1 

EFFECTOFPROJECTILENOSESHAPEONBALLISTICRESISTANCEOFINTERSTITIAL‐FREESTEELSHEETS

K.M.Kpenyigba1,T.Jankowiak2,A.Rusinek1*,R.Pesci3,B.Wang4

1NationalEngineeringSchoolofMetz,LaboratoryofMechanics,Biomechanics,Polymersand

structures,1routed'ArsLaquenexycs6582057078MetzCedex3,France

2InstituteofStructuralEngineering,PoznanUniversityofTechnology,Piotrowo5,Poznan,Poland

3ENSAM‐ArtsetMétiersParisTechCERofMetz,LEM3UMRCNRS7239,4rueAugustinFresnel

57078MetzCedex3,France

4SchoolofEngineeringandDesign,Bruneluniversity,UxbridgeUB83PH,UK

Abstract

In this paper an experimental and numerical work is reported concerning theprocess of perforation of thin steel plates using different projectile nose shapes. Themaingoalistoanalyzehowtheprojectileshapemaychangetheballisticpropertiesofmaterials.Awiderangeofimpactvelocitiesfrom35to180m/shasbeencoveredduringthetests.Alltheprojectileswere13mmindiameterandthetargetswere1mmthick,assuchtheprojectilecanberegardedasrigidandthetargetsheetswereofinterstitial‐free(IF)steel.Themassratio(projectilemass/steelsheetmass)andtheratiobetweenthespanof thesteel sheetand thediameterof theprojectilewerekeptconstant,equal to0.38 and 3.85 respectively. To define the thermoviscoplastic behaviour of the targetmaterial, the Rusinek‐Klepaczko (RK) constitutivemodel [1] was used. The completeidentification of the material constants was done based on a rigorous materialcharacterization. Numerical simulations of some experimental tests were carried outusinganon‐linearfiniteelementcodeABAQUS/Explicit.Itwasfoundthatthenumericalmodelsareabletodescribethephysicalmechanismsintheperforationprocesswithagoodaccuracy.

Keywords:perforation,failure,impact,ballisticproperties

1. Introduction

Structuralimpacthasbeenstudiedforalongtime,andsuchaproblemisknownto be complex, both from experimental, analytical and numerical points of view. Themost important parameters affecting the ballistic capacity of a target plate are theprojectile (size, shape, density and hardness), the target plate (hardness/strength,ductility, microstructure and thickness) and the actual impact conditions (such as

*Corresponding author: kpenyigbabenedict@yahoo.fr

Phone: +33 3 87 34 69 30 

 

 2 

impactvelocityandprojectileincidenceangle).Thispaperconcentratesontheballisticcapacityof targets,with thevarying factorswillbeing thenoseshapeof theprojectileandthenormalimpactvelocity.Thenoseshapeofhardprojectileshasa strong influenceon the failuremodeand theballisticlimitofatarget[2–4].Corranetal.[5]investigatedtheeffectofprojectilemass,noseshapeandhardnesson thepenetrationof steelandaluminumalloyplates.Bluntandconicalprojectilesof12.5mmdiameterwereimpactedonplates1.3to5.9mmthickandconsideringavelocityrangingfrom50to250m/s.Themassoftheprojectilevariedfrom 15 to 100 g. They observed that the ballistic limit of the plate changes withprojectilemassandnoseshape.Goldsmithetal.[6]experimentallyinvestigatednormalimpactofcylindro‐conicalandbluntprojectiles intoaluminumandsteelsheet targets.The thicknessofaluminumtargetsvaried from1.78mmto25.4mmand thatof steelplatesvariesfrom1mmto19mm.Itwasobservedthatthenoseshapeofprojectilehadinsignificant effect on the ballistic limit. Ipson and Recht [7] found that conicalprojectilespenetratedthetarget ina lessefficientwaythanbluntprojectileswhenthetargetthicknesswasmoderate.However, forathinandthicktarget,anoppositetrendwas observed by the authors. This is due to the different modes of failure involved.ExperimentsinvestigationcarriedoutbyZukasetal.[8]showedthattheprojectilewithblunt nose shape has higher ballistic limit velocity. A more recent work done byKpenyigbaetal.[9]ontheimpactofblunt,conicalandhemisphericalshapeprojectileson 1mm thick mild steel sheets indicated that the ballistic limit is higher forhemisphericalshapeprojectile,followedbyconicalandbluntrespectively.Theauthorsalsoshowedthatthebluntprojectilefailedthetargetbyplugejectionduetoaprocessofhighspeedshearing,theconicalprojectilecausedfailurebypetalingduetoaprocessofpiercing and the hemispherical shape projectile led to radial hole expansion inducingnecking and radial cracks. Backman and Goldsmith [3] concluded that blunt shapeprojectiles caused failure through plugging, wedge projectiles by ductile holeenlargement and sharp nosed projectiles by petaling. Borvik et al. [2,10] studiedexperimentally and numerically the perforation of 12 mm thick Weldox 460 E steelplates and showed that the blunt projectiles caused failure by plugging, which wasdominatedbyshearbanding,whilehemisphericalandconicalprojectilespenetratedthetargetmainlybypushing thematerial in frontof theprojectileaside.Guptaetal. [11]reportedthatthefailureinthinductiletargetsoccurredthroughshearpluggingbybluntprojectiles, petal forming by ogive nosed projectiles and tensile stretching byhemisphericalprojectiles. Thehighvelocityimpactsofthehighstrengthprojectilesintomonolith 6 mm and sandwiched plates 2x6 mm of 45 and of Q235 steels wereinvestigated[12].Theresultspresentthenoseshape(ogivalorblunt)influenceontheballisticresistanceofthestructure.Theauthorsalsopresenttheinfluenceoftheorderof the material in sandwiched structures. The available experimental and numericalinvestigations on the perforation and penetration of ductile metal targets by rigidprojectiles aremostly restricted to a few specific nose shapes such as hemispherical,conical,ogivalandblunt.Inthiswork,someexperimentaltestswerecarriedoutonthinIFsteeltargetsinordertostudyindetailtheeffectofprojectilenoseshapeinstructural

 

 3 

impact at sub‐ordnance velocities.Moreover, an additional investigationwasmade tostudy the influence of a double‐nosed stepped cylindrical projectile. Based on theexperimental results, the ballistic curves were plotted for each projectile shape.Numerical simulations using a finite element code ABAQUS/Explicit were performedandtheresultsobtainedallowpredictionproperlythecompleteprocessofperforation.

2. Experimentalmaterialbehaviordescription

Thematerial studied is a 1mm thick sheet of a deep‐drawingquality IF steel. The IFsteel was developed to obtain an optimized compromise between the mechanicalstrength and the formability due to specific metallurgy without interstitial elements(interstitial‐free). It ismainlyusedforstructuralcomponentssubjecttofatigueimpactloading.Thechemicalcompositionofthematerial(inweight%)isgiveninTable1.Thissteelhasaferriticstructurewithmoreorlessequiaxialgrainsabout25μmindiameter.

Table1.ChemicalcompositionoftheIFsteel(weight%).

C S Mn P Si Cu Ni Cr Al V Sn

0,0018 0,007 0,095 0,009 0,006 0,026 0,015 0,023 0,06 0,001 0,003

Quasi‐static tensile tests were conducted on a universal test machine (eg.

Instron)anddynamictensile testswereperformedona fastservo‐hydraulicuniversalmachine (eg. Zwick). Awide range of strain rateswas covered at room temperature,

4 1 110 250s s .Thegeometryanddimensionsofthetensilespecimensusedinthe

characterization are given in Fig. 1. The thickness of the specimens is 1t mm . To

analyzethepossibleanisotropyofthematerialintheplaneofthesheet,thetensiontestswereperformedintwodirectionscomplementarytotherollingdirection: 1 45 and

2 90 .

Fig.1.Geometryanddimensionsofthetensilespecimens,Rusineketal.[13].

 

 4 

Rusineketal.[13,14]showedthatthesedimensionsofthetensilespecimens(Fig.1)areoptimum,whichallowstoachieveaproperlevelofstresscoupledwithsufficientductility.Theauthorshighlightedthatthisspecimendesignallowstoavoidgeometricaldisturbancesthatfrequentlytakeplacewhensmallsamplesareusedtoreachveryhighstrain rates during testing. For all tests performed, the true stress‐strain curveswereobtaineduptoincipientnecking.

ThecurvesintheFig.2showthattheeffectofrollingdirectionisnotinsignificantonthemacroscopicbehaviorof thesteelsheet.Accordingtothisresultcoupledtothemicroscopic observation, the isotropy is assumed in the description of the materialbehavior.

0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Material: IF steel

T0 = 300 K

0°45°90°

Tru

e st

ress

,

(M

Pa)

True strain, (-)

s-1.

Fig.2.Comparisonofquasi‐statictensileflowcurvesinthreedirections:0°aswellas45°

and90°.The true stress‐strain curves at different strain rates are shown inFig.3. The

firstobservationisanincreaseoftheflowstressandtheyieldstressasthestrainrateincrease.Forahighstrainrate,thecurveschangeshapesparticularlybythepresenceofathermalsofteningduetolocaltemperatureelevationduringthetests.Theloadingtimeis short and the temperature does not have time to dissipate through the specimen.During these tests, neckingwhichdiffuses into the specimen to cause the final failure

occursmuchearlierthanfortestsat lowstrainrates.Forthestrainrateof1250 s ,

themacroscopicmaterialbehaviorissubstantiallydifferentatthebeginningofloading.A significant peak stress (450 MPa) is observed. This peak is related to themicrostructureofthematerialandmoreparticularlytothedislocationdensity.Similarbehavior has been reported in [15,16] or in [17] where the level of carbon on themacroscopicbehaviorhasbeenstudied. Itwasobservedthatadecreaseof thecarbonlevelwasdissipating thepeak stress at thebeginningof loading.There is a relatively

 

 5 

high hardening for quasi‐static solicitation andwhich decreaseswhen the strain rateincreases.TheIFsteelhasapositivesensitivitytostrainrateasmostmaterialswithbcccrystallographicstructure.Thissensitivitytothestrainrateoftheflowstressisdefinedby:

,log

T

(1)

0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

10-4 s-1

10-3 s-1

10-2 s-1

1 s-1

100 s-1

250 s-1

Tru

e st

ress

,

(M

Pa)

True strain, (-)

Material: IF steel

T0 = 300 K

Fig.3.ComparisonoftruestressversustruestraincurvesofIFsteelatdifferentstrainrates.

It can be observed that the strain rate sensitivity at room temperature is not

constant,Fig.4.ThreecharacteristicregionswithdifferentsensitivitytothestrainratewerereportedbyCampbellandFerguson[18].

Inthenextsection,theconstitutiverelationusetodescribethematerialbehavior,

the comparison between the experimental data and analyticalmodeling are reportedanddiscussed.

 

 6 

0

100

200

300

400

500

10-5 0.0001 0.001 0.01 0.1 1 10 100 1000

( s-1 ).

Tru

e s

tres

s,

(M

Pa)

Material: IF steel

T0 = 300 K

= 0.1

Fig.4.Flowstressevolutionasafunctionofthestrainrateatroomtemperaturefora

strainlevelimposedof 0.1 .

3. Thermoviscoplasticmodelingofmaterialbehavior

Several constitutive relations are available in the literature and used in finiteelement code to predict the distribution of plastic deformation and the failure instructures impact. Some of them are limited to perfectly plastic approximations withsomevisco‐plasticity.Moresophisticatedversions,suchasthosethattakeintoaccountthe effect of the strain rate, temperature and hardening on the flow stress are alsoavailable nowadays. In this work, the RK constitutive relation is used. The model ispartially based on the theory of dislocations and its phenomenological description isbasedontheadditivedecompositionofthetotalstress[19]:

0

( )( , , ) ( , , ) ( , )

E TT T T

E (2)

whereeachtermofthepreviousexpression,Eq.(2),isdefinedbelow.Themultiplicativefactor 0( )E T E definesYoung’smodulusevolutionwithtemperature

[20]:

0( ) 1 exp 1 0m

m

TTE T E T

T T

(3)

where 0E , mT and * arerespectivelytheYoung’smodulusat 0T K ,themelting

temperatureandthecharacteristichomologoustemperature.Thisexpressionallowsto

 

 7 

definethematerialthermalsofteningdependingonitscrystallattice[21].Inthecaseofbccmetalslikesteel * 0.6 .

Theinternalcomponentofthetotalstresscalledtheathermalstressisdefinedbythefollowingequation:

( , )0( , , ) ( , )( )n TT B T

(4)

where, ( , )B T   is  the modulus of plasticity, ( , )n T   the hardening coefficient (

( , )B T and  ( , )n T dependonthestrainrateandtemperature)and 0  is thevalueofstrain

correspondingtotheyieldstressatagivenstrainrateandtemperature.

The authors propose the following formulations to describe the modulus ofplasticityandstrainhardeningexponent:

max0( , ) log 0

m

TB T B T

T

(5)

0 2min

( , ) 1 logm

Tn T n D

T

(6)

where 0B isthematerialconstant, proportionaltotemperaturesensitivity, 0n

thestrainhardeningexponentat 0T K , 2D thematerialconstant, min thelowerstrain

ratelimitofthemodeland max themaximumstrainratelevelacceptedforaparticular

material.TheMacaulayoperatorisdefinedasfollows: 0, otherwise 0if .

Theeffectivestress ( , )T istheflowstresscomponentwhichdefinestherate

dependent interactions with short range obstacles. It denotes the rate controllingdeformation mechanism from thermal activation. At temperatures T > 0 K, thermalactivationassiststheappliedstress.

The theory of thermodynamics and kinetics of slip [22] is founded on a set ofequations which relate activation energy G , mechanical threshold stress (MTS),applied stress , strain rate , temperature T and determined physical materialparameters. Based on such understanding of the material behavior, Rusinek andKlepaczko [1] derived the following expression, Eq. (7). This formulation gathers thereciprocity between strain rate and temperature by means of an Arrhenius typeequation:

 

 8 

max0 1( , ) 1 log

m

m

TT D

T

(7)

where 0

istheeffectivestressat 0T K , 1D thematerialconstantand *m the

constantdefiningthereciprocitystrainrate‐temperature[19].

In the case of adiabatic conditions of deformation the constitutive relation iscombined with the energy balance principle, Eq. (8). Such relation allows for anapproximation of the thermal softening of the material by means of the adiabaticheating.

f

e

0 0p

T T T T dC

(8)

where istheQuinney‐Taylorcoefficient, thedensityofthesheetsteel, pC the

specificheatand thefailurestrainlevel,Table2.

SubsequentlyastraightforwardmethodwasproposedbyRusinekandKlepaczko[1,23] for themodelcalibration. Itallowsdefining themodelparametersstepbystep.Contrary to other constitutive descriptions, the procedure does not involve a globalfitting. It must be noticed that the constants are defined using physical assumptions[1,23].

Themainstepsnecessaryfordefiningthemodelparametersare:

It isassumedthatat lowstrainrate 10.001s , thestresscontributiondueto

thermalactivationisreducedandinthiscasethefollowingrelationis imposed,Eq. (9).Thus, it ispossible todefine theconstant 1D dependingon themelting

temperature mT .

1300 ,0.001

1

max1

( , ) 0

300log

0.001

K s

m

T

DT

(9)

Therefore,atroomtemperatureandunderquasi‐static loadingthecontribution

ofthethermalstresscomponenttothetotalstressiszero.Theoverallstresslevelis defined by Eq. (10). Fitting this equation to experiment results, a firstestimationofB andn canbefound.

 

 9 

00

(300)( ,0.001,300) ( ) 0

,

nEB

E

B n first approximation

(10)

 

Next, itisassumedthattheincreaseoftheflowstresscausedbythestrainrate

augmentisduetothethermalstress *( , )T .Thus,thestressincreaseisdefined

as follows, Eq. (11). Fitting of Eq. (11) to experimental results for an imposedstrain level,makes it possible to determine thematerial constants * and *m .The strain level should be assumed not more than 0.1 in order to guaranteeisothermal condition of deformation. For larger strain values, adiabaticconditionsmay inducea thermalsofteningonthematerialandinsuchacaseadecreaseofthestrainhardening.

1 10.001 0.001

0

( , )

,

s sT

m

(11)

 

Finally, combining the complete equation of the total stress (Eq. 2) with theexperimentalresultsandreplacingtheparametersdeterminedabove,itallowstodeterminetheparameters ( , )B T and ( , )n T dependingonthestrainrateand

temperature.

ThecurvesfromRKmodelingareplottedinFig.5.a.Itisnotedthatthehardeningdecreaseswiththestrainrate.

a) b)Fig.5.a‐RKmodelfordifferentstrainrates;b‐ComparisonbetweenRKmodelingand

experimentaldata.

0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25

RK model, 10-4

s-1

RK model, 10-3

s-1

RK model, 10-2

s-1

RK model, 1 s-1

RK model, 100 s-1

RK model, 250 s-1

True strain, (-)

Material: IF steel

T0 = 300 K

0

100

200

300

400

500

0 0.05 0.1 0.15 0.2 0.25

Experiments, 10-3

s-1

Experiments, 100 s-1

Experiments, 250 s-1

RK model, 10-3

s-1

RK model, 100 s-1

RK model, 250 s-1

True strain, (-)

Material: IF steel

T0 = 300 K

 

A

dynamideterm  

Table

 

Acier IF 

4. E

4

Bplatesuexperima gas lprocessprojectmeasurpositiondescrip

Analyticalic loading

minedforth

2.Constant

D1 (‐)  D2(‐

0.48  0.79

Experim

4.1Expe

Ballistic peusingrigidmentalappauncher, Fsingofdataile velocityre the initned behindptionofthe

( )

0.9

approximawith a go

hematerial

tsusedtod(2),and

‐)  n0(‐) 

9  0.35 

entalper

rimental

erforation tdprojectilearatususedFig. 6, andaprovidedy. The firsttial impactd the targperforatio

Fig.6.E

ations areood agreemIFaresum

describethetodescribe

B0 (MPa) 

636.9  1

rforation

lset‐upd

tests haveswithdiffedforthereauxiliarybythetestt sensor lot velocityet allowsonprocessi

Experimenta

pC

comparedment, Fig.mmarizedin

emechanicatemperatu

min  

(s‐1) 

(

410   1

ntests

descriptio

been carrferentnoseealizationoequipmentts:especialocated at thof the prto measurisavailable

aldeviceus

1p (JKg K

470

with expe5.b. The

nTable.2.

albehaviorureincrease

max  

(s‐1)  (

710   0.2

on

iedout oneshapesanofballisticits requiredllythetwohe end ofrojectile are the resiein[9,24].

sedforimpa

1)

riments froconstants

basedonthe,Eq.(8)

(‐)  0 (‐) 

25  0.018 

1mm thicndapneummpacttestd for the rsensorsusthe launchnd the sedual veloc

acttests.

om quasi‐sof the mo

theRKmod

*m (‐) 

1.51 

ck IF steelmaticgasgtsconsistmrecordingsedtomeasher tube alecond senscity. The co

3(kgm )

7800

 

static toodel RK

el,Eq.

*0

(MPa) 

313.24 

l targetsgun.Themainlyofand thesurethellows tosor justomplete

10 

 

Tnose shhemispwithaassumeconfigu

energy

p 13

TThe actsupportthe provelocitisheet.

Fig.Conic

Theprojechape suchpherical‐conheattreatmedasrigidurationthe

so that tmm .

Thedimentive part istallowingojectile inies was co

7.Projectilcalprojecti

ctilesusedas conical,nical (projementtorea(withoutmprojectile

he shape

nsionsof ths 100*100thereductthe centransidered f

a)

c)

eshapesusile;c‐Hemi

inthisstud, hemispheectileB),Fachayieldmushroomemasswask

effect can

he squarepmm2, theionofeffecal zone asfor a comp

sedduringpispherical+

(p

dyweredeerical, hemFig.7.Theydstressof2effect)durikeptconsta

be analyz

platesusedthicknessctduringthshown inplete defini

perforation+conical(prprojectileB)

esignatedamisphericalyareallma2GPa.Theingtheproant, pm 3

zed. The d

dduringexis 1mmanhetest.TheFig. 8. A wition of the

ntests.a‐HerojectileA)B).

accordingt+ conicalachined frorefore,theocessofper0 g ,tohav

diameter o

xperimentsnd it is embeplatehaswide rangee ballistic

b)

d)

emispherica;d‐Hemisp

totheirres(projectileomMaragieprojectilerforation.Fvethesame

of all proje

s are givenbedded onbeenimpae of initialcurve of th

calprojectilpherical‐con

 

spective A) andngsteelmaybeForeachekinetic

ectile is

n,Fig.8.n a rigidactedby impacthe steel

le;b‐nical

11 

 

 12 

Fig.8.Geometryofthesteelplatesusedduringperforationtests,thickness1mm.

4.2Experimentalresultsunderballisticimpact In this part, the influence of the projectile shape on the perforation tests isstudied.Theballisticcurves 0RV V forprojectilearereportedonFig.9.

Theconicalandhemisphericalshapesaregenerallyusedtoanalyzetheprocessof failure.The first one induces a failuremodebypetaling and the secondone aplugejection due to circumferential necking. A previous analysis has been published byKpenyigbaetal.[9].Inadditiontothesetwoshapes,projectilescouplinghemisphericaland conical design (projectile A and B) have been used. The conical end allows toperforate the plate more easily by a process of piercing. The projectile damages thetargetplatebyradialneckingandpetalforming.Thefailureisdominatedbytheprocessof high‐speed piercing as in the case of a conical projectile. For projectile B, amix offailuremodesisobserved:petalingandradialcracks. Itisobserved,basedontheexperimentalresultsthattheconicalprojectileshapeallowstodecreasetheballisticlimit BV ofthematerialconsidered,Fig.9,incomparison

with the hemispherical projectile. Moreover, keeping the same conical nose angle

p 72 coupledtoahemisphericalprojectile,Fig.7‐c‐d,theballisticlimitislowerthanConicalbV 6 m / s compared to the conical projectile,Fig.7‐b. The difference is close toHemisphericalbV 10 m / s between shapeA andB and thehemispherical projectile.Using a

reducedconicalshapeprojectilenoseallowstoinduceafailuremodebypiercing.After

 

 13 

that,whenaholeisinitiatedtheprocessofholeenlargementiseasierandthevelocityofthe projectile is less reduced. Thismechanism is discussed in details using numericalobservationsintermsoffailuretime.

0

50

100

150

200

0 50 100 150 200

Projectile A

Projectile B

Conical projectile

Hemispherical projectile

Res

idu

al v

elo

city

, VR (

m/s

)

Initial impact velocity, V0 (m/s)

IF steelPlate thickness: 1mmProjectile mass: 30Dry contactDiameter: 13 mm

Fig.9.Experimentalresultsintermsofresidualvelocitiesfordifferentprojectilesshapes.

Theballisticcurvesingeneralwereapproximatedusingtheproposedanalytical

equationofRechtetal.[25]whoseformulationforthinplatesisasfollows:

1

R 0 BV V V (12)

where BV istheballisticlimitand isafittingparameter.

Table3.ConstantsusedtofitexperimentsbasedonEq.(12).

Conical Hemispherical A B

2.21 2.39 2.25 2.28

86.5 /BV m s 90 /BV m s 78 /BV m s 79 /BV m s

 

 14 

Thedifferencebetweentheballisticcurvesisgreaterattheballisticlimitthanathigh impact velocities ( 0 150 /V m s ). When the residual velocity RV is higher, the

energy absorbed by the plate to perforate is lower, as seen in Fig. 10. During theperforationoftheplate,apartofthekineticenergyoftheprojectileisabsorbedbytheglobal deformation of the steel sheet, the localized plastic deformation in the impactzone and the elastic work. The residual kinetic energy is quite simply the residualenergy of the projectile after impact. If the impact velocity is lower than the ballisticlimit, theenergyabsorbedbytheplate isdirectly thekineticenergyof theprojectile(

201 2 pm V ). The balance of the energy absorbed by the plate during perforation is as

follows[26]:

2 20

1

2Total K E P F KPlate Plate Débris p BW W W W W W m V V (13)

where K

PlateW ,  EW ,  PW ,  FW  and  KDebrisW  arerespectivelytheenergyrelatedtothe

global deflection of the plate, the elastic deformation energy, the plastic deformationenergy, the energy related to the phenomenon of friction and the kinetic energytransferredtothefragmentsejectedduring impact(ejectionofplugforexamplewhenthe hemispherical projectile is used). As shown in [9], the loss of energy due to thefrictioneffectcanbedisregardedandtheenergytransferredtothefragmentsisminor(

0FW ,  0KDébrisW ).Theenergybalanceisthenreducedasfollows:

2 20

1

2Total K E PPlate Plate p BW W W W m V V (14)

Theevolutionof TotalPlateW asafunctionof 0V obtainedexperimentallyforeachkind

ofprojectileshapeispresentedinFig.10.

 

 15 

0

50

100

150

200

250

300

0 50 100 150 200

Projectile AProjectile BConical projectileHemis projectile

En

erg

y (J

)

Initial impact velocity, V0 (m/s)

Kinetic energy of the projectile

Limit for no failure

IF steelPlate thickness: 1mmProjectile mass: 30Dry contactDiameter: 13 mm

Fig.10.Experimentalresultsintermofabsorbedenergyfordifferentprojectilesshapes.

Adecreaseoftheabsorbedenergywiththeimpactvelocityisgenerallyobserved:

anincrementofimpactvelocityleadstoafastlocalizationofthedeformationandquickfailureofthetarget.

To estimate the time necessary for the projectile to perforate the plate, calledfailure time, the experimental perforation tests were recorded using a high speedcamera Phantom v710. The failure time is defined as the time elapsing between thebeginningoftheimpactandwhenthenoseoftheprojectilepassescompletelythroughtheplate,asshowninFig.11(caseofaconicalprojectile). 

 30μs ‐impact 

150μs ‐pénétration  182μs ‐perforation 

Fig.11.Perforationprocessstepsusingaconicalprojectile, 0 178 /V m s .

Theexperimental results in termsof failure timeas a functionof initial impact

velocityarereportedinFig.12.Forallprojectilesstudied,thefailuretimedecreasewiththeimpactvelocity.Thehigheristheprojectileinitialvelocity,thefasteristheprocessofperforation.

 

 16 

0

100

200

300

400

500

80 100 120 140 160 180 200

Projectile AProjectile BConical projectileHemispherical projectile

Fa

ilure

tim

e (

s)

Initial impact velocity (m/s) Fig.12Failuretimedependingontheinitialimpactvelocityfordifferentprojectiles.

Apartfromthelowerspeedrange,ingeneral,thefailuretimeisloweratthesame

impact velocity for projectile A, followed by projectile B, then the conical and thehemisphericalprojectiles,respectively.

5. Numericalmodelfordynamicperforation

The impactandperforationtestsusingarigidprojectileagainstthinsteelsheetwerenumericallyanalyzedforthepurposetopredicttheexperimentalobservation.ThefiniteelementcodeABAQUS/Explicitwasusedtosimulatetheprocess.

5.1DescriptionofthenumericalmodelIn all simulations, the projectile was modeled as a three‐dimensional non‐

deformablerigidbody(discrete)withareferencepointtoaffectvelocity.Themassandinertiamomentswereautomaticallycalculatedbasedontheshape,volumeanddensityoftheprojectileandassignedtoitsreferencepoint.Thetargetwasmodeledasathree‐dimensionaldeformablebody,Fig.13.Thecontactbetweentheprojectileandtheplatewas modeled using the penalty method with finite sliding formulation. A constantcoefficient of friction 0.2 was applied based on experimental studies made by

Jankowiaketal.[27]andRusineketal.[28].Inordertooptimizethemesh,aftertryingseveralapproachesandtakingintoaccounttheinfluenceoftheelementtype,themeshdensityandthecomputationtime,wechosentodividethegeometryoftheplateintwoparts:acircularcentralpartof30mmdiameter(morethandoubleof thediameterofthe projectile, allowing to begin the process of crack propagation accurately withoutsignificant effects on the energy balance) and an exterior part that complements the

 

 17 

structure tohavearectangular target.Thecentralpartof theplatewasmeshedusingC3D8R elements (8‐node linear brick, reducing integration with hourglass modecontrol)availableintheAbaquslibrary[29].Thistypeofhexahedralelementareusedwith success in nonlinear plasticity analysis [9,26,28,30,31]. The effect of hourglassmode is controlledusingenhancedmethod [26]andanoptimalmesh.Thestabilityofthe resultsand itsmesh sizedependencewerepreviously checked [8,24].TheaspectratioofelementswasmaintainedclosetounityasrecommendedbyZukasandScheffler[32].Aconvergencestudyusingalinearhexahedralelementintheimpactzonewiththeinitialsize 0.2x y z mm ensuresthestabilityofthenumericalsolutionwithout

mesh sensitivity effect and with optimal time computation. The central part of thenumericalmodel has 110390elements (5 elements through the thickness of the steelplate).TheexternalpartwasmeshedusingC3D8Ihexahedralelements(8‐node linearbrickwith incompatiblemode). These elements have an additional internal degree offreedom which enhances the ability to model an additional displacement gradientthroughtheelement,therebyimprovingthebendingbehaviorofthestructure.ThesizeofC3D8Ielementsusedwas 0.5x y z mm  (twoelementsalongthethicknessof

the steel sheet), leading to a total of 73 640 elements in the external part of thenumericalmodel,Fig.13.

Fig.13.Numericalmodelusedinthisworkandmeshdensitydistribution.

Centralpart

Exteriorpart

Zoomoncentralpart

 

 18 

ThematerialbehavioroftheIFsteelusedasatargetwasincorporatedthrough

RK thermoviscoplastic material model described previously (Eq. 2). Using the flowstress definition provided by the RK relation is possible to particularize the heatequation (Eq. 8). Thus, the temperature increase is decomposed in two contributions

duetotheinternalstress ( , , )T andtheeffectivestress ( , )T :

int

0

( )( , , ) ( , )

f

e

ernal stress effectivestress

p

T T T

E TT T T d

C E

(15)

Inordertoreproducetheperforationprocessitisnecessarytoconsiderafailure

criterion.Inthiswork,thefailurecriterionusedisbasedontheworkbyWierzbickietal.[33,34]whoshowedthattheequivalentstrainatfailureexpressedasafunctionofthestress triaxialitywould bemore appropriate for problems involving fast loading. Thegeneralformofthistypeoffailurestraincanbewrittenasfollows:

( ) ( )mf f f

(16)

where f is the effective plastic strain to failure and is the stress triaxiality

defined by the ratio of the mean stress m to the equivalent stress . This kind of

fracture model is expressed in the code Abaqus by erosive criterion inducinginstantaneousremovalofanelementwhenanimposedplasticstrainlevelisreachedintheelement.Thisfailurecriterionmodelbasedontheleveloffailurestrainisoftenusedfor dynamic applications [35–37]. The average value of the stress triaxiality can beestimated justbefore the failureof the target foreachprojectile studiedbasedon theworkofLeeandWierzbicki[38].Basedonaprocessofoptimizationforthewholerangeof impact velocities considered, the failure strain was estimated depending on theprojectile shape, given in Table 4. The process of numerical optimization was tominimizetheerrorontheresidualvelocitybasedonexperiments.Moredetailsontheoptimizationprocessareavailablein[9,24].

Table4.Failurestrainvaluesusedtosimulateperforationdependingontheprojectileshape.

ProjectileA ProjectileB Conicalshape HemisphericalshapeFailurestrainlevel, f (‐) 1.2 1.2 1.2 0.65Stresstriaxiality, 1/3 1/3 1/3 2/3

Numericalsimulationwereperformedforawiderangeofinitialimpactvelocities

( 020 / 200 /m s V m s )tocovertherangeofvelocitiesinvestigatedexperimentally,and

to provide an accurate and global description of the ballistic curves. All numerical

 

 19 

results in terms of ballistic curves, failuremodes, the energy absorption balance andfailuretimearepresentedanddiscussedinthenextsub‐section.

5.2NumericalresultsanddiscussionTheresultsobtainedfromthenumericalpredictions intermsofballisticcurves

R 0V V arecomparedtoexperimentsforeachprojectileandarepresentedinFig.14.A

smalldifference is observed for impact velocities close to theballistic limit.At impactvelocities greater than the ballistic limit ( 0 100 /V m s ), a good agreement with the

experimentalresultsisgenerallyobtained.

a) b)

c) d)

Fig.14.Definitionoftheballisticcurve,comparisonbetweenexperimentsandnumericalsimulations.a‐ProjectileA;b‐ProjectileB;c‐Conicalprojectile;d‐Hemisphericalprojectile.

0

50

100

150

200

0 50 100 150 200

Experimental

Numerical

Initial impact velocity, VR (m/s)

IF steelPlate thickness: 1mmProjectile A, m: 30 gDiameter: 13 mm

0

50

100

150

200

0 50 100 150 200

Experimental

Numerical

Initial impact velocity, VR (m/s)

IF steelPlate thickness: 1mmProjectile B, m: 30 gDiameter: 13 mm

0

50

100

150

200

0 50 100 150 200

Experimental

Numerical

IF steelPlate thickness: 1mmConical projectile, m: 30 gDiameter: 13 mm

Initial impact velocity, VR (m/s)

0

50

100

150

200

0 50 100 150 200

Experimental

Numerical

IF steelPlate thickness: 1mmHemispherical projectile, m: 30 gDiameter: 13 mm

Initial impact velocity, VR (m/s)

 

 20 

Numericalpredictionsoftheballisticlimitvelocitiesare85m/sforprojectileA,

86 m/s for projectile B, 90 m/s for the conical projectile and 93 m/s for thehemisphericalprojectile.TheballisticcurvesofprojectileAandBarealmostthesame,withtheirballisticperformanceimprovedbythedoublenoseshape.

The failuretimewasalsoestimatednumericallyusing thesamedefinitionas inthe experiments.The comparison is illustrated inFig.15, showing adecreased failuretimeinincreasedimpactvelocityforallshapesofprojectiles.Thisisreasonablebecausethehighertheimpactvelocityis,thelesstimetheprocessofperforationtakes.

a) b)

c) d)

Fig.15.Failuretimeasafunctionofinitialimpactvelocity,numericalresultscomparedtoexperimentalones.a‐ProjectileA;b‐ProjectileB;c‐Conicalprojectile;d‐

Hemisphericalprojectile.

0

100

200

300

400

500

600

80 100 120 140 160 180 200

Experimental

Numerical

Initial impact velocity (m/s)

IF steelPlate thickness: 1mmProjectile A, m: 30 gDiameter: 13 mm+5%

-5%

0

100

200

300

400

500

600

80 100 120 140 160 180 200

Experimental

Numerical

Initial impact velocity (m/s)

IF steelPlate thickness: 1mmProjectile B, m: 30 gDiameter: 13 mm

+5%-5%

0

100

200

300

400

500

600

80 100 120 140 160 180 200

Experimental

Numerical

Initial impact velocity (m/s)

IF steelPlate thickness: 1mmConical projectile, m: 30 gDiameter: 13 mm+5%

-5%

0

100

200

300

400

500

600

80 100 120 140 160 180 200

Experimental

Numerical

Initial impact velocity (m/s)

IF steelPlate thickness: 1mmHemispherical projectile, m: 30 gDiameter: 13 mm

+5%-5%

 

 21 

Thereisagoodagreementbetweenthenumericalpredictionsandexperiments.Theerrorsarelessthan±5%.Atafixedimpactvelocity,projectileBtakeslesstimethanallotherprojectilestoperforatetheplate,followedrespectivelybytheconicalprojectile,projectileAandthehemisphericalprojectile.

All failuremodesobtainedexperimentallywerereproducednumerically,showninFig.16.Numericalresultsshowafailuremodebypetalsformingforconicalprojectileaswell asprojectilesA andB. For thehemispherical projectile, aplug ejection failuremodewasobtained.ThepetalsformedbyprojectilesAandBarebroader,andthosebyaconicalprojectilearemorepointed.itisalsobelievedthatitistheconicalnoseoftheprojectilesAandBwhichgovernsthefailureofthetarget.

 a)  

b)

 c)

d)

 e)  

f)

 

 22 

 g) h) 

Fig. 16. Comparison between the failure modes obtained numerically and experimentally. a‐ Projectile A, numerical; b‐ Projectile A experimental; c‐ Projectile B, numerical; d‐ Projectile B, 

experimental; e‐ Conical projectile, numerical; f‐ Conical projectile, experimental; g‐ Hemispherical projectile, numerical; h‐ Hemispherical projectile, experimental.  The pictures in Fig. 16 clearly show that the numerical models qualitatively

reflect theoverall physical behavior of theplateduring the impact andperforation.Alocalization of the plastic strain is noted at the ends of the petals in the case of theconicalprojectileaswellasprojectilesAandB,but inthezonesoftheneckingforthehemisphericalprojectile.Themaximumequivalentfailurestraincorrespondstothatsetbythefailurecriterionforeachprojectilesshape.

Wealsomeasuredtheoveralldeflectionoftheplateforeachprojectileshape.Thecomparison between experimental and numerical results is presented in Fig. 17. Thenumerical simulation is in perfect agreement with the experiment. The maximumdeflectionmeasuredis14mmforthecaseoftheconicalprojectileduetothepetalsandtheminimum is 12mm for the hemispherical projectile. Because the thickness of theplateislow(1mm)comparedtothediameteroftheprojectile(13mm),thedeflectionoftheplatedoesnotchangesignificantlywiththeimpactvelocitywhenit isgreaterthantheballisticlimit.

a) b)

0

5

10

15

20

25

30

-60 -40 -20 0 20 40 60

Experimental

Numerical

Length of the plate (mm)

IF steelThickness: 1mmProjectile ADiameter: 13 mm

V0 = 100 m/s

0

5

10

15

20

25

30

-60 -40 -20 0 20 40 60

ExperimentalNumerical

Length of the plate (mm)

IF steelThickness: 1mmProjectile BDiameter: 13 mm

V0 = 100 m/s

 

 23 

c) d)Fig.17Globaldishingof1mmand1mmthicktarget.a‐ProjectileA;b‐ProjectileB;c‐

Conicalprojectile;d‐Hemisphericalprojectile.

6. Conclusions

Experimental and numerical investigations have been made on an IF steelsubjected to impact and perforation loading. Material tests were performed todeterminethematerialconstantsfortheconstitutiverelationproposedbyRusinekandKlepaczko[1]totakeintoaccounthardening,strainrateandtemperaturesensitivities.Four different shapes of projectile were considered in this work such as a conical, ahemisphericaland twodoublenose (combinationof conicalandhemispherical shape)projectile.Theballistic limitforeachprojectileshapewasdeterminedandtheballisticcurves plotted. It is found that the ballistic limit, the failure mode and the energyabsorptioncapacityofthetargetarestronglylinkedtotheprojectilenoseshape.Energyabsorbedby theperforated structure isdissipatedby itsoveralldeflection, theelasticdeformation and plastic deformation in the localized impact area. The hemisphericalprojectileistheleastefficienttoperforatethetargetfollowedrespectivelybytheconicalprojectile, projectile B and A. This observation is confirmed bymeasuring the failuretime of the target using a high speed camera. Indeed, at a giving impact velocity, theprojectileAtakeslesstimetoperforatethetarget,followedrespectivelybyprojectileB,conical and hemispherical projectile. A numerical analysis of the impact problem hasbeenmadeusingthefiniteelementcodeABAQUS/Explicit.ThemechanicalbehaviorofthetargetmaterialhasbeenimplementedinthecodeusingtheRKconstitutiverelation.Theresultsobtainedfromthenumericalpartwerecomparedtoexperimentsandagoodagreement is observe in terms of failuremode, ballistic limit, ballistic curves, energyabsorbedbythetargetandtheglobaldeflectionofthetarget.

0

5

10

15

20

25

30

-60 -40 -20 0 20 40 60

Experimental

Numerical

Length of the plate (mm)

IF steelThickness: 1mmConical projectileDiameter: 13 mm

V0 = 100 m/s

0

5

10

15

20

25

30

-60 -40 -20 0 20 40 60

Experimental

Numerical

Length of the plate (mm)

IF steelThickness: 1mmHemispherical projectileDiameter: 13 mm

V0 = 100 m/s

 

 24 

Acknowledgements

AuthorsthanktotheNationalCentreofResearchandDevelopmentforfinancialsupportunderthegrantWND‐DEM‐1‐203/00.

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