DTIC ADA595590: A Meso-Scale Unit-Cell Based Material Model for the Single-Ply Flexible-Fabric Armor

5.4 MB·English

by  Defense Technical Information Center

MaterialsandDesign30(2009)3690–3704
ContentslistsavailableatScienceDirect
Materials and Design
journal homepage: www.elsevier.com/locate/matdes
A meso-scale unit-cell based material model for the single-ply flexible-fabric armor
M. Grujicica,*, W.C. Bella, G. Arakerea, T. Hea, B.A. Cheesemanb
aInternationalCenterforAutomotiveResearchCU-ICAR,DepartmentofMechanicalEngineering,241EngineeringInnovationBuilding,ClemsonUniversity,
Clemson,SC29634-0921,UnitedStates
bArmyResearchLaboratory–SurvivabilityMaterialsBranch,Aberdeen,ProvingGround,MD21005-5069,UnitedStates
a r t i c l e i n f o a b s t r a c t
Articlehistory: Ameso-scaleunit-cellbasedmaterialmodelforaprototypicalplain-wovensingle-plyflexiblearmoris
Received29April2008 developedandimplementedinamaterialusersubroutineforuseincommercialexplicitfiniteelement
Accepted12February2009 programs. The main intent of the model is to attain computational efficiency when calculating the
Availableonline20February2009
mechanicalresponseofthemulti-plyfabric-basedflexiblearmormaterialduringitsimpactwithvarious
projectileswithoutsignificantlysacrificingthekeyphysicalaspectsofthefabricmicrostructure,architec-
Keywords: tureandbehavior.Tovalidatethenewmodel,acomparativefiniteelementmethod(FEM)analysisiscar-
Flexiblearmor
riedoutinwhich:(a)theplain-wovensingle-plyfabricismodeledusingconventionalshellelementsand
Meso-scaleunit-cellmaterialmodel
weavingisdoneinanexplicitmannerbysnakingtheyarnsthroughthefabricand(b)thefabricistreated
High-performancefibers
asaplanarcontinuumsurfacecomposedofconventionalshellelementstowhichthenewmeso-scale
Ballisticperformance
unit-cellbasedmaterialmodelisassigned.Theresultsobtainedshowthatthematerialmodelprovides
areasonablygooddescriptionforthefabricdeformationandfracturebehaviorunderdifferentcombina-
tionsoffixedandfreeboundaryconditions.
(cid:2)2009ElsevierLtd.Allrightsreserved.
1.Introduction arestillbeingusedtoday,high-performancepolymericfibers(typ-
icallyusedintheformofwovenfabrics)arenowthestandardin
Torespondtothenewenemythreatsandwarfaretactics,mil- most fiber-reinforced body-armor applications. To increase the
itarysystems,inparticularthosesupportingtheUSgroundforces, ballistic performance of the body-armor vests relative to up to
arebeingcontinuouslytransformedtobecomefaster,moreagile, 0.30 caliber threats, ceramic insert strike-plates are commonly
andmoremobilesothattheycanbequicklytransportedtooper- used[2].
ations located throughout the world. Consequently, an increased High-performance polymeric fibers used today are character-
emphasis is being placed on the development of improved light- izedbysubstantiallyimprovedstrength,stiffnessandballisticper-
weightbody-armorandlightweightvehicle-armorsystemsaswell formance.Amongthesehigh-performancefibersthemostnotable
on the development of new high-performance armor materials. are:(a)poly-aramids(e.g.Kevlar(cid:3),Twaron(cid:3),Technora(cid:3));(b)highly
High-performance fiber-based materials have been exploited for oriented poly-ethylene (e.g. Spectra(cid:3), Dyneema(cid:3)); (c) poly-benz-
both body-armor (e.g. as soft, flexible fiber mats for personal-ar- obis-oxazole, PBO (e.g. Zylon(cid:3)), and (d) poly-pyridobisimi-dazole,
mor vests) and for the vehicle-armor systems (e.g. as reinforce- PIPD(e.g.M5(cid:3)).Whentestedintension,allthesematerialsdiffer
ments in rigid polymer matrix composites, PMCs, for lightweight significantlyfromthenylonfibers,havingveryhighabsolutestiff-
vehicle-armorsystems). ness, extremely high density-normalized strength, and quite low
Throughout history, flexible lightweight materials have been (<4%)strains-to-failure.Thesefibersessentiallybehave,intension,
used in body-armor systems to provide protection against speci- asrate-independentlinear-elasticmaterials.Whentestedintrans-
fiedthreats,atreducedweightandwithoutcompromisingperson’s versecompression,however,thesefibersaresimilartonylonand
mobility. Early materials used included leather, silk, metal chain can undergo large plastic deformation without a significant loss
mail and metal plates. Replacement of metal with a nylon (poly- intheirtensileload-carryingcapacity.Thisbehaviorisquitediffer-
amide) fabric and an E-glass fiber/ethyl cellulose composite in entfromthatfoundincarbonorglassfibers,whichtendtoshatter
body-armor systems can be linked to the Korean War [1]. under transverse compression loading conditions. Among the
Although,primarilyduetotheirlowcost,nylonandE-glassfibers polymericfibersmentionedabove,thebestperformer,PBO,hasa
tensile-strength of ca. 5GPa (i.e. more than three times the
tensile-strength level of the strongest ‘‘piano-wire” steel but at a
one-fifth of the steel density, resulting in an (cid:2)15-fold higher
* Correspondingauthor.Tel.:[phone];fax:[phone].
density-normalizedstrength).
E-mailaddress:[email protected](M.Grujicic).
0261-3069/$-seefrontmatter(cid:2)2009ElsevierLtd.Allrightsreserved.
doi:10.1016/j.matdes.2009.02.008
Report Documentation Page Form Approved
OMB No. 0704-0188
Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and
maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,
including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington
VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it

does not display a currently valid OMB control number.
1. REPORT DATE 3. DATES COVERED
2009 2. REPORT TYPE 00-00-2009 to 00-00-2009
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
A meso-scale unit-cell based material model for the single-ply
5b. GRANT NUMBER
flexible-fabric armor
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
Celmson University,Department of Mechanical REPORT NUMBER
Engineering,Clemson,SC,29634
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)
11. SPONSOR/MONITOR’S REPORT
NUMBER(S)
12. DISTRIBUTION/AVAILABILITY STATEMENT
Approved for public release; distribution unlimited
13. SUPPLEMENTARY NOTES
14. ABSTRACT
A meso-scale unit-cell based material model for a prototypical plain-woven single-ply flexible armor is
developed and implemented in a material user subroutine for use in commercial explicit finite element
programs. The main intent of the model is to attain computational efficiency when calculating the
mechanical response of the multi-ply fabric-based flexible armor material during its impact with various
projectiles without significantly sacrificing the key physical aspects of the fabric microstructure,
architecture and behavior. To validate the new model, a comparative finite element method (FEM) analysis
is carried out in which: (a) the plain-woven single-ply fabric is modeled using conventional shell elements
and weaving is done in an explicit manner by snaking the yarns through the fabric and (b) the fabric is
treated as a planar continuum surface composed of conventional shell elements to which the new
meso-scale unit-cell based material model is assigned. The results obtained show that the material model
provides a reasonably good description for the fabric deformation and fracture behavior under different
combinations of fixed and free boundary conditions.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 15
unclassified unclassified unclassified Report (SAR)
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
M.Grujicicetal./MaterialsandDesign30(2009)3690–3704 3691
Asmentionedabove,fabricsbasedonhigh-performancefibers rial models have been developed by Boisse et al. [20] and
areextensivelyemployedinvarietyofballisticandimpactprotec- Peng and Cao [21]. Also, Shahkarami and Vaziri [22] pro-
tionapplications.Despitethefactthatoverthepasttwodecades, posedasimilarbutsimplermodeltothatintroducedbyKing
therehasbeenagreatdealofworkdoneonunderstandingthebal- etal.[19]andprovidedadetailedaccountofitsincorpora-
listicbehaviorofthesefabricsusingvariousanalyticalandnumer- tionintoamaterialmodelsubroutinewhichcanbereadily
icaltechniques,thedesignoffabricarmorsystemsremainslargely coupled with commercial dynamic-explicit finite element
based on the employment of extensive experimental test pro- codes.
grams,empiricismandoldpractices.Whilesuchexperimentalpro- (d) Theuseofhigher-ordermembrane/shellfiniteelementanal-
gramsarecriticalforensuringtheutilityandeffectivenessofthe ysestorepresentthedynamicresponseoffabricunderbal-
armor systems, they are generally expensive, time-consuming listic loadingconditions and overcomethe aforementioned
andinvolvedestructivetesting.Consequently,thereisacontinuing computationalcostassociatedwiththeuseoffull3Dfinite
efforttoreducetheextentoftheseexperimentaltestprogramsby element analyses of the yarn/fabric structure. Among the
complementingthemwiththecorrespondingcomputation-based studies falling into this category, the most notable is the
engineeringanalysesandsimulations. one carried out by Scott and Yen [23]. While the use of
Among the main computational engineering analyses used to higher-order membrane elements was found to be indeed
model ballistic performance of flexible armor the following main advantageouscomputationally,itwasneverfullyvalidated
classescanbeidentified: bycomparingitsresultsagainsteitherthoseobtainedexper-
imentally or those obtained using full 3D finite element
(a) Finite element analyses based on the use of pin-jointed analyses.
orthogonalbarstorepresentflexible-fabricyarns.Themost
notable studies falling into this category of analyses are Aspointedoutabove,whilemajoreffortshavebeenmadeinre-
thoseperformedbyRoylanceandWang[3],Shimetal.[4], cent years to develop sophisticated numerical models capable of
Limetal.[5],Shahkaramietal.[6],Johnsonetal.[7],andBil- elucidating the ballistic performance of fabric armors, most of
lonandRobinson[8].Whilethepin-jointedorthogonalbars thesemodelseitherlackcomputationalefficiencyorfailtocapture
based finite element analyses have proven to be very effi- many physical aspects of the yarn and fabric architecture and/or
cientinapproximatingthedynamicbehaviorofwovenfab- contact dynamic phenomena. Hence, the main objective of the
rics, thediscrete natureof theyarn models was associated present work is to develop an efficient shell-based meso-scale
with inherent oversimplifications that significantly limited mechanics unit-cell based model that captures the essential dy-
the predictive capability of the analyses. In particular, namic/ballisticbehaviorofplain-wovenfabricunderimpact-load-
importantcontributionsassociatedwiththeweavearchitec- ingconditions.Theterm‘‘meso-scale”isusedtodenoteyarn-level
ture, surface-finish and friction governed yarn-to-yarn and millimeterlengthscale detailsofthefabric microstructure/archi-
layer-to-layercontacts(inmulti-layerfabrics)couldnotbe tecture.Inotherwords,finer-scalemolecular-levelandfiber-level
accountedfor. materialdetailsarenotconsideredexplicitlyandinsteadonlytheir
(b) More-detailedfull-blown3Dcontinuumfiniteelementanal- lumpedcontributionsaretakenintoaccount.The‘‘unit-cell”term
ysessuchastheonescarriedoutbyShockeyetal.[9],Duan is used to denote the basic structural unit in a woven single-ply
etal.[10–13],Zhangetal.[14],etc.havealsobeeninvesti- fabric so that a fabric patch can be considered as an in-plane
gated. While these analyses have proven to be powerful assemblyofsuchunits.Thematerialmodeldevelopedinthepres-
tools for capturing and elucidating the detailed dynamic ent work is essentially an extension of the model recently pro-
response of single-layer fabrics, they are computationally posedbyShahkaramiandVaziri[22]andincludesallthecasesof
very demanding when applied to practical armor systems in-planeunit-celldeformationmodesand,foreachofthesemodes,
whichtypicallycontain30–50fabriclayers/plies. allthecontact/no-contactcasesofthecrossingyarns.Inaddition,
(c) Unit-cell based approaches have been used extensively in three-dimensionalsingleunit-cellfiniteelementanalysesofdiffer-
order to derive the equivalent (smeared) continuum-level entin-planeandtransverseloadingscenarioswerecarriedoutto
(membrane/shell) material models of textile composites assess the yarn/yarn-contact force vs. over-closure relationship,
fromtheknowledgeofthemeso-scalefiberandyarnprop- aswellasthein-planeandout-of-planeshearmoduli.Thecontact
erties, fabric architecture and inter-yarn and inter-ply fric- force vs. over-closure relationship and the in-plane and out-of-
tional characteristics. Among the most notable studies planeshear unit-cellmoduli are thenused asinput in themeso-
basedontheseanalysesarethosecarriedoutbyKawabata scaleunit-cellbasedmaterialmodel.
et al. [15–17] who introduced simple analytical models to Theorganizationofthepaperisasfollows:detailsregardingthe
capture the uniaxial, biaxial and shear behavior of fabrics. computationalproceduresemployedtodevelopanewmeso-scale
Furthermore, Ivanov and Tabiei [18] proposed a micro- unit-cellbasedmaterialmodelforaprototypicalplain-wovensin-
mechanical material model for a woven fabric (in which a gle-plyfabricandtheimplementationofthismodelintoamaterial
visco-elastic constitutive model was used to represent the usersubroutinesuitableforuseincommercialfiniteelementpro-
mechanicalbehavioroftheyarns)fortheuseinnon-linear gramsarepresentedinSection2.Theformulationofasimplepro-
finite element impact simulations. In deriving the material jectile-armor impact problem used to validate the new material
model, Ivanov and Tabiei [18] considered the motion of modelisdescribedinSection3.Mainresultsobtainedinthecur-
the yarn crossover point and developed a procedure for rentworkarepresentedanddiscussinSection4.Themainsum-
determining the equilibrium position of this point under marypointsandconclusionsresultingfromthepresentworkare
the applied unit-cell strains. Recently, King et al. [19] pro- listedinSection5.
posed a new approach for deriving the continuum-level
material model for fabrics based on the properties of the
yarns and the weave architecture which involves the use 2.Computationalprocedure
of an energy minimization technique to establish the rela-
tionship between the configurations of the fabric structure In this section, a simple meso-scale unit-cell based material
tothemicroscopicdeformationoffabriccomponents.Simi- modelforaplain-wovensingle-plyfabricisdevelopedandimple-
lar unit-cell based continuum-level membrane/shell mate- mentedintoamaterialusersubroutinesuitableforuseincommer-
3692 M.Grujicicetal./MaterialsandDesign30(2009)3690–3704
ment discretization is depicted in Fig. 2a. A comparison of the
modelsdepicted in Figs.1a and 2a shows that thetwo yarnsare
simplifiedinFig.1aastwotwo-membertrusselementseachwith
L01
H01 aconstantellipticalcross-section.Adeformedconfigurationofthe
simplifiedunit-cellmodelisdepictedinFig.1b.
y01 H02 Duetothetruss-characteroftheyarnelements,thecontactbe-
y02 tweentheyarnsofthecrossoverisreducedtoapointcontact.It
(a) L02 syhaornu/lydabrnenpooteindtthcaotndtaucettocothrreesfipnointedsthitcokneasnsoinfitthiael-ynaornn-sz,etrhoe,
H +H , distance between the summit points of the two two-
01 02
membertrusselements,whereH andH aretheinitialout-of-
01 02
plane distances of the two summit points. Furthermore, it is as-
d2 sumedthat,duringdeformation,thein-planelocationofthecon-
tact point (i.e. the projection of the summit points to the shell
d1 Fc1 surface)remainsconstantbutthesummitpointscanmoveinthe
out-of-planedirection.
T1 The initial lengths of the corresponding single truss members
areL andL .Whentheunit-cellissubjectedtoin-planebiaxial
H1 01 02
Φ1 y1 H2 ninoirtmialalvastluraeisn,sy,dimanednsyiontsooyft(hHe)uannitd-cyel(lHar)e,crehsapnegcetdivferloym.Wthheeinr
y2 T2 theunit-celliss0t1retched02inap1art1icularin2-pl2anedirection,thecor-
respondingyarniseitherde-crimped/straightenedorstretched.In
(b) Φ2 Fc2 the former case, no tension is built within the yarn, while in the
latter case tension is created within the yarn and, if sufficiently
high,cancauseyarnfailure.Also,theextentoftensioninagiven
Fig.1. Geometricalrepresentationofasingleunit-cellforaplain-wovensingle-ply yarnisaffectedbythefailurestatusoftheothercrossingyarn.If
fabric:(a)beforeand(b)afterapplicationofanormalbiaxialin-planedeformation.
thecrossingyarnisnotbrokenandisincontactwiththestretched
yarn,tensionwilldevelopinthetwoyarns,theextentofwhichis
cial finite element packages. A simple schematic of the unit-cell dependentoftheextentofcontactover-closure(i.e.thereduction
whichisusedtorepresenttheplain-wovensingle-plyfabricstruc- in distance of the two summit points relative to the initial dis-
ture/architecture allotted to a single-yarn crossover in its initial tance) and the functional relationship which relates the contact
(un-deformed) configuration is depicted in Fig. 1a. It should be over-closurewiththemagnitudeofthecontactforce.Duetotheir
noted that the schematic displayed in Fig. 1a is a simplification low buckling resistance, compression cannot be built within the
of the corresponding unit-cell model based on the three-dimen- yarns when the cell is subjected to in-plane normal compressive
sionalfabricstructure/architecturewhosesolid8-nodefiniteele- strains.
Fig.2. Three-dimensionalfiniteelementrepresentationoftheunit-cellinaplain-wovensingle-plyfabric:(a)initialconfiguration;(b)afterin-planebiaxialstretching;
(c)afterin-planeshear;(d)aftertransverseshear.
M.Grujicicetal./MaterialsandDesign30(2009)3690–3704 3693
2.1.Ameso-scaleunit-cellbasedmaterialmodelforplain-woven twoyarnsandtheextentofthisforcedependsontheyarn-contact
single-plyfabric over-closure and a functional relationship which relates the con-
tact force magnitude to the contact over-closure. To determine
Asexplainedearlierthemainobjectiveofthepresentworkisto thefunctionalrelationshipbetweentheyarn-contactover-closure
develop a meso-scale unit-cellbased material model for a proto- d þd ¼d and the resulting contact force F a three-dimen-
c1 c2 c c1
typicalplain-wovensingle-plyKevlar129fabricwhichcanbeused sionalsolidFEMmodelofasingleunit-cellisconstructed,Fig.2a
in large-scale computational analyses of multi-ply flexible armor and subjected to the balanced biaxial in-plane tensile loads and
systems.Thedevelopmentofthismodelispresented inthissec- the total surface contact force monitored is a function of the
tion.Sincethemodelis intendedforthe usein conjunctionwith yarn-contactover-closure.Anexampleofdeformedconfiguration
shell finite elements, its development is divided in three parts: oftheunit-cellisshowninFig.2bwhilethecorrespondingcontact
(a) the normal biaxial in-plane response of the unit-cell; (b) the forcevs.yarnover-closureresultsobtainedinthisanalysisaredis-
in-planeshearresponseofthecell;and(c)theout-of-plane(trans- playedinFig.3a.FollowingShahkaramiandVaziri[22],theF vs.d
c c
verse)shearresponseoftheshell.Inthefollowingthreesections, relationisdescribedusingthefollowingfunction:
thedevelopmentofeachifthesethreecomponentsofthematerial
bd
model is presented. This is followed by a brief discussion of the F ¼ c ð1Þ
c a(cid:3)d
implementationofthematerialmodelinausermaterialsubrou- c
tinewhichiscoupledwiththecommercialfiniteelementprogram Eq.(1)ensuresthatthecontactforceinitiallyincreaseslinearly
ABAQUS/Explicit[24]. withtheover-closureandastheover-closurebeginstoapproach
As mentioned above, when the two yarns are not broken, in the yarn thickness the contact force becomes excessively high.
contactandthedistancebetweenthetwosummitpointsissmaller Using a simple unconstrained optimization procedure, the three-
than the initial distance, contact force is developed between the dimensionalsolidFEMF vs.d resultsdisplayedinFig.3aarefitted
c c
2500 90
(a) (b)
NoFriction,FEM 80
NoFriction,Fitted
2000 a
WithFriction,FEM P 70
G
With Friction,Fitted
N s,
m u 60
e,1500 dul
c o 50
or M
F r
ct ea 40
a1000 h
nt S
o e 30
C n
a
pl NoFriction,FEM
500 n- 20 NoFriction,Fitted
I
WithFriction,FEM
10 With Friction,Fitted
0
0
0 0.02 0.04 0.06 0.08 0 0.1 0.2 0.3 0.4
ContactOverclosure,mm In-planeShearStrain
1.2
(c)
a
P 1.1
G
s,
u
ul 1
d
o
M
r
ea 0.9
h
S
e
n
a 0.8
pl NoFriction,FEM
of- NoFriction,Fitted
ut- 0.7 WithFriction,FEM
O
With Friction,Fitted
0.6
0 0.1 0.2 0.3 0.4
Out-of-planeShearStrain
Fig.3. Typicalresultsobtainedinthree-dimensionalFEManalysesofthedeformationoftheunit-cell:(a)underin-planebiaxialtension;(b)in-planeshear;(c)out-of-plane/
transverseshear.
3694 M.Grujicicetal./MaterialsandDesign30(2009)3690–3704
to the function defined by Eq. (1) and the two parameters are aNewton–Raphsonroutine.Onced andd aredetermined,
c1 c2
determinedas:a=0.12389mmandb=983.39mNintheabsence thetwoyarntensionscanbecomputedusingEq.(2)andin
ofyarn/yarnfrictionanda=0.12884mmandb=1298.1mNinthe turn,thetwoin-planeunit-cellnormalstressescanbedeter-
caseofyarn/yarnfrictionalcoefficientof0.5andthesevaluesfora minedas:
and bare used in the remainderofthe paper.It shouldbe noted r ¼T=A ði¼1;2Þ ð7Þ
thatthedatapertainingtothecaseofyarn/yarnfrictionalcoeffi- i i i
cient of 0.5 in Fig. 3a are shifted upward by 500mN to improve
clarityofthefigure. (d) Whenbothyarnsareintactandoneofthein-planeunit-cell
normalstrainsistensilewhiletheotheroneiscompressive,
2.1.1.In-planebiaxialbehavioroftheunit-cell twopossiblecasescanarise:(i)ifthecurrentsummit-point
Whentheunit-cellissubjectedtoin-planebiaxialstrains,sev- distance islarger than itsinitial value(H01+H02), then the
eralcaseshavetobeconsidereddependingonwhetherthestrains yarns are not in contact and the yarn which is aligned in
aretensileorcompressiveandwhetheranyoftheyarnsisbroken: the compression–strain direction (the ‘‘compressed yarn”)
does not carry any load and the normal stress in the com-
(a) Whenbothyarnsarebrokenwithinacell,thecorresponding pression–straindirectioniszero.Asfarastheotherdirection
finiteelementiseroded,i.e.itisremovedfromtheanalysis. is concerned, tension can be developed in the yarn (when
(b) Whenoneoftheyarnsisbroken,thentheotheryarnistrea- yi>L0i, where i pertains to the yarn aligned in the tensile-
tedasbeingtheonlyyarnintheunit-cell.Iftheun-broken straindirectionoftheunit-cell,the‘‘stretchedyarn”).Inthis
yarn is subjected to compression or the unit-cell length in case, the yarn tension is given by the single-yarn tension
the direction of this yarn, y (i=1 or 2 and pertains to the relation, Eq. (2). Otherwise, no tension is built within the
i
un-broken yarn) is smaller than the corresponding initial yarnandthecorrespondingin-planenormalunit-cellstress
yarn length L , then no load is supported by this yarn. In also becomes zero; (ii) when yarn-contact over-closure is
0i
other words, zero in-plane normal stresses exist along the present, then tension builds up in both yarns and the two
axis of both yarns. When y >L then tension is developed in-plane unit-cellnormal stressesare bothpositive/tensile.
i 0i,
withintheun-brokenyarn.Thistensionisdefinedbythefol- Todeterminethetensionsinthetwoyarns,asimilarproce-
lowingsingle-yarnexpressionas: durecanbedevelopedastheoneusedunderpoint(c)inthis
section.Themaindifferenceisthatthereferenceconfigura-
Ti¼EAiðLi(cid:3)L0iÞ=L0i ð2Þ tionsfromwhichthedevelopmentoftheyarn/yarn-contact
forcesismeasuredaredifferentinthetwocases.Inthecase
whereEistheyarnaxialYoung’smodulus,A isthecross-sec-
i ofbiaxialstretchingoftheunit-cell,thereferenceconfigura-
tionalarea,L =y andireferstotheun-brokenyarn.
i i tioncorrespondstotheinitialconfigurationofthefabricin
which the two yarns are in contact but no over-closure
(c) When both yarns are intact, their respective unit-cell in-
(i.e. the contact force) is present. In the present case, it is
planenormal strainsare both positive and thereis ayarn-
advantageous to temporarily decouple the behavior of the
contact over-closure, then the effect of the contact force
two yarns. The summit-point height of the ‘‘compressed”
hastobe takenintoaccountbeforetheyarn tensions(and
yarn has increased under unit-cell compression. The sum-
in turn, the corresponding in-plane unit-cell normal stres-
mit-point ofthe ‘‘stretched”yarn is decreased due to unit-
ses)canbecomputed.Asimpleprocedureforthecalculation
celltensionandthemagnitudeofthisdecreaseislargerthan
oftheyarntensionsandunit-cellin-planenormalstressesis
thesummit-pointheightincreaseofthe‘‘compressed”yarn.
providedbelow.Itshouldbenotedthatthetotalyarn-con-
Hence,thereferenceconfigurationinthiscasecorresponds
tactover-closureisdefinedas,i.e.dc=dc +dc .Usingsingle
1 2 totheoneinwhichthedecreaseinthesummit-pointheight
geometrical arguments and Fig. 1a and b, the current yarn
ofthe‘‘stretched”yarnistemporarilysetequaltothesum-
crimpheightscanbedefinedas:
mit-point height increase of the ‘‘compressed” yarn. Then,
Hðd Þ¼H (cid:3)d ði¼1;2Þ ð3Þ further decrease in the summit-point height of the
i ci 0i ci
‘‘stretched” yarn will cause a further increase in the sum-
Likewise, the current single-member truss element lengths mit-point height of the ‘‘compressed” yarn which leads to
canbedefinedas: acontactforcebuild-upandtothedevelopmentoftension
in both yarns. Once account is taken of the new reference
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lðd iÞ¼ y2þH2 ði¼1;2Þ ð4Þ configuration,yarntensionsandthecorrespondingunit-cell
i c i i
normalstressescanbecalculatedusingtherelationsidenti-
ThecorrespondingyarntensionsarethendefinedbyEq.(2). caltothosepresentedunderpoint(c)inthissection.
Using the force equilibrium condition at the yarn summit
points,thefollowingexpressioncanbederivedforthecorre-
spondingcontactforces: 2.1.2.In-planeshearbehavioroftheunit-cell
F ðd Þ¼2Tðd ÞHðd Þ=Lðd Þ ði¼1;2Þ ð5Þ Themeso-scalematerialmodelpresentedintheprevioussec-
ci ci i ci i ci i ci
tionprovidesa wayfor assessingin-plane normalstressesof the
Theyarn/yarnforcesatthetwosummitpoints,F andF ,
c1 c2 unit-cell.Sincenofrictionbetweenthetwotwo-membertrussele-
definedbyEq.(5)areequaltoeachotherandarealsoequal
mentsistakenintoaccountinFig.1,theunit-cellmodelcouldnot
tothecontactforce,F,definedbyEq.(1).Thusthefollowing
c beusedtoassessitsin-planeshearbehavior.Toovercomethislim-
systemoftwonon-linearalgebraicequationswithtwoun-
itation, the three-dimensional unit-cell finite element model dis-
knowns,d andd ,canbedefinedas:
c1 c2 played in Fig. 2a is subjected to an increase in-plane shear and
2T ðd ÞðH ðd Þ=L ðd ÞÞ¼2T ðd ÞðH ðd Þ=L ðd ÞÞ ð6aÞ thecorrespondingshearstressisdeterminedbydividingthetotal
1 c1 1 c1 1 c1 2 c2 2 c2 2 c2
in-planeshearforcebythecorrespondingyarncross-sectionalarea.
2T ðd ÞðH ðd Þ=L ðd ÞÞ¼bðd þd Þ=ða(cid:3)d (cid:3)d Þ ð6bÞ
1 c1 1 c1 1 c1 c1 c2 c1 c2
Anexampleofdeformedconfigurationoftheunit-cellisdisplayed
where T(d ), L(d ) and H(d ) are defined by Eqs. (2)–(4), inFig.2cwhilethecorrespondingshearstressvs.shearstrainre-
i ci i ci i ci
respectively. Eqs. (6a) and (6b) can be readily solved using sultsobtainedinthisanalysisaredisplayedinFig.3b.TheFEMre-
M.Grujicicetal./MaterialsandDesign30(2009)3690–3704 3695
sultsdisplayedinFig.3barenextfittedtoaninversetangentfunc- ItshouldbenotedthatwithintheVUMATonlythenormaland
tion,denotedbyasolidlineinFig.3btoobtainarelationshipbe- shearin-planeresponseoftheshellmaterialiscomputed.Trans-
tween the in-plane shear modulus, Gin-plane, and the in-plane verseshearstiffnessoftheshellelements,ascomputedusingthe
shear strain, ein-plane. The following functional relations were three-dimensionalFEMprocedureoutlinedintheprevioussection,
obtained: Gin-plane (GPa)=38.45+23.744atan(67.0ein-plane(cid:3)26.8) hastobedefinedaspartoftheoverallFEMmodeldefinitionout-
in the absence of yarn/yarn friction, and Gin-plane (GPa)= sidetheVUMAT.Usingtheprovidedvaluesforthetransverseshear
41.142+22.26atan(69.0ein-plane(cid:3)27.6) in the case of yarn/yarn modulus, ABAQUS/Explicit uses a simple procedure in which the
frictionalcoefficientof0.5.Itshouldbenotedthatthedatapertain- transverse response of a shell is approximated by the transverse
ingtothecaseofyarn/yarnfrictionalcoefficientof0.5inFig.3bare responseofananalogoussolidfiniteelement.
shiftedupwardby10GPatoimproveclarityofthefigure. Toprovideamoredetailedinsightintotheimplementationof
thecurrentmeso-scaleunit-cellbasedmaterialmodelintheVU-
2.1.3.Out-of-plane(transverseshear)behavioroftheunit-cell MAT,aflowchartisprovidedinFig.4.Itisimportanttonotethat
Followingaproceduresimilartotheonedescribedintheprevi- ABAQUS/Explicit provides tensorial shear strain rather than the
oussection,thethree-dimensionalFEMmodeloftheunit-celldis- engineeringshearstrainincrements.Theprocedureusedtocom-
played in Fig. 2a is subjected to an out-of-plane shear and the puteunit-cellstressesattheendofthecurrenttimestepisclearly
corresponding shear stress determined by dividing the applied dependentonfailurestatusofthetwoyarnsattheendofthepre-
transverse force by the yarn cross-sectional area. An example of vioustimestepandonthecurrent(tensionvs.compression)state
deformedconfigurationoftheunit-cellisdisplayedinFig.2dwhile oftheunit-celledges.Yarnscannotsupportcompressiveloadsand
the corresponding transverse shear modulus vs. transverse shear brokenyarnscannotsupporttensileloads.Inbothcasestheasso-
strain results obtained in this analysis are displayed in Fig. 3c. ciatedin-planenormalstresscomponentsarezero.
The results displayed in Fig. 3c are fitted to a constant relation
yieldingthetransverseshearmoduliof0.89and0.97GPaforthe
casesofyarn/yarnfrictionalcoefficientof0.0and0.5,respectively. 3.Validationofthematerialmodel
Itshouldbenotedthatthedatapertainingtothecaseofyarn/yarn
frictionalcoefficientof0.5inFig.3careshiftedupwardby0.1GPa Asstatedearlier,themainobjectiveofthepresentworkwasto
toimproveclarityofthefigure. develop a fabric-structure/architecture dependent and physically
based,computationallyefficientmaterialmodelwhichcanbeused
2.2.MaterialmodelimplementationinaUser-materialSubroutine inballistic-protectionperformanceanalysesofmulti-plyarmor.In
this section, a simple projectile/armor impact problem is de-
Themeso-scaleunit-cellbasedmaterialmodeldescribedinthe scribed.Theproblemisusedtocarryoutpreliminarytestingand
previoussectionisnextimplementedinthematerialusersubrou- validation of the meso-scale unit-cell based material model for
tine,VUMAT,ofthecommercialfiniteelementprogramABAQUS/ plain-wovenfabricdevelopedinthepresentwork.
Explicit[24].Thissubroutineiscompiledandlinkedwiththefinite Theinitialconfigurationsoftheprojectile/armorfiniteelement
elementsolverandenablesABAQUS/Explicittoobtaintheneeded systemsanalyzedhereareshownasFig.5aandb.Inbothcases,a
information regarding the state of the material and the material rigidsphericalprojectilewitha4mmradiusanda2.12gweightis
mechanical response during each time step, for each integration propelledataninitialvelocityof400m/sinthedirectionnormal
pointofeachelement.Inthepresentworkfirst-order4-nodegen- to the single-ply armor surface and toward the center-point of
eral-purposereduced-integrationshellelements(ABAQUS/Explicit the armor. The armor is modeled as a 36mm by 36mm square
designationS4R)areused.DuetotheuseoftheSimpson’snumer- patch. In Fig. 5a, the plain-woven fabric structure is modeled
ical-integration method for the calculation of through-the-thick- explicitlybysnakingthroughorthogonallyorientedwarpandweft
ness deformation response of the shell, an odd number (greater yarns.Thesquare-shapedfabricpatchcontained44 warpand 44
than 1) of integration points has to be used in the through-the- weft yarns. Yarns are considered to have a constant 0.040mm2
thicknessdirection.Theresultsobtainedinthepresentworksug- hexagonal cross-sectional area. The hexagonal cross-section was
gestthatselectingthreeintegrationpointsprovidesagoodcompro- used since this is the best approximation to the actual elliptical
misebetweencomputationalefficiencyandaccuracy. cross-section which can be obtained using two-element wide
TheessentialfeaturesofthecouplingbetweentheABAQUS/Ex- yarnsandthe‘‘NodalThickness”optionavailableinABAQUS/Expli-
plicitfiniteelementsolverandtheVUMATMaterialUserSubrou- cit. The yarn width is set to 0.615mm, the peak height to
tine at each time increment at each integration point of each 0.105mm,thecrimpheightto0.088mmandcrimpwave-length
elementcanbesummarizedasfollows: to 1.64mm. All these values are consistent with those used by
Duanetal.[10–13]andZhangetal.[14]and,foraneffectiveyarn
(a) The corresponding previous time increment stresses and densityof0.6g/cm3,theyyieldaneffectivefabricareal-densityof
material state variables as well as the current time step 3.31g/cm2.
strainincrementsareprovidedbytheABAQUS/Explicitfinite Theyarn-materialisassumedtobelinear-elasticorthotropic(or
element solver to the material subroutine. In the present more precisely, transversely isotropic) with the unique material
work, the unit-cell strains, warp and weft yarn summit- directionbeingalignedintheyarn-axisdirection).Theorthotropic
point heights, yarn lengths, as well as the flags pertaining linear-elastic yarn-material properties used are listed in Table 1
tothefailurestatusoftheyarnsandthedeletionstatusof and they relate to the corresponding properties of Kevlar(cid:3) 129.
theelementareusedasstatevariables. Low values for the transverse normal and shear moduli and for
(b) Using the information provided in (a), and the meso-scale thePoisson’sratioslistedinTable1arisefromthefactthatthefi-
unit-cellusedmaterialmodelpresentedintheprevioussec- bersbundledwithinyarnsareonlyweaklycoupledtoeachother.
tion,thematerialstressstateaswellasvaluesofthemate- InthecaseofFig.5b,thearmorismodeledasaconstant-thick-
rial state variables at the end of the time increment are nessshell-basedcontinuoussurface.Inotherwords,warpandweft
determined within the VUMAT and returned to the ABA- yarnsand their weavingare not modeled explicitly.Theeffect of
QUS/Explicitfiniteelement solver.In addition,the changes yarn weaving, however, is included implicitly through the use of
inthetotalinternalandtheinelasticenergies(whereappro- the meso-scale unit-cell based material model which is assigned
priate)arecomputedandreturnedtothesolver. toeachshellelementofthefabricinFig.5b.
3696 M.Grujicicetal./MaterialsandDesign30(2009)3690–3704
Unit-cell Strain Increments, Previous Time-step
Stresses and State Variables Provided by Abaqus
Current Unit-cell Edge Lengths/Strains are Calculated
Yes
Has Any of the Yarns Already Failed?
Conpute Tension in Un-failed If Either of Yarns is Stretched, Conmpute Yarn
Stretched Yarn. Otherwise Tension(s) Using Newton-Raphson Scheme.
Set Yarn Tensions to Zero Otherwise Set Yarn Tensions to Zero
Yes
Has Tension Caused Yarns to Failed?
Set Yarn Tensions to Zero In
Failed Yarn(s)
No
Use Yarn-tension Values to Compute Unit-cell
In-plane Normal Stresses
Activate Element Deletion Flag if Both Yarns Have
Failed
Calculate Unit-cell In-plane Shear Stress
Update Material State Variables
Return Updated Streess and Material State Variable
Arrays to Abaqus
Fig.4. Theflowchartusedduringimplementationofthemeso-scaleunit-cellbasedmaterialmodelinaUser-materialSubroutine.
Fig.5. Theinitialconfigurationsoftheprojectile/fabricsystemsfor(a)ashellFEManalysisinwhichyarnweavingismodeledexplicitly;and(b)ashellFEManalysisinwhich
theeffectsofyarnweavingareincludedimplicitlythroughtheuseofanmeso-scaleunit-cellbasedmaterialmodel.
Threedifferenttypesoffar-fieldboundaryconditionsappliedto A simple penalty-based algorithm is used to model yarn/yarn
the edges of the fabric patch were considered: (a) all four fabric and projectile/fabric interactions. As simple Coulomb model was
edges are clamped; (b) two opposite fabric edges are clamped used to analyze yarn/yarn and projectile/fabric friction. Two fric-
andtheothertwoarefree;(c)allfourfabricedgesaresetfree. tionalconditionswereconsidered:(a)boththeyarn/yarnfriction
M.Grujicicetal./MaterialsandDesign30(2009)3690–3704 3697
Table1 4.Resultsanddiscussion
Theorthotropiclinear-elasticmaterialdata(GPa)foryarn[11].
E11 E22 E33 G12 G13 G23 m12 m13 m23 Inthissection,selectedresultsarepresentedanddiscussed.As
mentioned earlier, the primary purpose of the present work was
164.0 3.28 3.28 3.28 3.28 3.28 0.0 0.0 0.0
to develop, implement and validate a computationally efficient,
meso-scale unit-cell based material model for plain-woven sin-
gle-ply fabric armor. Since the key functional requirement for
coefficientl andtheprojectile/fabricfrictioncoefficientl are an armor system is to absorb the kinetic energy carried by the
y/y p/f
setto0.5;and(b)noyarn/yarnorprojectile/fabricfriction. projectile, a quantitative comparison of the results pertaining to
Due to explicit account of the warp and weft yarns and their the temporal evolution of the absorbed energies (through yarn
weavingwhichentailstheuseoffinermeshes,theFEMmodeldis- deformationandfracture,fabricaccelerationandfrictional-sliding
played in Fig. 5a is typically computationally 3–4 times more based energy dissipation) obtained using the two FEM formula-
expensivethantheonedisplayedinFig.5b. tionsdiscussedintheprevioussectionispresentedinthesubse-
Tovalidatethemeso-scaleunit-cellbasedmaterialmodel,the quent sections. The results obtained for the case of far-field
interactionbetweentheprojectileandthearmorisanalyzedusing boundary conditions corresponding to the all four fabric edges
thetwoFEMmodelandtwoaspectsoftheseinteractionsareclo- beingclamped,andforthetwofrictionalconditionsarepresented
selyexamined:(a)theresidualvelocityoftheprojectileafterthe and discussed in the next section. The effects of the far-field
projectilehassuccessfullypenetratedthearmorand(b)theextent, boundary conditions are then discussed in the subsequent
thetemporalevolutionandthespatialdistributionoffabricarmor section.
damage.
Itshouldbenotedthattheimpactofaprojectilewiththefabric
4.1.Four-edgesclampedfar-fieldboundaryconditions
is associated with the initiation of several phenomena, the most
importantofwhichare:(a)aresistingforceisexertedbythefabric
Alltheresultspresentedinthissectioncorrespondtothecase
ontotheprojectilewhichcausesareductionintheprojectileveloc-
offixedboundaryconditionsappliedtoallfouredgesofthefabric.
ity;(b)atthesametime,thefabricisbeingdeformedandacceler-
Asdiscussedearlier,yarn/yarnfrictionandprojectile/fabricfriction
ated; (c) strain waves generated in the impact region propagate
are generally present in experimental and field tests of flexible-
alongtheyarnstowardthefabricedges,wheretheyarereflected.
fabricarmor,andasshownbyTanetal.[24]inaseriesofpost-im-
Intheabsenceofanyexternalforceactingontheprojectile/fabric
pactfabric-inspectionstudies,playanimportantroleinabsorbing
system,thetotalenergyofthesystemmustbeconserved.Ingen-
theprojectilekineticenergy.Forexample,inthepresenceoffric-
eral, the energy dissipation due to projectile deformation, fiber
tionyarncrossover domains near theimpactregion are foundto
intermolecular friction, wind resistance and acoustic losses are
be characterized by extensive fiber breakage, while reduced fric-
all assumed to be negligible. Consequently, any loss in projectile
tionpromotesslippagebetweenthewarpandweftyarnsandtyp-
kineticenergyDE canbemainlyattributedtothefollowingthree
pk ically the impact-induced fabric-perforation is smaller than the
energy-absorbingmechanisms:yarnstrainenergyE ,yarnkinetic
ys projectile size. Consequently, the first case analyzed here is the
energyE ,andtheenergylostduetofrictional-slidingE.
yk f one in which both the yarn/yarn friction coefficient l and the
Theenergyconservationprinciplethenrequiresthat: y/y
projectile/fabric friction coefficient l are set to a non-zero
p/f
DE ¼E þE þE ð8Þ (=0.5)value.
pk ys yk f
AsclearlyshownbyDuanetal.[10–13],thelossofprojectileki-
4.1.1.Fabricdeformationandyarnfractureinthepresenceoffriction
neticenergyDE isgoverned(throughthethreeaforementioned
pk Itiswell-establishedthatwhenaprojectilehitsthearmor,two
energy-absorption mechanisms) by several factors such as the
wavesaregenerated:(a)alongitudinalwavewhichtravelsaway
material properties of the constituent fibers, fabric structure,
from thepoint of impact alongthe principalyarns(the yarnsdi-
boundaryconditions,projectilegeometry,impactvelocity,friction
rectlyhitbytheprojectile)andalongthosesecondaryyarns(the
betweentheprojectileandthefabric,andyarn-to-yarnandfiber-
yarnsnotdirectlyimpactedbytheprojectile)whichareinteracting
to-fiberfrictionwithinthefabricitself.
withtheprincipalyarns.Propagationofthelongitudinalwaveout-
Traditionally, the overall decrease in projectile kinetic energy,
wardfromthepointofimpactenablesalargefractionofthefabric
DE , is determined using ballistic impact experiments in which
pk armortoundergodeformationand,thus,absorbthekineticenergy
only the projectile initial velocity t and the residual velocity t
i r oftheprojectile;and(b)atransversewavewhichpropagatesout-
are measured. The overall decrease in projectile kinetic energy,
ward from the point of impact at a velocity substantially lower
DE ,isthendefinedas:
pk thanthesoundspeedandisresponsibleforstretchingoftheprin-
1 cipalandthe‘‘engaged”secondaryyarns.Sincethesoundspeedis
DEpk¼2m½t2i (cid:3)t2r(cid:4) ð9Þ controlled by the yarn axial stiffness and density, and these two
quantitiesareidenticalinthetwoFEMmodels,thetemporaland
wheremisthemassoftheprojectile.Recently,Starrattetal.[25] spatial evolutions of the longitudinal wave front are found to be
developeda ballistic impact methodfor continuous measurement quite comparable in the two models (the results not shown for
ofprojectilevelocitytðtÞ.Thelossofprojectilekineticenergyasa brevity). The temporal evolution and the spatial distribution of
functionoftimetduringimpactisthendefinedas: the transverse wave front are affected to a greater extent by
yarn/yarninteractions.Therefore,acomparisonofthetransverse-
1
DE ¼ m½t2(cid:3)tðtÞ2(cid:4) ð10Þ deflectionwavefrontpropagationresultsobtainedusingthetwo
pk 2 i
FEM analyses can be considered as a good validity test for the
While this experimental technique may greatly help improve meso-scaleunit-cellbasedmaterialmodeldevelopedinthepres-
ourunderstandingoftheballisticimpactbehavioroffabrics,many entwork.
essential physical phenomena accompanying projectile/fabric Examplesofthetemporalevolutionofdeformationwithinthe
interactions can only be obtained using numerical analyses and fabricobtainedusingtheFEManalysisinvolvingdiscreteorthogo-
simulationsliketheonesusedinthepresentwork. nal yarns plain-woven into a single-ply fabric (referred to as the
3698 M.Grujicicetal./MaterialsandDesign30(2009)3690–3704
Fig.6. Thetemporalevolutionofdeformationinthefabricfortheyarn-levelFEMmodelundertheyarn/yarnly/yandprojectile/fabriclp/ffrictioncoefficientsof0.5.Contour
bandscorrespondtodifferentvaluesofthetransversedisplacement,i.e.thedisplacementsnormaltothefabricsurface.Allfourfabricedgesarefixed.
‘‘yarn-level FEM analysis”) and the FEM analyses based on the principalyarns (the yarnswhich are in directcontact with
meso-scale unit-cell material model (referred to as the ‘‘unit-cell theprojectile)propagatesoutwardand,throughtheirinter-
basedFEManalysis”)aredisplayedinFigs.6a–dand7a–d,respec- actionswiththesecondaryyarns(theyarnswhicharenotin
tively.Inthesefigures,sideandtopviewsofthefabricalongwith direct contact with the projectile), at the yarn crossovers,
superimposed contour plots of the transverse displacement (the cause the secondary yarns to also deflect in the transverse
displacement normal to the fabric surface) are shown. A simple direction. Consequently, the transverse wave front begins
comparisonoftheresultsdisplayedinFigs.6a–dand7a–dreveals toacquireanearsquareshape,withthesquarecentercoin-
that the temporal evolution of the deformation state of fabric is cidingwiththeimpact-zonecenter,Fig.6bandc.Inthecase
quite similar in the two analyses and can be summarized as oftheunit-cellbasedFEManalysis,thefabricismodeledasa
follows: continuous surface and, hence, the transverse wave can
readily propagate in each radial direction from its source
(a) Initially,theshapeofthetransverse-deflectionwavefrontin point (i.e. from the point of initial impact). Consequently,
thefabricisnearlycircularinbothanalysesand,thus,essen- thetransverse-deflectionwavefrontacquiresanearlycircu-
tiallyidenticaltotheshapeoftheprojectile/fabriccontact- lar(moreprecisely,apolygonal)shape,Fig.7bandc.
zone,Figs.6aand7a. (c) Despite the differences in the transverse-deflection wave
(b) Inthecaseoftheyarn-levelFEManalysis,asthetimepro- front shape discussed in (b), the overall size of the fabric
ceeds,thetransverse-deflectionwavegeneratedwithinthe regionwhichhasundergonetransversedeflectionsandthe