PUBSY JRC41337 - 2007

Simulation of the Effects of an Air Blast Wave

Martin Larcher

a) t=2e-5 b) t=6e-5

c) t=1e-4 d) t=1.4e-4

The Institute for the Protection and Security of the Citizen provides researchbased, systems-oriented support to EU policies so as to protect the citizen against economic and technological risk. The Institute maintains and develops its expertise and networks in information, communication, space and engineering technologies in support of its mission. The strong crossfertilisation between its nuclear and non-nuclear activities strengthens the expertise it can bring to the benefit of customers in both domains. European Commission Joint Research Centre Institute for the Protection and Security of the Citizen Contact information Address: Martin Larcher, T.P. 480, Joint Research Centre, I-21020 Ispra, ITALY E-mail: martin.larcher@jrc.it Tel.: +390332789004 Fax: +390332789049 http://ipsc.jrc.ec.europa.eu http://www.jrc.ec.europa.eu Legal Notice Neither the European Commission nor any person acting on behalf of the Commission is responsible for the use which might be made of this publication. A great deal of additional information on the European Union is available on the Internet. It can be accessed through the Europa server http://europa.eu/ JRC 41337 ISSN 1018-5593 Luxembourg: Office for Official Publications of the European Communities European Communities, 2007 Reproduction is authorised provided the source is acknowledged Printed in Italy

Distribution List Lechner S. Anthoine A. Casadei F. Dyngeland T. Gradin M. Giannopoulos G. Gutierrez E. Halleux J.P. Larcher M. Paffumi E. (JRC Petten) Pegon P. Solomos G. DG Tran External: Mr. Bung H. (CEA) Faucher V. (CEA) Galon P. (CEA) Kill N. (Samtech) Cheruet A. (Samtech) Potapov S. (EDF)

S. Lechner

The information contained in this document may not be disseminated, copied or utilized without the written authorization of the Commission. The Commission reserves specifically its rights to apply for patents or to obtain other protection for the matter open to intellectual or industrial protection.

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CONTENTS 1 Introduction ...................................................................................................................................5 2 Air Blast Waves.............................................................................................................................6

2.1 Introduction..........................................................................................................................6 2.1.1 Detonations ......................................................................................................................6 2.1.2 Air Blast Waves ...............................................................................................................7

2.2 Literature Data .....................................................................................................................8 2.2.1 Pressure-Time Distribution ..............................................................................................8 2.2.2 Maximum / Minimum Pressure .....................................................................................10 2.2.3 Impulse...........................................................................................................................11 2.2.4 Negative Phase ...............................................................................................................11 2.2.5 Wave Form Parameter ...................................................................................................13 2.2.6 Shock Front Velocity .....................................................................................................16 2.2.7 Specific Heat Ratio ........................................................................................................16

3 Numerical Loading of a Structure with Air Blast Waves............................................................18 4 Investigations with Explosives as a Charge (Solid TNT) ...........................................................20

4.1 Modelling of the explosive ................................................................................................20 4.2 Behaviour in the Explosive ................................................................................................24 4.3 Cone with two Symmetry Axes .........................................................................................29 4.4 Cubic Charge with two Symmetry Axes............................................................................38 4.5 Spherical Charge with two Symmetry Axes ......................................................................42 4.6 Comparison between the Different Models .......................................................................46

4.6.1 Maximum Pressure.........................................................................................................46 4.6.2 Impulse...........................................................................................................................47 4.6.3 Arrival Time...................................................................................................................49 4.6.4 Positive Phase Duration .................................................................................................49 4.6.5 Comparison with results of other authors ......................................................................50

4.7 Influence of several parameters .........................................................................................50 4.7.1 Specific heat ratio (CON15) ..........................................................................................50 4.7.2 Values for , E0, .........................................................................................................51 4.7.3 Parameters for the explosive ..........................................................................................51 4.7.4 Burn mass fraction .........................................................................................................51

5 Bubble model...............................................................................................................................52 6 Control Volume...........................................................................................................................58

6.1 Flux between the CL3D and the fluid element ..................................................................58 6.2 Several models ...................................................................................................................58

7 Implementation of an Air Blast Loading Function .....................................................................62 7.1 Motivation ..........................................................................................................................62 7.2 Used Function ....................................................................................................................62 7.3 Implementation ..................................................................................................................62 7.4 Verification with Examples................................................................................................63

8 Mesh generation for LS-DYNA ..................................................................................................64 9 References ...................................................................................................................................65 10 Apendix ..................................................................................................................................67

10.1 EUROPLEXUS Code ........................................................................................................67 10.2 Miscellaneous code ............................................................................................................71 10.3 Sample input files...............................................................................................................74

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1 Introduction This work is being conducted in the framework of the project RAILPROTECT, which deals with

the security and safety of rail transport against terrorist attacks. The bombing threat is only

considered, and focus is placed on predicting the effects of explosions in railway and metro stations

and rolling stock and on assessing the vulnerability of such structures.

The project is based on numerical simulations, which are carried out with the explicit Finite

Element Code EUROPLEXUS that is written for the calculation of fast dynamic fluid-structure

interactions. This program has been developed in a collaboration of the French Commissariat

l'Energie Atomique (CEA Saclay) and the Joint Research Centre of the European Commission (JRC

Ispra).

As the aim of this project is to calculate the behaviour of structures under a loading produced by air

blast waves, an indispensable starting point in this study is the ability to simulate the generation of

such waves from a given quantity of explosive, and to follow their propagation through 3D spaces

as they finally impinge onto the structures under consideration.

The results of such numerical tests of free air blasts are presented in this report and are compared to

experimental data available in the literature. In the absence of such data the EUROPLEXUS results

are compared to the results of other codes, in particular to LS-DYNA, which is run in collaboration

with the University of Karlsruhe. This analysis is preceded by an exposition of some basic concepts

on blast wave characteristics, explosives, and a description of the equation of state adopted herein

for the modelling of the detonation of a solid explosive.

6

2 Air Blast Waves

2.1 Introduction

2.1.1 Detonations

Explosions can be distinguished in detonations and deflagrations. The difference between detona-

tions and deflagrations is the velocity of the reaction zone in the explosive. Deflagrations have a

slower reaction zone than the sound speed. Examples for deflagrations are the burning of gas-air-

mixtures and slow explosives like gun powder.

Detonations have a faster reaction zone than the sound speed. The most common explosives react

with detonations.

To compare different explosives the TNT equivalent can be used. The TNT equivalent is a method

for quantifying the energy released in the detonation of an explosive substance, by comparing it to

that of an equal quantity of TNT. It is known that 1 kg TNT releases the energy of 4.520x106 J.

The TNT equivalent is available for standard explosives and for some of them it is summarized in

Table 1.

Explosive Mass Specific

energy [kJ/kg]

TNT

Equivalent

TNT 4520 1

Torpex 7540 1.667

Semtex 1A 4980 1.102

C4 6057 1.34

Table 1: TNT equivalent for different explosives

The effects of an explosion can be distinguished in three ranges:

Contact detonation: The explosive is in contact with the loaded material. The load-time function depends on the loaded material, which, in most cases, is destroyed. Occurrences are the blasting

of concrete (demolition etc.) or terrorist attacks where the explosive is located directly on the

structure.

Near zone of the explosion: In most cases he material is also directly damaged like in the contact zone.

7

Far zone. The blast wave resulting from the detonation dominates the effects on humans and structures.

The size of all these zones depends on the quantity of the explosive charge.

Additional parameters for a detonation, depending on the size of the explosive, can be defined. For

example, the radius in which debris from the explosion (not from the blast wave) are possible is

given by Kinney [15] as

1345r W= (1)

where, r is expressed in m and W is the TNT equivalent of the explosive in kg.

2.1.2 Air Blast Waves

The pressure that arrives at a certain point depends on the distance and on the size of the explosive.

p

p

p

p

0

min

max

ta

t

td tn

Figure 1: Pressure-time curve for a free air blast wave

The main characteristics of the development of this pressure wave are the following:

- The arrival time ta of the shock wave to the point under consideration. This includes the time

of the detonation wave to propagate through the explosive charge.

- The peak overpressure pmax . The pressure attains its maximum very fast (extremely short

rise-time), and then starts decreasing until it reaches the reference pressure po (in most cases

the normal atmospheric pressure).

- The positive phase duration td, which is the time for reaching the reference pressure. After

this point the pressure drops below the reference pressure until the maximum negative

8

pressure pmin. The duration of the negative phase is denoted as tn.

- The incident overpressure impulse, which is the integral of the overpressure curve over the

positive phase td.

The idealised (free air blast) form of the pressure wave of Figure 1 can be greatly altered by the

morphology of the medium encountered along its propagation. For instance, peak pressure can be

increased up to 8 times if the wave is reflected on a rigid obstacle. The effects of the reflection

depend on the geometry, the size and the angle of incidence. By setting 1.4 = (ratio of specific heats of air), it can be shown that the reflected overpressure pr is

0 maxmax0 max

7 427rp pp pp p

+= + (2)

All parameters of the pressure time curve are normally written in terms of a scaled distance

3

= dZW

(3)

where W is the mass of the explosive charge and d the distance to the centre of the charge.

2.2 Literature Data

2.2.1 Pressure-Time Distribution

There are available in the literature several pressure-time-curves for different kinds of explosions.

The effects of nuclear explosions here should be disregarded.

The pressure at a known point can be described by the modified Friedlander equation (from Baker

[2]) and depends on the time t from the arrival of the pressure wave at this point ( 0= at t t )

0 max( ) 1 = +

d

btt

d

tp t p pt

(4)

The other parameters involved are the atmospheric pressure p0, the maximum overpressure pmax and

the duration of the positive pressure td. The parameter b describes the decay of the curve. It can be

calculated with a known minimum pressure after the positive phase. Alternatively, the parameter b

can be calculated with the knowledge of the impulse. This will be done in chapter 2.2.5.

All parameters for the pressure-time curve can be taken from different diagrams and equations

(Baker [2], Kinney [15], Kingery [14], see e.g. Figure 2).

9

Figure 2: Model of Kingery [14] with scaled distances

10

2.2.2 Maximum / Minimum Pressure

Kingery [14] developed in 1984 curves for the description of the different air blast parameters by

using a rich body of experimental data, which had been properly homogenised. The parameters are

presented in double logarithmic diagrams with the scaled distance Z as abscissa, but are also

available as polynomial equations. These diagrams and equations enjoy the greatest overall

acceptance and are widely used as reference by most researchers. The parameters are also

implemented in different computer programs that can be used for the calculation of air blast wave

values. e.g. they are implemented in ConWep a program developed from the US-Army that

calculates conventional weapons effects. The same curves are also used for an easy air blast load

model (*LOAD_BLAST) in LS-DYNA. Also in [14] curves are provided for reflection effects

(surface burst of hemispherical charges) and free air conditions (spherical charge).

Another equation has been proposed by Kinney [15], in which the overpressure-distance relation for

chemical explosions can be written as

2

max2 2 2

0

808 14.5

1 1 10.048 0.32 1.35

Zpp Z Z Z

+ = + + +

(5)

Figure 3 shows the small differences between the two models.

Figure 3: Difference of the model of Kingery and the model of Kinney with 1kg TNT

11

2.2.3 Impulse

The impulse of the air blast wave has a big influence on the response of the structures. The impulse

is defined here as the area under the pressure time curve with the unit of pressure*sec. The impulse

can be calculated with (Kinney [15])

4

2 43

0.067 1 ( / 0.23)

1 ( /1.55)

ZI

Z Z+= + (6)

Another possibility is the polynomial equation of Kingery [14]. The comparison of the impulse

resulting from both equations shows that the equation of Kinney simplifies the curve of the impulse

between a scaled distance of 0.5 and 1.5 m/kg1/3.

0

200

400

600

800

0 0.5 1 1.5 2 2.5 3

scaled distance [m/kg1/3]

impu

lse

[Pa

sec] Kinney

Kingery

Figure 4: Different equations for the impulse (Kinney [15] and Kingery [14])

2.2.4 Negative Phase

Detonations produce an overpressure peak, and afterwards the pressure decreases and drops below

the reference pressure (generally the atmospheric pressure). The influence of the so-called negative

phase depends on the scaled distance. For scaled distances Z larger than 20 and especially for Z

larger than 50 the influence of the negative phase can not always be neglected. The size of the

positive impulse and of the negative impulse is then nearly the same. If the structure can react

12

successfully to the positive pressure but is more sensitive to a negative pressure, failure of parts of

the structure can result from this negative pressure phase (see Krauthammer [16]). However, in

several cases the negative phase is neglected e.g. in the air blast function of the CONWEP-Code.

Smith [22] presents the following equation to calculate the value of the negative pressure

5min0.35 10 Pa for 1.6p Z

Z= > (7)

The duration time of the negative pressure minp can be calculated with

1/30.00125 [sec]nt W= (8) Another possibility to get these parameters is a diagram (see Figure 5) in Krauthammer [16]. By

using this diagram the limitation of equation (7) can be overcome by assuming

5

min

4min

0.3510 Pa for 3.5

10 Pa for 3.5

p ZZ

p Z

= >= + < += > +

(12)

16

Figure 8: Pressure-time curve for a free air blast wave approximation of the negative phase

2.2.6 Shock Front Velocity

The arrival time of the shock front at different points can be used to calculate the velocity of the

shock front. With the knowledge of this velocity the pressure can be obtained with the Rankine-

Hugoniot relationship.

Kingery [14] calculates also the shock front velocity depending on the pressure as

1/ 2

max0

0

112

+= + pu cp

(13)

The parameter (ratio of specific heats of air) depends also on the overpressure and can be taken from a table in [13]; 0c is the sound velocity in air (331 m/sec); maxp is the peak overpressure and

0p is the atmospheric pressure (101.3 kPa).

2.2.7 Specific Heat Ratio

The specific heat ratio is defined as

pv

cc

= (14)

with cp being the specific heat at constant pressure and cv the specific heat at constant volume. Both

17

the specific heat ratio and the speed of sound depend on the temperature, the pressure, the humidity,

and the CO2 concentration. Kingery [14] defines the variation of the specific heat ratio with a range

of 1.402 to 1.176.

18

3 Numerical Loading of a Structure with Air Blast Waves There are several ways of numerical modelling in order to load a structure with an air blast wave.

These methods differ in the number of used elements and with them in the calculation time.

Model with the mechanical modelling of the explosive (JWL-equation (15)). A fine mesh is essential to get realistic results. The size of the element in the range around the explosive should

be approximately 1 mm. These calculations are very expensive. To reduce the computation time

partitioning can be used. This method reduces the calculation time for models with a large

variation of element sizes.

The method proposed by Clutter [9] is also a solid TNT model and uses only one element for the explosive. This is possible by different not specified methods in combination with the

Becker-Kistiakowsky-Wilson EOS for the explosive.

1D to 3D. This modelling is proposed in [4] and is also a solid TNT model. A 1D calculation is used until the wave reaches a surface. Then the values of the density, energy, velocity and

pressure are mapped into the 3D mesh. Rose [20] maps the 1D model to 2D when the wave

arrives the first surface and maps the 2D model to 3D when the wave arrives a second surface

with another direction. EUROPLEXUS allows the implementation of this method. The method

should also be validated with a calculation of the first model. The model is a mixture of the first

and the third model. The calculation time should be larger than for the second model.

Model with a compressed bubble. The pressure-time function resulting from a compressed bubble can not easily match the curve of an air blast wave. The size of the compression can be

calibrated with the maximum pressure or the impulse. The calculation time is smaller than for

the first model.

Control volume. A volume around the explosive is loaded by a pressure-time curve. This pressure time curve can be calculated with a model based on the modelling of the explosive.

Alternatively, the well known curves from Kingery [14] can be used. This method should be

validated through comparisons with calculations of the first model. The computation time is in

the range of the second model.

Load-time function. This is only usable for an estimation of the behaviour of a structure loaded by an air blast wave. The structure is loaded by a load-time function built with the pressure-time

function e.g. from Kingery [14]. This function is implemented in EUROPLEXUS (see chapter

7). The calculation is relatively inexpensive. Alternatively, the pressure-time function can be

19

determined with a fluid pre-calculation with fixed boundaries for the structure. The structure is

then loaded by the pressures resulting from this fluid calculation.

The choice among these methods depends on the scope of the analysis and on further investigations

about their advantages and shortcomings. Figure 9 shows different models for the simulation of an

air blast wave.

Solid TNT

Compressed bubble

Load on a control volume

Load-time function

Figure 9: Several models for air blast wave simulations

20

4 Investigations with Explosives as a Charge (Solid TNT) The aim of the RAILPROTECT project is to contribute to alleviating the vulnerability of Europe's

passenger transport infrastructures. The effects of a terrorist attack should be simulated numerically,

and for a numerical investigation the knowledge of the loading of the structures is necessary. There

are different approaches and possibilities for the calculation of a detonation inside buildings, as

discussed in chapter 3.

A detonation releases a large amount of energy in a very short time. This results in an air shock-

wave which is spread outwards from the charge. Then, the air blast wave reaches the structure,

which, depending on the size of the charge and on the distance, will respond to this wave loading.

A calculation of the behaviour of the air blast wave requires the knowledge of the behaviour of the

explosive and of the air around the explosive. The results of the numerical investigation can be

compared for the validity of the calculations with existing experimental-analytical data. As will be

shown in this chapter, the experimental-analytical results of Kingery [14] will constitute the basis

for these comparisons.

4.1 Modelling of the explosive The explosive for the numerical investigation can be built up e.g. with the JonesWilkinsLee

(JWL)-equation. This equation of state (EOS) is widely used because of its simplicity and due to the

fact that most high explosives are well modelled by this equation. According to it, the value of

pressure is given as

1 21 2

1 1R V R VEOSEp A e B e

RV R V V = + + (15)

In this equation A, B, R1, R2 and are the model parameters, V is the ratio sol/, where =current density and sol =density of solid explosive, and E is the internal energy per unit volume of the explosive. It is noted that E=sol eint, where eint is the current internal energy per unit mass. The parameters of this equation for most explosives are shown in Dobratz [10]. Different authors use

slightly differing parameter values for this equation, as shown in Table 2. Note that the equation

will be reduced only to its last term if the solid explosive is exhausted and the resulting gases fully

expanded. The last term of equation (15) is the EOS of an ideal gas that can be used e.g. for the air.

=EOS Ep V (16)

21

From this asymptotic form it can also be concluded that =-1 (=ratio of specific heats). Parameter Description ref.[6] AUTODYN

- manual

ref.[21] parameters used

for the air

A (Pa) 3.738e11 3.7377e11 3.712e11

B (Pa) 3.747e9 3.7471e9 3.21e9

R1 4.15 4.15 4.15

R2 0.90 0.90 0.95

sol (kg/m3) density 1630 1630 1630 1.3

eint (J/kg) current internal

energy per unit mass

3.68e6 3.68e6 4.29e6 2.1978E5

specific heat ratio 1.35 1.35 1.30 1.30

vdet (m/sec) detonation speed 6930 6930 6930

Table 2: Parameters for the JWL equation for TNT

For the air the same EOS will be used without a detonation and different starting density and

internal energy. By ignoring the explosion, the last part of the JWL-equation will prevail and

therefore, an ideal gas will be used.

Different FE-Codes smear the detonation front over different time steps. This procedure is called

burn fraction and its motivation is to control the release of the chemical energy for the simulation.

The effects of the combustion on the pressure can be considered with this formula

( )( )1 2min 1,max ,EOSp p F F= (17) The burn mass functions F1 and F2 are computed by (see LS-DYNA and [18])

( )1 ,max

11

1

2if

30 if

e

e

t t d At t

Ft t

>=

(18)

21

1 CJ

VFV= (19)

where t1 is the ignition time of the observed element (calculated with the detonation velocity d),

Ae,max is the maximum surface area and e the volume of the element. V is the actual specific

volume and VCJ the specific volume at the Chapman-Jouguet-pressure. The Chapman-Jouguet

22

pressure is reached if the sonic velocity of the reaction gases reaches the detonation velocity. The

volume at the Chapman-Jouguet-point is

20

1 CJCJPV

d= (20)

The term F1 of the burn mass function intends to spread the burn front over several elements. The

second term should control the releasing of the energy. Interestingly, MSC-Dytran uses only the

term (19), whereas ABAQUS uses only one burn mass function

( )1

1

1

if

0 if b S e

t t dt t

F B lt t

>= (21)

In this formula BS is the constant that controls the width of the burn wave (set to a value of 2.5) and

le is the element length. This function is very similar to (18).

A 3D model of explosive is used to show the influence of the different burn mass fractions (Figure

12, explosive4), whose different components can be compared in Figure 10 and Figure 11. The

influence of the term (21) is visible. The slope of the pressure peak decreases with Fb, and the

arrival of the pressure peak is later. The calculations with EUROPLEXUS do not show an influence

of the second term (F2). Both curves are almost identical.

LS-DYNA uses the functions (18) and (19) for calculations with the

MAT_HIGH_EXPLOSIVE_BURN model. The input syntax allows calculations with both

functions (beta=0) or only with the function (19) (beta=1). The model in LS-DYNA was built with

hexahedral with an element size of 0.001 m. The results of a 3D model show that the burn mass

fraction in LS-DYNA reduces also the slope of the pressure peak. Nevertheless, the difference

between the calculations with beta=0 and beta=1 is negligible. The differences of the calculations

between EUROPLEXUS and LS-DYNA are the smaller peak in LS-DYNA and the higher pressure

values behind the peak (here for a distance less than 0.04 m). The impulses are nearly the same (LS-

DYNA 2.07 108 Pa sec, EUROPLEXUS 2.00 108 Pa sec).

Therefore, the burn mass fraction is for the future work implemented only with the equation (21).

23

0.0E+00

4.0E+09

8.0E+09

1.2E+10

1.6E+10

2.0E+10

0 0.01 0.02 0.03 0.04 0.05 0.06

Distance [m]

Pres

sure

[Pa]

without burn mass fractionburn mass fraction Fbburn mass fraction Fb and F2LS-DYNA beta=0LS-DYNA beta=1

Figure 10: Influence of the burn mass fraction in EUROPLEXUS (t=7 10-6 sec)

0.0E+00

4.0E+09

8.0E+09

1.2E+10

1.6E+10

2.0E+10

0.04 0.05 0.06 0.07 0.08 0.09 0.1

Distance [m]

Pres

sure

[Pa]

EUROPLEXUS bmf=0EUROPLEXUS bmf=2.5LS-DYNA beta=0LS-DYNA beta=1

Figure 11: Influence of the burn mass fraction in EUROPLEXUS (t=1.4 10-5 sec)

Another EOS for explosive is the Ignition and Growth Reactive Model based on Tarver et al.

[23]. This model can also be used for the burning of propellant (deflagrations).

24

4.2 Behaviour in the Explosive The blast behaviour in the air is affected by the development of the pressure in the explosive.

Therefore, it should be investigated, whether the numerical simulation can sufficiently represent the

behaviour of the explosive.

The numerical model for the explosive calculates the pressure with the JWL-equation (15). An

accurate model for the development of the detonation front is used (See [1]). The detonation starts

at the initiation point, and the detonation front is moved with the given velocity of the detonation.

An element detonates if the detonation front reaches this element. From this time the JWL-equation

will be used for this element.

A spherical TNT charge of volume of 8000 cm3, i.e. with a radius of 0.124 m is considered. To

control the behaviour of the pressure in the explosive a conical model is used (similar to the model

of chapter 4.3, see Figure 12).

x

dpyradex,in

dex,end

length

opening

Figure 12: 3D simplified model for the behaviour in the explosive

The models use hexahedrons and a pyramid for the top of the model. The FSR-condition is used for

all surfaces of the model. The difference between the meshes, listed in Table 3, is the refinement.

25

Case opening dpyra dex,in dex,end Number of elements

explosive1 Eulerian 2e-2 0.001 0.001 0.01 33

explosive2 Eulerian 1e-2 0.0005 0.0005 0.005 65

explosive3 Eulerian 4e-3 0.0005 0.0002 0.002 159

explosive4 Eulerian 2e-3 0.0002 0.0001 0.001 318

explosive5 Eulerian 2e-3 0.0002 0.0001 0.0005 499

explosive6 Eulerian 2e-3 0.0002 0.0001 0.0002 859

explosive7 ALE 2e-3 0.0002 0.0001 0.0005 499

Table 3: Comparison of different models for the explosive

The pressures are here analysed in intervals of 1sec. The procedure to get the values in space

(SCOURBE command in EUROPLEXUS) uses the averaged elemental pressure values for the

nodal values. This results in a half value of the pressure at the node between an element inside and

an element outside of the detonation zone (see Figure 13). The vertical lines in this figure

correspond to the distances of the elements that are detonated at a certain time step (can be get from

the listing). The detonation front can be identified by the steep increasing of the pressure.

Therefore, it is important to consider the value of the last element of the detonation front and not the

value of the last node.

Figure 13: Distribution of pressures over distance (explosive1)

The maximum pressure of the detonation is increasing with the distance from the initial detonation

point. The Chapman-Jouguet-pressure is the maximum experimental resulted pressure. The

26

parameters of the JWL-equation should represent this limitation (see Shin [21]). Shin shows the

influence of the discretisation with a 1D model, where a finer mesh reproduces larger pressures.

The results with the models explosive1 to explosive4 show the same dependency of the element

sizes (Figure 14). The Chapman-Jouguet-pressure seems to be the limit in the convergence study.

0.00E+00

4.00E+09

8.00E+09

1.20E+10

1.60E+10

2.00E+10

2.40E+10

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Number of elements

Pres

sure

[Pa] t=7e-6

t=1.4e-6Shin t=7e-6Shin t=1.4e-5Chapman-Jouguet-pressure

Figure 14: Maximum pressures in the explosive depending on the number of elements

The rather unexpected behaviour of the models with 499 and 859 elements has to be clarified. The

reason could lie in the location of the detonation point in conjunction with the location of the

integration points.

Figure 15 shows the pressure depending on the distance from the initiation point. The curve does

not differ for models with fine discretisation. It is observed that the pressures behind the pressure

peak in the model explosive6 are definitely smaller than the results of Shin. Therefore, the area

under the pressure-distance curve is also smaller. This area reaches only 67 % of the area of Shin.

The reason could be the missing burn fraction (see chapter 4.1).

27

0.0E+00

4.0E+09

8.0E+09

1.2E+10

1.6E+10

2.0E+10

2.4E+10

0 0.02 0.04 0.06 0.08 0.1 0.12Distance [m]

Pres

sure

[Pa]

explosive1, t=7e-6explosive1, t=1.4e-5explosive4, t=7e-6explosive4, t=1.4e-5explosive6, t=1.4e-5Shin t=1.4e-5Shin t=7e-6

Area Shin 5.54e8Area explosive6 3.71e8(between 0.051 and 0.97)

Figure 15: Pressure distance curve in the explosive comparison with Shin [21]

The model explosive7 uses an ALE mesh instead of an Eulerian mesh. The results are almost

identical for the maximum pressure as well as for the bending of the curve. A calculation with a

Lagrangian mesh fails.

The next question is how the pressures are developed at the border of the explosive. At this time

experimental results for this region are not available. Here, the numerical results of EUROPLEXUS

could be compared with numerical results of LS-DYNA. For this calculation a conical model is

used (see chapter 4.3, CON20, Table 4). In LS-DYNA the boundary condition like the FSR

condition is relatively complex (BOUNDARY_SPC). To avoid an influence of the usage of this

boundary condition the model in LS-DYNA is built as a full 3D model. The LS-DYNA model has a

length of 15 cm, a height of 5 cm and a width of 5 cm. The boundaries are built as fixed. Therefore,

after the reflection of the wave at the boundaries the pressure patterns are no longer spherical. The

element size in LS-DYNA is chosen as 1 mm with an ALE multi-material formulation (see Figure

16).

28

Figure 16: Model in LS-DYNA

The calculations show that the numerical results of both programs are nearly the same inside the

explosive as well as in the air near the explosive (see Figure 17). The calculation with

EUROPLEXUS is done without the burn mass fraction.

0.0E+00

5.0E+09

1.0E+10

1.5E+10

2.0E+10

1.4E-5 1.6E-5 1.8E-5 2.0E-5 2.2E-5 2.4E-5 2.6E-5 2.8E-5 3.0E-5Time [sec]

Pres

sure

[Pa]

EUROPLEXUS x=0.105EUROPLEXUS x=0.125EUROPLEXUS x=0.136LS-DYNA x=0.105LS-DYNA x=0.125LS-DYNA x=0.136

Reflection

Reflection

Figure 17: Pressure time curve in the explosive and in the air near the explosive; length of the explosive = 0.124 m

29

0.0E+00

1.0E+09

2.0E+09

3.0E+09

4.0E+09

5.0E+09

1.4E-5 1.6E-5 1.8E-5 2.0E-5 2.2E-5 2.4E-5 2.6E-5 2.8E-5 3.0E-5Time [sec]

Pres

sure

[Pa]

EUROPLEXUS x=0.105EUROPLEXUS x=0.125EUROPLEXUS x=0.136LS-DYNA x=0.105LS-DYNA x=0.125LS-DYNA x=0.136

Reflection

Reflection

Reflection

Figure 18: Detail of the pressure time curve

4.3 Cone with two Symmetry Axes A two dimensional model does not consider the behaviour of the fluid in the third direction.

Therefore, a three dimensional model for the simulation of the detonation should be used. The

calculation costs are minimized by using a conical model (pyramid, see Figure 19). This model can

be built with one hexahedral element in the radial direction and symmetry axes at the edges of the

elements. The element on the top is then a pyramid. To get elements with similar geometry the

length of the elements should be increased with an increasing distance from the centre.

The size of the opening angle depends also on the size of the elements. This angle should be chosen

so that the aspect ratio of the elements is not too large. Normally, the time step size for this model is

relatively small because of the very small element on the top of the cone.

30

x

Explosive

dpyradex,in dex,end

dair,in dair,end

length

opening

Figure 19: 3D simplified model with hexahedrons or tetrahedrons

Alternatively four tetrahedral elements can be used instead of one hexahedral element. This results

in a higher number of elements. The meshing can be done with scripts that convert the hexahedrons

in tetrahedrons. These scripts are presented in [7]. The script pxhex2te converts the hexahedrons in

tetrahedrons; the script pxqua2tr converts the quadrangles to triangles (necessary for the surfaces).

For the simulation a model is used which has in most cases a TNT charge of a volume of 8000 cm3,

accordingly a cube of 20 x 20 x 20 cm or a sphere with radius of 12.4 cm. The mass of this charge

is 12.8 kg. An ALE-calculation is used for all further investigations. The explosive is build as an

Eulerian mesh, the air is build as an ALE mesh.

The symmetry at the surfaces can be considered by defining symmetry planes with the CONT

SPLA AUTO command. This command defines symmetry conditions orthogonal to the surface of

the defined nodes. Another possibility is the definition with the FSR-command as a sliding surface.

Both methods give the same results.

The calculations performed with the conical model are summarized in Table 4 and shortly described

hereafter.

31

Case dpyr/dex,in/dex,end/dair,in/dair,end

opening/

length

Charge Element type tend

CON1 0.05/0.05/0.08/0.2/0.2 0.2/4.0 12.8 FL34 (NF34) 4.0e-3

CON2 0.01/0.02/0.02/0.02/0.05 0.2/4.0 12.8 FL34 (NF34) 4.0e-3

CON3 0.008/0.01/0.01/0.01/0.05 0.1/4.0 12.8 FL34 (NF34) 4.0e-3

CON4 0.008/0.005/0.005/0.005/0.02 0.03/1.3 12.8 FL34 (NF34) 4.7e-4

CON5 0.008/0.005/0.005/0.005/0.02 0.03/1.3 12.8 FL35/FL38 4.7e-4

CON6 0.002/0.002/0.002/0.002/0.01 0.01/1.3 12.8 FL35/FL38 4.7e-4

CON7 0.001/0.001/0.001/0.001/0.005 0.01/1.3 12.8 FL35/FL38 4.7e-4

CON8 0.001/0.001/0.001/0.001/0.01 0.05/3.0 12.8 FL35/FL38 1.5e-3

CON9 0.001/0.001/0.001/0.001/0.01 0.05/3.0 1.0 FL35/FL38 1.5e-3

CON10 0.001/0.001/0.001/0.001/0.005 0.05/3.0 1.0 FL35/FL38 1.5e-3

CON11 0.0005/0.0005/0.0005/0.0005/0.01 0.025/1.5 1.0 FL35/FL38 1.5e-3

CON12 0.005/0.001/0.001/0.001/0.01 0.1/3.0 1.0 FL35/FL38 1.5e-3

CON13 0.005/0.001/0.001/0.001/0.04 0.2/3.0 1.0 FL35/FL38 1.5e-3

CON14 0.005/0.001/0.001/0.001/0.02 0.2/1.5 1.0 FL35/FL38 1.5e-3

CON20 0.0002/0.0001/0.001/0.001/0.005; 6.5e-3/0.4 1.0 FL35/FL38 7.0e-5

Table 4: Calculations with a conical mesh

CON1

This calculation uses the modified tetrahedral elements (see [7]) also for the tip of the model. The

mesh is relatively coarse. The development of the air pressure wave is presented in Figure 20. The

pressure-time curve for a distance of 1 m shows the increase of the pressure until a value of

1.83 106 Pa (time step size for evaluation 2 10-6 sec) which is equivalent to 18.3 times the

atmospheric pressure. The resulting pressure does not depend on the time step size for the

evaluation. By choosing every calculated time step for the evaluation (1.75 10-7 sec) the maximum

pressure is also 1.830958 106 Pa. After the pressure peak the pressure is decreasing up to 232.2 Pa.

This value of the negative pressure is relatively high. The pressure-time curve for other distances

follows also these trends.

32

EUROPLEXUS14 JUNE 2007

DRAWING 1

FLUID_3D_SF7-1- Distance = 1m -2- Distance = 2m -3- Distance = 3m

Time [s]

0.0 5.0E-04 1.0E-03 1.5E-03 2.0E-03 2.5E-03 3.0E-03 3.5E-03 4.0E-03-5.0E+05

0.0

5.0E+05

1.0E+06

1.5E+06

2.0E+06

1

1

11 1 1

2

2

2 23 3

3

Figure 20: Development of the pressure depending on the distance to the charge

CON2

This model uses smaller tetrahedral elements than the mesh CON1.

The finer mesh results in a steeper air blast wave at a distance of 1 m as well as in a distance of 2 m.

The steeper wave causes also a higher pressure (See Figure 21). Figure 22 shows the maximum

pressure depending on the distance to the charge. These curves can be obtained by using a

CAST3M macro that selects all nodes lying near a line. The macro pxpdroi1 (See [7]) requires a

tolerance in which the nodes are lying. Then, with the macro pxordpoi the selected nodes are

ordered. The line which is used to select the nodes is one of the edges of the CON models.

With EUROPLEXUS it is possible to get out the data depending on the space variable. This can be

done at a certain time step. From the results of different time steps the maximal values, the impulse

and the positive phase duration can be extracted with the FORTRAN-executable

AIRBLASTRESULT (see Appendix).

A comparison of the different discretisations (see Figure 22) with the results of Kingery [14] shows

that a finer tetrahedral mesh produces smaller maximum pressures (disregarding the model CON1).

Except for distances less than 25 cm, these pressures are smaller than the experimental pressures.

The experience shows that Eulerian meshes normally react smoother than the reality.

33

Figure 21: Comparison of a coarse with a fine mesh pressure time curve

CON3 and CON4

These calculations use finer meshes. To reduce the computation time the length of model CON4

(see Figure 19) is limited to 1.3 m.

For the results of the models CON1 to CON4 it is important to note that the definition of the pres-

sure at a certain point is arguable. There are several tetrahedrons at the same distance with different

orientations, and these elements have different pressures. Thus, the results with the tetrahedrons is a

field that requires further work, as probably, tetrahedrons will be used for the calculation of the air

inside the structures. For the employment of the tetrahedrons in a productive framework, these

elements should show safer and more reliable results.

CON5

This calculation uses hexahedral elements instead of the tetrahedrons. The mesh is relatively coarse.

In comparison to the tetrahedrons the maximum pressures versus the distance show a relatively

good correlation with the analytical results.

CON6 and CON7

Both models use finer meshes than the model CON5.

The behaviour of models with hexahedral elements regarding the dependency on the element size

shows the reverse trends to those of the tetrahedral elements (See also Figure 22). A finer mesh

results in a higher maximum pressure. This should be the normal behaviour of a refinement of the

elements. The results of the model CON7 show the best correlation, even though the differences

34

between the experimental and the numerical results are still quite big. Further work has to be done

to check this discrepancy.

0E+0

2E+7

4E+7

6E+7

8E+7

1E+8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Distance [m]

Max

. pre

ssur

e [P

a]CON1 CON2 CON3 CON4 CON5 CON6 CON7 Kingery

Figure 22: Comparison of max. pressure distance relationships from different conical models.

CON8

This calculation is done with a length of 3.0 m. The element sizes are nearly the same as in model

CON7. In comparison to the results of model CON7 the difference is small (see Figure 23).

0E+0

2E+6

4E+6

6E+6

8E+6

1E+7

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Scaled distance [m/kg1/3]

Max

. pre

ssur

e [P

a]

KingeryCON7CON8 CON9 1kgCON10CON11CON12

Figure 23: Comparison of peak pressures scaled distance relationships from different conical models.

35

CON9

This model has approximately the element sizes of the mesh CON8 but has a charge of only 1 kg.

The resulting pressures in a scaled size are smaller than those of model CON8.

CON10

This is a finer mesh for the air of the model CON9. However, all values are nearly the same.

CON11

CON11 uses a finer mesh in the explosive and near the explosive. The pressures near the explosive

are higher than in the previous models, but the difference in a larger distance is small.

CON12

Figure 24 shows the pressure versus the distance at a time of 1.5 10-3 sec for the model CON9. The

pressures in the first elements are approximately 10 104 times higher than expected. Controlling the

aspect ratio of the elements at the tip (pyramid and hexahedrons), it is realised that the elements are

not conformant. Therefore, model CON12 is tried, which uses a larger angle of the cone than the

model CON9 and has also a bigger pyramid at the tip. The largest aspect ratio of the hexahedrons is

now 3.0 and of the pyramid 15. The aspect ratio of the pyramid depends on the aspect ratio of the

whole model (2

lengthopening

= ). The pressures in the pyramid and in the three first hexahedrons are

larger than expected (blue rhombus points in Figure 24).

To calculate the aspect ratio of an element the procedure aspectra can be used (see appendix).

Figure 24: Pressure at the top of the conical model

36

CON13

The angle of this model is increased. The aspect ratio of the hexahedrons is 1.5, of the pyramid 7.5.

The results for the elements near the tip show that the pressure value in the hexahedrons is

represented better if the aspect ratio is smaller.

CON14

The pressures in the pyramid are decreasing with an aspect ratio of 3.75. The calculations do not

show better results.

CON22, CON23, CON24

These models use only cubic elements. Therefore, the element size near the explosive is similar to

the other models. The size of the elements increases with the distance to the explosive. Then, the

only parameter for this model is the opening of the cone (or the angle of the cone). The number of

elements is relatively small but the time step size depends on the smallest element.

Case opening/ length Number of

elements

Charge

CON22 0.1/4.0 386 12.8

CON23 0.05/4.0 711 12.8

CON24 0.02/4.0 1595 12.8

Table 5: Calculations with a conical mesh

The peak pressures show a big dependency on the element size. (see Figure 25). The influence of

the element size on the impulse is much smaller (see Figure 26). Therefore, for further calculations

it has to be checked if the element size is small enough.

37

0E+0

1E+7

2E+7

3E+7

4E+7

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Scaled distance [m/kg1/3]

Max

. pre

ssur

e [P

a]

Kingery 12.8 kgSPHE9CON8 CON22CON23CON24

Figure 25: Peak Pressure Models CON22, CON23, CON24

0.0E+0

5.0E+2

1.0E+3

1.5E+3

2.0E+3

2.5E+3

0.1 0.2 0.2 0.3 0.3 0.4 0.4

Scaled distance [m/kg1/3]

Impu

lse

[Pa

sec]

SPHE9CON8 CON22CON23CON24Kingery 12.8kg

Figure 26: Impulse Models CON22, CON23, CON24

38

Summary

The pressures resulting from a calculation with tetrahedrons are too small. The results with hexahe-

drons are also small, but the difference to the analytical-experimental results is less. The difference

depends on the size of the elements. Smaller elements result in higher pressure and a better

correlation with the experimental results. The smallest element sizes are of the order of 1 mm. It is

not possible to calculate a fluid-structure interaction problem of realistic dimensions with a mesh of

this size. So, the question which kind of simulation of the loading is most appropriate to be used is

important (see chapter 3).

4.4 Cubic Charge with two Symmetry Axes To check if the boundary condition of the cone model represents a full 3D model two different

models are used. Both models are one eighth of the geometry with FSR conditions at the symmetry

faces. Another advantage with such a model is the possibility to compare the calculations with other

finite element programs like LS-DYNA.

In the first model the charge and the air are built as cubes. A regular mesh with hexahedrons is

applicable with the same size for all elements. The second model uses a spherical charge (see

chapter 4.5).

An interaction with a structure is not of interest for these calculations. Therefore, the explosive and

the air are built with fluid meshes. An ALE mesh is used for the air, an Eulerian mesh is used for

the explosive (see Figure 27).

Figure 27: Model CUB1 explosive as Eulerian (red), air as ALE (grey)

The calculations performed are summarized in Table 6 and shortly described hereafter.

39

Case Size of the model Description tend Steps CPU [s] Elements

CUB1 0.5 X 0.5 X 0.5 m Charge 12.8 kg TNT,

element size 0.02 m

4.7e-4 415 181.5 15625

CUB2 1.0 X 1.0 X 1.0 m Charge 12.8 kg TNT,

element size 0.02 m

4.7e-4 270 790.6 125000

CUB3 0.5 X 0.5 X 0.5 m Charge 12.8 kg TNT,

element size 0.01 m

4.7e-4 911 2275.9 125000

CUB4 1.0 X 1.0 X 1.0 m Charge 1.0 kg TNT,

element size 0.0167 m

6.0e-4 291 1367.7 216000

Table 6: Calculations with cubic charges

CUB1

In this calculation a small model with a coarse mesh is used. The charge has a mass of 12.8 kg. The

development of the pressure can be shown in Figure 28. At t=2 10-5 the explosive has burned down

except for the explosive in the corners of the cube. The pressure in the air is the atmospheric

pressure of 105 Pa. After the explosion (e.g. t=6 10-5) the pressure of the gas produced from the

explosive is decreasing and the pressure wave is running into the air. At t=10-4 there can be

observed a discontinuous pressure along the three axes at the same distance from the detonation

initiation point. The reason could be the cubical shape of the charge. At t=1.4 10-4 the pressure

wave is reaching the surface (FSR condition) and is reflected.

40

a) t=2e-5 b) t=6e-5

c) t=1e-4 d) t=1.4e-4

Figure 28: CUB1 Pressure at different time steps

CUB2

This calculation uses a bigger model but also with the coarse mesh. Figure 29 shows the maximum

pressure versus the distance from the charge. The maximum pressures resulting from model CUB1

and from model CUB2 are the same in the range of the size of model CUB1. The pressures are

overestimated up to a distance of 0.46 m; in relation to the size of the explosive (0.13 m) this range

of overestimation is negligible. The resulting maximum pressures after the range of 0.4 m are

definitely smaller than the experimental results.

CUB3

The size of the model is the same as that used in model CUB1 but with a finer mesh. The pressures

resulting from this calculation are bigger than in model CUB1. For CUB1 and CUB2 only the pres-

sures along the orthogonal to the surface of the explosive are considered. The values are smaller if

the pressures along the diagonal of the cube are used. The difference between both locations is very

big. Therefore to compare the results in a region near the explosive with experimental results a

cubic charge is not suitable. The differences between experimental and numerical results can be

compared better with a spherical model.

41

CUB4

This model uses a finer mesh than model CUB2. The ending time of the calculation is longer so that

the wave can be also observed in regions with a bigger distance from the charge.

In comparison to the model CUB2 the pressures at CUB4 are smaller because of the smaller charge.

A better method to compare is the use of the scaled distance, and in fact the values of the pressure

are of the same order of magnitude in this case. There is also a big difference between the pressures

along the diagonal and the pressures along the orthogonal to the charge surface.

In comparison to the results with LS-DYNA (cub2) the results of EUROPLEXUS for the pressure

are higher and represent the experimental values better but still not in a very satisfactory manner.

Figure 29: Maximum pressures versus distance cubical models

42

Figure 30: Maximum pressures versus scaled distance cubical models

4.5 Spherical Charge with two Symmetry Axes The weaknesses of using a cubical charge can be avoided by using a spherical charge. There are no

problems with a spherical charge from the different ending times of the detonation at the corners

and with effects resulting from the different surfaces.

As mentioned initially, these first investigations disregard the interaction with the structure. So it is

possible to use a spherical volume for the surrounding air, too. This allows an easier modelling of

the mesh, which can be done in several ways. At this point, the first approach of a spherically

symmetric regular mesh with the centre of the charge is in CAST3M neither possible in an easy

way nor is it necessary in considering the behaviour of the explosive and the air.

The method used instead converts a cubical model via the INCL operator to a spherical model. The

resulting model seems to be relatively regular. The method can be done with the following steps:

1. Modelling of a cubic surface for the outer charge (half volume) (Figure 31a, green).

2. Modelling of a cubic surface for the inner charge. This part will be meshed with regular hexahe-

drons (half volume) (Figure 31a, blue).

3. Modelling of a cubic surface for the air volume around of the charge (half volume) (Figure 31a,

red).

4. Projection of the outer charge surface and of the air surface with the PROJ operator to spheres

(Figure 31b).

5. Filling the volumes between the surfaces with hexahedrons (Figure 31c).

43

6. Complete the volumes by adding the turned model.

7. Defining of bounding box for an one-eighth model.

8. Choosing the elements inside of the bounding box with the INCL operator (Figure 31d).

9. Defining the bounding surfaces with the POIN operator.

a) Cubic surface for outer charge (green)b) Projection to a spherefor inner charge (blue) and for air (red)

c) Filling the volume between the surfaces d) Elements inside the bounding box

Figure 31: Modelling of the spherical model

Alternatively an eighth of the whole spherical model can be used form the beginning. This saves the

required memory.

Another possibility is the modelling with two macros (pxhex2te and pxqua2tr, see [7]). With them it

is possible to rotate first a line around a point to get a part of a circle. A second macro is available to

turn this part of a circle around a line to get a part of a sphere. However, the resulting model seems

to be more irregular then the model built with the method described before.

The calculation cases performed are summarized in Table 7 and shortly described hereafter,

together with the modifications introduced in the code to achieve this type of calculations. The

regions with different element densities are shown in Figure 32. The charge of all models is 12.8 kg

TNT, the end of the calculation depends on the size of the model.

44

Figure 32: Modelling of the spherical model number of elements in the different models

Case Size of the model

nel0/nel1/nel2

Length

[m]

Distance with

aspect ratio = 10

Number of

elements

SPHE1 10/10/10 1 (5.55) 1625

SPHE2 10/20/70 1 0.79 6875

SPHE3 10/20/90 1 0.62 8375

SPHE4 10/20/150 1 0.37 12875

SPHE5 10/20/200 1 0.28 16625

SPHE6 10/40/200 1 0.28 18125

SPHE7 20/40/200 1 0.56 73000

SPHE8 10/10/70 2 1.70 6125

SPHE9 20/20/140 2 1.70 12125

Table 7: Calculations with spherical charges

SPHE1

In this calculation a small model with a coarse mesh is used. The results show a relatively low

maximum pressure (See Figure 33).

45

Figure 33: Maximum pressures versus distance spherical models

SPHE2, SPHE3, SHPE4 and SPHE5

These calculations use finer meshes. The maximum pressures are increasing. The models SPE2 and

SPE4 give realistic results. The models SPHE4 and SPHE5 give larger maximum pressure than the

experimental values. An Eulerian calculation should normally result in smaller values than the

experimental values (in contrast to a Lagrangian calculation which reacts stiffer with smaller

element sizes).

SPHE6

The larger values for the maximum pressure in model SPHE5 start near the charge. Therefore, the

model SPHE6 tests if a smaller element size in the outer charge (See d) red range) can result in

more realistic values.

The overestimating of the pressure begins in this model earlier than in the model SPHE5. The size

of the elements in the explosive should not be the reason for the overestimation.

SPHE7

This calculation uses a finer mesh in the inner charge (See Figure 31 d) green range). With the

smaller elements in the inner charge also the number of the elements in tangential direction is

increased.

The results are quite better. For every model with small elements there is a limit from which the cal-

46

culation fails. This limit depends on the radial and the tangential size of the elements. It seems that

the elements are not usable when they are too thin. The pressures are overestimated if the aspect

ratio is about 10. When exceeding this ratio, the pressure is also not symmetric. It appears to be

higher near the edges and smaller in the diagonal of the model. This difference between the edge

and the diagonal begins approximately at a distance of 0.45. The aspect ratio is there 8.0. It seems

that the boundary conditions can not act with this exceptional element sizes. Respecting this, also

the values for the calculation SPHE3 would not be applicable over a distance of 0.60 m. This has to

be considered for further calculations with the conical model and the spherical model. The aspect

ratio reaches the critical value 10 at a certain distance. This distance is calculated in Table 7.

SPHE8

This model is a coarse spherical model with a size of 2.0 m. The results follow the results of the

models with a size of 1 m. The pressure is approximately half the experimental value.

SPHE9

This calculation uses a finer mesh than model SHPE8. The pressures are a little bit larger than the

pressures in model SHPE8.

4.6 Comparison between the Different Models

4.6.1 Maximum Pressure

The comparison will be done with the fine meshes of the different models summarized in Table 8.

Case Element size Charge

[kg TNT]

Size of the model

[m]

Steps CPU [s] Elements

CON4 (Tet) 0.02-0.005 12.8 1.3 8956 9 914.3 1582

CON8 (Hex) 0.001-0.01 12.8 3.0 2205 13 3625 860

CON9 (Hex) 0.001-0.01 1.0 3.0 2205 13 3482 805

SPHE3 0.01 12.8 1.0 698 98 8375

SPHE9 0.013 12.8 2.0 917 691 49000

Table 8: Comparison of different models

The summery of the different curves (Figure 34) shows that the conical model with the tetrahedral

elements results not in a realistic maximum pressure-distance curve. The closest agreement with the

47

experimental values is obtained with the finest conical mesh. This mesh has a smallest element size

of 1 mm! The calculation time is very high because of the small time step sizes that are necessary

with the small elements. The difference between the spherical and the conical model shows that the

element size has a big influence for the maximum pressures.

0E+0

2E+6

4E+6

6E+6

8E+6

1E+7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Scaled distance [m/kg1/3]

Max

. pre

ssur

e [P

a]

KingerySPHE3SPHE9CON4 CON8 CON9 1kg

Figure 34: Maximum pressures versus scaled distance

4.6.2 Impulse

The impulse is the parameter which has a capital importance for the loading of a structure. The

numerical results are shown in Figure 36. The impulse has been calculated with an integration of

the pressure- time curve over the time. The impulse is larger for the models with an explosive of

12.8 kg and is smaller for an explosive of 1.0 kg. This depends on the positive phase duration which

has also a big difference in the scaling. After the detonation the compressed combustion gas needs

time to expand. The dashed line for the model CON9 (Z

48

0.0E+0

4.0E-4

8.0E-4

1.2E-3

1.6E-3

2.0E-3

0 0.2 0.4 0.6 0.8 1 1.2

Scaled Distance [m/kg1/3]

Tim

e of

dur

atio

n t d

[sec

]SPHE9CON8 CON9 1kgKingery 1kgKingery 12.8kg

Figure 35: Positive phase duration versus scaled distance

0E+0

2E+2

4E+2

6E+2

8E+2

1E+3

0.1 0.3 0.5 0.7 0.9 1.1

Scaled Distance [m/kg1/3]

Impu

lse

[Pa

sec]

SPHE3SPHE9CON4 CON8 CON9 1kgKingery 1kgKingery 12.8kg

.

Figure 36: Impulse versus scaled distance

49

4.6.3 Arrival Time

The arrival time also depends on the distance. The results for the different models are shown in

Figure 37. The arrival time for the numerical investigations is defined by the arrival of the increased

pressure. The arrival time of Kingery is defined as the time from the initiation of the detonation to

the arrival of the pressure wave at this point. The arrival times for all calculations are higher than

the experimental arrival times. The numerical results show only a small dependence on the element

size.

0.0E+0

5.0E-4

1.0E-3

1.5E-3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Scaled Distance [m/kg1/3]

Arr

ival

tim

e t a

[sec

]

SPHE3SPHE9CON4 CON8 CON9 1kgKingery 1kgKingery 12.8 kg

Figure 37: Arrival time versus scaled distance

4.6.4 Positive Phase Duration

This parameter is shown in Figure 35, and is defined by the length of positive pressure phase. The

results of model SPHE9 represent the experimental results rather well. Discrepancies occur because

after a distance of 1.4 m the reflected wave at the boundary condition arrives before the positive

pressure is ended. Also the conical model CON8 represents sufficiently the positive phase duration.

The conical model with 1 kg TNT has a very small time td. The scaling rule for td is an issue to be

clarified like for the time of arrival.

50

4.6.5 Comparison with results of other authors

Experiments and calculations with spherical explosions without reflections are relatively rare in the

literature.

Clutter [9] uses only one element with a different EOS for the explosive. He compares the results of

the impulse and the peak pressure in a logarithmic scale. The values for the peak pressure are higher

for small scaled distances and are smaller for large scaled distances. The impulses for scaled

distances smaller than 0.8 m/kg1/3 are definitely too large (up to 2.5 times of the Kingery data). Also

the values for larger distances are overestimated. A fluid calculation should results in smaller peak

pressures and smaller impulses.

Fairlie [12] presents calculations with AUTODYN for a small amount of TNT (8 g). The

comparison of the peak pressures shows a good correlation with the values of Kingery (3%

difference). The difference for the impulse is 22%, in the same range as the herein presented

calculations.

The investigation from Alia [1] uses LS-DYNA with a multi material formulation. The amount of

454 g C-4 (equivalent to 609 g TNT) is modelled with the JWL-equation. The overpressure at a

certain point is smaller than the pressures resulting from Kingery (25%). The impulse is also

smaller (40%).

In the calculations presented the difference between the numerical results and the experimental

results of Kingery are in a range of 30-40% for the peak pressures as well as for the impulses. The

smallest elements in the CON-models have a length of 1 mm due to the small angle of the cone.

This raises the question whether the modelling of large structures is possible with a JWL-equation.

Therefore, the next chapter presents several possibilities for the simulation of air blast waves.

4.7 Influence of several parameters As seen above, there are differences between the experimental results and the numerical

calculations. The computed maximum pressure is too small, the arrival time longer, and the impulse

higher. The potential influence of the several parameters involved should be determined.

4.7.1 Specific heat ratio (CON15)

Kingery defines the parameter in the range of 1.176 to 1.402. The cited literature is still not avail-able, but as stated, varies with the overpressure. Without the knowledge of the dependence on the overpressure it is not possible to define a more accurate material law for the air. Nevertheless, the

influence of a varying can be tested with a calculation which chooses to be 1.176 (smallest

51

value in [13]). These results show that the maximum pressures are smaller, and the arrival time is

higher. To conclude, it seems that the influence of the parameter towards a better material law, would not lead to better results.

4.7.2 Values for , E0, There are different values for the parameters , E0 and of the ideal gas equation (16) in the literature. These differences should be considered and assessed.

Case Used values (CON8) (CON15 CON16

1.35 1.176 1.4

E0 2.1978e5 4.37e5 2.5e5

1.300 1.300 1.293

Table 9: Parameters , E0 and

The maximum pressure resulting from the model CON16 is higher, the arrival time is smaller but

the difference with model CON8 is very small. The difference from model CON15 to model CON8

is very small, too. Therefore the choice of the parameter values for the air doesnt have a big

influence.

4.7.3 Parameters for the explosive

The parameters for the JWL equation (15) could also have an influence on the results. The

calculations are made with the parameters used by AUTODYN and Shin [21] (see Table 2)

(CON17, CON18). It is found that, overall, the influence of the chosen parameters is small. The

results for the standard parameters of AUTODYN are nearly the same. The maximum pressure with

the values of Shin is at a distance of 0.40 m approximately 10 % higher. The arrival time is 5 %

smaller; the impulse is 1 % higher.

4.7.4 Burn mass fraction

The burn mass fraction has an influence on the behaviour in the explosive (see chapter 4.1). The

model CON8 is calculated with the burn mass fraction (BS = 2.5, model CON19) to test the

influence on the behaviour of the air. The difference between the calculations with and without burn

mass fraction is very small and represents for the overpressure, the impulse and the duration of the

positive phase only 1%. The arrival time is approximately 10 % higher.

52

5 Bubble model Several models with a compressed bubble are used to test whether a bubble model can represent the

air blast waves resulting from an explosion (see Figure 38). The 3D models are similarly built as the

models in chapter 4.5. The region with the explosive is increased and replaced with the compressed

bubble. The different parameters of the model are described in Table 10. The model bubble1 uses

20 elements in the circumference direction (for the eighth of the sphere); all other models use 40

elements. Model bubble1 has 13000 elements; all other models have 104000 elements.

Figure 38: Model compressed bubble

The resulting values (particularly the maximum pressure and the impulse) can be compared with the

experimental values resulting from an explosion. This is done here with the values of Kingery [14].

Those values show for the impulse a maximum at a scaled distance of 0.8 m. The curve from

Kinney does not consider this maximum and therefore, the impulse is smaller and with them the

corresponding TNT equivalent (see also Figure 4).

The impulse is the most important parameter for the calculation of the interaction between the air

blast wave and the structure. Therefore, an impulse-distance curve should be found by varying the

charge that represents the impulse-distance curve of the calculation with the compressed bubble.

53

Model lair

[m]

lb

[m]

Bubble

pressure

[Pa]

Bubble

density

[kg/m3]

Bubble

eint

[J/kg]

TNT equivalent

of the energy of

the bubble

Corresponding

blast wave with

TNT equivalent

Element length

at the border of

the bubble [m]

bubble1 3 0.5 1e7 13.0 2.16e6 3.22 2.0 0.083

bubble2 3 0.5 1e7 13.0 2.16e6 3.22 2.0 0.042

bubble3 5 0.5 1e7 13.0 2.16e6 3.22 2.0 0.042

bubble4 5 0.5 1e9 130.0 2.16e7 325 n/a 0.075

bubble5 5 0.5 2.5e6 6.5 1.10e6 0.78 0.6 0.075

bubble6 5 0.5 1.97e7 18.4 3.06e6 6.48 3.7 0.075

bubble7 5 0.5 3.93e7 26 4.32e6 13.0 6.5 0.075

bubble8 5 0.5 4.9e6 9.19 1.52e6 1.59 1.1 0.075

bubble9 10 1.0 1.0e7 13 2.2e6 26.2 16 0.15

bubble10 10 1.0 2.5e6 6.5 1.1e6 6.36 5 0.15

bubble11 10 1.0 1.25e6 4.6 7.77e5 3.05 2.5 0.15

bubble12 10 1.0 4.9e6 9.19 1.52e6 12.7 9 0.15

bubble13 10 1.0 3.6e6 7.8 1.32e6 9.28 6.8 0.15

Table 10: Compressed bubble models

The comparison of the maximum pressures shows that the maximum pressure is smaller in the

compressed bubble model than in the equations of Kingery. The peak pressures of the models

bubble1 and bubble3 show nearly the same behaviour because the element size is nearly the same.

The model bubble3 is bigger. The model bubble2 uses a finer mesh. Therefore the pressure peak is

higher and represents the experimental values of Kingery better. The difference in the impulses

between the models with different element sizes is relatively small.

54

0E+0

2E+2

4E+2

6E+2

8E+2

1E+3

0 0.5 1 1.5 2 2.5 3

Scaled Distance [m/kg1/3]

Impu

lse

[Pa

sec]

bubble1bubble2bubble3Kinney 2kgKingery 2kg

.

Figure 39: Comparison of several compressed bubble models (impulse)

0E+0

4E+5

8E+5

1E+6

2E+6

2E+6

0 0.5 1 1.5 2 2.5 3 3.5 4

Scaled distance [m/kg1/3]

Max

. pre

ssur

e [P

a]

Kingerybubble1bubble2bubble3

Figure 40: Comparison of several compressed bubble models (maximum pressure)

55

Several calculations are done with models with a radius of 0.5 m of the compressed bubble and with

different overpressures. The best fitted curve for the model bubble1 is the impulse-distance curve

with approximately 2 kg TNT (see Figure 39). All the resulting impulse-distance curves are fitted to

curves from Kingery to get a corresponding TNT equivalent. The values of the TNT equivalents for

different overpressures in the bubble are shown in Figure 41.

The bubble with the overpressure corresponds to an amount of energy that can be calculated with

5( 10 )

1bub bubV pE

= (22)

This energy can also be converted in a TNT equivalent. 1kg TNT has the energy of 4520 kJ. This is

the energy that is introduced in the model.

The corresponding energy and the introduced energy (both in TNT equivalent) are shown in Figure

41. The curves show in the logarithmic scale of the pressure an increasing dependency on the

pressure. The reference pressure in Figure 41 is not the overpressure in the bubble. The TNT

equivalent corresponding to the impulse-distance curve can also be transformed in a bubble

overpressure with the same energy. This pressure is used in the Figure 41 due to a better calculation

of the necessary bubble overpressure by a given charge.

0

2

4

6

8

10

12

14

1E+6 1E+7 1E+8Pressure = energy of the corresponding TNT equivalent (resulting pressure wave) [Pa]

Ener

gy [k

g TN

T eq

uiva

lent

]

Resulting pressure wave in TNT

Energy of overpressure in TNT

Figure 41: Comparison of the energy of the fitted curve with the energy of the bubble (models bubble3-bubble8)

56

The corresponding TNT-equivalent is divided by the TNT equivalent of the overpressure. This

division gives a value bub shown in Figure 42. This value bub shows a linear dependency also for

different sizes of the bubble, tested here with the radius of 0.5 m and 1.0 m. With this straight line it

should be possible to calculate an overpressure for a bubble with the volume Vbub for a given charge

of TNT. The idea is to describe an equation with the following form

( , )bub bubp f V W= (23) The equation of the line in Figure 42 can be described by this trend line

0.263 log( ) 2.41bub TNTp = + (24)

0

0.2

0.4

0.6

0.8

1

6 6.2 6.4 6.6 6.8 7 7.2 7.4Pressure = energy of the corresponding TNT equivalent [log Pa]

Ener

gy o

f ove

rpre

ssur

e/En

ergy

resu

lting

pre

ssur

e w

ave

lbub=0.5 mlbub=1.0 m

Figure 42: Factor bub

The following procedure can be used to get the overpressure of the bubble for a given TNT

equivalent.

1. Calculation of the energy of the detonation with

4520 /TNTe W kJ kg= (25) 2. Calculation of the overpressure energy of this amount of TNT with

( )1TNTTNTbub

epV

= (26)

57

3. Calculation of the factor bub with equation (24).

4. The pressure of the bubble can be calculated with

0TNTbubbub

pp p= + (27)

5. The values for the internal energy eint,bub and the density bub can be calculated by

multiplying the values of the uncompressed air with the factor fbub

0

bubbub

pfp

= (28)

With this method it is possible to calculate by a given bubble size and a given charge the

overpressure in the bubble.

58

6 Control Volume This model foresees the loading of a cut surface with a pressure depending on the AIRB-function

(see chapter 7). The difference to the model in chapter 7 is the loaded element which is here a fluid

element. Additional to the external forces (continua element) the flux between the CL3D element

and the fluid element is to be considered.

6.1 Flux between the CL3D and the fluid element The flux calculation for a fluid element is done by using the differences in the pressure, density and

the energy between the fluid element and the adjoined elements. Therefore, it is obligatory in the

case of AIRB to calculate the density and the energy for the CL3D elements, if they are adjoined

with a fluid element. As an assumption, an adiabatic equation of state will be used to calculate the

values.

00

1 1p p

=

(29)

The initial density 0 is used from the input. The pressure p in equation (29) can be calculated with the ideal gas equation resulting in

0 00

1 ( 1)p i = (30)

Then, the current density can be computed by

1/1

0

0( 1)p

i

=

(31)

The pressure p that exists at the surface is known and is calculated in the subroutine CL_AIRB by

adding the initial pressure (atmospheric pressure). Then, the current energy can be calculated with

the ideal gas equation.

( 1)

pi = (32)

6.2 Several models The model for the investigations with the compressed bubble is modified for calculations with a

control volume. Instead of the compressed bubble the air is loaded by a pressure-time function (5 kg

TNT) with the AIRB command. The models cv8 and cv9 are built so that all elements are almost

59

quadratic. Therefore, the size of the elements increases with the distance to the charge. Model cv6

uses the same thickness for all elements.

Figure 43: Model control volume (cv8)

Model lair

(m)

lb

(m)

TNT

equivalent

Elements in radial

direction

cv6 2.5 1 5 30

cv8 4.0 1 5 30

cv9 4.0 1 5 40

Table 11: Control volume models

The model cv6 has a coarse mesh and shows in the investigated region a sufficient representation of

the experiments (see Figure 44 and Figure 45). The models cv8 and cv9 are longer and the elements

are more conform. The coarse mesh cv8 gets very small pressures and impulses. The finer model

cv9 represent the experimental peak pressures from Kingery better. The peak pressure is

overestimated for larger scaled distances. No model of the models tested herein can represent the

impulse.

60

These models have also another problem if they are used for realistic calculations. The volume that

is cut away is missing in the model. The cut surfaces are additional boundaries with special

properties. Therefore, these models are here not more investigated.

0E+0

1E+6

2E+6

3E+6

4E+6

0.6 0.8 1 1.2 1.4 1.6 1.8 2

Scaled distance [m/kg1/3]

Max

. pre

ssur

e [P

a]

Kingerycv6cv8cv9

Figure 44: Maximum pressure for several control volume models

61

0

100

200

300

400

0.6 0.8 1 1.2 1.4 1.6 1.8 2

Scaled Distance [m/kg1/3]

Impu

lse

[Pa

sec]

Kingery 5 kgKinney 5 kgcv6cv8cv9

.

Figure 45: Impulse for several control volume models

62

7 Implementation of an Air Blast Loading Function

7.1 Motivation Calculations using the explosive with the JWL-equation may produce accurate results but need also

small elements. This results in small time step sizes and large computation times. There are several

possibilities to reduce this cost, as shown in chapter 3. One of these is the use of a load-time

function instead the fluid-structure interaction.

This load-time function can be used if reflections of the air blast wave do not have an influence and

if there is no need to consider the shadowing of the structure. The load-time function depends on

the size of the charge, the distance from the charge and some other conditions. The advantage of

this method is the reduction of the computation time.

Calculations with this method can be used to get an idea of the behaviour of the structure and to val-

idate results of a fluid-structure-interaction for special cases.

7.2 Used Function The modified Friedlander equation from Baker [2] (see equation(4)) can be used for the implemen-

tation. The parameters of this equation can be chosen from Kingery [14] or from Baker [2]. The

parameter b for the (decay) can be calculated with the knowledge of the maximum negative

pressure or with the knowledge of the impulse. The value of b, as calculated in chapter 2.2.5, will

be used here.

7.3 Implementation The load-time function is implemented as a new impedance. Impedances enable the input of bound-

ary conditions for special elements (CLxx) lying over the common elements. The command for the

impedance air blast wave (IMPE AIRB) allows the definition of a size of the charge, the origin of

the detonation and the starting time of the detonation. Different conditions with different parameters

are possible. Spherical and hemispherical explosion can be considered, also with reflection

conditions. The following conditions are possible by choosing the CONF parameter:

1 = spherical detonation (full space), reflection conditions, equation form Kingery

2 = spherical detonation (full space), no reflection conditions, equation form Kingery

3 = spherical detonation (full space), no reflection conditions, equation form Baker

4 = hemispherical detonation (half space), reflection conditions, equation form Kingery

63

5 = hemispherical detonation (half space), no reflection conditions, equation form Kingery

Changes are made in the following files

material_i_airb. This file includes two subroutines: MI_AIRB to read and check the input parameter and CL_AIRB to calculate the pressure at a certain time step for the chosen element.

In addition the maximum time step size is specified equal to the twentieth part of the duration of

the positive phase.

fl38. This file calculates for a FL38 element the flux between the fluid elements. The flux for the IMPE AIRB elements is calculated also in this file by using the formula in chapter 6.1.

cl3d and cl3i. The nodal forces resulting from the detonation pressure are calculated in these files.

7.4 Verification with Examples The implemented function has to be verified with different examples. The function for an air blast

wave in free air (without reflection) can be used for the control volume model (see chapter 3) and

can be compared with a model with the explosive implemented with the JWL-equation.

The reflected pressures can be validated with experimental and numerical data from the literature.

64

8 Mesh generation for LS-DYNA To compare the results of EUROPLEXUS with LS-DYNA an output is written that converts objects

in EUROPLEXUS towards an LS-DYNA input file. This file contains the nodes and the elements

of the objects. Different objects can be written in the same file with different part numbers for the

elements. There is not an output of the materials and the loads.

The mesh is written in a file called PXTOLS-DYNA.k.

65

9 References [1] Alia, A.; Souli, M.: High explosive simulation using multi-material formulations, Applied

Thermal Engineering 26, pp. 1032-1042, 2006. [2] Baker, Wilfrid E.: Explosions in the Air, University of Texas Pr., Austin, 1973. [3] Baker, W.E.; Cox, P.A.; Westine, P.S.; Kulesz, J.J.; Strehlow, R.A.: Explosion Hazards and

Evaluation, Elsevier, Amsterdam, 1983. [4] Birnbaum, Naury K.; Clegg, Richard A.; Fairlie, Gerg E.; Hayhurst, Colin J.; Francis, Nigel

J.: Analysis of blast loads on buildings, Preprint from Structures under Extreme Loading Conditions 1996, Montreal, Quebec, Canada, 1996.

[5] Brode, Harald L.: Numerical solutions of spherical blast waves, Journal of Applied Physics 26, No 6, pp. 766-775, 1955.

[6] Casadei, F.: Use of EUROPLEXUS for Building Vulnerability Studies. Progress Report 1, Technical Note I.05.50, July 2005.

[7] Casadei, F.; Anthoine, A.: Use of EUROPLEXUS for Building Vulnerability Studies, Progress Report 2, Technical Note I.006972, March 2007.

[8] CASTEM 2000, Guide d Utilisation, CEA, France, 1990. [9] Clutter, J.; Keith, Stahl, Michael: Hydrocode simulations of air and water shocks for facility

vulnerability assessments, Journal of Hazardous Materials, 106A, pp. 9-24, 2004. [10] Dobratz, B.M.; Crawford, P.C.: LLNL Explosives Handbook: Properties of Chemical Explo-

sives and Explosive Simulants, University of California, Lawrence Livermore National Laboratory, Report UCRL-5299, Rev. 2; 1985.

[11] EUROPLEXUS, Users Manual, online version. [12] Fairlie, Greg E.: Efficient analysis of high explosive air blast in complex urban geometr