Transient Response of a Plate-Liquid System Under an Aerial Detonation : Simulation and Experiments

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Transient Response of a Plate-Liquid System Under an Aerial Detonation : Simulation and Experiments

André Langlet, Mame William-Louis, Grégory Girault, Olivier Pennetier

To cite this version:

André Langlet, Mame William-Louis, Grégory Girault, Olivier Pennetier. Transient Response of a Plate-Liquid System Under an Aerial Detonation : Simulation and Experiments. Computers and Structures, Elsevier, 2014, 133, pp.18-29. �hal-00942232�

Aerial Detonation : Simulations and Experiments

André Langlet a,∗

, Mame William-Louis a

, Grégory Girault b

, OlivierPennetier a

a

Univ. Orleans,Enside Bourges, PRISMEEA4229

F45072,Orléans,Frane

b

Univ. BretagneSud,LIMATB,RueSaint-Maudé, Centre de Reherhe

BP92116,56321 Lorient,Frane

Abstrat

This paper presents a mixed numerial approah to model the blast waves gener-

atedbythe detonation ofaspherialstoihiometrimixtureof propane andoxygen,

impatingaplate-liquidsystem. Theproblemissplitintotwoparts. Therstalu-

lationpartreliesonthemodelingoftheblastloadanditspropagation. Over-pressure

distribution,inthispart,ispresented andrevealsaverygoodlevelofagreementwith

experimentalresults. Thetime andspae salesof theblast loaddata must beom-

patiblewiththeplate-liquidsystem. Thisompatibilityisensuredbyanappropriate

spatio-temporal interpolation tehnique. This tehnique is presented and its ee-

tiveness and auray are demonstrated. The seond part onsists in modeling the

response of the oupled plate-liquid system under the numerial blast load model.

Experimentsatredued sale are arriedoutintwoongurations inordertoassess

the eetiveness of this mixed numerialapproah. Convining results are obtained

and disussed.

Keywords: Blast wave, Plate-liquidsystem, Fluid-struture interation, Cartesian

methods, Redued sale experiments

∗

orrespondingauthor,63Av. deLattredeTassigny,F18020BourgesCedex, Frane

Emailaddress: andre.langletuniv-orleans.fr(AndréLanglet)

t

^{: Time}

[s]

ρ

ℓ ^{: Fluid mass density}

[kg.m

⁻³

]

χ

^{: Fluid bulkmodulus}

[Pa

⁻¹

]

c

ℓ ^{: Speed of aoustiwaves inthe liquid}

[m.s

⁻¹

]

ρ

^{: Plate mass density}

[kg.m

⁻³

]

E

^{: Plate Young modulus}

[Pa]

ν

^{: Plate Poisson's ratio}

h

^{: Plate thikness}

[m]

I = h

³

/12

^{: Moment of inertiaof the} ross-setion

[m

⁴

] G = E/[2(1 + ν)]

^{: Shear modulus}

[Pa]

D = EI/(1

−

ν

²

)

^{: Flexural modulus}

[Pa

·

m

⁴

]

r

plate ^{: Maximum radiuson the plate}

[m]

1. Introdution

Inthis workwestudythe mehanialeet ofanexplosioninairoveraatplate

restingon aquiesent uid. The response of the plate - liquidsystem isdetermined

bytheuidstrutureinterationwhihdevelopsveryrapidlyowingtotheblastwave

of the explosion.

In pratie, blast loads arise when solid or gas explosives detonate due to the

ignitionofhighexplosive materials. There isarealneed tounderstandthe eetsof

suhloads onstruturesoronpersons,forexample,inthe eldofriskandindustrial

safety, risksprevention against terroristattaks, orin militaryappliations.

Coneptually, the explosion phenomenon an be broken down intothe following

phases: (i)thedetonationproessintheexplosivemedium,(ii)theshokpropagation

in the surrounding environment, (iii) the shok reetion by an obstale wall, (iv)

theresponseoftheimpatedstrutureandoftheuidsand/ormaterialsonnedby

thestruture. These4phasesorrespondto4modelingstepsinvolvingmultiphysial

simulations: phase(i) isareative ow; phase(ii) dealswith unsteady ompressible

uid ow; phases (iii)and (iv) involve uid struture interations (FSI).

Blasts are reated by underwater explosion (UNDEX) and in air explosions

(INEX). The major dierene between UNDEX and INEX is due to the dynamis

ofthe gas oreproduedbythe detonationof highexplosives. In INEX the pressure

of the gas ore dereases (as the detonation produts expand) until it reahes the

atmospheripressure. InUNDEX,thegasprodutsformabubblewhihexperienes

tion is unavoidable; it has been studiedby Geers & Hunter (2002),Sprague (2002),

Galiev (1996), among others. Cavitation must be onsidered at the gas-liquid and

at the uid-struture interfaes. Experimental tehniques dediated to avitation

studiesare presented by Herbert et al.(2006). The modelingof UNDEX and INEX

must desribe the diering nature of the phenomena due to the diering properties

of the media inwhihthe explosiontakesplae.

Explosionsinairand theireets onstrutures havebeenwidelyinvestigated. A

reviewonerningvariousaspetsoftheresponseofblastloadedplateswaspublished

by Rajendran&Lee (2009). Thereare two majorapproahes forinvestigatingblast

eets onstrutures.

Firstly,studiesaddresstheexplosionphenomenonanditsouplingwiththestru-

ture. Numerial methods are elaborated to desribe the shokwave ignition and

propagation. For example, the equations of the reative ow an be solved using

the Eulerian multimaterialformulation with a nite element disretization (Alia &

Souli, 2006). Thus, the interation between the blast and the struture an be de-

sribed within long durations afterthe beginning of the explosion (Zakrisson et al.,

2011). However, these methodologiesrequire alarge amount ofoptimized numerial

parameters as well as very long omputational times. Consequently high frequeny

phenomena are diult to apture aurately. Simpliations might be hosen, as

done by Kambouhev et al. (2007) who applied the rigid-body assumption for the

plate but, nevertheless, fully solved the FSI ina Lagrangian frame.

Seondly, only the mehanial response is sought without modeling the blast

dynamis. Therefore, the loads are given as input funtions suh as deaying expo-

nential, onstant pulses, the parameters of whih are tuned to math experimental

data. Another kind of input data is the well-known US Army Tehnial Manual

ConWep ode providing empirial blast loading funtions Neuberger et al. (2007),

Longère et al. (2013). Here, the key point is to ompute the response under suh

loads inluding nite transformations see for example Langdon et al. (2013). The

simulations inlude user-dened materials (or UMAT) Longère et al. (2013) pro-

grammed in ommerial odes, mainly ABAQUS, LSDYNA, EUROPLEXUS. Dif-

ferentstrutures an be studied, rangingfroma simple plate, Jaintoetal. (2001)

and Neuberger et al.(2007), or a sandwih panel (Karagiozova et al.,2009) to very

omplex assembly suh as a soldier helmet with omposite and polymer materials

(Grujii etal.,2010) orlaminatedglass (Larher et al.,2012).

Theaforementionedommerialodesare indispensabletoolsforsolvingdynam-

is problems, espeially with blasts and FSI, in omplex real systems (suh as ve-

hiles, planes, ships, plants) for whih a long time and global response is sought,

as willbe seen inthis work, if very speializedaspets of the dynami response are

investigated, suh as the early response, it might be more appropriate to develop

fully ontrolled numerial odes whih allow fousing the model on high frequeny

waves. In addition, fully ontrolled odes (or white box) are better options than

ommerial odes for areful omparison with deliate and diult experiments, as

isthe ase inthe present work.

The interation of the impated struture deformation with the blast must be

taken intoaountif the solidwallexperienes largedisplaements,whihan inter-

atsigniantly withthe ow, (Børvik etal.(2009)). Suh loaddurations may exist

if explosions our in onned zones and generate planar blast waves (e.g. tunnels,

losed rooms). On the ontrary, a wall exposed to an aerial explosion is loaded by

a moving pressure front. In this ase, the rst movements of the target are small

inamplitude, unableto modify the shok reetion; large displaements may our

when the loadingis over.

The exlposive used in the present work reates a soure-explosion. Therefore,

the inident waves are spherial, and the wave reetions are due mainly tooblique

inident waves. Aording tothe studiesby Bakeret al. (1973),Kinney (1962), the

mehanism of this reetion an be aurately desribed. When the wave reahes

the plate, the inident angle is zero. Kinney has shown that if this angle is lower

than a ertain limit, the reetion is regular. Beyond this limit the reeted wave

annotmaintain the owparallelto the wall. Then, itfollowsthat the inidentand

the reeted waves oalese in a triple point, and form a third shok wave whih

is detahed from the wall the Mah reetion. This shok is stronger and faster

thanthe inidentshok. Thedistanebetweenthetriplepointandthewallinreases

as the reetion phenomenon goes on. For spherial shok waves, the lous of the

triple point forms a urve away from the wall. The reetion of a shok wave on

a struture is a omplex phenomenon. Reetion oeients are inuened by the

shok harateristis and the properties of the atmosphere in whih the reetion

takes plae Wadleyet al.(2010).

From the pointof viewof strutural dynamis,the onsideredblast pressure isa

movingload, from itsonset toits end. When the blast sweeps a wall, the rise time

of the pressure is very short (a few

µ

^{s for small sale} detonations assoiated with over-pressure about

10

^{5 Pa) and ours over a very narrow distane. This is why}

the moving pressure front is usually approximated by a disontinuity in analytial

studies. Thefrontstartstomovewithsupersoniveloities(relativelytotheaousti

wave inthe uid or inthe struture) whih rapidly derease to subsoni veloities.

The rst partiularity of the present work is that the transient response of the

wavesareanalyzedbeforeanyreetionoursattheboundaries. Theseondparti-

ularityisthattheresponseisstronglyinuenedbytheouplingwiththeunderlying

uid. Indeed, in suh very short times the uid reats on the struture due to its

ompressibility,andalsowithanaddedmasseet. Inthetwomedia,the smallper-

turbationstheorymaybeapplied,namely,elastiwavesandaoustiwavesformthe

present response observed withoutboundary inuenes. Researhing early time re-

sponsesmayrelyonsomehypotheses. Forexample,Sprague &Geers(1999)applied

partial series losure for solving the response of a spherial shell under a spherial

shok. Here, theearlyresponseis separatedintoalosed-formportion(representing

a planar wave approximation for the uid-shell interation), and a omplementary

mode-sum portion. Unlike suh an approah, we have made a diret simulation,

whih beneted from some spei features of the fast dynami response, as it will

appear in setion6. Whileavitation is anunavoidable issue in UNDEX, it is a re-

mainingquestiontodetermine wheter avitationoursbehind the plateonsidered

in this study. In fat the pure aousti uid model may lead to negative pressure

whih may be less than the hydrostati pressure; this suggests going further in the

modeling. However, in the present work, we have foused the analysis on the very

earlystagesof thesystem response observed inlaboratoryexperimentswith redued

sale explosions. The understanding of the oupled plate response and the model-

ingboth rely on previous works we have done onanalytial stationary responses of

the plate system(Renard et al. (2003), Renard & Langlet (2008)), with an aousti

modelfor the uid. This is why avitation was not onsidered in the paper. When

omparingexperimentswithsimulations,arewastaken toverifythatthe numerial

uidpressure neverfell belowthehydrostatipressure withmoderateexplosions. In

the experiments, the explosive energy was limited to that used in the modeling. In

thetime onsideredthe response takesthe formofwavesundisturbed by thebound-

aries. This is why real omplex uid struture systems may be simplied sine

only elements of them are set into movement. This is an additional argument for

designing in-house numerial odes rather than engaging full diret modeling with

heavy ommerial odes.

In the present work, the numerial simulation deals with both the explosion

and the response of a plate-liquidsystem. Experimental results with redued sale

detonationshave onrmedthe resultsofthis simulation. Theexplosionissupposed

to our at a given height over the plate resting on the liquid. Phases (i), (ii) and

(iii) are solved numerially by diret simulations (CFD modeling) of the hemial

energy release during the detonation, and of the propagation of the shok wave in

the atmosphereabovethe plate. Phase(iv)is solved by anexpliit shemebased on

and the uid. The blast pressure load is applied to the plate-liquid model as the

omputationsproeed,so that the load faithfullyreproduesthe pressure variations

oftheexternaloweld. Theobjetiveistoapture the highfrequeny omponents

of the waves with aeptable auray and reliability. One assumption is that the

aousti oupling with the atmosphere ahead of the load front is disregarded: only

the ouplingwith the uid supporting the plate ismodeled.

The hoie of this partiular experimental onguration, onsisting of an aerial

explosion over a plate, is motivated by the useful results it provides, both for the

modelingapproah and for the engineers.

2. Numerial model for the blast load

The dynami and thermal behaviors of the propagation phenomena of a blast

wave are governed by the unsteady transport equations for mass, momentum and

energy. Visous and thermal diusion proesses may play a signiant role in the

overalltransport phenomena,butthey arenot inludedinthe onventionaldiusion

termsleadingtotheNavier-Stokesequations,asthiswould requireveryne spatial

disretization in the regions with strong veloity and temperature gradients. The

orresponding grid sales would be extremely small and, thus, would involve mesh

sizes beyond the apaity of the omputers urrently available. Consequently, the

eetsofvisousandthermaldiusionareimplementedasglobalsouretermsadded

tothe transportequations. ThisunsteadyEuler three-dimensionalproblemissolved

by a software developed in-house. The numerial method involves an unstrutured

nite-volume ell-entered approah that ouples the lassi seond-order upwind

sheme with the two-step Van Leer time-expliitintegration sheme. This oupling

yields a seond-order aurate-in-spae-and-time method. In order to prevent the

numerial osillations that an our in regions with strong gradients, the lassial

minmod limiter is used. Initially, the ow is assumed to be at rest throughout the

three-dimensional domain, exept inside the sphere (of radius

r

b^{) that ontains the}

explosiveharge. Inthissphere,the owisdisturbedbytheblastwavealulatedby

the use of a one-dimensional spherial proedure that is made ompatible with the

three-dimensional mesh thanks to the 1D - 3D remapping algorithm. In this zone,

the hot detonation gases obey the Jones - Wilkins- Lee (JWL) law. Details of this

mixedmethod,ombining1Dand3Dartesianmethods, aredesribedinBenselama

etal. (2009).

As an illustration, a blast wave impating a rigid wall is depited in Fig. 1 for

dierent times. This gureshows theinidentwavearriving attime

t

A

(0)

^{aswell as}

point (T).

3. Numerial model of the plate-liquid system

3.1. The oupled system

Thephysialsystemisrepresented inFig. 2. Theliquidinontatwiththeplate

is assumed to be invisid and ompressible. Its ompressibility must be taken into

aountsine itis submitted tofastloading. The veloity of the soundwaves inthe

uid is:

c

ℓ

=

p

1/(ρ

ℓ

χ)

^{. The uid governing equations are derived under the small}

perturbations hypothesis. The Helmoltz equations an be derived by ombining (i)

the onstitutive equation of the uid, (ii) the equation of motion, (iii) the mass

onservation equation,and by using the veloity potential

ϕ

^:

∂

²

ϕ

∂t

²

= c

²_ℓ

∆ ϕ

⁽¹⁾

The uid pressure (denoted by

p

i^)is:

p

i

= ρ

ℓ

∂ϕ

∂t

⁽²⁾

TheequationsofmotionfortheplatemaybelassiallyderivedfromtheHamilton's

priniplewith appropriateformulationsof the virtualworksof respetively: internal

fores (tensile and ompressive fores, bending moments, shearing fores), and of

the external loads. The Mindlin Reissner's assumption Mindlin (1951) is retained

for the plate dynamis, whihinludes shear deformations. Also, the angularross-

setion rotation,

ψ

^{(Fig. 2)is assoiated with its properinertia. This} improvement of the plate theory is neessary for small wavelengths, Mindlin (1951). Under this

assumption, Girault (2006) wrote an appropriate formulation in order to take into

aountthe nonlinear eets ofthe membranestresses inthe plate, whihmay arise

for high amplitudeloads.

Sine the blast load is axisymmetri, the response of the plate needs to be sought

too. Therefore the equations of motion are expressed in ylindrialoordinates, (

r

^,

θ

^, respetively the radialand polaroordinates) retainingonlythe radial distane

r

forthe spatialdependene. Then, inapolaroordinatesystem, foranaxisymmetri

plateandunder theMindlin-Reissnerhypothesis, threeomponentsare neessary to

desribe the displaement of partiles belonging to the plate middle-surfae: radial

displaement

u

^{, out of plane} displaement

w

^{, and rotation}

ψ

^{of the normalvetors}

tothe undeformed middle surfae in the (

r

^,

θ

^{) plane. The} axisymmetri plate with thikness

h

^{is submitted to the external load}

p

e ^{and to the uid pressure}

p

i^{. Thus,}

the motionof apartilebelongingtothe plate neutralsurfae is modeledby:

rρ h ∂

²

u

∂t

²

= ∂(rN

rr

)

∂r

−

N

θθ

1

12 rρ h

³

∂

²

ψ

∂t

²

= ∂(rM

rr

)

∂r

−

M

θθ

+ rQ rρ h ∂

²

w

∂t

²

= ∂(rQ)

∂r +

rN

rr

∂w

∂r

−

p

e

r + p

i

r

(3)

where (per unit length):

N

rr^,

N

θθ^{, are} respetively the radial and hoop membrane fores,

M

rr^,

M

θθ ^{stand forthe radial and hoopmomentsand}

Q

^{designates the shear}

fore. These fores and moments are obtained by integration of the stress ompo-

nents over the plate ross setion.

It has been shown by Girault (2006) that the ombination of the uid added mass

eet withthe uid elastiresponse onsiderablyenhanes the stiness ofthe plate-

liquid system. Therefore, the oupling with the uid does not allow the onset of

nonlinear eets during the loading by the blast wave. So, only the linear terms of

the plate equations are retained in the present study. Negleting terms for mem-

branestresses

N

rr^,

N

θθ^{,theprevious equationsareexpressed using the}displaement omponents:

M

rr

= D ∂ψ

∂r + ν ψ r

M

θθ

= D ψ

r + ν ∂ψ

∂r

Q = κGh ∂w

∂r

−

ψ

(4)

It results inthe followingseond order partialdierentialequations:

ρ I ∂

²

ψ

∂t

²

= D ∂

²

ψ

∂r

²

+ 1 r

∂ψ

∂r

−

ψ r

²

+ κ Gh ∂w

∂r

−

ψ

ρ h ∂

²

w

∂t

²

= κ Gh ∂

²

w

∂r

² −

∂ψ

∂r + 1 r

∂w

∂r

−

ψ r

−

p

e

+ p

i

(5)

Ashear fator

κ = 5/6

^{isintrodued,whihisrelated toanon onstantshear stress}

distributionovertherosssetion. Forthepresentplatetheorythetwoharateristi

veloities are

c

s ^and

c

p ^{the veloity of} longitudinal waves and of shear waves, respetively:

c

p

=

s

E

ρ(1

−

ν

²

)

⁽⁶⁾

c

s

=

s

κ G

ρ

⁽⁷⁾

The oupling onditions between the plate and the liquid are ensured by the onti-

nuity of normal fores and displaements at the interfae. Therefore, the following

ondition must beveried atany time:

∂w

∂t =

−

∂ϕ

∂z

_z=0

(8)

Introduing the radius of gyration

r

0

= h/

√

12

^{as the} harateristi length, and

t

0

= r

0

/c

p ^{as the} harateristi time, all variables and funtions are made non- dimensional. Nondimensionaltermswillbewritteninapitalletters. Notethat

ψ = Ψ

^{athomologouspoints. Three}non-dimensionalratios aresuienttoparameterize the ouplingbetween the plate and the liquid:

θ = c

s

c

p

(9)

δ = c

ℓ

c

p

(10)

µ = ρ

ℓ

ρ

√

12

(11)

Then,the unknown funtions are

W = W (R, T )

^,

Ψ = Ψ(R, T )

^and

Φ = Φ(R, Z, T )

^.

The axisymmetri problem to be solved an be expressed in the following non-

dimensionalform:

∂

²

Ψ

∂T

²

= ∂

²

Ψ

∂R

²

+ 1 R

∂Ψ

∂R

−

Ψ R

²

+ θ

²

∂W

∂R

−

Ψ

(12)