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Experimental and numerical study of isotropic circular plates' response to underwater explosive loading, created by conic shock tube: Experimentelle und numerische Untersuchung zum Einfluss einer explosiven Belastung unter Wasser durch konische Stoßrohre auf isotrope runde Platten

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Today, to reduce the cost and increase the safety, test devices like conic shock tube have been widely used to investigate the underwater explosion phenomenon and its impact on structures. A shock tube is designed, manufactured and utilized in the mechanic of explosion laboratory of mechanic faculty of K.N. Toosi University of Technology to study the effect of isotropic metal plates' material in the present study. The source which creates shock in the utilized shock tube is an explosive material and the positive point is that in such a tube a high pressure is produced with a tiny explosive charge. In order to investigate the effect of the material and the geometry of the utilized metal plate, three materials are considered with two different thicknesses in the experimental tests. The behavior of the plate can be measured if the amount of the pressure produced by the explosive charge and the amount of plate's transformation is specified. To present a semi-experimental equation of the behavior of the plate which is under the explosion loading with the water interface in the experimental tests, numerical simulation is performed with LS-Dyna software. At the end, by combining the experimental and simulation results, the effect of the material and the thickness changes is studied specifically and an explosive charge is provided by adding weight parameter to anticipate the transformation of these metal plates.
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Experimental and numerical study of isotropic circular
platesresponse to underwater explosive loading,
created by conic shock tube
Experimentelle und numerische Untersuchung zum Einfluss einer
explosiven Belastung unter Wasser durch konische Stoßrohre
auf isotrope runde Platten
M. Heshmati1, J. Zamani A1, A. Mozafari2
Today, to reduce the cost and increase the safety, test devices like conic shock tube
have been widely used to investigate the underwater explosion phenomenon and its
impact on structures. A shock tube is designed, manufactured and utilized in the
mechanic of explosion laboratory of mechanic faculty of K.N. Toosi University of
Technology to study the effect of isotropic metal platesmaterial in the present
study. The source which creates shock in the utilized shock tube is an explosive
material and the positive point is that in such a tube a high pressure is produced with
a tiny explosive charge. In order to investigate the effect of the material and the geo-
metry of the utilized metal plate, three materials are considered with two different
thicknesses in the experimental tests. The behavior of the plate can be measured if
the amount of the pressure produced by the explosive charge and the amount of
plates transformation is specified. To present a semi-experimental equation of the
behavior of the plate which is under the explosion loading with the water interface in
the experimental tests, numerical simulation is performed with LS-Dyna software. At
the end, by combining the experimental and simulation results, the effect of the ma-
terial and the thickness changes is studied specifically and an explosive charge is
provided by adding weight parameter to anticipate the transformation of these metal
plates.
Keywords: Shock tube / underwater explosion / LS-Dyna / shock wave / explosion /
numerical simulation
Schlüsselwörter: Stoßrohr / Explosion unter Wasser / LS-Dyna / Schockwelle /
Explosion / numerische Simulation
1 Introduction
The underwater explosion phenomenon has been no-
ticed by many of the researchers and scholars for its
huge function in forming metals and floating struc-
Corresponding author: Mehran Heshmati, Modern Me-
tal Forming Laboratory, Department of Mechanical En-
gineering, K.N. Toosi University of Technology, No. 19,
Pardis St., Mollasadra St., P.O. Box 19395-1999, Teh-
ran, Iran, E-Mail: heshmati@dena.kntu.ac.ir
1Modern Metal Forming Laboratory, Department of
Mechanical Engineering, K.N. Toosi University of
Technology, No. 19, Pardis St., Mollasadra St., P.O.
Box 19395-1999, Tehran, Iran
2Department of Aerospace Engineering, K.N. Toosi
University of Technology, No. 1, East vafadar St., 4th
Tehranpars Square., P.O. Box 16765-3381, Tehran,
Iran
Mat.-wiss. u. Werkstofftech. 2017,48, No. 2 DOI 10.1002/mawe.201600578
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tures. The study of the physical phenomenon is per-
formed in three areas of experimental tests, analyti-
cal solution and numerical solution. It is impossible
to utilize the analytical solution due to the complex-
ity of equations of state and the geometry of studied
structures in the survey of the underwater explosion
phenomenon and its effects on the floating struc-
tures. Therefore, the studies are conducted the areas
of experimental tests and numerical solution. Due to
the complexity of the condition and to use the nu-
merical solution, it is needed to do the experimental
tests first.
To have specific conditions and facilities (like ex-
plosion pond with non-reflective walls), high costs
and the dangers of the explosion in the open envi-
ronment are the barriers against the area of experi-
mental tests. Therefore, utilizing the shock tube is a
logical solution to resolve the restrictions. The shock
tube provides advantages like repetitive and control-
lable front plats wave and parameters [1].
Some basic works are done in the area of plates
transformation under explosion and by analyzing the
previously performed experimental tests and the
platesbehavior in underwater explosion; they pro-
posed models for anticipating the constant transfor-
mation of the plate with air at one side and the plate
with water in both sides during underwater explo-
sion [24]. Moreover, they have scrutinized the elas-
tic transformation margin and the plates transforma-
tion speed. Other results present an equation to an-
ticipate the center displacement to the thickness of
sphere and rectangular plates ratio [5]. The effects of
underwater explosion on aluminum plates have been
studied experimentally and numerically [6]. Many
scholars have worked on presenting an energy-based
shock factor to describe the structure loading in un-
derwater explosion, one tried to give a new factor
[7]. In demonstrating the previous factors of spheri-
cal characteristics, the shock wave front and the rela-
tive location between the detonation and the struc-
ture were not considered. An inquiry was made with
another modeling of the simple plates subjected to
the initial invariable non-contact load by the
ANSYS/LS-DYNA software [8]. In the elastic solu-
tion, the simulation results are a good match for the
experimental data while the anticipated non elastic
transformation is half the amount in the experimental
tests which is due to the renewed loading.
As the experiment tool in this research is the
shock tube, it is necessary to address the results of
the studies conducted in this area. The shock tubes
are of three types:
Diaphragm tube
In its simplest form, the diaphragm tube includes an
invariable circular area which is divided to a stimu-
lant and ambulant parts. When the diaphragm is sud-
denly destroyed, a shock wave is created inside the
tube. This kind of tube is not used to propagate wave
in water [9].
Gas gun shock tube
This tube consists of a gas gun and a conic tube. Ex-
plosive charge causes a high pressure and the sudden
stroke of flyer plate which creates impulse in water
environment inside the conic area [10].
Explosive conic shock tube
The geometry of conic shock tube is designed to re-
present a part with a cone shaped head angel by a
radial extension caused by a blast of tiny explosive
sphere in open water.
A vast research was made to develop the under-
water explosion shock tubes. The shock tube is me-
chanized in a way that with a strike to the ambulant
plate, a shock wave is spread inside the conic case
which is full of water and then, its effects on the me-
tal plates are investigated [11]. The mechanical be-
havior and carbon composite fracture caused by the
explosion wave with diaphragm shock tube were
studied [12]. The primary conic shock tube which
creates shock wave using an explosive charge was
built. This machine was named the shock gun. Re-
searches demonstrated that with a tiny explosive
charge it can create a shock wave the creation of
which in free mode demands a thousand times big-
ger mass [13]. Various compositesattributes against
the shock wave applying the same kind of shock
tube were examined. A factor was introduced called
reinforcement theory factor and announced that there
is a huge difference between the real and the theore-
tical amount [1, 14, 15]. The effect of the shock tube
geometry was studied. At first was investigated the
effect of the angel and the length of the explosive
shock tube and then, by changing the weight of the
explosive charge, and its effect on the maximum
generated pressure. At the end, the amplification fac-
tor was modified and presented an equation to calcu-
late the equivalent mass [16].
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In the present paper, at first the performed experi-
mental tests are described and explained. In order to
investigate the effect of explosive loading on the me-
tal plates, a shock tube is utilized. According to the
results of the research, the geometry of the shock
tube is chosen. In order to examine the effect of the
material on the transformability of the circular
plates, 3 materials of aluminum, copper and steel are
chosen. Another parameter which is involved with
the transformability of the plates is thickness. The
results are also studied by considering different
thicknesses for each substance. Then, using the non-
linear LS-Dyna code, the phenomenon of explosion
under water in a shock tube is simulated. The results
of the simulation are compared with the results of
the experimental tests. The comparison shows a high
accuracy of the chosen numerical simulation method.
At the end, the results are combined with the results
of the experiments performed with different weights
to present an equation of transformation of plates ac-
cording to the weight of the used explosive charge.
2 Conic shock tube
If the shock tube wall is assumed rigid and limits the
extended pressure field, then a main explosive sphere
can be equal to a tiny conic part in the cone head,
Figure 1 [1]. If the shock waves generated by the
conic shock tube are compared with the waves gen-
erated by a similar explosive charge in an open en-
vironment, there will be important results which are
discussed in this essay. One of the outcomes reveals
that the shock wave produced by the conic shock
tube is much bigger than the one in the open envi-
ronment as discussed above. This increase in pres-
sure depends on the geometry of the cone. A theo-
retical amplification factor for a shock tube can be
calculated by dividing the explosive sphere volume
by a tiny conic volume of an explosive charge, the
result of which is called the amplification factor. In
another way, it is the explosive sphere weight in
open environment divided by the semi cone sphere
sector in shock tube which results equal pressure
theoretically. The amplification factor is shown in
Eq. (1) [17].
AF ¼1
sin2α=4ð1Þ
The experimental results show that the real ampli-
fication factor is much smaller than the theoretical
one. The investigation of the experimental and nu-
merical results leads to the correction of the theore-
tical amplification factor and by adding the correc-
tion factor to it, the real amplification factor is pre-
sented to calculate the generated pressure, Eq. (2).
This equation is used to find the equivalent mass
[16].
AFreal ¼0:21
sin2α=4ð2Þ
3 The basic equations
3.1 The basic equations of the experimental tests
3.1.1 The experimental equations of the explosion
underwater
The maximum pressure of the initial wave is given
in MPa by Cole and Swisdak in Eq. (3) [18, 19]:
Pmax ¼K1
W1
3
S
!
A1
ð3Þ
K1and A1are experimental constants the T.N.T.3
amounts of which are 52.16 and 1.16.
Figure 1. Amplifying the explosive charge in a conic shock
tube 3Trinitrotoluene
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Cole has also defined the impulse amount per sur-
face area (I) in kPa s1in Eq. (4).
I¼K3W1
3W1
3
S
!
A3
ð4Þ
K3and A3are the experimental invariants for T.N.T.
and its amounts are 5.76 and 0.891.
The scaling rules in explosion leads to defining
the non-dimensional parameters by which pressure
effects are calculated:
Z¼S=w1=3ð5Þ
Scaled Impulse ¼I=w1=3ð6Þ
In this equation W is equivalent mass of the T.N.T.
for the explosive charge in kg and S is the distance
from the center of the charge in m.
3.1.2 The plastic response of the plate
According to the conducted studies the absorbed en-
ergy during the dynamic plastic transformation of
the plate is proportionate to the caused square of the
curve of the plate. The plates transform between
sphere and conic forms during the underwater explo-
sion. When the absorbed energy for the sagittal
transformation is equal with the one for sphere trans-
formation, for conic and hyperbolic transformation
the absorbed energy is half the energy needed for
transforming the equal depth of the plates convex-
ity. The energy absorption for the transformation
caused by the convexitys depth is obtained by mul-
tiplying materials yield tension by the depth and in-
crease in surface. The profile of the transformation is
usually sphere like in static pressure. Transformation
profile changes from conic to sphere under the influ-
ence of underwater explosion loading if the sphere
pressure pulse is converted to the wave plate.
3.1.3 Anticipating the plates curvature
There are various theoretical anticipations for plates
curvature under the influence of impulse loads.
Johnson suggests a guideline to assess the platesbe-
havior in relation to strike loading using the non-di-
mensional number αj, Eq. (7).
αj¼ρpV2
σdð7Þ
Where: V is the plates speed, ρpis density of the
plates material and σdis the damage tension. John-
sons damage number is only predictable when the
plates have similar dimensions. Johnsons number
can be written with impulse term, Eq. (8).
αj¼Itot
A2t2ρpσdð8Þ
Where Itot is the total impulse, σdis damage ten-
sion and A is the area of the plate on which the im-
pulse is entered. σdis considered σyfor simplifica-
tion. The corrected damage parameter Φis defined
by Nurick and it includes plates dimensions and
loading parameter [2]. This parameter is like Eq. (9)
for circular plates. Using the damage parameter, the
ratio of circular platestransformation to their depth
is given in Eq. (10).
Φc¼Itot
πRt2ðρpσyÞ1=2ð9Þ
δ
t

c¼0:425Φcþ0:227 ð10Þ
Where: δis the displacement of the center of the
plate and t is the plates depth. For under water explo-
sion, impulse per unit area of the displacement shows
a linear equation with the main depth of the plate.
Moreover, the impulse needed for the liner rupture in-
creases with the main depth of the plate. This equation
was corrected based on experimental tests [20] to give:
δ
t

c¼0:541Φcþ0:433 ð11Þ
Eq. (11) shows that plates will have less curvature
for a reloading influence in an explosive depth less
than twice the distance from the center of the charge.
Predictions for the amount of the displacement by
the depth is given in Eq. (12) [5]:
δ
t

c¼0:817Φcð12Þ
For the circular plates, the Eq. (12) is modified as
Eq. (13) by considering the strain rate:
δ
t

c¼0:817 Φc
ffiffi
n
pð13Þ
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In which n is as follows:
n¼1þI2
3ρ2
pt2DR
ρp
3σy

1=2
!
1=q
ð14Þ
In which D and q are the materials parameter (D
= 40 and q = 5 for steel).
By comparing the results of these scholarsre-
searches, it is clear that Nurics results are less than
the experimental results because the renewed load-
ing effect has not been considered for them. When
the strain rate is not calculated, Jonesanticipations
have greater amounts than the real results. Other-
wise, the results will be better.
3.2 The Basic equations of simulation
3.2.1 The equation of state of explosive products
The equation of state has described the energy-vo-
lume-pressure behavior of explosion and is the most
effective element in accuracy of the calculations.
Many of equations of states are suggested in this re-
gard. Among the equations of states, the Jones
WilkinsLee equation of state by JonesWilkins
Lee is suggested which is able to accurately describe
the state of the explosive material of explosive
charge and is widely used. JonesWilkinsLee is an
equation of a standard state which is used to describe
the pressure-volume-energy behavior of the explo-
sive material worldwide and is almost used in all dy-
namic computing fluid codes (dynamic numerical
analyzing software) like ABAQUS, Autodyn, LS-
Dyna. For different explosive materials, the specific
factors of JonesWilkinsLee equation of state are
determined by adaption of the equation with the ex-
perimental C-J condition, the explosion speed and
pressure data. This data is usually taken from the cy-
linder test and is associated with high costs.
The JonesWilkinsLee function is an experimen-
tal equation of state. This equation is based on an
equation first suggested by Jones and Miller in 1984
and the one which was suggested by Wilkins in 1964.
These equations were presented by Lee in 1968. The
JonesWilkinsLee equation of state is used in LS-
Dyna software as it can be seen in Eq. (15):
P¼A1ω
R1V

eR1VþB1ω
R2V

eR2Vþω
VE
ð15Þ
Where: A, B, C, R1,R
2,ωare the fixed material
and P and V are non-dimensional pressure and vol-
ume respectively [2123].
The factors of equation of state parameters of the
T.N.T. explosive charge which is used in this study
are given in Table 1 [24].
3.2.2 Gruneisen equation of state
This equation of state gives pressure for compressed
material with cubic shock velocity- particle velocity
as follows [25]:
P¼
ρ0C2μ1þ1γ0
2

μa
2μ2
hi
1S11ðÞμS2
μ2
μþ1S3
μ3
μþ1ðÞ
2
"#
2
þγ0þaμðÞE
ð16Þ
And for expanded material:
P¼ρ0C2μþγ0þaμðÞEð17Þ
Where: C is the intercept of νsνpcurve, Sis are
the factors related to the slope of νsνpcurve, γ0is
the Gruneisen Gama and
μ¼ρ
ρ01ð18Þ
This equation of state is used in null material
model to simulate the water interface. Thus, the fac-
tors related to this equation are presented in Table 2
[24].
Table 1. The specifications of T.N.T explosive and the
coefficients used in JonesWilkinsLee equation of state
Invariant Unit Amount
Density g/cm31.63
Detonation speed D cm/μs 0.6930
Chapman-jouguet pressure Pcj Mbar 0.21
Specific energy per volume unit 0.07
R14.15
R20.95
ω0.3
A Mbar 3.712
B Mbar 0.03231
V01
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3.2.3 Linear polynomial equation of state
This equation of state is linear in internal energy on
E mass unit. In this equation, pressure is given as
follows:4
p¼C0þC1μþC2μ2þC3μ3þEðC4þC5μþC6μ2Þ
ð19Þ
Where: Cis are fixed elements defined by the user.
If μ< 0 is:
C2μ2¼C6μ2¼0ð20Þ
μ¼ρ
ρ01ð21Þ
Where: ρand ρ0are density and initial density re-
spectively [26].
This kind of equation has two parts in the software.
In first part, the linear polynomial factors and the
other part the initial thermodynamic state of the mate-
rial is determined. E0&V
0Parameters in the second
part define the internal energy on the reference speci-
fic volume and initial relative volume. This equation
of state is also used to model the ideal gas with Gam-
ma Law. This case is addressed as follows:
C0¼C1¼C2¼C3¼C6¼0C
4¼C5¼γ1
ð22Þ
Where: γis the specific heat factor. In fact, pres-
sure is presented in Eq. (23):
p¼γ1ðÞ
ρ
ρ0
Eð23Þ
4 Experimental tests
4.1 Shock tube geometry
In order to do the experimental tests, an explosive
shock tube with the length of 3.105 m is utilized.
This shock tube is conic with a 3.2° internal angel.
Schematic view of the shock tube is given in Fig-
ure 2. Inside the cone of the tube is filled with
water and a pressure sensor is placed in 0.5 m dis-
tance from the metal plate. The tube length is di-
vided to smaller parts due to the restrictions of
manufacture and these parts are linked together. A
schematic of the shock tube and its holding base is
shown in Figure 3. In order to access inside the
tube, a mechanism is designed so that an exit bore of
the stuffing wire is filled with specific paste, Fig-
ure 4. Moreover, the small diameter of the bore
makes no bad influences on the process of the test.
4.2 The utilized explosive charge
In order to do the experimental tests, three different
weights are considered for the explosive charge. The
explosive charges are equal to 0.5, 1 and 1.5 g
T.N.T. and are filled in aluminum covered capsules.
The charge is fixed with the paste at the end of the
Table 2. Coefficients used for water medium [24]
Amount Unit Invariant
Density g/cm31.025
4Pressure cut off Mbar 1e-6
μMbar μs 1.13e-11
C cm/μs 0./1480
Factor S12.56
Factor S21.986
Factor S30.2268
GAMAO 0.5
Primary internal energy Mbar 1.89e-6
Primary relative volume 1
Figure 2. The schematic view of the explosive shock tube
Figure 3. The real schematic view of the explosive shock
tube
4PC
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capsule. The utilized capsule and charge are illu-
strated in Figure 5.
4.3 The platesspecifications
Three different kinds of plates are used in these ex-
periments. By considering these three materials, one
can presents an equation for the loading effect of the
explosive load in the shock tubes on the transforma-
tion amount. Copper, aluminum and steel are three
different utilized materials. Mechanical specifica-
tions of each kind which are resulted from the ten-
sion test are given in Table 3. Another factor which
has influenced the metal platesresponse is the
platesthickness. For the copper and aluminum
plates four different thicknesses of 0.5, 1, 1.5 and
2 mm are considered. Moreover, for steel plates
three different thicknesses of 0.5, 1 and 1.5 mm are
regarded. Figure 6 illustrates a view of a copper
plate before the loading. Table 4 gives the specifica-
Figure 4. The location of explosive charge and charge output
Figure 5. The capsule and charge used for explosive charge
Table 3. Mechanical properties of the used metal plates in the experiments
Row Material Density kg/m3Relative
elongation
% A50
Final strength
Rm(MPa)
Strength proof
0.2 % Offset Rt
(MPa)
Surface area
(mm2)
1 Copper 8890 44.5 156 140 12.05
2 Aluminum 2720 4 175 139 12.06
3 Steel (St12) 7872 46 320 198 10.46
Figure 6. Circular copper plate before the loading
Tabelle 4. Designed and performed tests
Row Plates material The amount of the explosive
charge (g)
Plates thickness (mm) Number
of the tests
1 Copper 3 weights (0.5, 1.0, 1.5) 4 thicknesses (0.5, 1.0, 1.5, 2.0) 12
2 Aluminum 3 weights (0.5, 1.0, 1.5) 4 thicknesses (0.5, 1.0, 1.5, 2.0) 12
3 Steel 3 weights (0.5, 1.0, 1.5) 3 thicknesses (0.5, 1.0, 1.5) 9
Total number of the
tests
33
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tions of the designed experiments. The total number
of the experiments which are done in this regard is
33.
5 Simulation
5.1 Discretization method
Traditionally, the two main methods of Lagrangian
and Eulerian are used in numerical solution techni-
ques and each can be chosen according to their own
advantages, disadvantages and the anticipated beha-
vior of the material under the analysis. In this re-
search, the Lagrangian method is used to model the
shock tube and the metal plate. In Lagrangian view-
point, the coordinate system depends on the material
such that it moves and circulates with the material.
This viewpoint is the basis for the limited classic ele-
ments analysis. Accordingly, the mesh is the repre-
sentative of the material during the analysis and each
element follows the movement of the material indi-
vidually. The main advantages of this viewpoint are
that the elements are the representative of material
volumes and spots and that the outer surfaces of the
border elements can be considered the materials
outer surface, the geometrical borders of different
materials can be determined by a total outer surfaces
of the martials and accurately define the contact sur-
faces of different parts [27].
In order to simulate the fluids, ALE method is
used. This method is a combination of Eulerian and
Lagrangian and includes the advantages of both
methods. There is no need to present the contact sur-
faces between the materials. Like Eulerian method
there is the possibility to simulate huge transforma-
tions. As in this method the mass flow passes
through the problem mesh, the anticipation of every
materialsfree surfaces is not as easy as Lagrange
solution. To solve this problem, it is necessary to
make the mesh finer.
It is necessary to have a systematized solution
therefore, this methods is more time consuming than
the other methods. This method is both mono mate-
rial and poly material and here ALE poly material
solution according to the material behavior is ap-
plied. Each element is specifically for a material in
mono material solution. In this method, if a sphere is
meshed with quadrangular mesh, the square helps to
create the sphere and the sphere border is drawn by
them so the sphere border is not curved. As a result,
the bigger are these quadrangular elements, the less
is the accuracy of meshed form. In poly material
method, an element is filled with many different ma-
terials at the same time. This means that in this
method, a sphere can be filled with quadrangular
elements such that the sphere borders shape does
not change [28].
5.1.1 The utilized equation factors
The numerical simulation is done through three gen-
eral levels in the limited component method. In the
first level, the geometrical model consist of the ex-
plosive charge, water, steel case and the air around
the model is simulated by the LS-Prepost. The simu-
lated model and its mesh are shown in Figure 7.In
the second level, the non-linear dynamic analysis
and then in the third level, post analyzing of the ana-
lysis is done by the LS-Prepost to interpret the re-
sults. To simulate the explosive charge the high ex-
plosive burn model and the Jones-Wilkins-Lee equa-
tion of state are utilized and its factors are presented
in Table 1. Null model and Gruneisen state are used
for water interface and its factors are presented in
Figure 7. The simulated shock tube by axial symmetry
Table 5. The specifications of the steel used in the body of
the shock tube [29]
Mechanical
properties
Density (kg/m3) 7850
Yield stress (MPa) 350
Youngs module (MPa) 210
Poissons ratio 0.3
Elongation 28 %
Plastic model
cowper symonds
D
q
6400
0.25
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Table 2. The specifications of the steel used in the
body of the shock tube is given in Table 5. The si-
mulation is done two- dimensionally with a 1/2 axial
symmetry.
6 Results and discussion
The results of the experimental tests and the numer-
ical simulation along with the simulation errors are
given in Table 6. As the final goal of the analysis is
to investigate the transformation of the metal plates
under the influence of the explosive load and present
an equation to anticipate the results, the experiments
are designed to avoid any rupture. According to the
results, no failure has happened in other experiments
except for one which is shown in Figure 8. More-
over, Figures 9 to 12 illustrate some of the metal
plates after loading.
The comparison between results of the experimen-
tal tests and the simulation confirmed the modeling
process. The process of shock wave dissemination
Table 6. The results of simulation and experimental tests
Test
num
Explosive
material
amount
(gr)
Plates
material
Plates
thickness
(mm)
Z I (Pa.s) Deformation
of the center
of plate
(simulation)
Deformation
of the center
of plate
(experimental)
error Φ
1 0.5 Copper 0.5 38.11 2295.1 21.9 24 8.75 871.78
2 1 0.5 30.25 3820 31.5 30.6 2.94 1451.01
3 1.5 0.5 26.43 5282.8 38.4 34.73 10.57 2006.65
4 0.5 1 38.11 2295.1 15.9 16.92 6.03 217.95
5 1 1 30.25 3820 21.4 26.68 19.79 362.75
6 1.5 1 26.43 5282.8 25.9 30.29 14.49 501.66
7 0.5 1.5 38.11 2295.1 15.4 15.27 0.85 96.86
8 1 1.5 30.25 3820 21.7 23.61 8.09 161.22
9 1.5 1.5 26.43 5282.8 25.6 29.02 11.78 222.96
10 0.5 2 38.11 2295.1 13.1 9.32 40.56 54.49
11 1 2 30.25 3820 18.9 17.61 7.33 90.69
12 1.5 2 26.43 5282.8 22.9 23.92 4.26 125.42
13 0.5 Aluminium 0.5 38.11 2295.1 18.8 15.7 19.75 1582.62
14 1 0.5 30.25 3820 27.6 23.75 16.21 2634.13
15 1.5 0.5 26.43 5282.8 fail fail 3642.82
16 0.5 1 38.11 2295.1 15.3 13.22 15.73 395.65
17 1 1 30.25 3820 20.08 19.66 2.14 658.53
18 1.5 1 26.43 5282.8 26.6 25.33 5.01 910.71
19 0.5 1.5 38.11 2295.1 11.8 9.94 18.71 175.85
20 1 1.5 30.25 3820 19 18.79 1.12 292.68
21 1.5 1.5 26.43 5282.8 25.1 21.54 16.53 404.76
22 0.5 2 38.11 2295.1 9.13 7.27 25.58 98.91
23 1 2 30.25 3820 15.2 12.91 17.74 164.63
24 1.5 2 26.43 5282.8 18.9 16.81 12.43 227.68
25 0.5 Steel 0.5 38.11 2295.1 14.3 12.8 11.72 779.46
26 1 0.5 30.25 3820 22.1 22.26 0.72 1297.34
27 1.5 0.5 26.43 5282.8 27.3 25.44 7.31 1794.13
28 0.5 1 38.11 2295.1 10.2 9.05 12.71 194.86
29 1 1 30.25 3820 15.4 16.03 3.93 324.34
30 1.5 1 26.43 5282.8 21.7 18.34 18.32 448.53
31 0.5 1.5 38.11 2295.1 8.23 7.02 17.24 86.61
32 1 1.5 30.25 3820 12.9 12.11 6.52 144.15
33 1.5 1.5 26.43 5282.8 19.8 17.24 14.85 199.35
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along the shock tube is shown in Figure 13. The
analysis of shock wave dissemination shows that the
shock wave turns to a flat wave in a short time.
With the help of the charges, explosive material
turn to high pressure gasses after the explosion. Due
to the closed environment, these gasses are sur-
rounded inside the tube and this leads to the high
pressure inside the tube. Compared with the pressure
resulting from the shock wave, one can betake this
pressure. However, this little pressure causes a form
of transformation in plates with less thickness.
Figure 8. 0.5 mm aluminum plate rupture during loading 1.5 g
T.N.T.
Figure 9. Transformation of 1.5 mm steel plate during loading
1.5 g T.N.T.
Figure 10. Transformation of 1.5 mm aluminum plate during
loading 1 g T.N.T.
Figure 11. Transformation of 0.5 mm copper plate during
loading 1 g T.N.T.
Figure 12. Transformation of 1.5 mm copper plate during
loading 1 g T.N.T.
Figure 13. Shock wave dissemination inside the shock tube
with 400 microseconds intervals
Figure 14. The range of fluids and plate transformation at the
end of the test (0.5 g explosive charge and copper plate with
1.5 mm thickness)
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The effect of the pressure which is caused by the
explosion gasses is discussed in the next part. Fig-
ure 14 illustrates the range of these gasses and the
other fluids at the end of the explosion. Firstly
here, the effective factors on the transformation are
discussed distinctly and then, combining these fac-
tors leads to the anticipation of platescurvature.
6.1 The effects of changing the weight of explosive
charge
Figures 118 address the effect of changing the
weight of explosive charge on the diagram of trans-
formation of copper circular plate with different
thicknesses. As it is expected, with the increase in
the weight of the explosive charge, the loading
amount on the plate increases therefore, the maxi-
mum transformation of the center of the plate in-
creases. The investigation of most diagrams shows
that by increasing the thickness of the plate, the
transformation diagram changes and instead of being
a two level with small thicknesses, it becomes a sin-
gle level with bigger thicknesses. As mentioned be-
fore, the reason is the pressure which is caused by
the outcome gasses of the explosion. This gasses are
rapidly expanded and because the explosion area is
sealed, it causes a pressure inside the shock tube.
Compared with the shock wave pressure, this pres-
sure is less and negligible however, this pressure
causes a primary transformation in the thinner plates
of 0.5 and 1 mm before the shock wave hits. More-
over, the study of Figure 15, shows that by the in-
crease in the explosive charge, the primary transfor-
mation increases. The reason of this phenomenon is
justified by the amount of the gas generated by the
explosive charges of different weights. By compar-
ing Figures 15 and 16 one can find that the lower
the thickness is, the faster the transformation would
take place. This means that with the increase in
thickness, the expansion of the gas must reach a
point that the thickness is yielded to the created pres-
sure by the volume increase. Figures 17 and 18
clearly illustrate that more thickness increase causes
the lower pressure caused by other gasses such that
they are unable to make the primary transformation
in the metal plate.
6.2 The influence of thickness changes
As it is expected, by the increase in the thickness of
the plate its resistance against the loading increases
which results in the general decrease in the transfor-
mation. The diagrams in Figures 1921 analyze the
Figure 15. Diagram of the transformation of the circular cop-
per plates center with 0.5 mm thickness under the loading of
explosive charges with different weights
Figure 16. Diagram of the transformation of the circular cop-
per plates center with 1 millimeter thickness under the load-
ing of explosive charges with different weights
Figure 17. Diagram of the transformation of the circular cop-
per plates center with 1.5 mm thickness under the loading of
explosive charges with different weights
M. Heshmati, J. Zamani A, A. Mozafari Mat.-wiss. u. Werkstofftech. 2017,48, No. 2
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effect of thickness parameter on the diagram of
transformation of circular copper plates center over
time. The study of these diagrams revealed unex-
pected controversial points. The final transformation
which is resulted by loading in a specific weight of
an explosive charge are very close to each other in 1
and 1.5 mm thicknesses while it was expected that
the transformation of the 1 mm plate would be very
different from the 1.5 mm plate.
In order to explain this phenomenon considerate
is necessary to consider the primary pressure which
is caused by the increase of the gas volume of the
explosion. As the thickness increases the primary
transformation begins in a longer period of time and
after a certain point it cannot lead to the transforma-
tion of the initial plate. The fact that the transforma-
tion of the plates with 1 and 1.5 thicknesses are close
has the same reason. When the primary pressure is
unable to transform the initial metal plate, it is added
to the shock wave pressure which causes the loading
to strike the metal plate with more pressure. Loading
with more pressure causes a bigger transformation
such that the transformation of 1.5 thickness ap-
proaches 1 millimeter thickness.
6.3 The influence of material changes
A principle reveals that with the increase in a metals
yields resistance, the transformation of a fixed load-
ing decreases. It should be noted that the transforma-
tion is different in higher loading rates. Higher rate
loading causes a phenomena which is called stiffness
strain and makes the yield resistance increase. Fig-
ures 22, 23 illustrate the same subject. Studying
these figures shows that because of the higher yield
stress, the transformation of steel plate is less than
the other two plates. Yield stress in aluminum and
copper is close therefore, it should be noted that
due to their different stiffness strain, the dynamic
Figure 18. Diagram of the transformation of the circular cop-
per plates center with 2 mm thickness under the loading of
explosive charges with different weights
Figure 19. Diagram of transformation of circular copper
plates center with different thicknesses under the 1.5 g T.N.T.
loading
Figure 20. Diagram of transformation of circular copper
plates center with different thicknesses under the 0.5 g
T.N.T. loading
Figure 21. Diagram of transformation of circular copper
plates center with different thicknesses under the 1 g T.N.T.
loading
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yield stress of these materials is different such that
it is more in aluminum than copper. Another point
that can be inferred from these diagrams is from the
primary transformation of the plates. Due to the
high yield resistance of steel, the primary transfor-
mation is not observed however in the other two
materials, this transformation can be seen.
Figures 2427 also compare the maximum trans-
formation of the metal platescenter with different
materials in the experimental tests. In these dia-
Figure 22. Diagram of transformation of circular plates center
with different materials and the 1.5 mm thickness under the
1 g T.N.T. loading
Figure 23. Diagram of transformation of circular plates center
with different materials and the 1.5 mm thickness under the
0.5 g T.N.T. loading
Figure 24. Maximum transformation of circular plates center
with 0.5 mm thickness by different explosive charges in differ-
ent materials
Figure 25. Maximum transformation of circular plates center
with 1 mm thickness by different explosive charges in differ-
ent materials
Figure 26. Maximum transformation of circular plates center
with 1.5 mm thickness by different explosive charges in differ-
ent materials
Figure 27. Maximum transformation of circular plates center
with 2 mm thickness by different explosive charges in differ-
ent materials
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grams, the thickness is fixed and the effect of the
weight parameter is studied along with the effect of
material changes on the maximum persistent trans-
formation of the center of plates.
In Figures 2830 the maximum experimental per-
sistent transformation of the center of the plate with
different thicknesses and materials are presented.
They illustrate the fact that with the increase in the
thickness, the persistent transformation of the plates
decreases. The slope of this decline in thickness
changes from 1 to 1.5 mm is less than the one in
thickness changes from 0.5 to 1 and 1.5 to 2 mm.
Moreover, the persistent transformation of copper
plate is more than aluminum and the aluminumsis
more than steel.
In order to anticipate the maximum transformation
of the center of the plate the modified damage para-
meter is utilized. This non-dimensional parameter is di-
rectly related with non-dimensional parameter of trans-
formation ratio to the thickness of the plate. It is useful
to find the linear equation of this relation to anticipate
the curvature which is caused by a specific amount of
the explosive charge. Figure 31 is plotted by utilizing
the transformations of copper plates in experimental
tests and simulations. The linear equation obtained by
the interpolation of the spots is given in Eq. (24).
δ
t
¼0:0364Φcþ7:6436 ð24Þ
Similar diagrams are plotted for aluminum and
steel. For the diagrams of aluminum and steel are
shown in Figures 32, 33 respectively and after inter-
polation between the spots in these diagrams,
Eq. (25) and Eq. (26) are given to anticipate the non-
dimensional parameter in relation to the transforma-
tion of aluminum and steel platesthicknesses.
δ
t
¼0:0194Φcþ6:1058 ð25Þ
δ
t
¼0:0298Φcþ4:1435 ð26Þ
Figure 28. Maximum experimental transformation of circular
plates center with different thickness and materials under the
loading of 0.5 g explosive charges
Figure 29. Maximum experimental transformation of circular
plates center with different thickness and materials under the
loading of 1 g explosive charges
Figure 30. Maximum experimental transformation of circular
plates center with different thickness and materials under the
loading of 1.5 g explosive charges
Figure 31. Changing the ratio of the deflection to the thickness
as a function of dimensionless number Φfor the copper plate
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7 Conclusion
1. With the change in the thickness, the maximum
transformation of the metal plate changes line-
arly.
2. When the thickness reaches more than 1 milli-
meter, the primary transformation of the thinner
plates is reduced and it is disappeared in thicker
plates. More investigations indicated that the in-
itial transformation that has happened before the
shock wave caused by hydrostatic pressure is due
to the expansion of explosive materials.
3. As the thickness of the plate increases, the re-
quired energy for transformation increases as
well. If this energy is stable, the observed trans-
formation decreases.
4. Unlike the last casesresults, when the thickness
changes from 1 mm to 1.5 mm it makes some-
how similar transformations the reason of which
is discussed earlier.
5. Although the static yield tension of the copper
and aluminum utilized in this research is close,
they have different density and dynamic yield
tensions which leads to more transformations of
the copper plate.
6. The maximum transformation of the center of the
plate is anticipated by the modified damage
dimensionless parameter which is given in
Eq. (2426).
8 List of symbols
A the fixed parameter in JonesWilkinsLee
equation of state which is determined experi-
mentally
AF theoretical amplification factor
AFreal real amplification factor
B fixed parameter in JonesWilkinsLee equa-
tion of state which is determined experimen-
tally
C fixed parameter in JonesWilkinsLee equa-
tion of state which is determined experimen-
tally
D explosion speed
E Youngs Module
E0internal energy per specific reference vo-
lume unit
fð_
εÞparameter of sensitivity to strain rate
m meter
MPa mega pascal
Kg kilogram
P pressure
Pcj chapman jouguet pressure
Pmpressure peak
R1fixed parameter in JonesWilkinsLee equa-
tion of state which is determined experimen-
tally
R2fixed parameter in JonesWilkinsLee equa-
tion of state which is determined experimen-
tally
V volume
V0relative primary volume
S distance from the center of the charge
Sifactors related to the slope of the curve
νsνp
W explosive charge weight
Z non-dimensional number to scale the explo-
sion
ϑPoisson factor
Figure 32. Changing the ratio of the deflection to the thick-
ness as a function of dimensionless number Φfor the alumi-
num plate
Figure 33. Changing the ratio of the deflection to the thick-
ness as a function of dimensionless number Φfor the steel
plate
M. Heshmati, J. Zamani A, A. Mozafari Mat.-wiss. u. Werkstofftech. 2017,48, No. 2
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Greek Symbols
αcone head angle of the shock tube
_
εstrain rate
γspecific heat factor
Φmodified damage parameter
σYstatic yield strength
σYd dynamic yield strength
sμmicroseconds
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Received in final form: July 22nd 2016 T 578
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