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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

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 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 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

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 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

plates’behavior 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 [2–4]. Moreover, they have scrutinized the elas-

tic transformation margin and the plate’s 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 composites’attributes 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 s–1in 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 plate’s convex-

ity. The energy absorption for the transformation

caused by the convexity’s depth is obtained by mul-

tiplying material’s 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 plate’s curvature

There are various theoretical anticipations for plates’

curvature under the influence of impulse loads.

Johnson suggests a guideline to assess the plates’be-

havior in relation to strike loading using the non-di-

mensional number αj, Eq. (7).

αj¼ρpV2

σdð7Þ

Where: V is the plate’s speed, ρpis density of the

plate’s material and σdis the damage tension. John-

son’s damage number is only predictable when the

plates have similar dimensions. Johnson’s 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 plate’s dimensions and

loading parameter [2]. This parameter is like Eq. (9)

for circular plates. Using the damage parameter, the

ratio of circular plates’transformation 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 plate’s 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

ﬃﬃﬃ

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 material’s parameter (D

= 40 and q = 5 for steel).

By comparing the results of these scholars’re-

searches, it is clear that Nuric’s 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, Jones’anticipations

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–

Wilkins–Lee equation of state by Jones–Wilkins–

Lee is suggested which is able to accurately describe

the state of the explosive material of explosive

charge and is widely used. Jones–Wilkins–Lee 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 Jones–Wilkins–Lee 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 Jones–Wilkins–Lee 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

Jones–Wilkins–Lee 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 [21–23].

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 Jones–Wilkins–Lee 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

R1–4.15

R2–0.95

ω–0.3

A Mbar 3.712

B Mbar 0.03231

V0–1

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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 S1–2.56

Factor S2–1.986

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 plates’specifications

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 plates’response is the

plates’thickness. 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 Plate’s material The amount of the explosive

charge (g)

Plate’s 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

materials’free 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 border’s 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

Young’s module (MPa) 210

Poisson’s 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)

Plate’s

material

Plate’s

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 microsecond’s 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 plates’curvature.

6.1 The effects of changing the weight of explosive

charge

Figures 1–18 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 19–21 analyze the

Figure 15. Diagram of the transformation of the circular cop-

per plate’s 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 plate’s 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 plate’s center with 1.5 mm thickness under the loading of

explosive charges with different weights

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effect of thickness parameter on the diagram of

transformation of circular copper plate’s 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 metal’s

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 plate’s center with 2 mm thickness under the loading of

explosive charges with different weights

Figure 19. Diagram of transformation of circular copper

plate’s center with different thicknesses under the 1.5 g T.N.T.

loading

Figure 20. Diagram of transformation of circular copper

plate’s center with different thicknesses under the 0.5 g

T.N.T. loading

Figure 21. Diagram of transformation of circular copper

plate’s 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 24–27 also compare the maximum trans-

formation of the metal plates’center with different

materials in the experimental tests. In these dia-

Figure 22. Diagram of transformation of circular plate’s 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 plate’s 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 plate’s center

with 0.5 mm thickness by different explosive charges in differ-

ent materials

Figure 25. Maximum transformation of circular plate’s center

with 1 mm thickness by different explosive charges in differ-

ent materials

Figure 26. Maximum transformation of circular plate’s center

with 1.5 mm thickness by different explosive charges in differ-

ent materials

Figure 27. Maximum transformation of circular plate’s 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 28–30 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 aluminum’sis

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 plates’thicknesses.

δ

t

¼0:0194Φcþ6:1058 ð25Þ

δ

t

¼0:0298Φcþ4:1435 ð26Þ

Figure 28. Maximum experimental transformation of circular

plate’s center with different thickness and materials under the

loading of 0.5 g explosive charges

Figure 29. Maximum experimental transformation of circular

plate’s center with different thickness and materials under the

loading of 1 g explosive charges

Figure 30. Maximum experimental transformation of circular

plate’s 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 cases’results, 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. (24–26).

8 List of symbols

A the fixed parameter in Jones–Wilkins–Lee

equation of state which is determined experi-

mentally

AF theoretical amplification factor

AFreal real amplification factor

B fixed parameter in Jones–Wilkins–Lee equa-

tion of state which is determined experimen-

tally

C fixed parameter in Jones–Wilkins–Lee equa-

tion of state which is determined experimen-

tally

D explosion speed

E Young’s 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 Jones–Wilkins–Lee equa-

tion of state which is determined experimen-

tally

R2fixed parameter in Jones–Wilkins–Lee 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

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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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