Explicit Non-Linear Finite Element Analysis
with Fluid Structure Coupling of a High Velocity Impact
by
Jonathan Edward Homesley
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
August 6, 2011
Approved by
Winfred A. Foster, Jr., Chair, Professor of Aerospace Engineering
Gilbert L. Crouse, Associate Professor of Aerospace Engineering
Robert S. Gross, Associate Professor of Aerospace Engineering
ii
Abstract
A computational investigation was undertaken to study the pressure
wave generated from a high velocity, multi-material penetrator impacting a
compartmented cylindrical container. Variations of the penetrator material
stacking order are studied to evaluate the efficiency of delivering energy to the
target. The relevant application of this research, some previous work done in
high velocity impact dynamics, a comparison of the different penetrator
configurations, and the overall results of the analysis are discussed.
iii
Acknowledgments
The author would like to thank the members of the advisory committee
for their guidance throughout this work. Special thanks go to Dr. Winfred
Foster. Without his continual advice, support, and guidance this work could not
have been accomplished. The author would also like to thank his parents,
Dennis and Redonda Homesley, his sister, Rachael Waddle, his brother, Nathan
Homesley, and all of his friends and family for the love and encouragement they
have given along the way.
iv
Table of Contents
Abstract.............................................................................................................................. ii
Acknowledgments .......................................................................................................... iii
List of Figures .................................................................................................................. vi
List of Tables ................................................................................................................... xv
Nomenclature ................................................................................................................ xvi
Chapter 1: Introduction ................................................................................................... 1
1.1 The Objective .................................................................................................... 1
1.2 Background ....................................................................................................... 1
1.2.1 LOSAT ........................................................................................................... 2
1.2.2 CKEM ............................................................................................................. 3
1.3 Previous Work .................................................................................................. 3
1.3.1 Johnson-Cook Constitutive Model ............................................................ 4
1.3.2 Failure Modes ............................................................................................... 7
1.3.3 Shock Wave Propagation ............................................................................ 8
Chapter 2: Model Theory and Setup ........................................................................... 12
2.1 Explicit Non-Linear Solver ........................................................................... 12
2.2 Lagrangian Solver .......................................................................................... 13
v
2.3 Eulerian Solver ............................................................................................... 14
2.4 Geometry ......................................................................................................... 15
2.4.1 Penetrator .................................................................................................... 17
2.4.2 Target ........................................................................................................... 18
2.5 Elements .......................................................................................................... 19
2.5.1 Penetrator .................................................................................................... 19
2.5.2 Target ........................................................................................................... 22
2.5.3 Coupling Surfaces ...................................................................................... 27
2.5.4 Fluid ............................................................................................................. 28
2.6 Solid Material Models and Properties ........................................................ 30
2.7 Fluid Material Models, Properties, and Initial Conditions ...................... 32
2.8 Boundary Conditions and Initial Velocity ................................................. 33
2.9 Contact ............................................................................................................. 34
2.10 Coupling Algorithm ...................................................................................... 35
2.11 Failure Models ................................................................................................ 36
Chapter 3: Results .......................................................................................................... 38
3.1 Lagrangian Results ........................................................................................ 38
3.2 Euler ................................................................................................................. 60
References ....................................................................................................................... 69
Appendix ......................................................................................................................... 71
vi
List of Figures
Figure 1: LOSAT Launch [5] ........................................................................................... 3
Figure 2: Comparison of Computed Shapes and Test Results for Cylinder Impact
Tests at Various Velocities [9] ........................................................................................ 6
Figure 3: Failure Modes in Impacted Plates [11] ......................................................... 7
Figure 4: Regions of Elastic, Elasto-Plastic, and Shock Wave Propagation [12] ..... 9
Figure 5: Propagating High-Pressure Wave [12] ....................................................... 11
Figure 6: Buildup of a Pressure Wave to a Shock Wave [12] ................................... 11
Figure 7: Explicit Time Step Loop [13] ........................................................................ 12
Figure 8: Explicit vs. Implicit Efficiency and Cost Comparison [14] ...................... 13
Figure 9: Lagrangian Mesh [13] ................................................................................... 14
Figure 10: Eulerian Mesh [13] ....................................................................................... 15
Figure 11: Penetrator Drawing [15] ............................................................................. 16
Figure 12: Target Drawing [15] .................................................................................... 16
Figure 13: Complete Penetrator ................................................................................... 17
Figure 14: Target............................................................................................................. 18
Figure 15: Tungsten Section of the Penetrator ........................................................... 20
Figure 16: Steel Section of the Penetrator ................................................................... 20
vii
Figure 17: Aluminum Section of the Penetrator ........................................................ 21
Figure 18: Rear View of Zytel Penetrator Shell.......................................................... 21
Figure 19: AST Penetrator Configuration Excluding Zytel Shell ............................ 22
Figure 20: TSA Penetrator Configuration Excluding Zytel Shell ............................ 22
Figure 21: Front Plate ..................................................................................................... 23
Figure 22: Rear Plate ...................................................................................................... 23
Figure 23: Bulkheads ..................................................................................................... 24
Figure 24: Inside Walls .................................................................................................. 25
Figure 25: Outside Wall ................................................................................................. 25
Figure 26: Front Plate, Bulkheads, and Rear Plate .................................................... 26
Figure 27: Plates, Bulkheads, and Inside Walls ......................................................... 26
Figure 28: Penetrator and Target ................................................................................. 27
Figure 29: Coupling Surfaces, View 1 ......................................................................... 28
Figure 30: Coupling Surfaces, View 2 ......................................................................... 28
Figure 31: Interior Euler Regions and Coupling Surfaces ........................................ 29
Figure 32: Euler Regions ............................................................................................... 30
Figure 33: Johnson-Cook Yield Model [13] ................................................................ 32
Figure 34: Boundary Conditions on Plates ................................................................. 33
Figure 35: AST Energy of Distortion vs. Time ........................................................... 39
Figure 36: TSA Energy of Distortion vs. Time ........................................................... 40
viii
Figure 37: Target Energy of Distortion vs. Time ....................................................... 41
Figure 38: Penetrator Energy of Distortion vs. Time ................................................ 41
Figure 39: AST Internal Energy vs. Time .................................................................... 42
Figure 40: TSA Internal Energy vs. Time .................................................................... 43
Figure 41: Target Internal Energy vs. Time ................................................................ 44
Figure 42: Penetrator Internal Energy vs. Time ......................................................... 44
Figure 43: AST Kinetic Energy vs. Time ..................................................................... 45
Figure 44: TSA Kinetic Energy vs. Time ..................................................................... 45
Figure 45: Target Kinetic Energy vs. Time ................................................................. 46
Figure 46: Penetrator Kinetic Energy vs. Time .......................................................... 47
Figure 47: AST X - Momentum vs. Time .................................................................... 48
Figure 48: TSA X - Momentum vs. Time .................................................................... 48
Figure 49: AST Y - Momentum vs. Time .................................................................... 49
Figure 50: TSA Y - Momentum vs. Time .................................................................... 49
Figure 51: AST Z - Momentum vs. Time .................................................................... 50
Figure 52: TSA Z - Momentum vs. Time .................................................................... 50
Figure 53: AST Penetrator, 1.0e-5 sec. ......................................................................... 51
Figure 54: TSA Penetrator, 1.0e-5 sec. ......................................................................... 51
Figure 55: AST Penetrator, 2.0e-5 sec. ......................................................................... 51
Figure 56: TSA Penetrator, 2.0e-5 sec. ......................................................................... 51
ix
Figure 57: AST Penetrator, 3.0e-5 sec. ......................................................................... 52
Figure 58: TSA Penetrator, 3.0e-5 sec. ......................................................................... 52
Figure 59: AST Penetrator, 4.0e-5 sec. ......................................................................... 52
Figure 60: TSA Penetrator, 4.0e-5 sec. ......................................................................... 52
Figure 61: AST Penetrator, 5.0e-5 sec. ......................................................................... 52
Figure 62: TSA Penetrator, 5.0e-5 sec. ......................................................................... 52
Figure 63: AST Penetrator, 6.0e-5 sec. ......................................................................... 52
Figure 64: TSA Penetrator, 6.0e-5 sec. ......................................................................... 52
Figure 65: AST Penetrator, 7.0e-5 sec. ......................................................................... 52
Figure 66: TSA Penetrator, 7.0e-5 sec. ......................................................................... 52
Figure 67: AST Penetrator, 8.0e-5 sec. ......................................................................... 53
Figure 68: TSA Penetrator, 8.0e-5 sec. ......................................................................... 53
Figure 69: AST Penetrator, 9.0e-5 sec. ......................................................................... 53
Figure 70: TSA Penetrator, 9.0e-5 sec. ......................................................................... 53
Figure 71: AST Penetrator, 1.0e-4 sec. ......................................................................... 53
Figure 72: TSA Penetrator, 1.0e-4 sec. ......................................................................... 53
Figure 73: AST Penetrator, 1.1e-4 sec. ......................................................................... 53
Figure 74: TSA Penetrator, 1.1e-4 sec. ......................................................................... 53
Figure 75: AST Penetrator, 1.2e-4 sec. ......................................................................... 53
Figure 76: TSA Penetrator, 1.2e-4 sec. ......................................................................... 53
x
Figure 77: AST Penetrator, 1.3e-4 sec. ......................................................................... 54
Figure 78: TSA Penetrator, 1.3e-5 sec. ......................................................................... 54
Figure 79: AST Penetrator, 1.4e-4 sec. ......................................................................... 54
Figure 80: TSA Penetrator, 1.4e-4 sec. ......................................................................... 54
Figure 81: AST Front Plate Displacement, 1.0e-5 sec. ............................................... 54
Figure 82: TSA Front Plate Displacement, 1.0e-5 sec. ............................................... 54
Figure 83: AST Front Plate Displacement, 2.0e-5 sec. ............................................... 55
Figure 84: TSA Front Plate Displacement, 2.0e-5 sec. ............................................... 55
Figure 85: AST Front Plate Displacement, 3.0e-5 sec. ............................................... 55
Figure 86: TSA Front Plate Displacement, 3.0e-5 sec. ............................................... 55
Figure 87: AST Front Plate Displacement, 4.0e-5 sec. ............................................... 55
Figure 88: TSA Front Plate Displacement, 4.0e-5 sec. ............................................... 55
Figure 89: AST Front Plate Displacement, 5.0e-5 sec. ............................................... 55
Figure 90: TSA Front Plate Displacement, 5.0e-5 sec. ............................................... 55
Figure 91: AST Front Plate Displacement, 6.0e-5 sec. ............................................... 56
Figure 92: TSA Front Plate Displacement, 6.0e-5 sec. ............................................... 56
Figure 93: AST Front Plate Displacement, 7.0e-5 sec. ............................................... 56
Figure 94: TSA Front Plate Displacement, 7.0e-5 sec. ............................................... 56
Figure 95: AST Front Plate Displacement, 8.0e-5 sec. ............................................... 56
Figure 96: TSA Front Plate Displacement, 8.0e-5 sec. ............................................... 56
xi
Figure 97: AST Front Plate Displacement, 9.0e-5 sec. ............................................... 56
Figure 98: TSA Front Plate Displacement, 9.0e-5 sec. ............................................... 56
Figure 99: AST Front Plate Displacement, 1.0e-4 sec. ............................................... 57
Figure 100: TSA Front Plate Displacement, 1.0e-4 sec. ............................................. 57
Figure 101: AST Front Plate Displacement, 1.1e-4 sec. ............................................. 57
Figure 102: TSA Front Plate Displacement, 1.1e-4 sec. ............................................. 57
Figure 103: AST Front Plate Displacement, 1.2e-4 sec. ............................................. 57
Figure 104: TSA Front Plate Displacement, 1.2e-4 sec. ............................................. 57
Figure 105: AST Stress, 1.0e-5 sec. ............................................................................... 58
Figure 106: TSA Stress, 1.0e-5 sec. ............................................................................... 58
Figure 107: AST Stress, 6.0e-5 sec. ............................................................................... 58
Figure 108: TSA Stress, 6.0e-5 sec. ............................................................................... 58
Figure 109: AST Stress, 1.1e-4 sec. ............................................................................... 58
Figure 110: TSA Stress, 1.1e-4 sec. ............................................................................... 58
Figure 111: AST Stress, 1.6e-4 sec. ............................................................................... 58
Figure 112: TSA Stress, 1.6e-4 sec. ............................................................................... 58
Figure 113: AST Stress, 2.1e-4 sec. ............................................................................... 59
Figure 114: TSA Stress, 2.1e-4 sec. ............................................................................... 59
Figure 115: AST Stress, 2.6e-4 sec. ............................................................................... 59
Figure 116: TSA Stress, 2.6e-4 sec. ............................................................................... 59
xii
Figure 117: AST Stress, 3.1e-4 sec. ............................................................................... 59
Figure 118: TSA Stress, 3.1e-4 sec. ............................................................................... 59
Figure 119: AST Stress, 3.6e-4 sec. ............................................................................... 59
Figure 120: TSA Stress, 3.6e-4 sec. ............................................................................... 59
Figure 121: AST Stress, 4.1e-4 sec. ............................................................................... 59
Figure 122: TSA Stress, 4.1e-4 sec. ............................................................................... 59
Figure 123: AST Interior Euler 1, Pressure vs. Z-Coord. .......................................... 60
Figure 124: TSA Interior Euler 1, Pressure vs. Z-Coord. .......................................... 61
Figure 125: AST Interior Euler 2, Pressure vs. Z-Coord. .......................................... 61
Figure 126: TSA Interior Euler 2, Pressure vs. Z-Coord. .......................................... 62
Figure 127: Euler Contour Regions ............................................................................. 63
Figure 128: Euler 1 Contour Regions Excluding Hole .............................................. 63
Figure 129: AST Euler Region 1 Pressure, 1e-4 sec. .................................................. 64
Figure 130: TSA Euler Region 1 Pressure, 1e-4 sec. .................................................. 64
Figure 131: AST Euler Region 1, 1.5e-4 sec. ................................................................ 64
Figure 132: TSA Euler Region 1 Pressure, 1.5e-4 sec. ............................................... 64
Figure 133: AST Euler Region 1 Pressure, 2.0e-4 sec. ............................................... 65
Figure 134: TSA Euler Region 1 Pressure, 2.0e-4 sec. ............................................... 65
Figure 135: AST Euler Region 1 Pressure, 2.5e-4 sec. ............................................... 65
Figure 136: TSA Euler Region 1 Pressure, 2.5e-4 sec. ............................................... 65
xiii
Figure 137: AST Euler Region 1 Pressure, 3.0e-4 sec. ............................................... 65
Figure 138: TSA Euler Region 1 Pressure, 3.0e-4 sec. ............................................... 65
Figure 139: AST Euler Region 1 Pressure, 3.5e-4 sec. ............................................... 65
Figure 140: TSA Euler Region 1 Pressure, 3.5e-4 sec. ............................................... 65
Figure 141: AST Euler Region 1 Pressure, 4.0e-4 sec. ............................................... 66
Figure 142: TSA Euler Region 1 Pressure, 4.0e-4 sec. ............................................... 66
Figure 143: AST Euler Region 2 Pressure, 8e-4 sec. .................................................. 66
Figure 144: TSA Euler Region 2 Pressure, 8e-4 sec. .................................................. 66
Figure 145: AST Euler Region 2 Pressure, 9.0e-4 sec. ............................................... 66
Figure 146: TSA Euler Region 2 Pressure, 9.0e-4 sec. ............................................... 66
Figure 147: AST Euler Region 2 Pressure, 1.0e-3 sec. ............................................... 67
Figure 148: TSA Euler Region 2 Pressure, 1.0e-3 sec. ............................................... 67
Figure 149: AST Euler Region 2 Pressure, 1.1e-3 sec. ............................................... 67
Figure 150: TSA Euler Region 2 Pressure, 1.1e-3 sec. ............................................... 67
Figure 151: Foster - AST Energy of Distortion vs. Time [11] ................................... 71
Figure 152: Foster - AST Internal Energy vs. Time [11] ............................................ 71
Figure 153: Foster - AST Kinetic Energy vs. Time [11] ............................................. 72
Figure 154: Foster - AST Z-Momentum vs. Time [11] .............................................. 72
Figure 155: Foster - TSA Energy of Distortion vs. Time [11] ................................... 73
Figure 156: Foster - TSA Internal Energy vs. Time [11] ............................................ 73
xiv
Figure 157: Foster TSA Kinetic Energy vs. Time [11] ............................................... 74
Figure 158: Foster TSA Kinetic Energy vs. Time [11] ............................................... 74
xv
List of Tables
Table 1: Constitutive Constants for Various Materials [9] ......................................... 5
Table 2: Misc. Material Properties ............................................................................... 31
Table 3: Material Properties and Constants for Johnson-Cook Yield Model ........ 32
Table 4: Air Properties ................................................................................................... 33
xvi
Nomenclature
A ? Static Yield Stress
B ? Hardening Parameter
C ? Strain Rate Parameter
cm ? Centimeter
CP ? Specific Heat
E ? Modulus of Elasticity
EPS0 ? Reference Strain Rate
G ? Shear Modulus
in ? Inch(es)
K ? Bulk Modulus
lb ? Pound-Force
m ? Temperature Exponent
mph ? Miles per Hour
MPS ? Maximum Plastic Strain
m/s ? Meters per Second
n ? Hardening Exponent
PMIN ? Minimum Spallation Pressure
psi ? Pound-Force per Square Inch
sec ? Second(s)
TMELT ? Melt Temperature
TROOM ? Room Temperature
U ? Wave Propagation Velocity
v = Poisson?s ratio
Y0 ? Static Yield Strength
? ? Density
0? ? Reference Density
? ? Stress
HEL? - Hugoniot Elastic Limit
yield? ? Yield Stress
1
Chapter 1: Introduction
1.1 The Objective
The goal of the analysis is to lend understanding to situations encountered in
certain military applications, particularly kinetic energy, penetrator systems for
penetrating armored vehicles. To accomplish this goal an explicit non-linear
finite element analysis was done to determine not only the amount of energy
transferred to a compartmented cylindrical container (target) impacted by a high
velocity penetrator, but also to model the resulting pressure wave transmitted
through the target. The penetrating projectile is made of three equal mass rods.
The three rods are each made of a different metallic material encased in a
polycarbonate shell. The metallic materials used are aluminum, steel, and
tungsten. More details concerning the target and penetrator dimensions,
material properties, and initial conditions will be discussed in up-coming
sections.
1.2 Background
Kinetic energy missile systems have been considered by the U.S. Army for
use in heavy anti-armor threats since the 1980s. There are several potential
advantages of a kinetic energy missile (KEM) over one dependent on chemical
explosions to destroy the target. A KEM system is simple. It has no warhead,
fuse or onboard sensor to risk malfunction. With less complicated components
2
the system has the potential of a reduced cost per expended round. The missile
is required to travel at hypervelocity speed to impart the necessary kinetic
energy to the target. The requirement for this velocity makes for a reduced travel
time to target reducing the target?s capabilities of intercepting the missile or
maneuvering and returning fire. Since 1997 Lockheed Martin has developed two
types of kinetic energy missiles, the Line-of-Sight Anti-Tank Weapon (LOSAT)
and the Compact Kinetic Energy Missile (CKEM). Neither of the two systems
that were developed moved to production, but they are important demonstrators
of the capabilities of kinetic energy warheads [5].
1.2.1 LOSAT
The LOSAT warhead is characterized as a ?High-density rod armor
penetrator.? The missile is 112 inches long with a diameter of 6.4 inches. It has a
weight of 177 lb. It uses a solid rocket motor developed by Alliant Techsystems
(ATK) to propel it to a velocity of 5,000 feet per second and a range of over 4
kilometers. This hypervelocity allows it to reach its maximum range in less than
five seconds. Tests of the system proved that it has the capability to accomplish
the design goals. During testing it destroyed an M-60 tank moving at 22 mph at
2,400 meters, an incoming tank at 2,400 meters, a moving target in night-time
conditions at 4,300 meters, a moving tank at relatively short range, and a
3
reinforced urban structure. An example of a LOSAT being launched from a
modified Humvee is given in Figure 1 [5].
Figure 1: LOSAT Launch [5]
1.2.2 CKEM
The CKEM was a spiral project from the LOSAT. It was designed to be
much smaller with an extended range while still delivering devastating effects to
the threat. The CKEM is 60 inches long and weighs less than 100 pounds. At a
velocity of Mach 6.5+ it has demonstrated extreme lethality and a high
probability of first-round kill against all advanced threat armor and hardened
bunkers, fortifications, and urban structures. It has an extended range beyond
that which tank main guns are capable (5+ kilometers) which gives it a range
overmatch [6].
1.3 Previous Work
4
During the last 5 decades there has been extensive research and work done in
developing and refining the finite element method [8]. Due to the abundance of
information on the subject much of the past work done in the field will not be
covered, only a brief introduction to some of the major breakthroughs in the
areas that are relevant to the topic at hand.
1.3.1 Johnson-Cook Constitutive Model
Johnson and Cook recognized that the capabilities of computer codes were
becoming limited by not having a simple, accurate model to define the material
characteristics for both strength and fracture. They developed their material
model to be used in situations of high strains, high strain rates, and high
temperatures. The intent was to have a constitutive model that would require
only a minimal amount of material constants that can be determined by few
experiments. The material constants were determined by purely empirical
means. They conducted torsion tests over a wide range of strain rates, static
tensile tests, dynamic Hopkinson bar tensile tests, and Hopkinson bar tests at
elevated temperatures to gather the data required to determine the appropriate
material constants for their constitutive model. The constitutive constants for the
tested materials during the model development are given in Table 1.
5
Table 1: Constitutive Constants for Various Materials [9]
After completing the model development they evaluated the model and data by
comparing their results with data from cylinder impact tests. The evaluation
tests conducted provided strains exceeding 2.0 and strain rates exceeding 105 sec-
1. The results from the cylinder impact tests are given in Figure 2 below [9, 10].
6
Figure 2: Comparison of Computed Shapes and Test Results for Cylinder Impact
Tests at Various Velocities [9]
7
1.3.2 Failure Modes
Different loading conditions can lead to varied failure mechanisms.
Typically one failure mechanism will dominate, though several may contribute
to the overall failure of the material. The failure mechanisms or modes that will
be discussed are plugging, petaling, and spalling. Examples of these failure
modes and others are given in Figure 3 below [11].
Figure 3: Failure Modes in Impacted Plates [11]
Spalling is defined as a tensile failure that is a result of the interaction of
one or more rarefaction waves near any surface except the one the loading is
applied. Spallation is a common failure mechanism in hypervelocity impacts,
especially when the material is stronger in compression than in tension [11, 12].
8
Plugging is common when the penetrator has a blunt or hemispherical-
nose shape and impacts the target at a velocity close to its ballistic limit. The
ballistic limit is the minimum velocity required for penetration. This set of
parameters can lead to the formation of a nearly cylindrical slug with a similar
diameter to the penetrator being pushed through and ejecting from the target. If
the impact velocity exceeds the ballistic limit by more than 5 ? 10%, multiple
fragments instead of an intact plug will result [11].
Petaling is most common in thin targets being impacted near the ballistic
limit. It is caused by high radial and circumferential tensile stresses after the first
stress wave passes. Petaling is characterized by star shape cracks that develop
around the penetrator nose. The sections separated by the cracks are then
pushed back by the penetrator as it passes through the target. The result is a
petal shape to the fractured surface [11].
1.3.3 Shock Wave Propagation
A uniaxial stress-strain curve that will be used to describe the motion and
development of shock waves in a solid is given in Figure 4.
9
Figure 4: Regions of Elastic, Elasto-Plastic, and Shock Wave Propagation [12]
Uniaxial strain can be described as deformation that is restricted to one
dimension. The importance of the uniaxial stress-strain curve given is that it
indicates the stress at which multiple stress waves exist in the material. This
stress is referred to as the Hugoniot elastic limit which is defined as the
maximum stress for 1D elastic wave propagation in plate geometries and is
written as HEL? . Above HEL? the elastic wave will move with a speed
)1)(21(
)1(
0
2 vv vEC
E ??
??
?
(8)
Following the elastic wave will be one or more plastic waves each with a velocity
10
?
?
? d
dC
p
0
1? (9)
Above c? strong shock waves can occur. In this region materials display
characteristics similar to fluids [12].
The high-pressure wave builds up to a shock wave due to the differences
of the pressure-density relationships in different regions. The pressure and
density are linearly related in the elastic region. Above the elastic region they are
not linearly proportional and the wave velocity increases with pressure or
density. Figure 5 is an illustration of a propagating high-pressure wave. At
point A in this illustration, the pressure is low and the wave velocity is low. At
point B the velocity is higher because above the elastic limit the wave velocity
increases with increasing pressure. At point C the wave velocity is even higher
than at point B.
11
Figure 5: Propagating High-Pressure Wave [12]
Figure 6 shows the wave continuing to steepen until it eventually reaches a
vertical orientation. When the wave becomes a vertical front it is considered a
shock wave. Instead of the once present smooth transition of matter in front of
the wave to matter past the wave, now there is a discontinuity between the
shocked and un-shocked material [12].
Figure 6: Buildup of a Pressure Wave to a Shock Wave [12]
12
Chapter 2: Model Theory and Setup
2.1 Explicit Non-Linear Solver
MSC-Dytran was the explicit non-linear finite element analysis solver
used in the current simulations. An explicit solver has an advantage over the
more commonly used implicit solver in simulations when a very small time step
is required. A small time step may be required for one of the following reasons:
material nonlinearity, large geometric nonlinearity, when it is necessary to
recover stress wave effects, and when the body undergoes large displacements.
Explicit solutions are better suited than implicit solutions for analysis requiring
small time steps as well as large problems because matrix decompositions or
matrix equation solutions are not necessary. The flow diagram as seen on Figure
7 illustrates the loop that is carried out for each explicit time step [13].
Figure 7: Explicit Time Step Loop [13]
13
Figure 8 gives an indication of the differences in efficiency and costs between an
implicit and explicit solution due to nonlinearities and problem size [14].
Figure 8: Explicit vs. Implicit Efficiency and Cost Comparison [14]
2.2 Lagrangian Solver
The Lagrangian solver calculates the motion of elements which have
constant mass. This is accomplished by fixing grid points, also referred to as
nodes, to locations on the geometric body that is to be analyzed. By connecting
the nodes together, elements are created. A collection of elements is a mesh. The
nodes move and the elements distort as the body deforms. The Lagrangian
solver is used to model the solid materials required for the simulations being
investigated. An example of a Lagrangian mesh undergoing distortion is given in
Figure 9 [13].
14
Figure 9: Lagrangian Mesh [13]
2.3 Eulerian Solver
The Eulerian solver calculates the motion of material through elements of
constant volume. Instead of the nodes moving with the body they are fixed in
space. The mesh then becomes a ?fixed frame of reference? that the mass,
momentum, and energy of the material of the body being analyzed moves
through. The Eulerian solver is used to model the fluid materials for the
simulations being investigated. An example of the transportation of material
through an Eulerian mesh is given in Figure 10 [13].
15
Figure 10: Eulerian Mesh [13]
2.4 Geometry
The geometry for the finite element analysis was created with MSC.Patran
which is the Pre- and Post-Processor used. The target and penetrator were
modeled as closely to the experimental specifications as possible. Drawings of
the penetrator and target can be seen in Figure 11 and Figure 12 respectively [15].
16
Figure 11: Penetrator Drawing [15]
Figure 12: Target Drawing [15]
17
2.4.1 Penetrator
The cylindrical penetrator was made of three different metallic sections of
equal mass encased in a polycarbonate shell. The penetrator is given in Figure 13
below.
Figure 13: Complete Penetrator
The diameter of each section is 0.908 inch. The metallic sections are tungsten,
steel, and aluminum. The tungsten section has a length of 0.407 inch. The steel
section has a length of 0.915 inch. The aluminum section has a length of 2.57
inches. The polycarbonate shell was made of Zytel. The zytel case has
maximum outside diameter of 1.375 inches and a length of approximately 4.563
inches. The order of the metallic sections was varied to identify the
configuration that distributes the most energy to the target. Two configurations
will be discussed in this work. These are referred to as Aluminum-Steel-
Tungsten (AST), where the aluminum section impacts first followed by the steel
18
then the tungsten, and Tungsten-Steel-Aluminum (TSA), where the tungsten
section impacts first followed by the steel then the aluminum [15].
2.4.2 Target
The cylindrical target was made of steel and was designed to represent
three separate sections which are segmented with a bulkhead with a hole in the
center and a solid bulkhead. The target is given in Figure 14 below.
Figure 14: Target
It consists of a front plate which at 2.5 inches thick is five times as thick as the
interior bulkheads and two and a half times as thick as the rear plate. This
differential in the thicknesses simulates the difference between interior walls and
the exterior plating found in many armored vehicles. The plates and bulkheads
are bound by two tubes which will be referred to as walls. These two walls are
separated by a distance of 0.0625 inch. The inner wall with a thickness of 0.25
inch attaches the bulkheads to the front plate and back plate. The outer wall
19
with a thickness of 0.3125 inch attaches the front plate to the rear plate. The
overall length of the target is 19.75 inches and has a diameter of 6.75 inches [15].
2.5 Elements
CHEXA elements are used throughout the model to construct the solid
Lagrangian and Eulerian elements whenever possible. CPENTA elements are
used as necessary to fill in the gaps left by the CHEXA elements. CHEXA is a
six-sided solid element with eight grid points. CPENTA is a five-sided solid
element with six grid points. The coupling surfaces require the use of shell
elements. For these, CQUAD4 elements were used whenever possible and
CTRIA3 were used when necessary to fill the gaps left by the CQUAD4 elements.
CQUAD4 is a quadrilateral shell element with four grid points. CTRIA3 is a
triangular shell element with three grid points. The next sections will highlight
the steps taken to mesh each part of the model.
2.5.1 Penetrator
The metallic sections of the penetrator are all meshed with 20 elements on
the circumference and 5 elements in the radial direction. The tungsten section
which was meshed with 5 elements along the length of the rod and contains 500
elements and 606 nodes is given in Figure 15.
20
Figure 15: Tungsten Section of the Penetrator
The steel section which has 10 elements along the length of the rod and contains
1000 elements and 1111 nodes is given as Figure 16.
Figure 16: Steel Section of the Penetrator
The aluminum section which has 35 elements along the length of the rod and
contains 3500 elements and 3636 nodes is given in Figure 17.
21
Figure 17: Aluminum Section of the Penetrator
The zytel was meshed to match the corresponding metallic section encased and
consists of 2560 elements and 3809 nodes. A rear view of the zytel section is
given below as Figure 18.
Figure 18: Rear View of Zytel Penetrator Shell
Figure 19 and Figure 20 given below shows the AST and TSA penetrator
configurations. It is important to note that the direction of travel is along the
positive Z axis.
22
Figure 19: AST Penetrator Configuration Excluding Zytel Shell
Figure 20: TSA Penetrator Configuration Excluding Zytel Shell
2.5.2 Target
The front and rear plates were created using four different solids and then
meshed separately. This was to ensure the appropriate walls mated correctly
and the distance between the walls is properly accounted for. Each piece of the
target was meshed with 20 elements along the circumference. The center section
of the front plate, rear plate, and solid bulkheads are meshed with 15 elements in
the radial direction. The bulkhead with the hole in the center was meshed with
23
10 elements in the radial direction. All of the outer rings of the target pieces are
meshed with 1 element in the radial direction. The front plate which was
meshed with 10 elements along its length or thickness and consists of 3600
elements and 3971 nodes is given in Figure 21.
Figure 21: Front Plate
The rear plate which was meshed with 5 elements along its length or thickness
and consists of 1800 elements and 2166 nodes is given in Figure 22.
Figure 22: Rear Plate
24
The bulkheads which were each meshed with one element along their length or
thickness and consist of a total of 860 elements and 1764 nodes are given in
Figure 23.
Figure 23: Bulkheads
The inside and outside walls were meshed with one element in the radial
direction and twenty elements along the circumference. The inside wall was
made of three separate sections, each meshed with 10 elements along their
length. This results in a total of 600 elements and 1320 nodes for these sections.
The inside walls are given in Figure 24.
25
Figure 24: Inside Walls
The outside wall is one solid piece and was meshed with 32 elements along its
length giving it a total of 640 elements and 1320 nodes. It is given below in
Figure 25.
Figure 25: Outside Wall
To give a perspective on the section sizes relative to the complete model, the
target pieces assembled are shown in Figure 26 through Figure 28. Figure 26
shows the front plate, rear plate, and bulkheads. Figure 27 shows the front plate,
26
rear plate, bulkheads, and inside walls. Figure 28 shows the complete target and
penetrator
Figure 26: Front Plate, Bulkheads, and Rear Plate
Figure 27: Plates, Bulkheads, and Inside Walls
27
Figure 28: Penetrator and Target
2.5.3 Coupling Surfaces
To couple the Euler elements to the Lagrangian elements, ?dummy?
surfaces were used. Coupling surfaces have no properties or thickness assigned
to them. This is a feature of the solver to allow surfaces to be defined for the
transfer of the energy, mass, and momentum from one element type to the other.
The coupling surfaces were grouped into 2 separate sections to correspond to the
2 interior Euler regions. The first section coupled the two target compartments
connected by the bulkhead with a hole in the center to their corresponding Euler
region. The first section was also used to couple the interior Euler regions to the
exterior Euler region. The second section coupled the target compartment
separated by the solid bulkhead and rear plate to its corresponding Euler region.
Each section was meshed to match the Lagrangian solid it coupled. There were a
total of 2240 elements and 2264 nodes that made up the coupling surfaces. A
28
further discussion of the Euler regions will be given in the coming sections. Two
views of the coupling surfaces are given in Figure 29 and Figure 30.
Figure 29: Coupling Surfaces, View 1
Figure 30: Coupling Surfaces, View 2
2.5.4 Fluid
The Euler regions were constructed using the mesh box feature of the solver.
This control allows the user to construct the Euler region by specifying the
29
origin, length of the region in each coordinate direction, and the number of
elements in each coordinate direction. The first interior Euler region began at 0.1
inch before the target front plate ends and extends to 0.1 inch after the solid
bulkhead begins. The second interior Euler region began at 0.1 inch before the
solid bulkhead ends and extends to 0.1 inch after the rear plate began. The
exterior Euler region began at 27.4 inches before the front plate began and
extends to 0.1 inch after the front plate ends. The first interior Euler region was
meshed with a total of 16,800 elements and 18,963 nodes. The second interior
Euler region was meshed with 8,000 elements and 9,261 nodes. The exterior
Euler region is meshed with 32,000 elements and 35,301 nodes. The interior
Euler regions covering their corresponding coupling surfaces are given in Figure
31. The exterior and interior Euler regions together are given in Figure 32.
Figure 31: Interior Euler Regions and Coupling Surfaces
Interior
Euler 1
Interior
Euler 2
30
Figure 32: Euler Regions
2.6 Solid Material Models and Properties
Four different materials were used to model the penetrator and target. 4340
steel was used for the steel sections of the penetrator and for the entire target.
The other metallic sections of the penetrator consist of tungsten (0.07 Ni, 0.03 Fe)
and 6061-T6 aluminum. The polycarbonate shell that holds the penetrator
together is constructed from zytel, a material manufactured by Dupont. All
material values which will be discussed in more detail in the following
paragraphs were given by Foster [15].
The material properties were input into the solver via a constitutive
model. A complete constitutive model consists of a combination of an equation
of state, a shear model, a yield model, a failure model, and a spallation model [4].
The equation of state, shear model, yield model, failure model, and spallation
model used will be discussed in more detail in the coming sections.
31
The polynomial equation of state was used for all solid materials to define the
hydrodynamic volume limit. This was necessary due to the hydrodynamic
volume failure being used. More details concerning the hydrodynamic volume
failure will be discussed in the failure model section.
The elastic shear model, SHREL, which was used for each material has a
constant shear modulus. The shear modulus used in this analysis is the slope of
the linear relationship between the shear stress and strain.
The von Mises yield model was used for the zytel. This model when used
with solids is considered an elastic perfectly plastic yield model, no strain
hardening present, therefore only the yield stress is required. The Zytel yield
stress, shear modulus, and other material properties are given in Table 2.
Table 2: Misc. Material Properties
Material
G
(psi) yield? (psi)
K
(psi)
MPS
(in/in)
PMIN
(psi)
?
(lb_sec^4/in^4)
Aluminum (6061-T6) 4.0000E+06 N/A N/A 1.00 -1.74E+05 2.530E-04
Steel (4340) 1.1618E+07 N/A N/A 1.25 -4.64E+05 7.320E-04
Tungsten (0.07 Ni, 0.03 Fe) 2.3206E+07 N/A N/A 0.75 -1.31E+05 1.589E-03
Zytel 1.1100E+05 6.9E+03 2.4E+05 0.33 N/A 1.002E-04
The Johnson-Cook yield model was used for every material with the
exception of the zytel. This yield model is denoted in the solver by the YLDJC
entry. The yield stress is calculated as a function of the plastic strain, strain rate,
and temperature. The Johnson-Cook yield stress equation and variable
32
definitions are given below in Figure 33 [13]. The Johnson-Cook constants and
other material properties used for each material are given in Table 3.
Figure 33: Johnson-Cook Yield Model [13]
Table 3: Material Properties and Constants for Johnson-Cook Yield Model
Material
A
(psi)
B
(psi) n C m
EPS0
(in/in/sec)
CP
(in?/sec?-?R)
TMELT
(?R)
TROOM
(?R)
Aluminum
(6061-T6) 4.700E+04 1.650E+04 0.42 2.0E-03 1.34 1.0 7.1717E+04 1666 530
Steel (4340) 1.149E+05 7.390E+04 0.26 1.4E-02 1.03 1.0 4.1140E+05 3228 530
Tungsten
(0.07 Ni,
0.03 Fe) 2.184E+05 2.560E+04 0.12 1.6E-02 1.00 1.0 1.1580E+05 3102 530
2.7 Fluid Material Models, Properties, and Initial Conditions
Air is the material used in the fluid sections. A single material, HYDRO
constitutive model was utilized. This material model uses the gamma law
equation of state to calculate the pressure. The material is defined by the specific
internal energy and density. From these two values and the ratio of specific
heats, pressure can be calculated using the following formula:
33
P = (?-1)?e (10)
where e is the specific internal energy per unit mass, ? is the overall material
density, ? is the ratio of specific heats [13]. The property values used as well as
the initial conditions for the air that were used can be found in Table 4.
Table 4: Air Properties
?
CV
(in^2/s^2/R)
e
(in^2/sec^2)
?
(lbf_s^2/in^4)
1.4 6.12901E+05 3.30208386192E+08 1.12297E-07
2.8 Boundary Conditions and Initial Velocity
The target is constrained at a point on the outer radius, centered between the
two edges of the front plate and at a point on the outer radius, centered between
the two edges of the rear plate. These constraints restrict translation in any
direction. Figure 34 shows the constraints placed on the front and rear plates.
The penetrator was given an initial velocity of 70,866 in/sec in the positive Z
coordinate direction [15].
Figure 34: Boundary Conditions on Plates
34
2.9 Contact
The solver?s master-slave contact algorithm was used to define the contact
between the penetrator and target. The monitoring side, search algorithm,
adaptive setting, initial penetration, and slave node deactivation were all enabled
in the contact algorithm.
The monitoring side was set to BOTH. This allows the solver to set the
penetration side of the element to the side which the slave node approaches the
master face.
The FULL search algorithm was used. This setting results in all the faces
of the master surface being considered when the search for the closest master
face for a slave node is performed. The slave node?s search for the closest master
surface is done by a normal projection of the slave node on the faces of the
master surface. The FULL option is the most reliable of the available options, but
is the most computational expensive, however, the computational expense was
not a burden in this simulation.
The adaptive setting was set to YES. In The solver the adaptive setting
determines how the master surfaces react to element failure. For Lagrangian
solids the free faces are active initially and the internal faces are deactivated.
When an element fails, it is possible that some of the internal faces may become
free faces. If that is the case the newly defined free faces then become active.
35
After all the elements connected to a master face have failed, it then is
deactivated throughout the rest of the analysis. This logic is ideal for modeling
penetration phenomena, and is also referred to as ?eroding contact.?
The initial penetration allows the slave node to penetrate the master
surface up to a set tolerance at the beginning of an analysis. The solver?s default
of 1.E20 was used as the tolerance for this analysis.
METHOD3A was the slave node deactivation method used in this
analysis. This method sets the nodes active as slaves from the beginning of the
analysis regardless if they reside on the inside or the outside of the mesh. This
also allows for the nodes of the master surface to act as slaves once they reside on
the outside of the mesh. After all the connected elements have failed the nodes
are deactivated as slaves [16].
2.10 Coupling Algorithm
The solver?s general coupling algorithm was used with the addition of the
failure definition that allows the Euler regions to interact and transfer material
between regions when the coupling surfaces fail. The model was created with all
coupling surfaces having a failure definition, but the only one that utilized this
function in the simulation was the coupling surface attached to the back of the
front plate as this was the only region of failure in the target.
36
2.11 Failure Models
The maximum plastic strain and hydrodynamic volume limit failure models
were used for all materials. The spallation pressure failure model was used for
all materials except the zytel.
The maximum plastic strain failure model was defined in the material
constitutive model via the FAILMPS entry. When using this model the material
fails completely when the plastic strain exceeds the given limit [13]. The
assumed maximum plastic strains for aluminum was 1.0, steel was 1.25, tungsten
was 0.75, and zytel was 0.33 [15].
The solver?s default hydrodynamic volume limit of 1.1 was used for each
material. The hydrodynamic volume failure model will cause the elements with
a relative volume, defined as the reference density divided by the actual density,
greater than the limit to fail completely. Once the element fails the stress state is
set to zero [16].
The solver uses a constant minimum pressure method to determine if an
element spalls. The solver defines the positive pressures as compression,
therefore the minimum spallation pressure must be less than or equal to zero. If
the pressure in an element falls below the minimum set limit, the element spalls
and the pressure and yield stress are set to zero. At this point the material acts
like a fluid. At the point the pressure rises above zero, the material is no longer
37
in a spalled state. The cycle then can repeat with the pressure decreasing to the
specified minimum and spallation occurring again [13].
38
Chapter 3: Results
Due to the different nature of the fluid that was modeled in the Euler
region and the solid materials modeled in the Lagrangian sections these results
will be discussed and presented in separate sections.
3.1 Lagrangian Results
There are three different types of results presented for the Lagrangian
elements. These consist of time history graphs, energy of distortion contour
plots, displacement contour plots and effective stress fringe plots. An
explanation of each will be given in the following paragraphs.
The first set of results given is the material time history plots for each
penetrator configuration (AST and TSA). These plots show the energy of
distortion, kinetic energy, internal energy, and momentum in all three coordinate
directions for each material as a function of time. The time history plots given
below were compared to Foster?s results to gain confidence the Lagrangian
sections were properly constructed before the Euler regions and fluid structure
coupling were added. It was found through this comparison that the trends
were similar. Foster?s results are shown in the Appendix.
The energy of distortion for a material is the sum of the distortional energy of
all the elements that are in that material. The distortional energy for one element
is defined as the sum (in time) of deviatoric stresses (a measure of the shearing
39
exerted by a state of stress [17]) multiplied by the strain increments multiplied by
the change in time [18]. Figure 35 and Figure 36 shows the Energy of Distortion
vs. Time plot of each material for both penetrator configurations. The small
increase in the target energy of distortion at approximately 0.00025 second was
found to be the result of a piece of the aluminum that survived the impact with
the front plate striking the solid bulkhead.
Figure 35: AST Energy of Distortion vs. Time
40
Figure 36: TSA Energy of Distortion vs. Time
For ease of comparison of the penetrator configurations with and without the
fluid the distortional energy curves are given in the following plots. Figure 37
shows the distortional energy of the target. Figure 38 shows the distortional
energy of the complete penetrator. This is calculated by summing the energy of
distortion for the tungsten, steel, aluminum, and zytel sections. It can be seen
from these figures that the target incurs a larger distortional energy when the
penetrator is in the TSA configuration, but the penetrator incurs a larger
distortional energy when it is in the AST configuration. This work also shows
that there is minimal difference in the penetrator distortional energy when the
fluid is applied to the model.
41
Figure 37: Target Energy of Distortion vs. Time
Figure 38: Penetrator Energy of Distortion vs. Time
The internal energy for a material is the sum of the internal energy of all the
elements that are in that material. The internal energy for one element is defined
42
as the sum (in time) of the total stresses multiplied by the strain increments
multiplied by the change in time [18]. Figure 39 and Figure 40 show the Internal
Energy vs. Time plot of each material for both penetrator configurations.
Figure 39: AST Internal Energy vs. Time
43
Figure 40: TSA Internal Energy vs. Time
For ease of comparison the internal energy curves for both configurations are
given in the following plots. Figure 41 shows the internal energy of the target.
Figure 42 shows the internal energy of the complete penetrator. This as with the
distortional energy comparison charts given previously is calculated by
summing the internal energy for the tungsten, steel, aluminum, and zytel
sections. It can be seen from these figures that the target incurs a larger internal
energy when the penetrator is in the TSA configuration, but the penetrator incurs
a larger internal energy when it is in the AST configuration.
44
Figure 41: Target Internal Energy vs. Time
Figure 42: Penetrator Internal Energy vs. Time
The kinetic energy time versus time plots of each solid material are given for
both penetrator configurations in Figure 43 and Figure 44.
45
Figure 43: AST Kinetic Energy vs. Time
Figure 44: TSA Kinetic Energy vs. Time
It is shown in Figure 45 that the kinetic energy gain in the target occurs
substantially quicker when impacted with the TSA configuration penetrator, but
46
the AST configuration penetrator imparts more kinetic energy to the target as
seen by the higher peak in the AST curve. It is also shown in this figure that the
difference in the kinetic energy of the target when the fluid is included is
substantial. Figure 46 shows the kinetic energy versus time for each penetrator
configuration. The TSA configuration?s loss of kinetic energy occurs much
quicker than the AST configuration, but the AST configuration has a higher total
loss of kinetic energy. This is as we would expect after seeing the quicker gain of
the TSA configuration and the higher peak of the AST configuration in the target
kinetic energy comparison chart.
Figure 45: Target Kinetic Energy vs. Time
47
Figure 46: Penetrator Kinetic Energy vs. Time
The plots showing the X, Y, and Z momentum versus time are given in
Figure 47 through Figure 52. It is shown that the momentum profiles are similar
between the two penetrator configurations and the largest momentum occurs in
the Z direction as expected since this is the penetrator?s direction of travel.
48
Figure 47: AST X - Momentum vs. Time
Figure 48: TSA X - Momentum vs. Time
49
Figure 49: AST Y - Momentum vs. Time
Figure 50: TSA Y - Momentum vs. Time
50
Figure 51: AST Z - Momentum vs. Time
Figure 52: TSA Z - Momentum vs. Time
The second set of results given is the energy of distortion contour plots of the
penetrator for both configurations. These plots highlight the energy of distortion
51
in the metallic rods of the penetrator at select time steps. The failed elements
were excluded from the result which allows visualization of the penetrator
disintegration that occurs. The last time step before the AST penetrator is
completely disintegrated is 0.00014 second. The plots for both penetrators are
given side by side in Figure 53 through Figure 80. The plots are given in time
increments of approximately 0.00001 second beginning at the closest time step to
0.00001 second and ending at the closest time step to 0.00014 second. Note the
color spectrums on these plots are different scales. The scale also varies with
each time step.
Figure 53: AST Penetrator, 1.0e-5 sec.
Figure 54: TSA Penetrator, 1.0e-5 sec.
Figure 55: AST Penetrator, 2.0e-5 sec.
Figure 56: TSA Penetrator, 2.0e-5 sec.
52
Figure 57: AST Penetrator, 3.0e-5 sec.
Figure 58: TSA Penetrator, 3.0e-5 sec.
Figure 59: AST Penetrator, 4.0e-5 sec.
Figure 60: TSA Penetrator, 4.0e-5 sec.
Figure 61: AST Penetrator, 5.0e-5 sec.
Figure 62: TSA Penetrator, 5.0e-5 sec.
Figure 63: AST Penetrator, 6.0e-5 sec.
Figure 64: TSA Penetrator, 6.0e-5 sec.
Figure 65: AST Penetrator, 7.0e-5 sec.
Figure 66: TSA Penetrator, 7.0e-5 sec.
53
Figure 67: AST Penetrator, 8.0e-5 sec.
Figure 68: TSA Penetrator, 8.0e-5 sec.
Figure 69: AST Penetrator, 9.0e-5 sec.
Figure 70: TSA Penetrator, 9.0e-5 sec.
Figure 71: AST Penetrator, 1.0e-4 sec.
Figure 72: TSA Penetrator, 1.0e-4 sec.
Figure 73: AST Penetrator, 1.1e-4 sec.
Figure 74: TSA Penetrator, 1.1e-4 sec.
Figure 75: AST Penetrator, 1.2e-4 sec.
Figure 76: TSA Penetrator, 1.2e-4 sec.
54
The front plate displacement plots for both configurations are given in the
figures below with time increments of 0.00001 second beginning at 0.00001
second and ending at 0.00012 second (time at which AST configuration
completely penetrated). The failed elements are removed in order to show the
penetration of the front plate. Note the color spectrums on these plots are
different scales. The scale also varies with each time step.
Figure 77: AST Penetrator, 1.3e-4 sec.
Figure 78: TSA Penetrator, 1.3e-5 sec.
Figure 79: AST Penetrator, 1.4e-4 sec.
Figure 80: TSA Penetrator, 1.4e-4 sec.
Figure 81: AST Front Plate Displacement,
1.0e-5 sec.
Figure 82: TSA Front Plate Displacement,
1.0e-5 sec.
55
Figure 83: AST Front Plate Displacement,
2.0e-5 sec.
Figure 84: TSA Front Plate Displacement,
2.0e-5 sec.
Figure 85: AST Front Plate Displacement,
3.0e-5 sec.
Figure 86: TSA Front Plate Displacement,
3.0e-5 sec.
Figure 87: AST Front Plate Displacement,
4.0e-5 sec.
Figure 88: TSA Front Plate Displacement,
4.0e-5 sec.
Figure 89: AST Front Plate Displacement,
5.0e-5 sec.
Figure 90: TSA Front Plate Displacement,
5.0e-5 sec.
56
Figure 91: AST Front Plate Displacement,
6.0e-5 sec.
Figure 92: TSA Front Plate Displacement,
6.0e-5 sec.
Figure 93: AST Front Plate Displacement,
7.0e-5 sec.
Figure 94: TSA Front Plate Displacement,
7.0e-5 sec.
Figure 95: AST Front Plate Displacement,
8.0e-5 sec.
Figure 96: TSA Front Plate Displacement,
8.0e-5 sec.
Figure 97: AST Front Plate Displacement,
9.0e-5 sec.
Figure 98: TSA Front Plate Displacement,
9.0e-5 sec.
57
The final result sets given for the Lagrangian elements will be contours
plots of the stress wave as it travels through the front plate, rear plate, and inside
walls. The plots for both configurations will be presented at an initial time of
approximately 0.00001 second with an increment of 0.00005 second. The last
time step given will be approximately 0.00041 second. These plots are shown
side by side for easy comparison in Figure 105 through Figure 122. Note the
Figure 99: AST Front Plate Displacement,
1.0e-4 sec.
Figure 100: TSA Front Plate Displacement,
1.0e-4 sec.
Figure 101: AST Front Plate Displacement,
1.1e-4 sec.
Figure 102: TSA Front Plate Displacement,
1.1e-4 sec.
Figure 103: AST Front Plate Displacement,
1.2e-4 sec.
Figure 104: TSA Front Plate Displacement,
1.2e-4 sec.
58
color spectrums on these plots are different scales. The scale also varies with
each time step.
Figure 105: AST Stress, 1.0e-5 sec.
Figure 106: TSA Stress, 1.0e-5 sec.
Figure 107: AST Stress, 6.0e-5 sec.
Figure 108: TSA Stress, 6.0e-5 sec.
Figure 109: AST Stress, 1.1e-4 sec.
Figure 110: TSA Stress, 1.1e-4 sec.
Figure 111: AST Stress, 1.6e-4 sec.
Figure 112: TSA Stress, 1.6e-4 sec.
59
Figure 113: AST Stress, 2.1e-4 sec.
Figure 114: TSA Stress, 2.1e-4 sec.
Figure 115: AST Stress, 2.6e-4 sec.
Figure 116: TSA Stress, 2.6e-4 sec.
Figure 117: AST Stress, 3.1e-4 sec.
Figure 118: TSA Stress, 3.1e-4 sec.
Figure 119: AST Stress, 3.6e-4 sec.
Figure 120: TSA Stress, 3.6e-4 sec.
Figure 121: AST Stress, 4.1e-4 sec.
Figure 122: TSA Stress, 4.1e-4 sec.
60
3.2 Euler
To evaluate the resulting pressure wave that occurs in the interior and on the
exterior of the target, plots of the pressure versus Z of the Euler regions for the
nodes centered along the X and Y axis as well as contour plots for the faces of
select slices of the Euler regions are given for both configurations. Figure 123
shows a pressure wave for the AST configuration as it moves through the first
interior Euler region from time 0 to 0.0004 second. Figure 124 shows the
corresponding pressure wave for the TSA configuration.
Figure 123: AST Interior Euler 1, Pressure vs. Z-Coord.
61
Figure 124: TSA Interior Euler 1, Pressure vs. Z-Coord.
The second AST configuration, interior Euler region had minimal pressure
change until approximately 0.0006 second. Due to this Figure 125, which shows
the pressures as it fluctuates along the Z-axis at the X and Y-axis centered nodes
in this region, begins at 0.0006 second and ends at 0.0010 second. Figure 126
shows the pressure of the second interior Euler region for the TSA configuration.
Figure 125: AST Interior Euler 2, Pressure vs. Z-Coord.
62
Figure 126: TSA Interior Euler 2, Pressure vs. Z-Coord.
The contour plots for the Euler regions are given next. Before these results
are presented the region definitions segmented for viewing the results will be
explained. The Euler regions in each of the three compartments (target sections
between front plate, bulkheads, and rear plate) are segmented into three slices
(Front, Middle, and Back) each. In addition the hole between the two
compartments in the first Euler region is given as a separate segment. Figure 127
shows each labeled contour segment in relation to the front plate, rear plate, and
solid bulkhead. So the reader may see the section of the target that contains the
Euler 1 hole contour segment, Figure 128 shows the first Euler region with the
Euler 1 hole contour segment removed.
63
Figure 127: Euler Contour Regions
Figure 128: Euler 1 Contour Regions Excluding Hole
The pressure contour plots of the front and center slices of compartment 1,
hole slice, and the center and rear slices of compartment 2 from the first interior
Euler region given below begins at approximately 0.0001 second, increments
approximately 0.00005 second, and ends at approximately 0.0004 second. Figure
Euler 1, Compartment 1:
Front, Center , and Rear
Euler 1, Compartment 2:
Front, Center, and Rear
Rear Plate Front Plate Bulkhead Euler 1: Hole
Euler 2: Front,
Center, and Rear
Compartment 1:
Front, Center, and Rear
Compartment 2:
Front, Center, and Rear
64
129 shows the pressure increase in the front slice of the first Euler region for the
AST configuration at 0.0001 second. Figure 130 shows the pressure increase in
the front slice of the first Euler region for the TSA configuration at 0.0001 second.
The plots for the other 6 time steps are given in Figure 131 through Figure 142.
Note the pressure begins to build on the front slice of the TSA configuration at
approximately 0.000025 second sooner than on the AST configuration. This is
likely due to the TSA configuration penetrating the front plate faster than the
AST configuration.
Figure 129: AST Euler Region 1 Pressure,
1e-4 sec.
Figure 130: TSA Euler Region 1 Pressure,
1e-4 sec.
Figure 131: AST Euler Region 1, 1.5e-4 sec.
Figure 132: TSA Euler Region 1 Pressure,
1.5e-4 sec.
65
Figure 133: AST Euler Region 1 Pressure,
2.0e-4 sec.
Figure 134: TSA Euler Region 1 Pressure,
2.0e-4 sec.
Figure 135: AST Euler Region 1 Pressure,
2.5e-4 sec.
Figure 136: TSA Euler Region 1 Pressure,
2.5e-4 sec.
Figure 137: AST Euler Region 1 Pressure,
3.0e-4 sec.
Figure 138: TSA Euler Region 1 Pressure,
3.0e-4 sec.
Figure 139: AST Euler Region 1 Pressure,
3.5e-4 sec.
Figure 140: TSA Euler Region 1 Pressure,
3.5e-4 sec.
66
The pressure contour plots given below of the front, center, and rear slice
from the second interior Euler region begins at approximately 0.0008 second,
incremented at approximately 0.0001 second, and ends at approximately 0.0011
second. Note the pressures are much less in the second Euler region compared
to the first Euler region. This indicates that the solid bulkhead separating the
two regions effectively absorbed or reflected the pressure wave.
Figure 141: AST Euler Region 1 Pressure,
4.0e-4 sec.
Figure 142: TSA Euler Region 1 Pressure,
4.0e-4 sec.
Figure 143: AST Euler Region 2 Pressure, 8e-
4 sec.
Figure 144: TSA Euler Region 2 Pressure, 8e-
4 sec.
Figure 145: AST Euler Region 2 Pressure,
9.0e-4 sec.
Figure 146: TSA Euler Region 2 Pressure,
9.0e-4 sec.
67
Figure 147: AST Euler Region 2 Pressure,
1.0e-3 sec.
Figure 148: TSA Euler Region 2 Pressure,
1.0e-3 sec.
Figure 149: AST Euler Region 2 Pressure,
1.1e-3 sec.
Figure 150: TSA Euler Region 2 Pressure,
1.1e-3 sec.
68
Conclusions
The main objective of this work was to calculate the over-pressure inside a
target generated from a high velocity impact. This work shows that both
penetrator configurations both produce a substantial over-pressure inside the
target.
It was found that the AST penetrator transfers more kinetic energy to the
target resulting in a higher over-pressure than the TSA penetrator. A factor in
this is that a fragment of the aluminum from the TSA penetrator survives the
initial impact and passes through the front plate into the interior of the target.
For the AST penetrator there are no fragments which pass through the front
plate, thus imparting all of its energy to the target upon impact. The deeper
penetration of the TSA configuration might be favorable in some applications, if
the mission profile calls for penetrating a hardened target without completely
destroying the penetrator.
This simulation shows the distortional energy and internal energy peaks
in the target to be higher in the TSA than in the AST. It also shows that though
the AST produced a higher internal pressure in the target. The pressure rise
occurred earlier for the TSA than it did for the AST.
69
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71
Appendix
Figure 151: Foster - AST Energy of Distortion vs. Time [11]
Figure 152: Foster - AST Internal Energy vs. Time [11]
72
Figure 153: Foster - AST Kinetic Energy vs. Time [11]
Figure 154: Foster - AST Z-Momentum vs. Time [11]
73
Figure 155: Foster - TSA Energy of Distortion vs. Time [11]
Figure 156: Foster - TSA Internal Energy vs. Time [11]
74
Figure 157: Foster TSA Kinetic Energy vs. Time [11]
Figure 158: Foster TSA Kinetic Energy vs. Time [11]