Spall Fracture of Multi-Material Plates Under Explosive
Loading
By
James Danyluk
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
Master of Engineering
Major Subject: Mechanical Engineering
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Thesis Adviser
Rensselaer Polytechnic Institute at Hartford
Hartford, CT
December, 2010
(For Graduation May 2011)
© Copyright 2010
by
James Danyluk
All Rights Reserved
ii
CONTENTS
Spall Fracture of Multi-Material Plates Under Explosive Loading .................................... i
LIST OF TABLES ............................................................................................................ iv
LIST OF FIGURES ........................................................................................................... v
LIST OF SYMBOLS ........................................................................................................ vi
ACKNOWLEDGMENT ................................................................................................. vii
ABSTRACT ................................................................................................................... viii
1. Introduction.................................................................Error! Bookmark not defined.
2. METHODOLOGY ...................................................................................................... 5
2.1
Theory of Shock Wave Interactions ................................................................... 5
2.2
Hydrocode Model Description ........................................................................... 9
2.3
Material Model Descriptions............................................................................ 11
2.3.1
Equation of State Material Model Descriptions ................................... 12
2.3.2
Strength Material Model Descriptions ................................................. 13
2.3.3
Failure Material Model Descriptions ................................................... 14
3. RESULTS AND DISCUSSION ................................................................................ 15
3.1
Hand Calculations ............................................................................................ 15
3.2
Single Plate Hydrocode Results ....................................................................... 15
3.3
Dual Plate Hydrocode Results ......................................................................... 17
3.4
Three Plate Hydrocode Results ........................................................................ 19
4. CONCLUSIONS ....................................................................................................... 22
5. REFERENCES .......................................................................................................... 23
6. APPENDIX................................................................................................................ 24
iii
LIST OF TABLES
Table 1: Solid Material Trials .......................................................................................... 11
Table 2: Material Models................................................................................................. 12
Table 3: Hugoniot parameters for materials analyzed ..................................................... 15
Table 4: Single plate hydrocode results summary ........................................................... 17
Table 5: Dual plate hydrocode results summary ............................................................. 18
Table 6: Three plate hydrocode results ............................................................................ 20
iv
LIST OF FIGURES
Figure 1: Typical P-υ (Pressure and Specific Volume) Hugoniot ..................................... 1
Figure 2: Pressure wave traveling through solid approaches a free surface ...................... 2
Figure 3: Pressure wave reaches the free surface .............................................................. 2
Figure 4: Reflected Wave and Moving Surface generated a Tensile Stress ...................... 3
Figure 5: Spall crater of a metal fragment ......................................................................... 4
Figure 6: Sample U-u Hugoniot Plane for Al6061-T6 ...................................................... 6
Figure 7: Example of a P-u Hugoniot ................................................................................ 6
Figure 8: P-u Hugoniot of an impact problem ................................................................... 7
Figure 9: General model setup ........................................................................................... 9
Figure 10: Solid material setup ........................................................................................ 10
Figure 11: Explosive shock wave entering an Aluminum Plate ...................................... 15
Figure 12: Shock wave reaching the free surface ............................................................ 16
Figure 13: Rarefaction wave is generated ....................................................................... 16
Figure 14: Pressure vs. Time in an Aluminum Plate ....................................................... 16
Figure 15: Tungsten pressure reaches minimum before polyethylene ............................ 18
Figure 16: Trial 10 pressure wave in explosive ............................................................... 19
Figure 17: Pressure wave enters the polyethylene plate, rarefaction wave generated in
tungsten ............................................................................................................................ 19
Figure 18: Pressure wave reaches free surface, rarefaction wave generated in tungsten 19
Figure 19: Trial 15 pressure wave in explosive ............................................................... 20
Figure 20: Pressure wave enters the polyethylene plate, rarefaction wave generated in
tungsten ............................................................................................................................ 20
Figure 21: Rarefaction wave generated in the polyethylene, pressure wave travels
through second tungsten plate ......................................................................................... 21
Figure 22: Rarefaction wave generated in second tungsten plate .................................... 21
v
LIST OF SYMBOLS
Name
U
ρ
u
p
co
s
υ
e
C
PCJ
uCJ
D
Y
A
B
R1
R2
ω
γ
Γ
G
T
β
εp
AJC
BJC
CJC
m
n
σ1,2,3
Description
shock velocity
density
particle velocity
pressure
slope of the U-u Hugoniot relationship
y intercept of the U-u Hugoniot relationship
specific volume
internal energy
Material sound speed
Chapman-Jouget Pressure
Chapman-Jouget Velocity
Detonation Velocity
Yield Strength
JWL Parameter A
JWL Parameter B
JWL Parameter R1
JWL Parameter R2
JWL Parameter ω
Ideal gas constant
Gruneisen coefficient
Shear Modulus
Temperature
Steinberg-Guinan hardening constant
Effective Plastic Strain
Johnson-Cook initial yield Strength
Johnson-Cook hardening constant
Johnson-Cook strain rate constant
Johnson-Cook temperature exponent
Johnson-Cook hardening exponent
Principal Stresses
vi
Units
km/s
g/cc
km/s
kPa
km/s
cc/g
kJ
km/s
kPa
km/s
km/s
kPa
kPa
kPa
kPa
K
kPa
kPa
kPa
ACKNOWLEDGMENT
TBD
vii
ABSTRACT
Plates undergoing high compression shock loading can experience a fracture that
is known as spall. This phenomenon occurs when the compressive shock wave travels
through the plate and reflects off of the free boundary. This reflection is now a tensile
wave that can tear and fracture the material. The mating material at the free boundary
directly affects the strength of the reflected tensile wave. Similar materials, such as
metals against metals, will have a low tensile wave reflection while dissimilar materials,
such as metal to plastic or air, results in a high tensile wave reflection.
This
phenomenon is seen often in warhead technology, where an explosive accelerates
fragments. If the explosive/fragment/boundary impedance mismatch is great enough,
the resulting reflected tensile wave will tear the fragments apart before they have begun
to accelerate.
viii
1. INTRODUCTION
One output from explosives is a high pressure wave that exists at the front of the
shock wave. This pressure wave will travel through air, liquid and solid materials,
slowly reducing in value until the pressure wave reaches ambient conditions.
A
relationship, known as the Hugoniot-Rankine Jump Equations, dictates how a material
changes in state when interacting with a traveling shock wave. These equations relate
the particle velocities to shock velocities shock wave pressures to particle or shock
velocities.
The parameters of these relations are material and density dependent
properties. These relationships are often plotted for a specific density/material. This
plot is known as a type of Hugoniot Plane.
Figure 1: Typical P-υ (Pressure and Specific Volume) Hugoniot
The Hugoniot-Rankine equations can also be used to determine how a shock wave
changes when moving across different materials. The relationships can be used to
determine the type of shockwave traveling through a material, which was generated by
an explosive being detonated. The relationship also dictates how the shockwave changes
when traveling between two different materials.
When a shock wave begins to interact with the free boundary, a rarefaction wave,
also known as a relief wave or unloading wave, is created. This rarefaction wave moves
in the opposite direction of the shock wave and with a scaled pressure value. When the
1
rarefaction wave interacts with the compression wave or other rarefaction waves, it can
create a tensile stress in the material. In the figures below, a constant pressure shock
wave is traveling through a thick plate of polystyrene. When the shock wave reaches the
free surface, a rarefaction wave is created, which moves in the opposite direction. At the
same time, the free surface is moving in the opposite direction of the rarefaction wave
and at the same speed of the of the rarefaction wave. These opposite velocities produced
by the wave interaction leads to a tensile load in the material.
Incoming pressure wave
Initial velocity = Vo
Free Surface
Zero initial velocity
Distance
Figure 2: Pressure wave traveling through solid approaches a free surface
Pressure wave reaches
the free surface
Free Surface
Distance
Figure 3: Pressure wave reaches the free surface
2
Pressure wave is reflected
Velocity is reversed, V=-Vo
After interaction from the
pressure wave, the free surface
now moves at Velocity V=Vo
Tensile stress
generated
due to
opposite
moving waves
Distance
Figure 4: Reflected Wave and Moving Surface generated a Tensile Stress
A free surface is not always needed to generate a rarefaction wave. A change in
material can also cause a relief wave to be generated, although it is not as high in the
case of a free surface interaction. This relief wave is based on the material's properties.
A high density metal mated up against a low denisty polymer will have a significant
relief wave generated. A low density metal mated up against a similar high density
polymer will have a lower rarefaction wave generated. The rest of the shock wave
continues to travel through the mating material.
If the tensile stress due to the rarefaction wave exceeds the dynamic tensile
strength, also known as the spall strength, the material can fracture or tear. This is seen
often in explosive applications such as a warhead. In this case an explosive is being
used to accelerate a set of fragments. If the warhead was designed improperly, the relief
wave generated due to free surface interaction can spall the fragments. This can shatter
the fragments before they even begin to accelerate, effectively rendering the warhead
useless. The figure below show the effect of a tungsten cube being accelerated by an
explosive loading. The rarefaction wave generated in this case spalled the surface of the
cube off, creating a crater.
3
Figure 5: Spall crater of a metal fragment
The theory of the phenomena is described in detail in Cooper’s Explosive
Engineering and Meyers Dynamic Behavior of Materials. . The book Spall Fracture, by
Tarabay Antoun, contains the spall strengths of various materials, which will be used as
a criterion for all of the calculations done. Finally, the articles referenced from the
Journal of Impact Engineering will be used as a supplement from Explosive Engineering
book. In addition to the hand calculations, ANSYS AUTODYN can be used to track
shock waves traveling through materials and determine their reactions. AUTODYN is
an explicit dynamic Finite Element Analysis (FEA) software, also known as a
hydrocode.
It specializes in high strain rate, short time duration and energetic
interactions.
4
2. METHODOLOGY
2.1 Theory of Shock Wave Interactions
The shock phenomenon can initially be described by using the Rankine-Hugoniot
Jump Equations. These equations can be separated into three distinct relationships, as a
mass, momentum and energy balance. These balances dictate how a shock front travels
through a material. The mass balance implies that mass is neither created nor destroyed.
The momentum balance implies that the force required to bring a material before the
state of shock to the state after shock must be equal. Finally the energy balance implies
that the energy increase in the mass must be equal to the work being done to it. These
three relationships can be expressed using the following equations respectively.
𝜌1 𝑈 − 𝑢𝑜 𝜈𝑜
=
=
𝜌𝑜 𝑈 − 𝑢1 𝜈1
𝑃1 − 𝑃𝑜 = 𝜌𝑜 (𝑢1 − 𝑢𝑜 )(𝑈 − 𝑢𝑜 )
𝑒1 − 𝑒𝑜 =
𝑃1 𝑢1 − 𝑃𝑜 𝑢𝑜 1 2
− (𝑢 − 𝑢𝑜2 )
𝜌𝑜 (𝑈 − 𝑢𝑜 ) 2 1
Where P is the pressure, U is the shock velocity, u is the particle velocity, rho is the
density and e is the specific internal energy, with the o and i subscripts representing the
two shock states of the material, before and after the shock front.
The Hugoniot
equation is the relationship between two of the variables in the balances previously
mentioned. One common relationship is the relationship between the shock velocity and
particle velocity, which is known as the U-u equation. This equation is often determined
from experimental results, where the s and co parameters are the slope and y-intercept
respectively of the trend line.
𝑈 = 𝑐0 + 𝑠𝑢𝑝
5
Figure 6: Sample U-u Hugoniot Plane for Al6061-T6
Another Hugoniot equation can be developed between the particle velocity and
pressure.
This equation is determined by substituting the U-u equation into the
momentum equation. This equation assumes that the initial velocity of the mass is not
zero. This equation is referred to as the P-u Hugoniot equation.
𝑃1 = 𝜌𝑜 𝑐𝑜 (𝑢1 − 𝑢𝑜 ) + 𝜌𝑜 𝑠(𝑢1 − 𝑢𝑜 )2
Figure 7: Example of a P-u Hugoniot
6
When a shock wave passes between two different materials, the P-u and U-u
Hugoniot equations can be used to determine the shock velocity and pressure traveling in
both materials. This can be done when the shock wave reaches the boundary between
the two materials. At this point, the P-u Hugoniot for both materials intersect. Using
this condition, the P-u Hugoniot equation for both materials can be set equal to one
another, as the pressure will be the same in both materials at this point.
𝜌𝑜𝐵 𝑐𝑜𝐵 𝑢1 + 𝜌𝑜𝐵 𝑠𝐵 𝑢12 = 𝜌𝑜𝐴 𝑐𝑜𝐴 (𝑢𝑜𝐴 − 𝑢1 ) + 𝜌𝑜𝐴 𝑠𝐴 (𝑢𝑜𝐴 − 𝑢1 )2
Figure 8: P-u Hugoniot of an impact problem
Where the A and B subscript represents the two different materials. Setting these two
equations equal to each other, a quadratic equation can be formed. Solving this equation
for u1 will give the particle velocity for each material. Substituting either of these
particle velocities back into their corresponding P-u Hugoniot will allow the pressure of
the shockwave to be known.
The Hugoniot equations can also be used to determine the shockwave pressure
and velocity induced in a solid material due to an explosive being detonated behind the
material. The shockwave pressure of the explosive is known as the CJ pressure (PCJ), a
material parameter. The particle velocity of the explosive, known as the uCJ, can be
determined by the relationship:
7
𝑢𝐶𝐽 =
𝑃𝐶𝐽
𝜌𝑜 𝐷
Where PCJ is the CJ pressure, rho is the density of the explosive and D is the detonation
velocity of the explosive. As done with the shockwave traveling between two materials,
the P-u Hugoniots of the explosive and solid can be set equal to one another. Solving
the resulting quadratic yields the particle velocity in the solid and, as done before, the
shockwave pressure in the explosive can be determined by substituting the solved
particle velocity back into the P-u Hugoniot.
When the shockwave interacts with a free surface, part of the shock wave is
reflected into what is known as a rarefaction wave, or relief wave. As previously shown,
the P-u Hugoniot equation can be used to solve for the velocity of the shock wave
traveling back into the original material. The reflected shock wave will be tensile and
that the slope of the P-u Hugoniot will have the slope of –ρoCL. At the point where the
rarefaction waves meet, the tensile stress will reach a maximum and the velocity will be
reduced to zero. At this point, the tensile stress can be calculated using the equation:
∆𝑃
= 𝜌𝑜 𝐶𝐿
∆𝑢
Where CL is defined as the material sound speed. The final pressure, assuming that the
initial pressure was zero, will be the max tensile stress seen in the plate due to the
rarefaction wave. A rarefaction wave isn’t just generated when reacting with a free
surface. When a shockwave moves through different materials, a rarefaction wave is
generated, although this wave is not as strong as the free surface interaction. The
strength of the rarefaction wave is generally controlled by the equation:
𝑃𝑅 𝜌𝐵 𝐶𝐵 − 𝜌𝐴 𝐶𝐴
=
𝑃𝐼 𝜌𝐵 𝐶𝐵 + 𝜌𝐴 𝐶𝐵
Where A and B are the two different materials, C is the respective material sound speed,
PI is the pressure at the material interface and PR is the rarefaction wave pressure. Using
these concepts, the rarefaction wave strength can be calculated for any material for a
shock wave that is produced by an explosive load.
8
2.2 Hydrocode Model Description
The explicit dynamics code ANSYS AUTODYN will be used for all analyses on
this project. Due to this problem’s short time frame and high strain rate possibilities, the
2D Euler Multi-material solver was selected. Although the 2D solver will be used, the
model will essentially be setup as a 1D system, as only a single cell will be used in the
width.
The model will be made up of an explosive component, a solid material
component and an air component. The explosive will be modeled as HMX using the
Jones-Wilkins-Lee (JWL) equation of state (EOS). The air component will be modeled
using the Ideal Gas EOS. For the solid material component, several configurations will
be analyzed. All of the solid component configurations will be modeled using the Shock
EOS and various strength models.
Detonation front
HE (HMX)
Air
Selected Solid Materials
Figure 9: General model setup
The solid material configurations will be modeled as a single, double or triple
layer, consisting of different materials. Several materials will be investigated such as;
304L steel, 6061 aluminum, a tungsten alloy and polyethylene material. The double and
triple layers will be setup so that different materials are between each layer. The amount
of HMX and Air will be constant for all solid material configurations. The HMX was
modeled as a 1.5” thickness and each solid material layer was modeled as ¼”. A flowout condition was applied to all boundaries of the Euler grid. This boundary condition
allowed material to freely leave the grid. This assumed that an infinite volume was
behind HMX, preventing an overpressure from building behind the solid layers. The
flow-out boundary also prevented relief waves from being generated off the lateral
9
boundaries. The final boundary condition employed was a line detonation condition.
This condition fully reacts the HMX at the detonation line and propagated the explosive
further.
The analysis did not consider any input into starting the reaction of the
explosive.
Single Layer
Two Layers
Three Layers
Figure 10: Solid material setup
The analysis was done in several stages. The first set of analyses had the solid
material modeled as purely elastic. This first set of analyses will also be compared with
the classic hand calculations. The second set of analyses will add plastic deformations to
the model. The third set of analyses will add a failure model to the solid materials. This
failure model will be defined as the hydrodynamic tensile failure, which is defined as the
spall strength of the material. For each configuration, the pressure will be tracked in
each layer of the solid materials. The maximum tensile pressure, due to the reflected
shockwave, will be compared with the material’s spall strength in order to determine if
the solid material would fail. The set of analyses with the failure model will show spall
planes being developed if the spall strength has been exceeded.
10
Table 1: Solid Material Trials
Trial
Layer 1
Layer 2
Layer 3
1
6061-T6
2
Steel v250
3
304L
4
Tung
5
6061-T6
Tung
6
Tung
6061-T6
7
304L
Tung
8
Tung
304L
9 Polyethylene
Tung
10
Tung
Polyethylene
11
Steel v250
Tung
12
Tung
Steel v250
13
Tung
304L
Tung
14
Tung
6061-T6
Tung
15
Tung
Polyethylene
Tung
16
304L
Tung
304L
17
Steel v250
Tung
Steel v250
18
6061
Tung
6061
19 Polyethylene
304L
Tung
20
Tung
304L
Polyethylene
Gauges were placed along the thickness of each solid layer, at a spacing of
0.00254” (0.1mm). The gauges recorded pressures seen in the solid material throughout
the transient. The data extracted from these gauges were the main results used to
determine the strength of the rarefaction wave. Contour plots of the pressure waves
were also extracted during the transient to supplement these results.
2.3 Material Model Descriptions
Multiple material models were used for the hydrocode analyses. Two of the
materials only used an EOS for modeling. HMX used the JWL EOS and the air material
used the Ideal Gas EOS. All of the solid material components used an EOS, Strength
and Failure models.
As mentioned previously, the first set of iterations, the solid
materials used an Elastic strength model, which was defined by the material’s shear
modulus. The second set of iterations implemented plasticity in the strength model.
Multiple plasticity models were used, the Steinburg-Guinan, Von Mises and JohnsonCook models. The final set of iterations used implemented the Hydro (Pmin) failure
11
model, also known as the hydrodynamic tensile strength. The first two sets of iterations
did not use a failure model.
Table 2: Material Models
Material
EOS
Strength
Failure
HMX
JWL
NA
NA
Air
Ideal Gas
NA
NA
SS 304L
Shock Steinburg-Guinan Hydro (Pmin)
Al6061-T6
Shock Steinburg-Guinan Hydro (Pmin)
Maraging 250 Steel Shock Steinburg-Guinan Hydro (Pmin)
Polyethylene
Shock
Von Mises
Hydro (Pmin)
Tungsten Alloy
Shock
Johnson-Cook Hydro (Pmin)
2.3.1
Equation of State Material Model Descriptions
The JWL EOS is used to define the rapid expansion of gases, which is usually
caused by the detonation of a high explosive. The pressure generated by the high
explosive is generated by multiple parameters, which are defined using empirical data.
𝑃 = 𝐴 (1 −
𝜔𝜂 −𝑅𝜂1
𝜔𝜂 −𝑅𝜂2
)𝑒
+ 𝐵 (1 −
)𝑒
+ 𝜔𝜌𝑒
𝑅1
𝑅2
A, B, R1, R2 and ω are the empirically determined parameters, ρ is the current density, η
is the ratio of the current density to the original density and e is the internal energy of the
explosive, derived from the CJ energy. The pressure produced by the explosive is
limited to the CJ pressure and the initial velocity of the explosive is determined by the
CJ detonation velocity.
The CJ detonation velocity, energy and pressure are all
empirically derived parameter in addition to the previously mentioned JWL parameters.
The Ideal Gas EOS is defined by:
𝑃 = (𝛾 − 1)𝜌𝑒
Where γ is the ideal gas constant, ρ is the density and e is the internal energy of air. The
internal energy of air at room temperature/sea level was defined as 2.068e5kJ, this sets
the ambient pressure of 1atm (101kPa) to the air.
The shock EOS uses the concept of the P-u Hugoniot, mentioned previously, in
order to determine how shock waves travel through a specific material. This equation of
state uses the base relationship of particle velocity to shock velocity and the MieGruneisen pressure relationship:
12
𝑈 = 𝑐0 + 𝑠𝑢𝑝
𝑝 = 𝑝𝐻 + Γ𝜌(𝑒 − 𝑒𝐻 )
Where p is the pressure, ρ is the density, e is the internal energy, Γ is the Gruneisen
coefficient, and co and s are the Hugoniot parameters. pH and eH are defined as:
𝜌𝑜 𝑐𝑜2 𝜇(1 + 𝜇)
𝑝𝐻 =
[1 − (𝑠 − 1)𝜇]2
𝑒𝐻 =
2.3.2
1 𝑝𝐻
𝜇
(
)
2 𝑝𝑜 1 + 𝜇
Strength Material Model Descriptions
The Steinburg-Guinan strength model is used for modeling material undergoing
very high strain rates. The yield strength for this model is a function of temperature and
pressure and also assumes that the shear modulus is a function of temperature and
pressure. The equations for the shear modulus and yield strength respectively are:
𝐺𝑝′ 𝑝
𝐺𝑇′
𝐺 = 𝐺𝑜 (1 + ( ) 1/3 + ( ) (𝑇 − 300))
𝐺𝑜 𝜂
𝐺𝑜
𝑌𝑝′ 𝑝
𝑌𝑇′
𝑌 = 𝑌𝑜 (1 + ( ) 1/3 + ( ) (𝑇 − 300)(1 + 𝛽𝜀)𝑛 )
𝑌𝑜 𝜂
𝑌𝑜
Where T is the temperature, η is the compression ratio, ε is the effective plastic strain, G
is the shear modulus, Y is the yield strength, β is the hardening constant and n is the
hardening exponent.
The Johnson-Cook strength model is a plasticity model that incorporates
parameters for large strains, different strain rates and temperature conditions. The yield
strength is determined by a combination of these parameters. This model assumes that
the Von Mises cylinder’s radius changes as determined by these parameters.
𝑌 = [𝐴𝐽𝐶 + 𝐵𝐶𝐽 𝜀𝑝𝑛 ][1 + 𝐶𝐽𝐶 ln 𝜀𝑝∗ ][1 − 𝑇𝐻𝑚 ]
Where Y is the calculated yield strength, εp is the effective plastic strain, εp* is the
normalized effective plastic strain and TH is the homologous temperature (defined as (TTroom)/(Tmelt-Troom)).
‘AJC’ is the initial (zero stress) yield strength, ‘BJC’ is the
hardening constant, ‘n’ is the strain hardening exponent, ‘CJC’ is the strain rate constant
and ‘m’ is the temperature softening exponent.
13
The Von Mises strength model is an elastic/perfectly plastic strength model. It’s
defined using the materials shear modulus and yield strength. This model does not
incorporate any strain hardening parameters. The yield criterion is determined using the
classic Von Mises equation:
(𝜎1 − 𝜎2 )2 + (𝜎2 − 𝜎3 )2 + (𝜎1 − 𝜎3 )2 = 2𝑌 2
Where σ1, σ2, σ3, are the principal stresses and Y is the defined yield strength.
2.3.3
Failure Material Model Descriptions
The Hydro (Pmin) model, also known as the hydrodynamic tensile failure model,
defines a spall stress for a given material. As previously mentioned, when a rarefaction
wave travels through a material, a tensile stress can be generated. If this tensile stress,
seen as a negative pressure, exceeds the predefined spall strength, the material is
considered to have ‘failed’. At the failed locations, the stresses in the material are set to
zero and the material is turned into void cells in the Euler grid. The void cells are now
considered a free surface for future compression/rarefaction waves.
14
3. RESULTS AND DISCUSSION
3.1 Hand Calculations
Table 3: Hugoniot parameters for materials analyzed
Material
density (g/cc)
304L
7.9
Steel v250
8.12
6061-T6
2.703
Tungsten
17
Polyethylene
0.915
Material
density (g/cc)
HMX
1.891
C (km/s)
4.57
3.98
5.24
4.03
2.90
D (km/s)
9.11
s
1.49
1.58
1.4
1.237
1.481
Pcj (GPa)
42
3.2 Single Plate Hydrocode Results
As expected from the shock wave interaction theory and hand calculations, a
tensile stress was generated in the material after the shock wave interacted with the free
surface of the plate. In general, the shock wave became tensile after the free surface
interaction, where the minimum tensile stress was achieved. The tensile rarefaction
wave became a compression wave when interfacing with the plates other free surface,
which was next to the explosive. The wave again switched signs when reacting with the
free surface of the plate. This interaction repeated, with the overall maximum and
minimums of the pressure reducing as time continued.
Figure 11: Explosive shock wave entering an Aluminum Plate
15
Figure 12: Shock wave reaching the free surface
Figure 13: Rarefaction wave is generated
Trial 1 - Plastic Material Model
4.00E+07
3.50E+07
3.00E+07
Pressure (kPa)
2.50E+07
2.00E+07
1.50E+07
1.00E+07
5.00E+06
0.00E+00
0.00E+00 2.00E-03
-5.00E+06
-1.00E+07
4.00E-03
6.00E-03
8.00E-03
1.00E-02
1.20E-02
1.40E-02
1.60E-02
Time (ms)
Figure 14: Pressure vs. Time in an Aluminum Plate
The single plate configurations were ran with an elastic material model and a
plastic material model. When comparing the rarefaction stress between two different
material conditions, it can be seen that the plastic material models produced higher
tensile stresses. This is directly due to the additional deformation that is allowed by the
16
plastic material model. As the plate deforms plastically, the free surface is allowed to
move at a faster velocity. This increase in velocity leads to a greater rarefaction wave.
The elastic material model prevents this additional deformation, leading to a slower free
surface velocity.
Table 4: Single plate hydrocode results summary
Trial
1
2
3
4
1
2
3
4
Plate 1
Material
Min Pressure
Condition
(kPa)
Elastic
-7.42E+06
Elastic
-1.29E+07
Elastic
-1.50E+07
Elastic
-2.04E+07
Plastic
-7.42E+06
Plastic
-1.29E+07
Plastic
-1.65E+07
Plastic
-2.76E+07
Time
(µs)
4.34
4.77
4.71
7.29
4.34
4.93
4.75
5.4
3.3 Dual Plate Hydrocode Results
The addition of a second plate showed that with the addition of a second plate, the
overall tensile stress seen in the first plate was usually lower. Trial 6 consisted of a
tungsten plate and aluminum plate.
With the addition of the aluminum plate, the
minimum pressure seen in the tungsten plate reduced from -2.76e7kPa, from the one
plate set of iterations, to -1.55e7kPa. This decrease in tensile stress is most likely due to
the distance that the shock wave travels before the minimum pressure is seen in the
tungsten plate, as the overall pressure of the shock wave will decrease with time.
17
Table 5: Dual plate hydrocode results summary
Plate 1
Material
Min Pressure
Condition
(kPa)
Plastic
-7.80E+06
Plastic
-1.55E+07
Plastic
-1.86E+07
Plastic
-2.30E+07
Plastic
-9.77E+05
Plastic
-2.38E+07
Plastic
-1.71E+07
Plastic
-2.00E+07
Trial
5
6
7
8
9
10
11
12
Time
(µs)
6.83
8.06
6.71
8
12.6
5.49
6.79
7.98
Plate 2
Min Pressure
Time (µs)
(kPa)
-2.76E+07
5.81
-7.42E+06
6.14
-2.76E+07
6.27
-1.65E+07
6.45
-2.35E+07
6.36
-7.70E+05
6.16
-2.76E+07
6.37
-1.29E+07
6.69
When the tungsten plate material was paired up with an extremely low Hugoniot
parameter, a polyethylene plastic, a completely different reaction was seen. Referencing
trial 10 in the above table, a polyethylene plate was placed in front of the tungsten plate.
The tungsten plate reaches a minimum pressure at 5.49µs. The polyethylene plate
reaches a tensile stress at 6.16µs. Since the tungsten plate reached a minimum pressure
before the polyethylene plate did, it can be concluded that the minimum pressure in the
tungsten was due to the rarefaction wave between the two materials, not due to the free
surface.
Trial 10 - Plastic
6.00E+07
Tungsten
5.00E+07
Polyethylene
4.00E+07
Pressure (kPa)
3.00E+07
2.00E+07
1.00E+07
0.00E+00
0.00E+00
2.00E-03
4.00E-03
6.00E-03
8.00E-03
1.00E-02
1.20E-02
1.40E-02
1.60E-02
-1.00E+07
-2.00E+07
-3.00E+07
Time (ms)
Figure 15: Tungsten pressure reaches minimum before polyethylene
18
Figure 16: Trial 10 pressure wave in explosive
Figure 17: Pressure wave enters the polyethylene plate, rarefaction wave generated in tungsten
Figure 18: Pressure wave reaches free surface, rarefaction wave generated in tungsten
3.4 Three Plate Hydrocode Results
The addition of the third plate showed similar results to the two plate design. The
overall minimum pressure in the first plate decreased further when the rarefaction wave
had to travel through all three plates. In cases where the tungsten plate was paired with a
plate polyethylene, a similar reaction occurred as in the two plate configuration. The
tungsten plate reached a minimum pressure due to the rarefaction wave at the material
interface, not the free surface. The three plate configuration also showed that when the
19
tungsten plate was paired with an aluminum plate, the minimum pressure was generated
by the rarefaction wave at the material interface. When switching the tungsten plate to
the next highest density, steel, it was seen that the minimum pressure generated in the
steel was due to the free surface reaction, not the material interface.
Table 6: Three plate hydrocode results
Plate 1
Material
Trial
Min Pressure
Condition
(kPa)
13
Plastic
-1.65E+07
14
Plastic
-1.23E+07
15
Plastic
-2.38E+07
16
Plastic
-1.17E+07
17
Plastic
-8.47E+06
18
Plastic
-6.22E+06
19
Plastic
-7.70E+05
20
Plastic
-2.08E+07
Time
(µs)
6.83
8.06
6.71
8
12.6
5.49
6.79
7.98
Plate 2
Min Pressure
(kPa)
-1.58E+07
-7.57E+06
-4.42E+05
-2.15E+07
-1.94E+07
-1.71E+07
-1.65E+07
-1.65E+07
Time
(µs)
9.37
11.3
15
8.75
8.97
6.3
7.68
6.6
Plate 3
Min Pressure
(kPa)
-1.53E+07
-1.63E+07
-2.49E+06
-1.65E+07
-1.29E+07
-7.42E+06
-2.76E+07
-7.70E+05
Time
(µs)
7.76
10.74
13.4
7.45
7.72
6.79
7.44
7.28
Figure 19: Trial 15 pressure wave in explosive
Figure 20: Pressure wave enters the polyethylene plate, rarefaction wave generated in tungsten
20
Figure 21: Rarefaction wave generated in the polyethylene, pressure wave travels through second
tungsten plate
Figure 22: Rarefaction wave generated in second tungsten plate
21
4. CONCLUSIONS
TBD
22
5. REFERENCES
1. Cooper, Paul W. 1996, Explosive Engineering, Wiley-VCH, Inc., NY, pp 203-250
2. Antoun, Tarabay, Seaman, Lynn, Curran, Donald, Kanel, Gennady I., Razorenov,
Sergey V., Utkin, Alexander V. 2003, Spall Fracture, Springer-Verlag NY, pg. 142
3. Graff, Karl F. 1975, Wave Motion in Elastic Solids, Dover Publications Inc., NY,
pp. 378-379
4. Yellup, J.M., “The Computer Simulation of an Explosive Test Rig to Determine
the Spall Strength of Metals”, International Journal of Impact Engineering Vol.2,
No.2, 1984, pp.151-167
5. Rybakov, A.P., “Spall in non-one-dimensional shock waves”, International
Journal of Impact Engineering 24, May 2000, pp.1041-1082
6. Meyers, Marc., Dynamic Behavior of Materials, 1994, Wiley-Interscience
7. Hu, Lili, Miller, Phillip, and Wang, Junlan, “High strain-rate spallation and
fracture of tungsten by laser-induced stress waves”, Department of Mechanical
Engineering, University of Washington, 2008
8. Baumung, Kurt, Bluhm, Hansjoachim, Kanel, Gennady I., Muller, Georg,
Razorenov, Sergey V., Singer, Josef, Utkin, and Alexander V., “Tensile strength of
five metals and alloys in nanosecond load duration range at normal and
elevated temperatures”, International Journal of Impact Engineering 25, 2001,
pp. 631-639
9. Povarnitsyn, M.E., Khishchenko, and K.V., Levashov, P.R., “Simulation of shockinduced fragmentation and vaporization in metals”, International Journal of
Impact Engineering 35, 2008 pp. 1723-1727
10. Dobratz, B.M and Crawford, P.C., "LLNL Explosives Handbook", UCRL-52997
Rev.2 January 1985
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6. APPENDIX
24