at room
temperature can be stretched repeatedly to at least twice their
original length and,
immediately upon release of the stress, return with force to
their approximate original
length [101]. Table 1 lists the manufacturer’s specifications
for the polyurethane.
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27
Property PU 1 Manufacturer Smooth-On (Easton, PA) Product Name
PMC-780 Dry Tensile Modulus (MPa) 2.76 Tensile Strength (MPa) 6.21
Elongation to Break (%) 700 Shore Hardness 80 A
Table 1. Manufacturer reported properties for the polyurethane
system. The hardness testing of plastics is most commonly measured
by the Shore (Durometer)
test or Rockwell hardness test. Both methods measure the
resistance of the plastic toward
indentation by a spring-loader. Both scales provide an empirical
hardness value that
doesn't correlate to other physical properties or fundamental
characteristics such as
strength or resistance to abrasion. Shore hardness, either the
Shore A or Shore D scale, is
the preferred method for rubbers/elastomers and is also commonly
used for 'softer'
plastics such as polyolefins, fluoropolymers, and vinyls. The
Shore A scale is used for
'softer' rubbers while the Shore D scale is used for 'harder'
ones. The Shore A hardness is
the relative hardness of elastic materials such as elastomers or
soft plastics can be
determined with an instrument called a Shore A Durometer. If the
indenter completely
penetrates the sample, a reading of 0 is obtained, and if no
penetration occurs, a reading
of 100 results. Shore hardness is a dimensionless quantity. A
full description of the test
method can be found in ASTM D2240, or the analogous ISO test
method is ISO 868.
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28
3.5 Polymer Characterization
A characterization of the dynamical properties of the polymer is
necessary due to their effect on the ballistic response. Three
properties were characterized, the glass transition temperature,
the storage modulus and the loss modulus. The glass transition
temperature indicates the amount of crosslinking in the polymer and
affects the elongation to yield. This property can be ascertained
by measuring the heat capacity as a function of temperature with a
differential scanning calorimeter (DSC). The storage modulus and
loss modulus are related to a parameter, δTan , that indicates a
materials ability to absorb and dissipate energy. These rheological
properties can be ascertained by the use of a dynamic mechanical
analyzer (DMA).
3.5.1 DSC Analysis
A DSC analysis of PU 1 was conducted by M. Aronson et al. of the
University of
Virginia. The glass transition temperature, Tg, of the
polyurethane was determined with
modulated differential scanning calorimetry (MDSC®) using a
Q1000 Modulated DSC
(TA Instruments-Waters, LLC). The polymer was heated over a
temperature range of -80
to 240 °C with a heating rate of 3 °C/min and a modulation of ±
0.5 °C/60 sec period.
With traditional DSC, the heat flow curve is a superimposition
of the Tg, endotherms and
exotherms. Due to this superimposition, it is difficult to make
an accurate determination
of the Tg with traditional DSC. With modulated DSC, the
reversing heat flow curve
associated with the Tg is separated from the non-reversing heat
flow curve associated
with endotherms and/or exotherms, thus enabling an accurate
determination of the Tg
[103].
Figure 12 is a plot of the heat capacity, Rev Cp, of PU 1 as a
function of temperature.
The Cp of the polyurethane was determined by dividing its
reversing heat flow value,
J/(sec·g), by the heating rate, °C/sec.
-
29
Figure 12. Heat capacity (Rev Cp) of PU 1 as a function of
temperature (°C). The Tg of each sample was taken to be the
inflection point of the step-change in Cp.
Based on this definition, and the information included in Figure
12, the Tg of PU 1 was
-56 °C. The small step-change in Cp, around 70 °C, is believed
to be an experimental
artifact and not associated with a second Tg of this sample.
3.5.2 DMA Analysis
A DMA analysis of PU 1 was also conducted by M. Aronson et al.
of the University of
Virginia. The rheological properties of PU 1 were characterized
with dynamic
mechanical analysis (DMA) using a Q800 DMA (TA
Instruments-Waters, LLC).
Measurements were made on each sample at three different
frequencies, 1, 10 and 100
Hz, over a temperature range of -100 to 40 °C in 5 °C
increments. The data over the
entire temperature range were transformed using time-temperature
superposition (TTS)
with a reference temperature of 25 °C [106,491H107]. The result
of this data manipulation is a
master curve of predicted storage modulus, E ′ , and loss
modulus, E ′′ , values over a
frequency range of 10-1 to 1010 Hz for each sample at the
reference temperature.
-
30
Figure 13 is a plot of the storage modulus of PU 1 at a
frequency of 1 Hz over a
temperature range of -100 to 40 °C. As the temperature is
increased from -70 to 0 °C, the
storage modulus of PU 1 decreases by three orders of magnitude.
This difference is due
to the fact that the Tg of PU 1 is -56 °C, Figure 12.
Figure 13. Storage modulus at a frequency of 1 Hz for PU 1 as a
function of temperature. The predicted storage and loss modulus
values for PU 1 over a frequency range of 1 to
106 Hz at a reference temperature of 25 °C was computed, Figure
14. As previously
discussed, these predicted values were obtained by transforming
the measured E’ and E”
values obtained over the temperature range of -100 to 40 °C at
the three different
frequencies using TTS (data obtained at low temperatures
corresponds to the high
frequency data in Figure 14, while data obtained at high
temperatures corresponds to the
low frequency data).
-
31
Figure 14. Predicted values for the storage and loss modulus of
PU 1 at a reference temperature of 25 °C as a function of
frequency. The ratio of E ′ to E ′′ , which is referred to as Tan
δ, is a parameter that is often used to
assess the ability of a material to absorb and dissipate energy.
For materials with
comparable storage moduli, the greater the Tan δ value, the more
efficient the material is
able to absorb and dissipate energy. Figure 15 is a plot of Tan
δ of PU 1 over the same
frequency range covered in Figure 14.
-
32
Figure 15. Tan δ ( EE ′′′ / ) of PU 1 as a function of
frequency.
-
33
Chapter 4 Ballistic Testing Fifteen pyramidal lattice truss
structures of each alloy, with contrasting mechanical properties,
were tested using impact velocities between approximately 225 m/s
and 1225 m/s. Eleven 304 stainless steel monolithic plates, 3 mm
thick, with an equivalent areal density to 304 stainless steel
lattice truss sandwich panels, were tested as a comparison to
evaluate the ballistic resistance of the lattice truss sandwich
system. Both stainless steel systems had the same (as-received)
mechanical properties and neither underwent heat treatment prior to
testing.
4.1 Stage One Powder Gun
Ballistic testing was conducted by A. Zakraysek et al. at the
Indian Head Division, Naval
Surface Warfare Center, MD, using a powder gun shown
schematically in Figure 16.
Figure 16. Illustration of the single stage powder gun used for
ballistic studies. The sabot carried a 12.5 mm spherical
projectile.
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34
A cable connected the firing switch to an electric solenoid,
Figure 16. Upon closing the
circuit, the electric solenoid activated a firing pin. The
firing pin then struck a 0.38
caliber blank cartridge supplied by Western Cartridge Company
(East Alton, IL) which
ignited a gun powder charge whose mass determined the projectile
velocity. Gun powder
3031, manufactured by IMR (Shawnee Mission, KS), and cotton,
placed in front of the
gun powder, was contained in the middle region of the breech in
Figure 16. The purpose
of the cotton was to ensure the initiated shock wave remained
uniform throughout
propagation of the detonation. A sabot was located within the
2.54 cm bore gun barrel,
Figure 16. A series of holes placed along the gun barrel were
used to dissipate the shock
wave and maintain a smooth acceleration of the sabot until it
exited the barrel.
4.2 Sabot and Projectile
The plastic sabot was composed of four quarters that, upon
mating, surrounded a 12.5
mm diameter spherical projectile. The sabot plugs had an inner
diameter of 1.25 cm, an
outer diameter of 2.54 cm, a height of 3.50 cm and weighed
18.60±0.12 g. A 40° bevel at
the sabot opening facilitated separation of the sabot from the
projectile by air drag,
shortly after initiation of free flight. Figure 17 shows a
photograph of both the fully
assembled and separated sabot. The projectiles had a diameter of
1.25 cm and weighed
8.42±0.02 g. The spherical projectiles were manufactured by
National Precision Ball
(Preston, WA). They were made from 1020 plain carbon steel with
an ultimate tensile
strength of 365 to 380 MPa [99].
-
35
Figure 17. Illustration of the sabot used to carry the
projectile.
4.3 Test Fixture
The test sample fixture was located within a blast chamber,
Figure 18.
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36
Figure 18. Illustration of setup used in the blast chamber to
mount the samples, measure velocities and record via high-speed
photography. A square steel plate 41.28 cm long, 2.86 cm thick, was
located one meter from the end of
the barrel with a 3.8 cm diameter hole located in the center. A
square wood plate 41.28
cm long, 2.22 cm thick, with a 6.35 cm diameter hole, was
located 30.48 cm successively
after the steel plate. On the back of the wood plate, covering
the hole, was the first of
four brake screens to measure entry and exit velocities. A brake
screen is a piece of
paper with a thin, silver mask that is connected into a circuit.
These circuits were
connected into a four channel oscilloscope. Upon penetration by
the projectile, the
circuit is broken. Using four brake screens, two for entry and
two for exit, velocities can
be calculated by knowing the distance between and the time
difference between circuit
closures. The oscilloscope was precise to ±1 μs therefore
causing decreasing error in the
velocity measurement as the projectile velocity was increased
(i.e. time is inversely
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37
proportional to length in velocity). The mounting steel plate,
2.86 cm thick, was located
30.86 cm after the wood plate. In the center of plate, was a
13.34 cm square aperture to
house the sample. On each side of the sample, a brake screen was
attached. An iron
angle bracket 17.78 cm long, with 1.19 cm overlap over the top
and bottom was used to
clamp the sample into the fixture with bolts. Adjacent to the
fixture, a Fresnel lens was
used to collimate the light onto the sample to provide adequate
contrast for the high-
speed photography located on the opposing side. Located 30.48
cm, successively after
the mounting steel plate, was another square wood plate 41.28 cm
long, 2.22 cm thick,
with a 12.7 cm diameter hole located in the center. On the back
of this plate was the final
brake screen in the series. A final square steel plate 41.28
long, 2.86 cm thick, was
located 28.21 cm successively after the second wood plate. This
plate was used as
backstop to catch any fragments from the sample or remaining
projectile remnants.
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38
Chapter 5 Empty Lattice Resistance Both alloy systems, 304
stainless steel and AA6061 aluminum alloy, were fixtured as
described in Chapter 4.3 and impacted using the aforementioned
projectiles as described in Chapter 4.2. Exit velocity of the
projectile through the target was recorded and the impact velocity
was gradually increased from 200-1200 m/s. Fifteen sandwich
structure samples of both alloy systems and eleven 304 stainless
steel plate samples were tested.
5.1 304 Stainless Steel Panel Response
Table 2 displays the data acquired for the 304 stainless steel
pyramidal truss lattice
sandwich structure. The mass of each 304 stainless steel sample
was 432.2±1.1 g.
Shot #
Impact Velocity
(m/s)
Exit Velocity
(m/s)
Nodal Disbond of Distal FS
Penetration of Distal FS
1 339.2±0.1 N/A No No 46 290.8±0.1 N/A No No 56 227.1±0.1 N/A No
No 2 506.9±0.3 310.0±0.1 No Yes
48 481.3±0.3 266.1±0.1 No Yes 133 493.8±0.3 276.5±0.1 No Yes
3 810.8±0.8 551.1±0.4 No Yes 50 768.4±0.8 491.9±0.3 No Yes
134 812.3±0.9 653.5±0.6 No Yes 4 1029.9±1.4 721.5±0.7 No Yes
52 992.4±1.3 744.0±0.7 No Yes 135 1001.0±1.3 653.5±0.6 No Yes 54
1206.1±1.9 868.4±1.0 No Yes
136 1214.9±1.9 882.7±1.0 No Yes 137 1221.9±1.9 851.9±0.9 No
Yes
Table 2. The impact and exit velocities of the projectile, nodal
disbonding of the distal facesheet and whether the projectile
penetrated the distal facesheet for the 304 stainless steel
pyramidal truss lattice sandwich structure. Figure 19 depicts the
exit velocity of the projectile (m/s) as a function of the
impact
velocity (m/s) for the 304 stainless steel pyramidal truss
lattice sandwich structure. The
error bars for the impact and exit velocities are not
illustrated on the graph because they
are smaller than the size of the plotted data points.
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39
Figure 19. Plot of exit velocity of the projectile (m/s) as a
function of the impact velocity (m/s) for the 304 stainless steel
pyramidal truss lattice. The results for the system can be fitted
to a linear equation after the critical velocity
region (i.e. the velocity at which penetration begins to
occurs), R2 = 0.98,
xy 81.08.100 +−= (29).
Figure 20 depicts the energy absorbed (J) as a function of the
impact velocity (m/s) for
the 304 stainless steel pyramidal truss lattice system. As
described previously, the error
bars for the impact and exit velocities are not illustrated on
the graph because they are
smaller than the size of the plotted data points.
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40
Figure 20. Plot of energy absorbed (J) as a function of the
impact velocity (m/s) for the 304 stainless steel pyramidal truss
lattice system. The left parabaloid in Figure 20 represents the
target preventing penetration of the
projectile. After the critical velocity, the energy absorbed
begins to decrease until it
reaches a minimum, and then begins to increase monotonically
with the initial velocity.
The first 304 stainless steel pyramidal truss lattice sandwich
structure sample to be
penetrated occurred with an impact velocity of 481.3 m/s. The
projectile exited the
lattice structure at 266.1 m/s, approximately 55% of the impact
velocity. The critical
velocity is approximately 400±25 m/s.
Figure 21 shows a cross sectional view of a 304 stainless steel
sample that was impacted
by a spherical projectile at 339.2 m/s.
-
41
Figure 21. a) Cross section of the 304 stainless steel sample
that was impacted at 339.2 m/s and was not fully penetrated, shot 1
b) Cross section of the entry hole of shot 1 c) Cross section of
the exit hole of shot 1. Bulging and dishing are apparent on the
distal facesheet and a crack began to initiate petaling as the
energy from the impact was fully absorbed. Physical examination
after the test indicated a center-cell impact (i.e. equidistant
from
four nodes), Figure 21. The projectile impacted a truss/distal
facesheet node causing a
truss to separate and plastically deform. Penetration of the
proximal facesheet resulted in
an entry hole of 12.5 mm wide and deflected 4.5 mm. Dishing was
approximately 3 cm
in diameter. Full penetration of the distal facesheet did not
occur and fully arrested the
projectile resulting in a deflection of 12.5 mm. A star-shaped
crack began to initiate
forming sectors, but no petaling occurred. There was no nodal
disbonding of the trusses
and facesheets except for the impact location.
-
42
Figure 22. a) Cross section of the 304 stainless steel sample
that was impacted at 810.8 m/s, shot 3 b) Cross section of the
entry hole of shot 3 c) Cross section of the exit hole of shot 3. A
small amount of ductile hole enlargement occurred on the proximal
facesheet and bending/dishing exceeded the ductile limit of the 304
stainless steel to form petaling. Figure 22 shows a center-cell
impact on shot 3 of 810.8 m/s and brake screens recorded
an exit velocity 551.1 m/s, approximately 68% of the impact
velocity. The projectile
impacted a truss/distal facesheet node causing trusses to
separate and plastically deform.
Penetration of the
