Energy Storage and Dissipation in Polyurea Composites
by
Carl Bodin
B.S. Chemical Engineering
University of Illinois at Champaign-Urbana, 2002
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degrees of
ARCIVES
Naval Engineer
IMASSACHUSETTS
INSTWTE
OF TECHNOLOGY
and
MAY 3 0 2113
Master of Science in Mechanical Engineering
at the Massachusetts Institute of Technology
UBRA RIES
June 2013
0 Carl Bodin. All Rights Reserved
The author hereby grants to MIT and the US Government permission to reproduce and to
distribute publicly paper and electronic copies of this thesis document in whole or in part
in any medium now know or hereafter created.
A uthor..........................................................................
.....
......
Department of Mechanical Engineering
May 14, 2013
Certified by...........................................
lry C. Boyce
Ford Professor-of Mechanidal Engineering
Accepted by...........................
..............................
David E. Hardt
Ralph E. and Eloise F. Cross Professor of Mechanical Engineering
Chair, Department Committee on Graduate Students
.
Energy Storage and Dissipation in Polyurea Composites
by
Carl Bodin
Submitted to the Department of Mechanical Engineering
on May 14, 2013, in partial fulfillment of the
requirements for the degrees of Naval Engineer and
Master of Science in Mechanical Engineering
ABSTRACT
Polyurea composites have been of interest for a variety of engineering applications via
their highly dissipative yet resilient behavior under deformation. Polyurea composites
have been considered as a self-healing and anticorrosion coating in building applications,
and more interestingly, as a lightweight addition to steel armor. In combination with a
metal plate, a polyurea layer has been extensively studied under impact and blast loading.
In this research, the tunable performance of polyurea sandwich armor composites is
explored in modeling and experimentation. Cylindrical arrays comprised of polyurea, a
resilient yet dissipative material, enable improved load transmission by utilizing new
dissipation and storage pathways due to geometry. Experimentation and computational
modeling are used to quantify the dissipation features of the polyurea composite. This
research combines a new polyurea interlayer geometry with steel to improve the
composite armor blast performance by increasing energy dissipation.
Thesis Supervisor: Mary C. Boyce
Title: Ford Professor of Mechanical Engineering
3
Acknowledgements
I would like to thank my thesis supervisor, Professor Mary Boyce for her support
and guidance.
Thanks to Hansohl Cho who graciously allowed my use of his
constitutive polyurea model that enabled the accurate analysis of the materials
complicated response. Special thanks to all my lab mates and 2N students who
helped me get finish all the problem sets, presentations, and research.
I have been blessed with a wonderful family and I am so grateful for a wife and son
who unselfishly and without complaint sacrificed our time together during my
studies at MIT.
I would like to thank Calvary Baptist Church for encouraging my spiritual growth to
keep up with and exceed my scholarly development.
Finally, as a Christian, everything I am or ever will be I owe to Jesus Christ. This
degree was only possible through his gracious care and blessing.
4
Table of Contents
Introduction . .
11
...................-....--.....
..................................-.........-
11
.....................................
Traditional Steel Armor.....
Modern Composite Armor........................................12
12
12
13
. ........................................................................
P olym er Coatin gs.............................................................
Steel Armor with Polymer Coating ...............................................................................................
Steel-Polymer Sandwich Armor..................................................................................................
Overview ..............................-
.... .. ........
.......
.........
Background ...................
Concern............................-.................15
Threat of
.............
14
................................
..............
..........-
. 15
15
16
............................................................................
Bla st W av e .......................................................................
_................
_.......................
P rojectile Im p act............................................... .........................................
...... 16
Mechanisms of Energy Dissipation in Steel and Polymers.......................
Inelastic Deform ation .................................................................................................................................
17
Energy Absorption in Composite Armor ..............................
Bla st ................ ...................................................................................................................................................
17
17
P roje ctile Im p act...........................................................................................................................................1
8
..............
Material Properties of Armor Components..............
Ste e l....................................................................................................................................................................1
18
8
19
20
P S M4 ..................... . ........................................................................................................................................
. . ..................................
_.....
..................................
P o ly u rea .........................................................................
............ ......... 22
.....
Material Models of Armor Components.- - -.......
2
Ste e l....................................................................................................................................................................2
P SM4 ..................................................................................................................................................................
23
P oly u re a ............................................................................................................................................................
25
......
............
......
PSM4 Array Com pression....- - -.....
Materials and Methods.................................
Fabrication and Mechanical Testing .............................................................................................
A rray V oid Fraction .....................................................................................................................................
Fin ite Elem ent Mod el............................................................................................................................
Experimental and Simulated Response ...--......
29
29
... ........
.........
29
29
.. 3 1
32
.......................
........
................ ..........................
Small Cylinder Arrays............................................................
.........................................
.......................................................................
Experiments
Cylinder
Large
Comparison of Large Array and Small Array..............................................................................43
Effects of Lateral Array Constraints ................................................................................................
Effect of Compression on Void Fraction ........................................................................................
Conclusions ................
..........--...
--...........
Polyurea Array Com pression ...... - -......Materials and Methods......................
-......
-.....................
.............................
............................
32
41
44
47
48
.... .... 51
.................... 51
Fabrication and Mechanical Testing ................................................................................................
F in ite Elem ent Mod el..................................................................................................................................5
51
1
--............................
52
Experimental and Simulated Response ..................--.
.. .....
...........................................
Small Cylinder Arrays.............................._...............
...........................................................
Solid Equivalent Arrays ..............................................................
......................................
En ergy Dissip ation .............................................................................................
......... ..............................................
..............
Conclusions ...................
Improved Composite Armor Under Blast Loading............
5
52
65
68
71
......... 73
M aterials and M ethods.............................................................................................................
Finite Elem ent M odel..................................................................................................................................73
Sim ulated Com posite Arm or Response .............................................................................
73
Conclusions ......................................................................................................................................
81
Sum mary and Conclusions ................................................................................................
85
Conclusions ......................................................................................................................................
Future W ork.....................................................................................................................................85
A ppendix
p
...................................................................................
6
74
85
91
Table of Figures
Figure 1:Underwater explosion incident on steel plates with varying thickness of PU
13
c o a tin g ................................................................................................................................................
16
Figure 2: Explosive generated pressure wave from 1 kg TNT ......................................
Figure 3: PSM4 stress-strain behavior showing highly elastic large strain behavior19
Figure 4: Strain rate stiffening of polyurea reprinted with permission..................... 20
Figure 5: PU 1000 stress-strain behavior showing energy storage and dissipation. 21
Figure 6: DMA (1 Hz) Isochronal plots of PU between -100 and 150 *Creprinted
21
w ith p e rm issio n .............................................................................................................................
Figure 7: PSM4 experimental and simulated stress-strain using compressible Neo
25
H o o k ea n m o d el ..............................................................................................................................
Figure 8: PU Cylinder Compression Is Strain-Rate Dependent....................................... 26
Figure 9: Uniaxial compression using the constitutive model for PU and
28
experim ental data at 0.1/s ...................................................................................................
29
Figure 10: 1-row void dim ensions ..................................................................................................
29
Figure 11: 2-row void dim ensions ..................................................................................................
30
Figure 12: 3-row void dim ensions ..................................................................................................
Figure 13: PSM4 array finite element models for 1, 2,and 3-rows.............................. 32
Figure 14: PSM4 array of cylinders shown in uncompressed state............................. 33
Figure 15: PSM4 array of cylinders uncompressed with FE simulation overlay........33
Figure 16: PSM4 array of cylinders in fully compressed state......................................
33
.................
34
simulation
overlay
Figure 17: PSM4 array uncompressed state with FE
Figure 18: Experimental and simulated data for 1, 2 and 3-rows of PSM4.............. 34
35
Figure 19: PSM4 array effective stress-strain curve..........................................................
Figure 20: PSM4 1-row array strain field comparison (exx) of experiment and
36
simu la tio n .........................................................................................................................................
Figure 21: PSM4 1-row array strain field comparison (exy) of experiment and
37
simu la tio n .........................................................................................................................................
Figure 22: PSM4 1-row array strain field comparison (Eyy) of experiment and
sim ula tio n .........................................................................................................................................
38
Figure 23: PSM4 2-row array strain field comparison (Exx) of experiment and
39
sim ula tio n .........................................................................................................................................
Figure 24: PSM4 2-row array strain field comparison (Exy) of experiment and
40
sim ula tio n .........................................................................................................................................
Figure 25: PSM4 2-row array strain field comparison (Fyy) of experiment and
41
sim ula tio n .........................................................................................................................................
Figure 26: Experimental and simulated data for 1 and 2-rows of large PSM4
42
cylin d e rs ............................................................................................................................................
Figure 27: PSM4 large cylinder array--note PMMA spacer on right............................ 42
Figure 28: PSM4 large cylinder array uncompressed with FE overlay ...................... 43
43
Figure 29: PSM4 large array under full compression .......................................................
Figure 30: PSM4 large array under full compression with simulated deformed array
43
o v erla y ...............................................................................................................................................
Figure 31: Effective stress-strain comparison of large and small PSM4 arrays..........44
7
Figure 32: Contact radius resulting from flat plate compression based on Hertz
equ ation fo r a cylin der................................................................................................................
45
Figure 33: Experimental constrained array vs. unconstrained simulated array using
Hertz e q uatio n ................................................................................................................................
46
Figure 34: Deformation of fully constrained compression..............................................
46
Figure 35: Void and slope change compared with compressive force on 1, 2, and 3ro w a rray s ........................................................................................................................................
47
Figure 36: PSM4 experimental effective stress-strain curve ..........................................
48
Figure 37: Polyurea array simulation mesh for 1,2, and 3-rows................................... 52
Figure 38: Polyurea cylinder array uncompressed............................................................
53
Figure 39: Polyurea cylinder array uncompressed with FE overlay........................... 53
Figure 40: Polyurea cylinder area in the compressed state ...........................................
53
Figure 41: Polyurea cylinder array compressed with FE overlay ................................
54
Figure 42: 2-row polyurea array in compressed state with FE overlay.....................54
Figure 43: 3-row polyurea array in compressed state with FE overlay..................... 54
Figure 44: Polyurea array compression experiments.......................................................
55
Figure 45: Polyurea array compression experiments with FE simulations..............56
Figure 46: Effective stress-strain curves for PU array compression for 1,2, and 3ro w s .....................................................................................................................................................
57
Figure 47: Polyurea 1-row array strain field .22 ..................................................................
58
Figure 48: Polyurea 1-row array strain field xy..................................................................
59
Figure 49: Polyurea 1-row array strain field yy..................................................................
60
Figure 50: Polyurea 2-row array strain field .22 ..................................................................
61
Figure 51: Polyurea 2-row array strain field xy..................................................................
62
Figure 52: Polyurea 2-row array strain field Eyy..................................................................
63
Figure 53: Polyurea 3-row array strain field .22 ..................................................................
64
Figure 54: Polyurea 3-row array strain field yy..................................................................
65
Figure 55: Polyurea 1-row equivalent solid shown uncompressed and compressed
.............................. .................... ... .... ........ ................................................ . 6 6
Figure 56 Polyurea 2-row equivalent solid shown uncompressed and compressed
66
........................
........ ... .................. ...... ........... ...................................................................
Figure 57: Polyurea 1 and 2-row equivalent solid samples under compression........67
Figure 58: Polyurea 1 and 2-row equiv. solid tests with 1-row equiv. FE ................ 67
Figure 59: Two loading and unloading cycles for polyurea under uniaxial
co m pre ssio n ....................................................................................................................................
68
Figure 60: Two loading and unloading cycles for 1-row constrained array............69
Figure 61: Two loading and unloading cycles for 2-row constrained array............69
Figure 62: Two loading and unloading cycles for 3-row constrained array............70
Figure 63: Polyurea array dissipation density for 1,2 and 3-row arrays .................. 71
Figure 64: Array effective stress-strain response for 1, 2 and 3-rows....................... 71
Figure 65: Simulation geometry of solid Polyurea composite armor ......................... 74
Figure 66: Deflection of solid polyurea composite armor under blast loads...........75
Figure 67: Solid polyurea interlayer with 1/2 kg TNT ......................................................
75
Figure 68: Solid polyurea interlayer with 1 kg TNT ............................................................
76
Figure 69: Solid polyurea interlayer with 2 kg TNT ............................................................
77
8
Figure 70: Simulation geometry of cylindrical array polyurea composite armor......78
Figure 71: Deflection of 1-row polyurea composite armor under blast loads........ 78
Figure 72: Cylinder polyurea interlayer with 1/2 kg TNT..............................................
79
Figure 73: Cylinder polyurea interlayer with 1 kg TNT ....................................................
79
Figure 74: Cylinder polyurea interlayer with 2 kg TNT ....................................................
80
Figure 75: Plastic dissipation and strain energy of the top steel layer in two
composite armor configurations .......................................................................................
80
Figure 76: Comparison of top plate deflection of two sandwich composite armor
g e o m etrie s ........................................................................................................................................
82
Figure 77: Comparison of bottom plate deflection of two sandwich composite armor
g e o m etrie s ........................................................................................................................................
82
F igu re 7 8 : P SM 4 Sh eet..........................................................................................................................9
1
92
Figu re 79 : P U Sam ple Stock ................................................................................................................
92
Figure 80: Top down view of PU sample stock ....................................................................
Figure 81: Polyurea sample preparation...............................................................................
92
93
Figure 82: Front fixture face ........................................................................................................
94
Figure 83: Fixture with metal wire bracing and C-clamp.................................................
95
Figure 84: Large sample fixture...................................................................................................
96
Figure 85: Zwick Testing Machine..............................................................................................
Figure 86: Material characterization with and without barreling................................ 96
9
10
Chapter 1
Introduction
Traditional Steel Armor
From the days of sail until World War II, naval combatants were armored to reduce
the damage caused by ballistic trajectory projectiles fired from cannons and guns
(Corbett, 1918). Warships were designed to trade damage with the enemy. The
current use of much higher power precision guided missiles reduces the
effectiveness of armor. The response to these overmatching threats has been to
merely isolate the damaged and flooded compartments, and attempt to keep the
ship afloat.
Consequently, modern naval combatants do not have significant
armored personnel protection beyond the structural steel. The structure of the ship
is primarily designed with sufficient strength requirements to survive waves
crashing on the deck, and worse case deck/keel stresses. The most common naval
construction material is the low alloy steel designated DH-36 (Ships Design
Standard 100).
Attacks on US naval warships in the last 25 years have been conducted with low
technology and low cost bulk explosive charges.
USS Samuel B. Roberts was
operating in international waters and detonated a mine that caused significant
damage in 1988 (Peniston, 2006). A 5m hole was ripped in the underside of the
hull. Terrorists rammed the USS Cole during a port stop with an explosive filled
boat in 2000 (Stone, 2012). A 12m hole was cut into the hull of the Cole at the
waterline. Both attacks caused sufficient damage to the unarmored ships that they
narrowly avoided sinking (Peniston, 2006; Stone, 2012).
Sufficient steel armor can be added to prevent rupture of the hull but at a large
addition in weight. Multiple options exist for improving the blast protection for the
roughly 300 operational warships (OPNAV N8, 2013). Some options include adding
an explosive resistant polymer coating to the interior of the hull, adding exterior
steel to create a thicker solid hull, adding a double hull with an air/seawater gap, or
adding a double hull with a polymer interlayer. Doubling the hull thickness below
11
the waterline by adding a 1 cm thick addition to the hull of the USS Cole would
weigh 265 MT. This is only a 3% increase in the ships total weight, but the ship will
pay a fuel penalty to haul this extra weight over a 30-year operational life. The
lightest composite armor that can still defeat the simple blast threat results in a
lighter ship or more offensive capability.
Modern Composite Armor
Polymer Coatings
The military has studied polymer coatings for several years. Polyurea has emerged
as one of the best polymers due to its toughness, resilience, and dissipation. It is of
interest as a coating for concrete buildings to improve safety under blast loading
(Porter et al., 2011). Polyurea coatings are currently used in a Marine Corps body
armor and vehicle armor (Eubanks, 2012). It is considered as a self-healing tank
coating and has fire resistant properties. The Office of Naval Research (ONR) has
funded research on the use of polyurea as an explosive resistant coating for several
years (Barsoum et al., 2009).
Steel Armor with Polymer Coating
Composite armor, which includes layers of different materials, frequently shows an
improvement in ballistic resistance with a reduction in weight over solely metallic
armor. Roland et al conducted impact experiments comparing a single steel plate
against one coated with polyurea (Roland et al., 2010). They found that despite the
negligible increase in weight of the polyurea coating, there was an increase of over
60% in the ballistic limit. It was also found that the equivalent ballistic protection
would require twice the weight if using only steel. In another experiment it was
found that when 11.2mm of polyurea was added to a 4.8mm plate of DH-36 it
improved the ballistic limit for pointed impactors by 42% over a steel plate alone
(Xue et al., 2010). Interest in composite polymer armor has also grown in recent
years after numerous successful tests under blast loading. In several tests, polyurea
12
added to the backside of a metal plate showed improved fracture resistance from
blast by changing the failure mechanism of the metal (Ackland et al., 2013; Amini et
al., 2010)
The US Department of Defense through the Advanced Materials, Manufacturing, and
Testing Information Analysis Center (AMMTIAC) has tested polyurea coatings on
steel plates against blast loading underwater. Figure 1 shows the damage to three
steel plates sustained from an underwater blast load. An identical explosive charge
was used on each plate. The leftmost plate had no polyurea coating while the
middle and right plates had 5/8" and 1" respectively. The 5/8" coating reduced the
size of the rupture while the 1"coating prevented rupture (Barsoum et al., 2009).
Figure 1:Underwater explosion incident on steel plates with varying thickness of PU coating
Figure 1 courtesy of AMMTIAC: reprinted with permission
Steel-Polymer Sandwich Armor
Several tests have included a composite armor with a polyurea interlayer (Tekalur
et al., 2008; Bahei-El-Din et al., 2006) Tekalur et al. use;l E-glass vinyl ester outer
layers with a polyurea interlayer and showed the polyurea protected the stiff outer
layers and improved blast performance by 100% for a 60% weight addition over the
outer layers alone. Bahei-El-Din et al. showed that the addition of polyurea to other
ductile interlayers reduced the peak kinetic energy of the armor by 50%. Both tests
have been limited to the analysis of a solid layer of polyurea. There may be room to
improve on the energy absorption and dissipation of steel-polymer sandwich armor
by considering a new interlayer geometry.
13
Overview
This thesis presents a possible improvement to a composite armor panel with a
polyurea interlayer. A potential application of this interlayer configuration is the
improvement of energy absorption and dissipation in composite sandwich armor
for fragmentation and blast protection (Tekalur et al., 2008). Initially, experimental
and numerical analyses of polymeric materials that exhibit tunable large strain
mechanical behavior are considered. Specifically, the use of geometric structuring is
compared with a homogeneous solid material in a sandwich armor composite. A
simple elastomer, PSM4, is used to develop the basic response of close packed
cylinder arrays under compression.
PSM4 exhibits highly elastic large strain
deformation behavior with little hysteresis (Bertoldi et al., 2008), while polyurea
has rate sensitivity, viscoelasticity, viscoplasticity, and resilience (Yi et al., 2006).
Finally, a common low alloy steel used in ship construction, DH-36, is modeled in
simulations from extensive data in the literature for use in composite armor
simulations.
The energy absorption and dissipation characteristics of steel-polymer sandwich
armor are investigated using numerical simulations under several blast loads. A
finite element model was developed in which constrained arrays of cylinders were
compressed with varying void fractions. Experimental models were constructed
and tested to validate the numerical material and geometric models. The analysis of
these experimental and numerical models provide insight into the dissipation and
storage of energy in the sandwich armor panel and aid in finding the optimal
interlayer geometry.
14
Chapter 2
Background
Threat of Concern
Blast Wave
Most modern offensive weapons do not rely exclusively on kinetic energy to inflict
damage. The weapons magnify the damage by converting chemical energy stored in
the explosive into mechanical energy. When detonated, the explosive creates a
rapid expansion of gases that bursts the casing and creates a shock wave.
The peak pressure is reached very quickly as the shock wave passes, after which the
pressure subsides.
Steel is damaged from the pressure wave via plastic
deformation, and both ductile and brittle fracture.
Polymer damage can occur
through fracture or tearing from elongation. Most of the energy from the explosive
is released in both a pressure wave and high velocity casing fragments (Federation
of American Scientists, 1998). High velocity fragments can deform or perforate both
steel and polymers.
The pressure wave is most simply modeled with the Friedlander waveform (Dewey,
1963)
-t
P(t)=Pse7(1
-
)(1
Where Ps is the maximum pressure immediately behind the wave front, t is the time
as measured from the shock wave arrival, and t*is the time the pressure returns to
ambient. An explosive, such as TNT, has a super-sonic detonation velocity of 7000
m/s (Brode, 1965). The resulting pressure wave of 1kg of TNT measured 1m from
the detonation point produces the pressure wave shown in Figure 2.
15
Pressure wave from 1 kg TNT at 1m
a.
0.3-
2
0.20..
0.1 -
0-
-0.1
0
5
10
15
20
25
30
35
40
45
Time (ms)
50
Figure 2: Explosive generated pressure wave from 1 kg TNT
The peak pressure experienced falls off as 1/r 3 as the wave expands spherically in
space. The incident blast wave results in very high strain rates.
Projectile Impact
Warheads generate a quantity of fragments based on the warheads objective. The
velocity of the fragments can be calculated using equation 2.
V =
(2)
C/Al
1+K(j)
In this equation, AE = 2.715x10 6
J
kg
for TNT, C/M is the charge to fragmenting
weight ratio, and K is the shape value of 0.5 for a cylinder (Federation of American
Scientists, 1998). The most frequently used general-purpose bombs and warheads
can be estimated to have a fragmentation size and velocity comparable with a .50
caliber Browning Machine Gun (BMG) rifle round or 50-gram projectile at 1000 m/s.
Mechanisms of Energy Dissipation in Steel and Polymers
Armor is used to provide protection for personnel and equipment from blast waves
and high velocity projectiles. The armor is designed to absorb the kinetic energy of
the blast wave or fragment and reduce the remaining energy in order to mitigate
16
harm to the object being protected. The armor can absorb the impact energy and
dissipate it through a variety means.
Inelastic Deformation
The plasticity exhibited by steel and some polymers cause the material to undergo a
permanent deformation. In metals, it is a shear-dominated phenomenon that occurs
where the underlying deformation mechanism are dislocations that glide along
crystallographic slip planes. Polymer plasticity is also shear dominated. Plastic
deformation occurs when the intermolecular barriers to yield are overcome by the
stress. Plastic deformation is caused by shear stresses generated in the material as
it absorbs the ballistic or blast loading. The work done in plastic deformation
dissipates the energy of the impact. An additional form of dissipation in polymers is
the viscous component of viscoelastic deformation, which is also associated with
hysteresis loops. Hysteresis loops give the energy dissipated during loading and
unloading of an elastomer. The energy is lost through internal friction.
Energy Absorption in Composite Armor
Composite armor is designed to deform when attacked by a high energy impulse.
This type of armor has a lower density than a monolithic steel plate and requires a
greater deformation to absorb an equivalent amount of energy. The composite
armor considered here is designed for energy absorption from blast waves and nonpenetrating projectiles.
Blast
Energy is transferred from an incident pressure wave to the metal armor through
plastic deformation. For a given thickness and hardness of metal, lower energy
blasts tend to create dents while higher energies rip the metal through shearing.
Some polymers, such as polyurea, have a large failure strain and can undergo
significant elongation before failure (Roland et al., 2010).
If the blast first
encounters steel, the polymer behind it stretches and provides a cushion to improve
the failure response of the steel. As the elongation increases, the dissipated energy
17
also increases (Yi et al., 2006). The layering of the polymer with the steel also
distributes the loading to the steel differently, in a less localized manner, which
enables the steel to undergo plastic deformation over more the volume of the steel
and hence increase the ability of the steel to dissipate the energy of the threats,
whether ballistic or blast.
Projectile Impact
Non-penetrating projectiles result in a local or global dent in the outer metal plate.
Fracture and spalling can also occur in harder metals. More ductile metals like DH36 tend to deform plastically rather than experiencing brittle fracture or spalling
(Klepaczko et al., 2009). As the metal thins in the region of the dent, the polymer
layer beneath it acts as a shock absorber to spread the initial impact impulse over a
greater area of the steel. Composite armors that maximize polymer deformation
absorb and dissipate the most energy.
Material Properties of Armor Components
Steel
Ballistic steel is frequently used for vehicle armor. According to the standardized
procedures used to determine material properties as outlined by the American
Society for Testing and Materials (ASTM) A370, a typical ballistic steel has high
hardness (-500 HBS) but low ductility with only 8% elongation (Steel Warehouse
Plate, 2013). This results in good protection against penetration, but can result in
fragmentation and spalling of the opposite side of the plate. The steel considered
here is DH-36 and is classified by the American Bureau of Shipping (ABS) as a higher
strength carbon steel used for shipbuilding that offers improved fracture resistance
to impact loadings (American Bureau of Shipping, 2012). DH-36 is more ductile
then most traditional armor steels with 21% elongation (BEBON International,
2013). This characteristic results in a fracture resistance greater than most armor
steels (Xue et al., 2010). This type of steel exhibits good strength and toughness,
very good weldability, and reasonable cost. DH-36 is specified in ASTM A131 to
have a minimum yield strength of 355 MPa.
18
DH-36 was found to have (Klepaczko et al., 2009; Nemet-Nasser et al., 2003) a
Poisson ratio of 0.3, a Young's modulus of 210 GPa, a density of 7.8g/cm 3, and a
chemical composition as shown in Table 1.
Table 1: DH-36 Steel Alloy Composition in %
C max
Si max
Mn
P max
S max
Ti max
0.18
0.5
0.90-1.60
0.035
0.035
0.02
Cu max Cr max
0.35
0.2
Ni max
0.4
V
Nb
Mo max
0.08
0.02-0.05 0.05-0.10
PSM4
PSM4 is a photoelastic polymer frequently used for stress field visualization. In the
This polymer has a
following experiments its birefringence is not utilized.
relatively low Young's modulus and experiences essentially no plastic deformation
or hysteresis even during large strain deformation. This is considered a benefit that
permits repeated testing. The material exhibited very little rate dependence. PSM4
is utilized as a model elastic material for comparison with polyurea.
It was
produced by Vishay Precision Group was stated by the manufacturer to have the
following characteristics: a Young's modulus of 3.45MPa, a Poisson ratio of nearly
0.5, and a density of 1 g/cm 3 . Initial material characterization was completed by
uniaxial compression of an upright cylinder and is shown in Figure 3.
Characterization of PSM4
-,
-,-
PSM4 Cylinder
2.5-
C,,
I15
0.5 -
0
.-
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0. 9
1
True Strain
Figure 3: PSM4 stress-strain behavior showing highly elastic large strain behavior
19
Polyurea
Polyurea is primarily used as a waterproof coating for vehicles, boats, and buildings.
Here, polyurea refers to a specific formulation designated as polyurea 1000 (PU).
This polymer exhibits large dissipation and resilience with significant rate
dependence. The material's stiffening at high strain rates makes it ideal for armor
applications.
The uniaxial compressive stress-strain behavior of polyurea was
examined over 7 decades of strain rate by Sarva et al. Figure 4 shows the behavior
of polyurea at strain rates ranging from 10- 3 s-1 to 9000s-1 (Sarva et al., 2007).
A 70
A900081
6>.500su
56
Intermed ate Rate SHP8
2250 s'
SHPB
CL
1200 6*
14
016 s
0
02
04
0.6
0!8
True Strain
1.2
1-4
Figure 4: Strain rate stiffening of polyurea reprinted with permission
Chemically, PU is a cross-linked elastomer consisting of hard and soft segments. The
"hard" component is diisocyanate and exists in the glassy state at room
temperature. The "soft" component is diamine and it exists in the rubbery state at
room temperature.
The thermodynamic incompatibility results in a phase-
separated microstructure (Mock et al., 2011). The large dissipation exhibited under
loading, occurs due to a change in the microstructure of the polyurea. Specifically, a
breakdown in the hard domain aggregate network structure occurs and is the
governing mechanism for the large dissipation of the first loading cycle (Rinaldi et
al., 2011). The large dissipation can be seen in Figure 5.
20
PU 1000 Cylinder Compression
12
Strain Rate 0.1/s
10-
cu
0.
8-
0
0
e
6--
U)
e
4
0/
I-
2-
0.1
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
True Strain
Figure 5: PU 1000 stress-strain behavior showing energy storage and dissipation
The PU used for experimental testing was produced at Naval Surface Warfare Center
Carderock Division (NSWCCD). It was created by combining 80% oligomeric amine
and 20% of an isocyanate curative (Mock et al., 2011). It is composed of 36% of
"hard" isocyanate, which is bulky, stiff, and has a high glass transition temperature.
The remainder is "soft" diamine, which is long, flexible, and has a low glass
transition temperature. Rinaldi et al used dynamic mechanical analysis (DMA) to
show that PU has two glass transition temperatures and exhibits elastomeric-like
behavior at room temperature (Rinaldi et al., 2011).
10000
-
0.2
1000
0.15 <o
T-
E- 40MPa
0.1
E
LO
100
0.05 00
10
-100
-50
0
50
100
Temperature, T (oC)
-- 0
150
Figure 6: DMA (1 Hz) Isochronal plots of PU between -100 and 150 *Creprinted with permission
The initial Young's modulus of the PU was determined to be 70 MPa with a density
of approximately 1.1 g/cm 3 .
21
Material Models of Armor Components
Steel
The deformation characteristics of steel have been well studied and were not
reproduced. The properties of the steel used in armor simulations were taken from
experiments found in the literature (Dikshit et al., 1995; Klepaczko et al., 2009;
Nemat-Nasser et al., 2003). DH-36 is a high strength, low carbon alloy steel. The
ASTM defines DH-36 to have a yield strength of 355MPa and an elongation of 21%.
The flow stress of the DH-36 was determined using a Johnson-Cook constitutive
model with strain rate dependence. Johnson-Cook is a phenomenological model
that is matched to experimental data for accurate predictions. The material flow
stress is calculated using strain, strain rate, and temperature. The first part of the
equation is a hardening relationship that follows the power law where the constant
A represents the quasi-static yield stress at the reference strain rate. The constant B
provides the strain hardening modulus, while n is the strain-hardening exponent.
The second term accounts for the strain rate sensitivity of the material through the
constant C. The third term takes in temperature effects.
The constant m is
temperature sensitivity, T is the current temperature, To is the initial temperatures
and Tm is the melting temperature (Klepaczko et al., 2009).
y=
(A + BEp)(1 + Cln(-7)(1-
(3)
m
=T-*3
(Tm-To)
(4)
Table 2: DH-36 Johnson-Cook Model Parameters
A(MPa)
B(MPa)
n
o(s-')
C
m
To (K)
Tm(K)
1020
1530
.4
.1
.015
.32
50
1773
Where the metals flow stress, ay, is given by material constants, A,B,C,n,m, with
reference temperatures To, and Tm.
22
PSM4
PSM4 is a simple polymeric material that provides a basis of analysis to convey the
fundamental nature of elasticity and deformation. The study of PSM4 provides an
introduction of the mathematical models used in finite element simulations. This
material also demonstrates some of the basic response of the more complicated PU
polymer used later.
This same material was used by Bertoldi et al to analyze the compression
characteristics of PSM4 sheets with a lattice of holes (Bertoldi et al., 2008). They
used a second order Yeoh model in order to better capture small strains, which are
not as important in this analysis.
In this study, strains of 0.2 and higher are
common. They found the initial Young's modulus as 3.25 MPa. In contrast, this
study used a compressible Neo Hookean model that captures the majority of the
moderate and large strains very well. The PSM4 experiments were accurately
modeled with ABAQUS 6.11 using the material parameters of 4.35MPa as the initial
Young's modulus.
A strain energy density function is used to approximate the mechanical behavior of
the material. A compressible model give close results for uniaxial compression up to
0.8 strain.
Only the compressible Neo Hookean model was considered in this
analysis. The Neo Hookean model is based solely on the first invariant of the left
Cauchy-Green deformation tensor (B). The left Cauchy-Green strain tensor is based
on the deformation gradient (F). Assuming F is given in the principle directions, the
matrix B can be constructed as shown in equation 9.
1A 0
B 2 FFT = 0
A2
0
0
0
0
2
(5)
The first invariant (I1) is the trace of B.
tr(B)= A2+
(6)
2 + A2
The compressible strain energy density function (U) is also based on material
properties, E0 , the initial Young's modulus, and
C1=A2 =
K,
the bulk modulus.
(7)
6
23
D1= 2-(8)
The strain energy density function used for this model is
2C 1 (J-2 / 3 11 - 3)
UNH=
+ D1(J-1) 2
(9)
The volume ratio Jis defined as the determinant of F.
J 1e det(F) =1X213
(10)
Considering uniaxial compression in the 1-direction with symmetry
2=3=(11)
The relationship between the Cauchy stress and the left Cauchy-Green strain tensor
is shown below: (Ogden, 1984)
B
1ae
2
=
(12)
The Cauchy stress is found for a general energy density function after differentiation
as:
a=y
+I1j)B-yB -B+2J 'j
1
(13)
Where
1
1= 0
0
0
1
0
0
0
1
(14)
After completing the differentiation on the strain energy density function and
making use of the uniaxial symmetry, equation 17 is solved for the uniaxial case in
terms of stretch and material parameters as
G11NH
_-
2
-
(15)
+ 2D 1 (J - 1)
The value of J can be found for each value of X. It is determined from the boundary
conditions and the bulk modulus. Two equivalent equations for stress can be found
(G22=a33)
and when set equal to each other yield the next equation.
0=J(8 / 3)D1 - J(5/3D1+
(16)
--
The roots of this equation give J and now
G11NH
can be solved.
The material
parameters C1 and D1 used to fit the model to the experimental data were 0.725MPa
and 1GPa respectively.
24
Experimental data and the Neo Hookean model shown in equation 15 are compared
and shown in Figure 7.
PSM4 Characterization
A
--
Compressible NH Simulation
Experimental PSM4 Data
-
3.5 2-A
1.25
0.5-
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
True Strain
Figure 7: PSM4 experimental and simulated stress-strain using compressible Neo Hookean model
The compressible Neo Hookean model was used in finite element simulations and
produced an adequate fit for both uniaxial characterization and constrained cylinder
array compression.
Polyurea
PU exhibits a changing response based on its deformation history. It shows both a
viscous and an elastic response during loading. As a polymer it is composed of long
chains of repeating units. The long chains contain both hard and soft segments in
the repeating units.
These units can vary in the mixing process and are not
necessarily homogeneous. Many of the unique properties of PU occur due to the
morphological arrangement of soft and hard segments (Rinaldi et al., 2011).
Viscoelasticity and Viscoplasticity
Viscoelasticity causes PU to creep, to relax, and show rate dependence (Cho et al.,
2013). This viscoelasticity also results in hysteresis during loading and unloading.
When deformed, PU experiences some plasticity resulting in unrecoverable shape
change. This can be most easily seen in the load/unload curve and recognize the
25
start and finish strains are not equal. PU also undergoes a breakdown in the
morphological structure, which softens the behavior and also gives significant
hysteresis. The combination of viscoelasticity, viscoplasticity, and strain-induced
softening results in an even larger hysteresis than with viscoelasticity alone. The
plastic strain of the PU samples is generally 5-20% of the total strain depending on
the applied strain. For the maximum strain of 1.0 shown in Figure 8, the permanent
plastic strain is about 0.2.
Polyurea Cylinder Compression
12
1- - - - Strain Rate 0.001/s
Strain Rate 0.01/s
- - - Strain Rate 0.1/s
10
0
0.1
0.2
0.3
0.4
0.5
0.6
True Strain
Figure 8: PU
0.7
0.8
0.9
1
Cylinder Compression Is Strain-Rate Dependent
Rate Sensitivity
Small cylinders were tested in compression to determine the loading and unloading
characteristics. The hard segments in the polyurea structure govern the low strain
rate behavior. As shown in Figure 8, the PU has some rate sensitivity even at these
low strain rates. Previous experiments with PU show a substantial jump in strain
rate sensitivity around rates of 1/s. Above strain rates of about 1/s, the flow stress
of the material increases markedly because the soft segments now offer additional
resistance (Cho et al., 2013). At the higher rate, the soft domains are no longer fully
above their glass transition temperature and this provides additional resistance to
deformation.
26
Model
Many researchers have attempted to accurately model PU (Shim and Mohr, 2011; Li
and Lua, 2009; El Sayed et al., 2009; Amirkhizi, et al., 2006). The loading behavior
has a relatively stiff initial response followed by a rollover to a more compliant
slope. This type of rollover response has been successfully modeled (Boyce et al.,
2000). The unloading response, and related energy dissipation, is more challenging.
The model used in this work was developed by Boyce and Cho (Cho et al., 2013) at
MIT's Solid Mechanics Lab.
The full explanation of the model is found in Soft
Matter's 2013 article "Constitutive Modeling of the Rate-Dependent Resilient and
Dissipative Large Deformation Behavior of a Segmented Copolymer Polyurea". A
brief overview of the constitutive model is given here for general background into
its function. The model breaks down the response into 4 general contributions. The
hard domain is represented by a spring-dashpot for the intermolecular response
and a spring for the network response. The soft domain has a spring-dashpot for
the intermolecular response and another spring-dashpot for the network response.
The hard network response is best modeled as a compressible Neo Hookean
material. The remaining hard intermolecular response is modeled using an elastic
element in series with a nonlinear viscous element captured with an Eyring element.
The soft intermolecular response again follows a compressible Neo Hookean
material and the network response uses an elastic element in series with a
nonlinear viscous element.
The constitutive model was implemented through a user-defined material in
ABAQUS. The fit of the model with the experimental data for uniaxial compression
is shown Figure 9.
27
Polyurea material response
Experiment Load and Reload
Simulation Load and Reload
--
12
A
1
a.
AYA
4A0
2
I-
0
~
0.1
.--
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
True Strain
Figure 9: Uniaxial compression using the constitutive model for PU and experimental data at 0.1/s
28
Chapter 3
PSM4 Array Compression
Materials and Methods
Fabrication and Mechanical Testing
Testing was conducted on 4.8mm diameter PSM4 cylinders measuring 9mm in
length. Five cylinders were laid side-by-side with no gap between cylinders and the
sidewalls. The cylinder faces were smoothed and lubricated to minimize friction
between the face and the fixture. This allowed the plane strain assumption to be
used. The array was compressed by a smooth metal bar measuring 24mm long,
9mm wide and 50mm tall. Each array of 1, 2, or 3-rows was considered a unique
experiment.
Array Void Fraction
Based on the stacking arrangement, the void volume fraction of the array changes
with each layer added.
2r
T
Figure 10: 1-row void dimensions
r
43 r
r
T
Figure 11: 2-row void dimensions
29
I
r
43 r
43 r
r
T
Figure 12: 3-row void dimensions
This is shown graphically in Figure 10 through Figure 12 and is also listed in Table
3. The single row has the greatest initial void fraction because the flat base and
compressor do not fill any of the area between the cylinders. The second and
subsequent rows fall into the space in the row beneath it resulting in a reduced void
fraction. The total height of the 2-row array is 2r + 1Fr or 8.96mm. The height of
the 3-row array is 2r + 2-i3r or 13.11mm. The void fraction continues to decrease
as rows are added. Eventually, the impact of the top and bottom rows become
insignificant and as the array approaches an infinite number of layers, the void
fraction approaches a limit.
Initial Void Fraction
= Total rectangle area-Cylinderfrontal area
Total rectangle area
(17)
By letting
X = # Rows -1
(18)
The IVF equation becomes
IVF = 10r 2 (2+Xvfs(X+1)7rr2
10r 2 (2+XV3
(19)
or simplified to
IVF = 4-7r+X(2V3-7r)
4+ x(243)
(20)
As the number of rows grow (as X-+oo), the IVF equation approaches a limit.
IVF=
=
0.0931
(21)
30
............ ....
..
.. ..
..
..
. .....
..
....
Table 3: Void Fraction of Various Array Sizes
Number
Height (mm)
Initial Void Fraction
1 Row
4.8
0.215
2 Rows
8.96
0.158
3 Rows
13.11
0.138
o
oo
0.093
Finite Element Model
A two-dimensional plane strain model was developed using finite element analysis.
The PSM4 cylinders were modeled with 4 node bilinear plane strain quadrilateral
elements. The ABAQUS library defines these elements as CPE4R. The cylinders
were modeled with contact against rigid sides, top and bottom. The normal contact
was defined as "hard", while the tangential contact was modeled with friction. The
coefficient of static friction was 0.1. The mesh can be seen in Figure 13. The model
was compressed through displacement control and the reaction force was
measured.
31
1 Row mesh
2 Row mesh
3 Row mesh
Figure 13: PSM4 array finite element models for 1, 2,and 3-rows
Experimental and Simulated Response
Small Cylinder Arrays
In Figure 14, an array with 1-row in the uncompressed state is shown. A speckle
pattern was added to the face of the cylinders to allow strain measurement via video
extensometer. The array was compressed by % of the cylinder diameter, or by
1.2mm for an overall engineering strain of 0.25. The compressed array is shown in
Figure 16.
32
Figure 14: PSM4 array of cylinders shown in uncompressed state
The numerical simulation was carried out with the same sample sizes and boundary
conditions as the experiment. The FE simulation is shown in Figure 15 as an overlay
on the experimental photo to demonstrate the orientation of the model.
Figure 15: PSM4 array of cylinders uncompressed with FE simulation overlay
The FE model resembled the experiment to the fullest extent possible. In addition to
force vs. displacement data shown in Figure 18, the curvature change was also
compared with the experimental results.
Figure 16: PSM4 array of cylinders in fully compressed state
The array was fully compressed in the finite element model and produced the
flattened shapes shown in the shaded portion on the left of Figure 17. While the
simulation is close to the experimental results, the simulation slightly overestimates
the shape change.
33
Figure 17: PSM4 array uncompressed state with FE simulation overlay
The experiment was repeated for 1, 2, and 3-rows. Each array was compressed at a
constant strain rate of 0.01/s. The force vs. displacement plot with finite element
simulation is shown in Figure 18 with very good agreement.
Small PSM4 Array Compression for 1, 2, and 3 Rows
800
1
-'-
700
1RowExp
2 ROWS Exp
- 3 Rows Exp
A 1 Row FE
A 2 Rows FE
3 Rows FE
500-
A-
200300-
0
A
0.5
1
1.5
2
2.5
3
3.5
4
Displacement (mm)
Figure 18: Experimental and simulated data for 1, 2 and 3-rows of PSM4
As more rows are added, the force-displacement response changes and becomes
more compliant with the addition of each row, in part, due to the increase in overall
height of the sample as each row is added. This dependence on number of rows is
not only a height effect but also the void fraction. The effective stress vs. true strain
curve is shown in Figure 19.
The plots of array compression use effective
engineering stress using the initial overall area. The true strain, E, is shown in
equation 22.
34
E=
ln(A) = in (-)
(22)
H;T
The stress-strain curve shows the 1-row sample to be the most compliant, in part
due to the higher void fraction and, in part, due to the manner in which the cylinders
deform. The 2-row and 3-row arrays show nearly identical stress-strain behavior
due to their similar void fraction and manner in which the cylinders deform. This
suggests that the 3-row stress-strain behavior is close to that of an infinite array.
PSM4 Stress-Strain
S1.5
U)
0
0.05
0.1
0.15
0.2
0.25
0.3
Strain
Figure 19: PSM4 array effective stress-strain curve
The experimental strain fields of PSM4 arrays were captured by digital image
correlation (DIC).
The FE fields for compressive strain, lateral strain, and shear
strain are shown for comparison with experiment in Figure 20 through Figure 25.
35
-nky
en[ -11
0.104
PSM4 Array under
compression
0.151937
Exx
0.127913
0.10165
0.155375
0.06M75
0.103M32
LE, LE11
(Avg: 75%)
+2.528e-01
+2.039e-01
+1.549e-01
+1.059e-01
+5.694e-02
+7.961e-03
-4.102e-02
-8.999e-02
-1.390e-01
-1.879e-01
-2.369e-01
-2.859e-01
-3.349e-01
0.091625
0.109625
E= 0.1
0.06715
0014375
0.043375
E= 0.2
0.0313553
0.01M2
&--I I I %
0,0071875
4.7806W75
-02193
Figure 20: PSM4 1-row array strain field comparison (Exx) of experiment and simulation
36
LE, LE12
(Avg: 75%)
+2.990e-01
+2.528e-O1
+2.039e-01
+1.549e-01
+ 1.059e-01
+5.694e-02
+7.961e-03
-4.102e-02
-8.999e-02
-1.390e-01
-i.879e-01
-2.369e-01
-2.859e-01
-3.349e-01
-4.237e-01
0.103
PSM4 Array under
compression
0.0911875
0.079375
Exy
0.0675625
0.01075
0.0439370
0.032125
0.023125
0.0080
-0.003302n
I
E= 0.2
.0.01012
-0.009375
- A~
40.03975
E= 0.25
.0050525
4.062375
-. 0741875
Figure 21: PSM4 1-row array strain field comparison (exy) of experiment and simulation
37
PSM4 Array under
Co mnpression
0."
O.OD0n'
LE, LE22
(Avg: 75%)
+2.528e-01
+2.039e-01
+1.549e-01
+1.059e-01
+5.694e-02
+7.961e-03
-4.102e-02
-8.999e-02
-1.390e-01
Eyy
46"1
-1.879e-O1
-2.369e-01
-2.859e-01
-3.349e-01
413325
~~-0.2221B
42445
F= 0.2 5
F1ai311g25
Figure 22: PSM4 1-row array strain field comparison (eyy) of experiment and simulation
38
Exx
0.206
E= 0.1
0.1sM"
0.1435
0.127875
LE, LE11
(Avg: 75%)
+2.150e-01
+1.502e-01
+1.178e-01
+8.544e-02
+5.306e-02
+2.068e-02
-1.170e-02
-4.408e-02
-7.647e-02
-1.088e-O1
-1.412e-01
-1.736e-01
0.11225
O.oMS
E=
0.2
0.081
-
0.065375
04975
50.03412S
0.0355
0.00275
40.03375
4.0"4
Figure 23: PSM4 2-row array strain field comparison (exx) of experiment and simulation
39
I0.142
0.125125
E= 0.1
0.091375
0,0)s5
0.007625
LE, LE12
(Avg: 75%)
+6.237e-01
+5.302e-01
+4.366e-01
+3.430e-01
+2.494e-01
+1.558e-01
+6.226e-02
-3.132e-02
-1.249e-01
-2.185e-01
-3.121e-01
-4.056e-01
-4.992e-01
E= 0.2
0.007
-0.00167
-ohan
4.04362
4.060s
0.077375
.0942s
-o.11112s
- 2n
om
Figure 24: PSM4 2-row array strain field comparison (exy) of experiment and simulation
40
evy[(a 0.06
E0.
6OK"
LE, LE22
(Avg: 75%)
+7.049 e-02
+3.567 e-02
0.030US
e02
E= 0.1
-6.8824e-02
e0
5 e-01
-1.733 e-01
e-01
e-01
-2.7784
5'02.2s3
O.OW2U~l-3.3994
F.02.081 e
Ssf
c-2.430
E= 0.2
4388-125
-04069
feldcomprisn (sy) f exerient
rraystrin
25 PS4 2-ow
Figue
S=
nd
3imultio
fl2
Large Cylinder Experiments
In order to validate the small cylinder array results, a larger array was tested. The
larger array cylinders were approximately twice the diameter of the small array at
9.2mm and 9.6mm in length. The large cylinders were cut with a laser and did not
have tabs like the smaller PSM4 cylinders. The experimental and simulated loading
curves shown in Figure 26 show reasonable agreement.
41
Large PSM4 Array Compression
. 0
450
--400
o2
1 Row
1 Row FE
2 Row
Row FE
350
-
300
-p
0
250
0
IL
O
-
200
0
150
-00
100
-0---
50
0
0
000
0
-
-
-0
0
0.5
1
1.5
2
2.5
Displacement (mm)
Figure 26: Experimental and simulated data for 1 and 2-rows of large PSM4 cylinders
The finite element simulation produced a force vs. displacement plot that shows
good agreement with the 1-row experiment.
The array is shown in the
uncompressed state in Figure 27 and in the maximum compressed state in Figure
29.
17
Figure 27: PSM4 large cylinder array--note PMMA spacer on right
The finite element simulation is overlaid on the uncompressed array picture in
Figure 28. Only the left half of the array was modeled for computational efficiency.
The FE simulation used the experimental sample size and boundary conditions.
42
Figure 28: PSM4 large cylinder array uncompressed with FE overlay
The large PSM4 array was compressed to the same maximum strain as the small
PSM4 array. A maximum strain of 0.25 was used to capture significant shape change
while still enforcing the experimental boundary conditions.
The finite element
simulation is overlaid on the compressed array picture in Figure 30 with very good
agreement.
Figure 29: PSM4 large array under full compression
Figure 30: PSM4 large array under full compression with simulated deformed array overlay
Comparison of Large Array and Small Array
The large cylinders were tested to verify that sample fabrication techniques did not
affect the experimental results. The force-displacement curves did not match those
43
of the small array due to the different cylinder dimensions. The large array was
compared to the small array using the effective stress of each array vs. the true
strain of that array. The effective stress-true strain curves of the large arrays was
found and compared to the small arrays. The effective stress was found by taking
the force and dividing by the initial overall area of the large array, and the strain was
found using the displacement and dividing by the height of the large array. As
expected, both the 1 and 2-row large arrays matched the small arrays as shown in
Figure 31 below.
PSM4 Array Compression
-
1R
1R
- - - -2R
2R
- -
0.9 -0.8
Large PSM4
Small PSM4 Large PSM4
Small PSM4 .
-
C- 0.70.60.5
--0.4-
00
0.4
LU 0.3-
0.1 0
0.05
0.1
0.15
True Strain
0.2
0.25
Figure 31: Effective stress-strain comparison of large and small PSM4 arrays
Effects of Lateral Array Constraints
Lateral constraints play a major role in the overall array response. Initially, the
array shows a similar compliance with free cylinders under compression as shown
in Figure 33. As the material is compressed and the voids are filled, the array
responds in a similar fashion to a constrained block of material. This compression
regime simply tests the bulk modulus of the material. The initial response of a
single row of cylinders can be easily explained. The force that results from one
cylinder with no lateral constraint follows simple non-adhesive elastic contact
mechanics developed by Hertz as shown below.
44
(23)
F=(lT/4)E*LS
Where L is the cylinder length, 5 is the indentation depth. The effective Young's
modulus (E*) is found by
1 =1+ E*
El
(24)
E2
where E1 and vi are the Young's modulus and Poisson ratio of that material. The
effective radius is found by
1= +1
R
R1
(25)
R2
and the contact radius a, follows
(26)
a=VR
The force was calculated for one cylinder compressed between two plates as shown
in Figure 32.
F
2a
Figure 32: contact radius resulting from flat plate compression based on Hertz equation for a cylinder
The Hertz contact force for one cylinder was found using the equations above and
was multiplied by 5 to simulate the reaction force of the 1-row array when
unconstrained laterally.
This force was analyzed at several displacements and
plotted on Figure 33 as the black (unconstrained) line. As expected, the force, as
captured by the Hertz model, initially follows the equation in the small deformation
regime. However, at larger deformations, there is an increasing contact area. This
changing geometry gives the nonlinear stiffening and departure from the Hertz
solution.
45
Unconstrained vs. Constrained Array Compression Comparison
400
- - -
350
0
Experimental PSM4 Constrained
Hertz Contact PSM4 Estimate
Bulk Response
300
0
250200-
.-
0
150
0
100
-
..
00
50-
0
0
0.5
1
1.5
Displacement (mm)
Figure 33: Experimental constrained array vs. unconstrained simulated array using Hertz equation
The other extreme, the fully constrained case, was examined by assuming the
cylinders were compressed and fully deformed with horizontal and vertical
constraints.
The fully constrained and compressed case is shown in Figure 34.
Constant volume is assumed during the deformation until the cylinder has become a
block. The bulk modulus of the PSM4 material is approximately 2 GPa and results in
a nearly vertical line (blue circles) in the force vs. displacement plot.
The
experimental results show this vertical line is approached, but due to flexing of the
compression
fixture, a fully constrained condition could not be achieved
experimentally.
F
E
E
4.81 mm
Figure 34: Deformation of fully constrained compression
46
Effect of Compression on Void Fraction
As the array is compressed, the void fraction changes. The cylinders deform and
become more flat with sharper corners. The interstitial area is slowly filled, while
the density of the material also increases slightly. The resulting change in the void
fraction is shown in the middle plot of Figure 35. Additionally, the instantaneous
slope (k) of the force vs. displacement experimental curve was compared with the
simulated values in the bottom plot of Figure 35.
Small PSM4 Array Compression for 1, 2, and 3 Rows
800
III,
00 -----1 Row Exp
z 600 2 Row Exp
3 Row Exp
S400 --u 200
-,
...
--
0
0.5
1
1.5
2
2.5
3
3.5
0
0
0.5
1
1.5
2
2.5
3
3.5
3 0.2
0
U
800
I
-,d400 -
0
200-0
0
-0-!5
-
-0--
0.5
1
2
1.5
2.5
3
3.5
Displacement (mm)
Figure 35: Void and slope change compared with compressive force on 1, 2, and 3-row arrays
The initial void fraction partially governs the response of the array. As shown in the
middle plot of Figure 35, the initial void fraction slope of 1-row is much steeper than
that of the 2 and 3-rows. The 2 and 3-row void fraction curves are close together
indicating they are close to the infinite value. The effective stress vs. strain plot of
the PSM4 arrays in Figure 19 shows a significantly more compliant response for 1row and a nearly identical low strain response for 2 and 3-rows. For a given strain
on the array, the stress increases as the initial void fraction is reduced. The slope of
the effective stress vs. strain curves continues to increase as the initial void fraction
47
decreases. Eventually, as the initial void fraction goes to zero, the bulk modulus of
the material is tested when the fully constrained material is further compressed.
Conclusions
The preceding experiments demonstrate the impact of a new geometry on the array
response. The cylinder array response is based on the material itself, the initial void
fraction, and the cylinder contact area. The constrained array response of PSM4 is
shown in Figure 36 for 1, 2, and 3-rows. The digital image correlation in Appendix A
shows that the 1-row array has the largest contact area with the compression bar
and the highest peak local stresses. The single row has the largest initial void
fraction, and this results in the greatest compliance.
PSM4 Stress-Strain
2.5
-- -2
--
1 Row PSM4
Row PSM4
-3Row PSM4
2
0.5
0
0
0.05
0.1
0.15
Strain
0.2
0.25
0.3
Figure 36: PSM4 experimental effective stress-strain curve
The arrays initially deform almost freely and follow the unconstrained Hertz force
for small strains. The initial void volume of the array is reduced as the cylinders
deform and take up the interstitial space. The array response rapidly stiffens as the
material becomes more fully constrained. This initial compliance and subsequent
stiffening can be explained by analyzing the impact of the initial void fraction and
the material's constraints. The global tunable stiffness exhibited by the arrays can
have many uses.
48
Geometry can be used to create a unique global response with the same material.
One application for this type of response is composite armor.
If a constrained
polymer is combined with steel, such as in a steel-polymer-steel sandwich, the
steel's strength and large energy dissipation during plastic deformation could be
used capture the initial event. As the steel continues to deform and weaken, the
polymer increasingly stiffens and may limit further deformation of the steel to
prevent fracture.
49
50
Chapter 4
Polyurea Array Compression
Materials and Methods
Fabrication and Mechanical Testing
The PU samples obtained from NSWCCD were cut from a solid puck shown in Figure
79 using a water jet cutter. Material characterization was completed by uniaxial
compression of upright cylinders. Various height-to-diameter ratios were compared
with consistent results. Most samples were 4.8mm in diameter and 4.8mm tall.
Several different lubricants were used. A copper nanoparticle deposition provided
the most accurate results for the characterization experiments, and calcium
bicarbonate gave the best results for array compression. The samples were tested
at various strain rates from 0.001/s up to 0.1/s. The material exhibited significant
rate dependence as shown in Figure 8. Higher strain rates were tested, but the data
collection speed was not adequate for accurate results.
Finite Element Model
The polyurea model developed by Cho and Boyce was previously explained in
Chapter 2 and was used for the numerical analysis with the constrained arrays (Cho
et al., 2013). The solid material was discretized into 3d elements with eight nodes
and reduced integration defined as C3D8R by the ABAQUS library. The mesh for
each row can be seen in Figure 37. The modeled used displacement control and
generated loading and unloading curves.
51
1-Row Mesh
2-Row Mesh
3-Row Mesh
Figure 37: Polyurea array simulation mesh for 1,2, and 3-rows
Experimental and Simulated Response
Small Cylinder Arrays
The same fixture and experimental setup for the PSM4 was used for the PU array
testing to ensure plane strain conditions. Each array measured 5 cylinders wide.
Each cylinder measured 4.8mm in diameter and 9mm tall.
The array was
compressed by a smooth metal bar measuring 24mm long, 9mm wide and 50mm
tall and shown in Figure 83. The 1-row cylindrical array in the uncompressed state
is shown in Figure 38.
52
Figure 38: Polyurea cylinder array uncompressed
The FE model used symmetry to reduce the computation time. The uncompressed
FE model is overlaid in grey on the left half of the experimental photo in Figure 39.
Figure 39: Polyurea cylinder array uncompressed with FE overlay
The array was compressed to various displacements and compared with the FE
model force vs. displacement curve and shape change. The shape change can be
seen in Figure 40 below. The overlay of the 1-row FE model and the experiment in
their fully compressed states is shown in Figure 41 below.
Figure 40: Polyurea cylinder area in the compressed state
53
Figure 41: Polyurea cylinder array compressed with FE overlay
The deformed 2 and 3-row arrays are shown in Figure 42 and Figure 43 with FE
mesh for comparison.
In general, the FE deformation is more severe then the
experimental photograph. This is due to the flexibility of the compression fixture
and therefore less precise experimental constraints.
Figure 42: 2-row polyurea array in compressed state with FE overlay
Figure 43: 3-row polyurea array in compressed state with FE overlay
54
Each array of 1, 2, or 3-rows was considered a unique experiment.
Because the
microstructure of the PU physically changes under strain, the samples could not be
used again for another test. The exception to this was during reloading tests where
the permanent dissipation was studied. The constitutive polyurea model developed
by the Boyce group and explained in Chapter 2 was used to simulate array
compression for 1,2, and 3-row cylindrical arrays. The force vs. displacement plot is
shown in Figure 44. A comparison of the ideally constrained FE simulations and
experimental results can be seen in Figure 45. The FE data accurately captured the
peak loading and unloading behavior. The experimental force was likely reduced by
modest bending of the PMMA fixture under higher loads. As seen in Figure 40
above, the polyurea cylinders pushed the front PMMA face out and permitted some
expansion of the cylinders through elongation, reducing the peak experimental
force. If the cylinders were fully constrained in a stiffer fixture, the peak force
would be closer to the FE result.
Polyurea Array Compression
1800
1600-
--- 3R
14001200.-
1'1000 -
-
o 80-
I.
Figure 44: Polyurea array compression experiments
55
Polyurea Array Compression
1000 -
2 o
-
4--
600
.E
o
0
d.2
2
0.2
0.4
"
m
0.6
o
0.8
1
2
14
1
1.2
1.4
1.6
1.8
2
Displacement (mm)
Figure 45: Polyurea array compression experiments with FE simulations
The 1-row experiment and simulation general response match well, with a deviation
only a higher forces. The 2 and 3 row experiments are increasingly compliant
compared to the simulations. This compliance was primarily caused by an increase
in the constrained experimental volume through the elongation of the cylinders. As
the cylinders were compressed they created a lateral force stretching the
experimental fixture in the horizontal direction and increased the volume of the
experiment. The samples also elongated under compression and stretched the face
of the fixture in the out-of-the-page direction. This also increased the volume of the
samples occupied and further weakened the constant volume goal of the imposed
constraints.
When the volume increased, the testing of the bulk response became
delayed and the compliance of the fixture was tested.
For this reason, the
experimental tests showed excessive compliance. An additional observation was
made regarding the unloading curves, and therefore the dissipation. The simulated
unloading curves show more dissipation then the experimental curve due to the
reduced peak experimental force and shape change.
A speckle pattern was sprayed on the faces of the cylinders in the array tests. Digital
image correlation (DIC) was used to determine the strain field for the various tests.
56
All the arrays were compressed to a strain of 0.15 at a strain rate of 0.01/s. The true
strain fields for the y direction, labeled Eyy, and the x direction, Exx, and also the shear
strain, Ey can be seen in Figure 47 to Figure 54.
Differences between the
experimental and simulated strain fields occur due to the bending of the
compression fixture. The experimental constraints were not effectively maintained
at larger forces resulting in variation from the simulated results. The force vs.
displacement experimental data was converted to effective stress vs. true strain as
shown in Figure 46. The arrays exhibited dissipation and a unique response based
on the number of rows present.
PU 1000 Array Compression
3.5
3
2.5
W 1.5
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
True Strain
Figure 46: Effective stress-strain curves for PU array compression for 1,2, and 3-rows
57
x[1I - nar
M0.202
0.18s62s
LE, LE11
(Avg: 75%)
+1.607e-01
+ 1.349e-01
0.1S25
0.152875
0.1365
0.1U251
0.10375
0.05
0.087375
0.071
0.05625
0.1
0.03825
0.02875
0.0055
0.15
4.010875
-0.02725
-0.043625
-
-0.06
Figure 47: Polyurea 1-row array strain field Exx
58
+1.091e-01
+8.330e-02
+5.748e-02
+3.167e-02
+5.853e-03
-1.996e-02
-4.578e-02
-7.159e-02
-9.741e-02
-1.232e-01
-1.490e-01
exY013
1-Hen
flO 13
0.113875
0.09775
0.081625
0.0655
0.037
0.05
0.03325
0.017125
0.001
0.1
-0.015125
0.03125
-0.047375
-0.0635
0.15
0.079625
-0.09575
-0.111875
F -0.wa
Figure 48: Polyurea 1-row array strain field exy
59
LE, LE12
(Avg: 75%)
+2.353e-01
+1.877e-01
+1.402e-01
+9.256e-02
+4.496e-02
-2.630e-03
-5.022e-02
-9.782e-02
-1.454e-01
-1.930e-01
-2.406e-01
-2.882e-01
-3.358e-01
EyY
to Hwdv
00624999
1
4eus
-0.0=1
0.05
4.S0
4W)1
0.1
*0325
43223
4201
421675
Figure 49: Poiyurea 1-row array strain field ey
60
LE, LE22
(Avg: 75%)
+1.384e-01
+1.384e-01
+1.107e-01
+8.305e-02
+ 5.536e-02
+2.766e-02
-2.924e-05
-2.772e-02
-5.542e-02
-8.311e-02
-1.108e-O1
-1.385e-01
-1.662e-01
-1.939e-01
-1.939e-01
0.187
LE, LE11
(Avg: 75%)
+ 1.696e-01
+1.696e-01
+1.480e-01
+ 1.263e-01
+ 1.047e-01
+8.305e-02
+6.141e-02
+3.977e-02
+1.814e-02
-3.50le-03
-2.514e-02
-4.678e-02
-6.841e-02
-9.005e-02
0.a1712
0.1502s
0.144037
0.05
0.M125
0.111072
0. 0,W75
0.0876875
0.0735
0.1
0.0993125
0.045125
0,029375
0.01675
0.0025625
4.011625
4.0258125
.04
Figure 50: Polyurea 2-row array strain field exx
61
Exy
exy f14- hvky
0.085
0.07525
0.0655
0.046
-
0.05
-.am
0.0265
0.01675
0.1
-0.0257
-4.0075
0.15
E
0.0517
0.06125
-0.07t
Figure 51: Polyurea 2-row array strain field exy
62
LE, LE12
(Avg: 75%)
+2.819e-01
+2.359e-01
+1.898e-01
+1.438e-01
+9.776e-02
+5.172e-02
+5.677e-03
-4.037e-02
-8.641e-02
-1.325e-01
-1.785e-O1
-2.245e-01
-2.706e-01
E 00
0.05
LE, LE22
(Avg: 75%)
+5.513e-02
+3.278e-02
+1.044e-02
-1.190e-02
-3.424e-02
-5.658e-02
-7.892e-02
-1.013e-01
-1.236e-01
-1.459e-01
-1.683e-01
-1.906e-O1
-2.130e-01
40032
0.1
416135
0.15
Figure 52: Polyurea 2-row array strain field syy
63
Exx
0.05
0 164
0.12
0084
0.1
0o0
0.15
4*4
Figure 53: Polyurea 3-row array strain field exx
64
0.05
E
LE, LE22
(Avg: 75%)
+9.291e-02
+6.448e-02
+3.605e-02
HI-+7.615e-03
-2.082e-02
--
4.925e-02
-1.06le-O1
-1.345e-0O1
-1.630e-01
-1.914e-01
-2.198e-01
0.1
-2-483e-01
0.1
0144
0.15
Figure 54: Polyurea 3-row array strain field eyy
Solid Equivalent Arrays
In order to determine the impact of shape change on the response of a row of
cylinders, a solid block was tested in compression.
Blocks of a 1 and 2-row
equivalent mass were tested. The same fixture was used with the same plane strain
assumption. One row consisting of five 4.8mm diameter cylinders has an equivalent
solid height of 3.77mm. For 2 and 3-rows the equivalent height is 7.54mm and
11.31mm respectively. As expected, the fully constrained blocks are not able to
expand during compression and the bulk modulus is rapidly tested. Also of note is
significantly less dissipation compared to an identical mass cylinder array.
Compression of the solid blocks measuring 3.7mm and 7.5mm were compressed by
0.4mm as shown in Figure 55 through Figure 56.
65
I*I
Figure 55: Polyurea 1-row equivalent solid shown uncompressed and compressed
Figure 56 Polyurea 2-row equivalent solid shown uncompressed and compressed
The solid blocks must respond with the same force when compressed by an
identical amount. The near identical force vs. displacement curve is shown in Figure
57.
66
Solid Block Polyurea Compression
4000
U-
300200100 --
0
0.05
0.15
0.1
0.2
0.35
0.3
0.25
0.4
Displacement (mm)
Figure 57: Polyurea 1 and 2-row equivalent solid samples under compression
The slope of the experimental force vs. displacement curve of the solid slab is larger
then the arrays. The constrained solid should have the least compliance. For the
fully constrained boundary conditions, the bulk modulus of the material is tested.
During the compression experiment, the sample fixture flexed noticeably even when
mechanical screw-type clamps were added. Additionally, the sample was not cut to
precisely fit the fixture, allowing some lateral expansion during compression. The
force vs. displacement curves shown in Figure 57 demonstrate the compliance of the
fixture more then the response of the bulk material.
Solid Block Polyurea Compression
2000
1800-
-
1R Solid
A
2R Solid
1RFE
1600
14001200 -
8
1000-
0A
LL
--
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Displacement (mm)
Figure 58: Polyurea 1 and 2-row equiv. solid tests with 1-row equiv. FE
67
The FE simulation of the 1-row equivalent solid shown in Figure 58 is much stiffer
then the experimental response.
simulation.
The experimental data should match the FE
The fully constrained FE model shows no dissipation since it is
primarily sampling the elastic bulk volumetric behavior. The loading and unloading
curves were identical for the idealized constrained model. The experimental solid
response shows the least dissipation of any test. If the solid sample were fully
constrained, as in the FE model, no dissipation would be seen. Since dissipation
exists in the experimental data, it is confirmed that the fixture slipped and allowed
the solid sample to deform and dissipate energy; i.e. fully constrained conditions
were not achieved.
Energy Dissipation
As discussed earlier, PU exhibits significant dissipation and resilience. The array
response is different from the bulk material response.
The cylindrical array
structure does not change the material's response, but creates an inhomogeneous
strain profile that results in a different global response. The PU arrays should show
large dissipation, and ideally more than a constrained solid layer. The bulk material
load and reload response can be seen in Figure 59. The first load/unload cycle is
shown as a blue dot-dash, while the second cycle is shown as a black solid line.
PU 1000 Cylinder with Two Loading Cycles
12
1stLoad
2nd Load
10-
0
-
6-4-
2-
00
01
02
03
04
05
06
07
08
09
1
True Strain
Figure 59: Two loading and unloading cycles for polyurea under uniaxial compression
68
The first loading cycle of the 1-row array is shown in Figure 60 as the outer loop,
while the second loading cycle is the smaller inner loop. The permanent dissipation
is the difference in the area of the first cycle minus the area of the smaller second
cycle. The various PU arrays can be compared by their respective dissipation.
Each array showed a permanent change in the response due to microstructural
breakdown in the material on the first loading.
PU 1000 1 Row Array Compression
4.543.53--
LU
-.
1.5-
12.5--
-
0.5-0
0
0.02
0.04
0.06
0.1
0.12
0.08
True Strain
0.14
0.16
0.18
0.2
Figure 60: Two loading and unloading cycles for 1-row constrained array
The arrays with 2 and 3-rows showed an identical response at low strains. The
2-
loading and reloading responses are shown in Figure 61 and Figure 62.
PU 1000 2 Row Array Compression
5
1
9
0.1
0.12
4.54-
3.5-
- 2
S1.5-
0.50
0
'
0.02
0.04
0.06
0.08
0.14
0.16
0.18
0.2
True Strain
Figure 61: Two loading and unloading cycles for 2-row constrained array
69
PU 1000 3 Row Array Compression
4.54-0.5
-
-
1.5
**
,
A//
/0
O
00
-
02.5-
0
L
0
0.02
0.04
0.06
0.08
0.1
0.12
True Strain
0.14
0.16
0.18
0.2
Figure 62: Two loading and unloading cycles for 3-row constrained array
In order to quantitatively compare the dissipation between the various arrays, the
permanent dissipation and the permanent dissipation per volume were determined.
The dissipation values are shown in Table 4 below.
Table 4: Polyurea array energy dissipation
1st Cycle
Dissipation
(J)
2nd Cycle
Dissipation
(J
Permanent
Dissipation
(J
Permanent
Dissipation/
Array Volume
(MJ/m3)
1Row
2Row
33.0
47.0
3Row
51.11
18.8
20.3
19.1
14.2
26.7
32.0
17.4
16.4
13.1
The area of the first loading cycle was found and this dissipation was subtracted
from the area of the (smaller) reloading cycle. The permanent dissipated energy
from the microstructural breakdown was divided by the volume of PU in each array.
For a maximum of 0.16 true strain, it can be seen in Figure 63 that the 1-row array is
the most efficient.
70
PU Array Dissipation for Multiple Rows
17.
I
17
16.5
0
16
0
15.5
15
CL
14.5
E
14
13.5
-9
2
1
3
# Rows
Figure 63: Polyurea array dissipation density for 1,2 and 3-row arrays
Conclusions
Both the PSM4 and PU arrays result in an inhomogeneous strain profile with some
portions of the cylinder in compression and other portions in tension. This strain
profile and experimental constraints partially explains the PSM4 array response
shown in Figure 64.
Polyurea Array Compression
PSM4 Stress-Strain
E
/
-
1.5-
1LU
0.5.-
0
0
005
0.1
0.15
0.2
0.25
0
0.3
0.05
0.1
0.15
0.2
0.25
True Strain
Strain
Figure 64: Array effective stress-strain response for 1, 2 and 3-rows
The 1-row array has the greatest initial void fraction and therefore is the most
compliant. The 2 and 3-row arrays initially have an identical response. All the
71
PSM4 arrays become increasingly stiff with larger compressive strain. As the voids
become filled, the nearly incompressible material eventually stiffens to its bulk
response.
The PU array acts to moderate the initial stiffness and subsequent softening in the
material. Figure 8 shows the large initial stiffness followed by the pronounced
softening of the material in uniaxial compression. The array plots display a more
compliant initial behavior followed by a more gradual softening. Again, the 1-row
array is the most compliant due to the largest initial void fraction. This compliance
results in the largest deformation of each cylinder.
The larger deformation
produces more stretch-induced softening. The most pronounced softening was seen
in the 1-row array curve as shown in Figure 64. Additionally, dissipation is related
to the amount of deformation in the sample. The 1-row array was shown to provide
the greatest permanent dissipated energy density.
72
Chapter 5
Improved Composite Armor Under Blast Loading
Materials and Methods
The composite armor design presented here was not fabricated but rather modeled
in ABAQUS using the numerical tools established up to this point. The experimental
analysis previously conducted helped to guide the geometry configuration for the
interlayer.
Finite Element Model
A 3d finite element model was used to determine the response of steel-polyureasteel sandwich armor composite. The model is based on the cylindrical polymer
array compression tests with the modification of a 1mm thick steel plate above and
below the polymer layer. The boundary conditions were identical to that of the
earlier experimental and numerical compression tests. The half width model used
symmetric boundary conditions in the horizontal direction for computational
efficiency. The steel plates are 1mm thick, 9mm deep, and 12mm wide. The steel
plate is fixed to the left side of the model to represent the edge of the armor panel.
The PU is also constrained to the left side of the model to represent the edge of the
armor. The plates are in contact with the PU, but not adhered. The normal contact
between the steel and polyurea was defined as "hard", while the tangential contact
was modeled with friction. A coefficient of static friction of 0.1 was used. The plane
strain condition exists. The solid material was discretized into 3d elements with
eight nodes and reduced integration defined as C3D8R by the ABAQUS library.
The integral conventional weapons (CONWEP) module used TNT explosion data to
reproduce a spherical air blast with varying charges of TNT. The detonation point
was kept constant at Im from the center of the full armor panel as shown in Figure
65.
73
Simulated Composite Armor Response
The 3D model used a 1mm thick DH-36 top and bottom layer, with an equivalent
mass of PU in either an array or solid slab. The cylinders in the simulated array
have the same dimensions as the experimental arrays tested.
The cylinders
measure 4.8mm in diameter, while the solid slab measures 3.77mm thick. The
incident blast wave varies from a % to 2 kg of TNT detonated 1m from the sandwich
armor specimen. Figure 65 demonstrates the design of the numerical composite
armor simulation.
1m
Free
ARA
$ 1mm
[)H-30 ' tt'el
Fixed
Free
3.77mm
1mm
DH- 36 Y 2ei
Free
Figure 65: Simulation geometry of solid Polyurea composite armor
The resulting deformations can be seen in Figure 66 for the solid polyurea interlayer
composite. The displacement is shown in meters for each blast magnitude. The
displacement (U2) is in the y direction as shown on the axis label in Figure 65. The
same orientation was used for the deformed composite armor shown in Figure 71.
74
U, U2
+4.656e-06
-45e- 05
-7.376e-05
1.130e- 04
-1.522e-04
-1.914e-04
U, U2
-+3.883e-06
3.5306 05
5 489e-05
7.449e-05
-9.408e-05
-1.137e-04
1.333e-04
-1.529e-04
-1.724e-04
-1.920e-04
-2.116e-04
2.312e-04
-2.698e-04
-3.090e-04
-3.482e-04
-3.874e- 04
-4.26-04
0.5 kg
TNT
*4.65*-04
U, U2
+1 173e-06
-1.273e-04
1
-1.915e 04
-2.558e-04
-3.200e-04
-3.842e-04
:4.485e-04
5.127e-04
5.769e-04
-6.412e-04
-7.054e-G4
7.696e-04
kg
TNT
2 kg
TNT
Figure 66: Deflection of solid polyurea composite armor under blast loads
The strain and stress fields in the compressive, lateral, and shear directions are
compared for each blast magnitude. Figure 67 to Figure 69 show the response for
the solid polyurea interlayer.
S, S22
(Avg, 75%)
+2 200e+08
+1:851e8+08
+1.493e+08
+1.136e+08
+7 784e+07
+4:209e+07
+6.347e4-06
-6.314e107
1,0098+08
-1,366e+08
-724e+08
-081e4-08
+2.292e-03
:-5.989e-03
-1.01e-02
-1.25e-02
-1.4e-02
-3.083e-02
-3.4970-02
-3.911,-02
-4.3258-02
-4.739e-02
0.5 kg TNT
I-2.475e4-
F9,912
S. Sil
(Avg: 15%)
(Avg: 15%)
+ 1,007e+07
1.151e+-09
K+
+9.292e+08
1.187e+08
1.831e1-08
+7.076e+08
+4.859e4-08
+2.643e1-08
3.119e4-08
-33763.a08
4.4066+08
50508+08
5:694e+08
-6.3388+08
6,982e1-08
-1.790e+08
-4.006e4-08
-6.222e4-08
-8.438e+08
-1.065e4-09
-1.287e+09
7 AW4-fA
I+6:583e-0
LE,LE11
(Avg: 75%)
+1,315S-02
+1.21le-02
+9.066e-03
+6.026e-03
+2.9868 03
53363a-05
-6.134e-03
-9.1748-03
I
12258-02
1:575o802
-829a-02
-23
VP02
LE,LE12
(Avg; 7$%)
+1 943e.02
+1:515e.02
+1 0868-02
+2.301. 03
1980.-03
6:262e-03
:1.054e-02
1483.-02
1 911. 02
2.339e.02
-2,767. 02
It 19QU0
Figure 67: Solid polyurea interlayer with 1/2 kg TNT
75
r0
0.
Va'
-o
CL,
ME
-F
95
NiW.;
rJl
+ ;nr
BER
ER
S
..
list
..
I MUSS
+
.
..
......
+2.%65e+08
+2.560e+08
+2.15ft+08
I+
(Avg: 75%)
+-9.1950+-08
+-8.2600+08
+7.325e+08
+6.390R+08
2 kg TNT
-
h--+1.752e+08
5.4552+08
+1.348e+08
+5.391e+07
+1.348e+07
+4.5200k+08
+3.585&4-08
+2.6490+08
+ 1.7140+e08
+7.792e+07
-1.559&+07
1.091.4+08
-2.026e+08
-6.730e+07
- 1.07BLe+08
1.48U0+08
-187e+08
LE LE22
+2.166e.02
4 2.3e03
+4.38We 03
1.288.-02
3,016e-02
-3.e02
-4.74e-02
5.606.-02
6.470.-02
7.333e.02
8.197e-02
L.E,LE12
I
I+8.95
s, sil
75%)
+1.059e+09
(Avg:
+7.325Q+08
+ 5.694e+ 08
+4.0630+08
+2.432e+08
+8.011e+07
-8.300e4-07
-2.461e+08
-4.092e+08
-5.7238+08
-7.3540+00
-!.985e+08
L.E,tELlI
(Avg: 75%)
+7.54 .- 02
+6.316002
+5.084. 02
+3.8510-02
+56902
+~43 1.-0
+2.923002
+1,480o03
+2 619. 02
+1.387e-02
+1.552e-03
-2.6288-02
4.S02
. 406-02
6:791e-02
8179e002
.9-566e.02
-1.095.-Ol
-2,309e-02
-3.541o-02
4.773e-02
6.005e-02
7.237.-02
Figure 69: Solid polyurea interlayer with 2 kg TNT
The solid slab of constrained PU is quite stiff and allowed minimal deformation of
the steel plate above. The steel plate experienced a maximum deflection of only
0.76mm for the 2kg blast wave, resulting in little plastic deformation of the top steel
plate.
The same conditions were applied to the sandwich composite made of a cylindrical
array of PU shown in Figure 70.
The 1-row array was tested as the optimal
configuration based its initial compliance to maximize plastic deformation of the top
steel layer. The fully compressed state for each blast magnitude is shown in Figure
71.
The top steel layer of the cylindrical array attained a 1.4mm maximum
deflection from the 2kg blast wave, resulting in much greater plastic deformation of
the top steel plate and better energy absorption.
77
1m
Free
/
/
1mm
Fixed
Free
4.8mm
/
/
$
1mm
Free
Figure 70: Simulation geometry of cylindrical array polyurea composite armor
U, U2
+
- 7.31e-06
-2.252e-05
-5.235e-05
I---8.218e-05
-1.120e-04
-1.717e-04
-2.015e-04
-2.313e-04
-2.612e-04
-2.910e-04
-3.208e-04
-3.506e-04
U U2
+O.Oooe+oo
+0.000e-0
U, U2
+0.000e+00
K-7.062e-05
-1.412e-04
--
0.5 kg
TNT
-2.379e-04
-3.568e-04
-4.757e-04
-5.947e-04
-7.136e-04
-8.325e-04
-9.515e-04
-1.070e-03
-2.119e-04
-2.825e-04
3.531e-04
-4.944e-04
- 5.650e-04
-6.356e-04
-7.062e-04
-7.76e-04
1 kg
k
TNT
-8.475e-04
308e-03
S1427e-03
2 kg
TNT
Figure 71: Deflection of 1-row polyurea composite armor under blast loads
The strain and stress fields in the compressive, lateral, and shear directions are
compared for each blast magnitude. Figure 72 to Figure 74 show the response for
the cylindrical polyurea interlayer.
78
. .................
..
CD
e1
o9.
ot
A
m!i
PO-
?H
z
N
CD
-4
0
~1
N
-1
~1
m
WN"Wile"
Will
&000
8821§ 60 I..?.
1; Im...........m
++++++
l NN'lvaIw h
I'l
-4i
I+4.870e+08
S, S22
(Avg: 75%)
+7.431Le+O8
+6.577e+08
+5.724e+08
S, S12
S. SlS
(Avo: 75%)
(Avg: 75%)
2 kg TNT
:3.646e+08
+2.914e+08
+2.181e+08
+ 1.4490+08
i-7.1699+07
- 1.539e+06
-7.477e+07
-1.480e+08
-2.212e+08
-2.944e+08
-3.677e+08
-4409e+08
-5.141e+08
+4.0170+08
+3.163e+08
+2.310a+08
+1.456e+08
+6.026e+07
-1.958e+O
-2.8116+08
+1.237e+09
+1.0608+09
+8.838e+08
+7.0740+06
+5.310e+08
+3.545e+08
+1.781+08
+1.666e+06
-1.748&+06
-3.512e+06
-5.276e+08
:7.041e+08
-8:805e4+06
I+1.302e-01
I+2.75
I-2.935e-01
LE, LE22
(Avg: 75%)
+1.75le-01
LE, LE1L2
(Avg:75%)
+4.459e-01
+3.638e-01
+2.8169-0i
+1.994e-01
+ 1.173e-01
+3.513e-02
+8.4658-02
+3.9i1e-02
-6.440e-03
-5.19ge-02
-9.753e-02
* 1.431e-O1
1,886e-01
-2.342e-01
-2,79?e-01
-3,253-01
*3.70se-01
L.E,
LE11
(Avg: 75%)
+-2.352e-01
+ 1.9499-01
+1.544e-01
4-1.140e-01
+7.363-02
+3.325e-02
-7.130-03
-4.7514-02
-8.78%-02
14283e-01
-I.8e-01
-2.090-01
-1.292e-01
-2.1130e-01
-3,756e-01
-4.576e-01
5.400e-01
Figure 74: Cylinder polyurea interlayer with 2 kg TNT
The deformed models were analyzed in ABAQUS in order to compare energy
absorption of the composite armor configurations.
The energy stored and
dissipated in the top steel plate is much larger for the cylindrical interlayer as
shown in Figure 75.
045
0.A
Composite Armor Energy Transetr Solid Interley
Pasic Energy
SaMnEnergy
0.45
0.4
0.35
0.35
0.3
0.3
0.25
0.25
LN 02
S0.2
0.15
0.15
0.1
01
005
Fiur 75
1.8
005
lsi1.9
ispto
2
Time
2.1
n
taneeg
2.3
2.2
ftetpsellyri
w
Time
x 10,
opst
ro
x 10
Figure 75: Plastic dissipation and strain energy of the top steel layer in two composite armor
configurations
80
Conclusions
Several researchers have studied a composite armor with a solid layer of PU
sandwiched between two steel plates (Tekalur et al., 2008; Bahei-El-Din et al.,
2006). The presence of the void volume in the arrays allows the PU to deform more
than an equivalent solid layer.
This greater deformation results in greater
microstructural breakdown and more energy absorption. The objective of armor is
to absorb energy and prevent damage from passing through the armor. Sandwich
armor provides a hard outer layer to absorb energy through plastic deformation, a
softer interlayer to allow the metal to deform but provide some cushion to minimize
rupture of the outer metal layer. The top metal plate of the composite sandwich
armor must be thick enough to not rupture under a blast or projectile impact and
only dent the metal.
In this analysis, a blast load was applied to two types of PU composite armor. First,
a traditional PU sandwich with a solid PU slab between two metal plates, and a
cylinder PU array between the same steel plates. As the top steel plate undergoes
plastic deformation from the blast wave, the PU beneath it is permanently
compressed, and cannot unload.
This results in both dissipation through
microstructural breakdown in the polyurea, and strain energy to be stored in the
deformed array. The blast wave travels very rapidly and PUs significant strain-rate
stiffening was considered.
The cylindrical array of polyurea between the steel top and bottom layer allow
greater plastic deformation of the top steel layer and maximizes deformation and
therefore strain energy storage of the polyurea interlayer. The deflection of the top
steel layer can be seen in Figure 76. The deflection of the composite with the row of
cylinders is nearly twice that of the solid polyurea, resulting in significantly more
energy dissipation in the top steel layer as shown in Figure 75.
81
PU 1000 Composite Armor Blast Response Top Plate
o
*
PU Row Top Plate
PU Solid Top Plate
E
E -0. 4
C
0
6 -
-0.
4.
E
8-
0
-E
2
-1.4 0.5
1
P
1.5
KG of TNT
Figure 76: Comparison of top plate deflection of two sandwich composite armor geometries
The deflection of the bottom steel layer is shown in Figure 77 for the cylinder row
and solid PU interlayers. The cylindrical interlayer results in a smaller deflection of
the bottom plate. A larger part of the kinetic energy from the incident blast wave
was absorbed by the top plate and cylinder row interlayer. This demonstrates that
the top steel layer and cylindrical interlayer absorb and dissipate energy more
effectively then a solid interlayer configuration.
PU 1000 Composite Armor Blast Response Bottom Plate
A
PU Row Bottom Plate
PU Solid Bottom Plate
X
-0.1
-0.2
x
-0.3
E
0
-0.4
-2
-0.5
0
-0.6
-0.7
-0.81
5
1.5
1
2
KG of TNT
Figure 77: Comparison of bottom plate deflection of two sandwich composite armor geometries
82
The goal of this composite armor design is to maximize the absorbed energy. The
plastic deformation of steel dissipates more energy than that of the PU because of its
much higher flow stress. The most effective composite armor optimized for blast
protection must utilize the ductility and high flow stress of steel. The previous blast
experiments have noted that a PU layer changes the way the steel fails and can
result in the significant increase in the failure strain and deformation of steel
without rupture (Tekalur et al., 2008; Roland et al., 2010; Xue et al., 2010). A 1-row
array allows the steel to plastically deform while still providing a soft layer that
continues to stiffen as the steel deformation increases.
83
84
Summary and Conclusions
Conclusions
The experiments and simulations conducted here show that a cylindrical geometry
of a deformable material can offer a reduced initial compliance. The array behavior
is dependent upon the initial void fraction of the array, the compressibility of the
material, and internal characteristics such as stretch induced softening.
The
adjustment of these parameters can result a tunable response.
Constrained arrays permit greater deformation then solid slabs resulting in greater
dissipation. It is likely that more mechanisms for softening are introduced in PU
since the array experiences inhomogeneous strain with both compressive and
tensile forces acting on the material.
The low strain rate compression experiments demonstrated the energy absorption
and stiffening behavior of the constrained deformable polymer. These experiments
were designed to use the plane strain assumption. This allows the application of the
experimental results to a sandwich armor structure with long cylinders between the
metal plates.
Based on experimental array testing, it was determined that a single row array
provides the greatest permanent energy dissipation density. A single row array was
studied as the interlayer of a polyurea composite armor under blast loading. It was
found that arrays could be used to optimize sandwich composites experiencing blast
loads by maximizing the significant plastic dissipation in the steel top layer.
Future Work
Expand to 3D
The change in compliance for constrained cylinders may be even more pronounced
using spheres. Uniform spheres are difficult to produce from a solid sample block,
but are frequently produced industrially. Spheres have the advantage of easily
filling any void, specifically a pre manufactured steel box serving as the top and
bottom layers.
85
Conduct large-scale blast experiments
Only numerical simulations were conducted for the composite armor blast loading.
Blast loading of large-scale samples should take place to verify the numerical
results.
Interlayer fabrication
Develop prototyping techniques to fabrication polyurea in unique shapes. Consider
the use of injection molding to fill premade steel layers with polyurea cylinders or
spheres.
86
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89
90
Appendix A: Sample Production
The 3/8" (9.5mm) thick sheet of PSM4 was used to cut cylinders of 4.8mm diameter
using a water jet cutter with a thin metal backing. A thin tab of material was left
connecting the cylinder to the block.
This prevented losing the pieces during
cutting. The tabs were removed with a razor blade. The PSM4 cylinders were cut
down to 9mm with a laser cutter. The laser cutter is very effective at cutting thin
PSM4 (under 5mm). The PSM4 cylinders can be cut without a tab using a laser
cutter, but because the thicker polymer sheet melts during cutting, the precision is
poor.
Figure 78: PSM4 Sheet
The PU 1000 was obtained from NSWCC in the shape of a puck measuring
approximately 25mm thick. The PU was cut using a tilt compensated water jet
cutter. A thin sheet of steel was adhered to the top and bottom faces of the puck
before cutting. A tab was also required, however the tab was wider then the tab
used for the PSM4 because the PU was much thicker and harder. The water jet only
produced an accurate cut for the first 10-15mm. After that distance, the diameter of
the cylinder widened and became irregular. This irregularity can be seen in Figure
81. A razor blade was used to remove the tab and shorten cylinders to 9mm. The
experimental setup is based on a plane strain assumption, so uniform contact of the
cylinder faces to the fixture is important. Any irregularities on the cylinder faces
and bodies were removed using two grades of sandpaper, first medium followed by
emery cloth.
91
Figure 79: PU Sample Stock
Figure 80: Top down view of PU sample stock
Figure 81: Polyurea sample preparation
92
One improvement for future study could improve the sample yield. Perhaps a
different water jet setup could produce a more uniform cylinder and result in twice
as many test samples.
Fixture Production
The fixture was required to constrain the samples under compression. The small
fixture was constructed of 5.5mm thick PMMA. The front face, rear face, and sides
were cut and drilled using the laser cutter. The front and rear faces measured
35.2mm by 18mm.
Figure 82: Front fixture face
The sides measure 9.1mm by 18mm. The fixture was designed to hold three layers
of cylinder with -5mm of space to ensure the bar used for compression is properly
aligned. Holes measuring 1mm in diameter were cut to allow a 1mm thick metal
wire to pass from the front face through the side, and out the rear face. Four holes
were drilled on each side and wire was passed through and bent for strength. As
much lateral flexure as possible was prevented by the metal wire bracing and a
mechanical C clamp.
93
Figure 83: Fixture with metal wire bracing and C-clamp
An aluminum bar for compressing the samples was cut to the size of the fixture by
the central machine shop.
This provides a perfectly flat and rigid surface for
uniform compression of the arrays.
A larger fixture was made of the same PMMA to accommodate the larger PSM4
cylinders.
These larger samples were not shortened and measured 9.3mm in
diameter and 9.5 mm tall. This larger fixture was constructed in the same manner
as the smaller one. In order to reduce expense, a 5cm by 1cm bar was used to
compress the samples. This required a larger fixture than the samples alone. A
3mm wide shim was cut to constrain the cylinders laterally. The fixture with the
shim added is shown in Figure 27. The front and rear faces measured 61mm by
30mm. The sides measured 30mm by 9.6mm. The fixture with the bar used for
sample compression is shown in Figure 84.
94
Figure 84: Large sample fixture
Testing Procedure
Uniaxial and constrained compression testing of the samples took place on a Zwick
testing machine. The machine is shown in Figure 85.
95
Figure 85: Zwick Testing Machine
All compression testing used a screw-driven Zwick testing machine with a 2.5kN
load cell. Initial material characterization tests of PSM4 and PU were performed
using two flat plates and some sort of lubrication between the plate surface and the
sample. Lubrication is necessary to prevent barreling of the sample.
Figure 86: Material characterization with and without barreling
Lubrication ranged from petroleum jelly to powders to copper nanoparticles. In
addition to coatings, Teflon sheets were also tested.
96