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THEFAILUREOFNOTCHEDCOMPOSITELAMINATESUNDER
COMPRESSIONUSINGINTEGRATEDMACRO MICROMECHANICSMODEL
JunghyunAhn andAnthonyMWaas1 2
CompositeStructures
Laboratory
DepartmentofAerospaceEngineering,
UniversityofMichigan,AnnArbor,MI481092140
A combined micromechanicsmacromechanics analysis methodology is developed to predict
damage initiation in compressively loaded symmetric notched multidirectional laminates under
remoteuniaxial loading.Anovelhybrid localglobalfiniteelementmethod inconjunctionwith the
finiteelementcommercialcodeABAQUS isusedtoevaluatethegoverningsystemofequations.The
resultsobtained for thepredictionsarecomparedagainstasetofexperimental resultsforcrossply
laminates.A failuremechanismbasedunifiedmodel that capturesdamage initiation through fiber
kinkbanding
and
which
accounts
for
stress
gradient
effects
is
presented.
I.Introduction
Laminatedfiberreinforcedpolymermatrixcomposites(PMCs)arefindingincreaseduseinabroad
range of aerospace industrial applications. More importantly, PMCs are increasingly being used as
primary loadbearing components notjust as weight saving secondary structures. These applications
include onepiece fully cocured wing structures, gas turbine composite fan blades, and fuselage
reinforcementstructures.
Composite materials are superior to conventional monolithic materials for a number of reasons,
noteworthyamongstthesebeingtheirmechanicalproperties.AshbyandJones[(1994)]1haveformulated
performanceindicesthatcanbeusedtoselectamaterialforagivenapplication.PMCappearstobethe
material of choice for several applications. Topics such as damage tolerance, durability, low/high
cycle/high fatigue undermultiaxial loads, thermal cycle response and assemblyjoint technologies are
currently being researched with a view to expanding the confidence levels associated with PMC
applications.
ExperiencewithpreviousapplicationsofPMCsforrotorblades inthehelicopter industry,pressure
vessels, andother similar situations that call for superior tensile stiffness and tensile fatigue lifehave
shownthesuperiorperformanceofPMCsinatensileenvironment.Incontrast,compressivestrengthof
PMCsisknowntobelessattractive[MathewsandRawlings,(1994)]2,[WaasandSchultheisz,(1995)]3.
The response of composite laminates when subjected to mechanical loads is influenced by the
material type (fiber and matrix) and configuration (stacking sequence, layup). In addition to these
factors,geometrical
parameters
(cut
out,
notch,
thickness
change,
etc.)
and
loading
characteristics
(multiaxial,thermal,cyclicloading,etc)alsoaffecttheoverallperformanceofcompositelaminates.One
of thechallenging tasks in theanalysisanddesignwithPMCbasedstructural laminates is to improve
predictivecapability,particularly thedevelopmentofsuitableanalyses tools thatare robustenough to
1 Researcher, Aerospace Engineering, University of Michigan, Ann Arbor, Copyright 2005 by Junghyun Ahn,published by the American Institute of Aeronautics and Astronautics, Inc; with permission2 Professor of Aerospace Engineering, Associate Fellow, AIAA.
American Institute of Aeronautics and Astronautics1
46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference8 - 21 April 2005, Austin, Texas
AIAA 2005-195
Copyright 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
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incorporaterealisticgeometric imperfectionsandmanufacturingprocessrelatedmaterial imperfections
(whichwereferbroadlytoastheimperfectionsignature)andpredictstrengthanddamageinitiationload
levels.Thisrequiresthatcarefullyconductedexperimentsbecarriedoutsothattheresultscanbeusedto
guide the development ofphysics basedfailuremodels that canbe incorporated in finite elementbased
numerical codes. The present paper extends the authors previous work [Ahn and Waas, (2002)]5 on
failureprediction innotched laminates. In particular,a micromechanicsbased model thatconsistsofa
regionmodeleddiscretelyaslayersoffibersandmatrixmaterial(themicroregion)iscoupledtoaregion
thatismodeledasahomogeneousorthotropicelasticcontinuum.Damageisabsentinthelatterregion,
whichisremotefromthemicroregion.
Thedevelopmentofthemodelreliesonpreviousexperimentalworkreportedbytheauthors [Ahn
and Waas, (2002)]5. The procedure established using this model approach is found to provide an
improvedcapabilityforthepredictionofnotchedstrengthinmultidirectionallaminates.
II.ModelingofCompressiveResponseofaNotchedMultidirectionalLaminate
As shown in a companion paper [Ahn and Waas, (2002)]5, when a multidirectional laminate
containing a circular hole is subjected to remote compressive loads, failure in the form of a micro
structuralinstabilityisinitiatedinthevicinityofthehole.Forcrossplylaminates,thefailureistriggeredby zerodegreeply instability, characterizedby fiberkinking. For the caseofquasiisotropic laminates,
whichconsistofzeroand45degreeplies,twotypesoffailuremechanismscompeteagainsteachother.
Theseare failure initiationdueto fiberinstabilityinthezeropliesandfailure initiatedbymatrixshear
response,responsible for interface (fiber/matrix)crackingand/orshear failure in the45degreeplies.In
thepresentpaper,onlycrossply laminatesareconsidered.Thedominant failuremode for this typeof
laminateshasbeenreportedaszeroplykinkingfailure[AhnandWaas(1999)]6,[Soutisetal.(1991)]12.
The micromechanicsbased modeling approach used in previous papers, [Ahn and Waas (1999)]6,
[AhnandWaas,(2004)]15,employedseveralsimplifyingassumptionsaimedatreducingthesizeofthe
numericalproblem.Inparticular,symmetryconsiderationswereusedtomesharegionthatishalfofthe
micromodelregionusedinthepresentstudy,ineffecteliminatinganynonsymmetricinstabilitymodes.
The overall effect of this assumption is to introduce additional stiffening, thus generating larger thanmeasuredfailureinitiationloads.
Asanintermediatestepintheevolutionaryprocessofdevelopingaunifiedmodelthatspansseveral
lengthscales, themicroregionwasextended (referred toas the fullmodel) toeithersideof theholeas
showninFigure1andFigure2.AschematicofthenotchedcrossplylaminatesareshowninFigure1.In
Figure 1 and Figure 2, the region ABCD shows theboundary of the meshed area that represents the
microregionwithinwhich the fibersandmatrixaremodeledasdiscrete layers. Thismicroregionwas
decoupled from the remote areas through an analytical solution that was used to compute the remote
displacementfield.Thesedisplacementfieldswereusedtoloadthemicroregion,inordertostudyits
stabilitycharacteristics.Theresultsofthefullmodelwerepresentedinapreviouspaper[AhnandWaas,
AIAA20041844]15.
Afullyextendedmodel(microregionembeddedwithinahomogenizedorthotropiclamina,whichwe
refer to as the extendedmodel) is shown in Figure 3 and is the subject of this paper. In this extended
model, themicroregion ismodeledasacontinuumwithinwhich the fibersandmatrixaremodeledas
discrete layers. The region immediately outside the microregion is modeled as a homogenized
orthotropiclamina.TheLekhnitskiiorthotropicelasticity[Lekhnitskii,(1968)]6solutionforahomogenous
infinite plate containing a centrally placed cutout and subjected to remote uniaxial loading is used to
computethedisplacementfieldsalongtheedgesEE1F1F,FG,GHandHE(Figure3)ofthemicroregion.
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Becausetheentirelaminateismodeledaccordingtoclassicallaminationtheoryintheareasremotefrom
thehole,theLekhnitskiisolutionprovidestherequiredstrainfieldsfromwhichthedisplacementfields
along theboundaries EE1F1F, FG, GH and HE (Figure 3) of the microregion are computed. These
displacementfieldscorrespondtoaunitfarfielduniaxialcompressiveload.Forothermultiaxialloading
proportionalloadingsituations,asimilarprocedurecanbeadopted.Forthesakeofbrevity,theuniaxial
loadingcaseisusedtopresentthemethodologyanddemonstrateitsusefulness.
The failure initiation analysisbased on micromechanics is carried out in a manner similar to that
described in [Ahn and Waas, (1999)]6. For the FEA analysis, a rectangular mesh containing 16,006
elements(parabolicplanestrain)and48,533nodes,withtwodegreesoffreedom(uandv)pernodewas
chosen within the microregion. Convergence of the solutions with respect to microregion size and
microregionmeshdensitywerebothcheckedbyincrementallyincreasingthemicroregionsizeandmesh
density until no substantial change in the salient features (and values) associated with the response
resulted.Thesefeaturesincludedexaminationoftheinplanestressanddisplacementfields.
AflowchartoftheanalysisprocedureisindicatedinFigure4(a).First,anelasticeigenvalueanalysis
ofanisolated(decoupledfromtherestoftheplate)microregionmodeliscarriedoutinordertoobtain
inplaneeigenmodesthatcorrespondtotheentireplatehavingthesamegloballoadcorrespondingtothe
subsequentresponseanalysis that follows.Theeigenmodeshapeassociatedwith thesmallestnonzeroeigenvalue is used to perturb the FE mesh corresponding to the microregion. Because the eigenmode
shapes are localized, the outer boundaries of the microregion (ABCD) are unperturbed and they
smoothly attach to the outer homogenized orthotropic region of the laminate. That is, the interface
betweenthemicroregion(ABCD)andtheouterhomogeneous(orthotropic)regionismodeledasglued
contact interface, to avoid excessive mesh transitionby assuming the interface as a contact interface.
ResponseanalysesarecarriedoutusingthearclengthmethodoptionprovidedinABAQUS.Duringthe
responseanalysis,theboundariesEE1F1F,FG,GH,andHEaresubjectedtodisplacementfieldsthatare
computed from theorthotropicelasticitysolution.The tractionload,previouslyappliedalong theedge
AB in the fullmodel (see [Ahn and Waas, (2004)]15), isno longer necessarybecause the currentmodel
boundary is extended to the free edge of the hole. Simulations were carried out for a series of
imperfectionmagnitudes
for
ahole
size
(0.25R).
Theeigenmodesprovidetheperturbationshapefortheinitialfibermisalignmentbutnottheabsolute
magnitudeofperturbation.Thus,theusermustspecifytheimperfectionmagnitude.Inthepresentwork,
thisisachievedasfollows;asshowninFigure4(b),themaximumamplitude,,ofthelowesteigenmode(which occurs at the free notch edge indicated as pointJ in figure 2) is chosen such that the fiber
misalignment angle canbe controlled. Since the characteristic half wave length , of the localizedeigenmode shape is known, is chosen such that assumes the intended value. Several simulationscorrespondingtodifferentvaluesofwerecompleted.Ineachcase,themaximumloadassociatedwiththegloballoadproportionalityfactorappliedinproportiontotheinitialdisplacementfieldiscomputed.
In a typical run, the relation between the applied load and the axial displacement (average axial
displacement of the microregion along the edge BC) is as shown in Figure 4(a) inset response of
microregion.The
load
rises
almost
linearly,
followed
by
awell
defined
peak
load
and
subsequent
snap
backintheresponse.Thepeakloadcorrespondstotheonsetofinstabilitywhilethesnapbackpartofthe
response corresponds to the formation of a kinkband (in which the deformation is localized) with
simultaneous growth and rotation of fibers within the kink band. The farfield applied loads
correspondingtothepeakloadscanbeplottedasafunctionofimperfectionmagnitude(seeFigure4(a)
insetextrapolationtoperfectcase),fromwhich,byextrapolation,themaximumappliedremotestress
correspondingtoperfectlystraightfiberscanbeextracted.Thus,theprocessoutlinednotonlygivesthe
maximumstrengthofthelaminatecorrespondingtothegiven(ormeasured)fibermisalignment,butalso
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providesthemaximumattainablestrengthofthelaminate.Thelattercouldbeusedasanupperboundon
strength,whilethestrengthcorrespondingtoafibermisalignmentof1.5o(say)couldbeusedasthelower
bound. The amount of fiber misalignment is dependent on the manufacturing imperfection signature.
Thatis,differentmanufacturingprocessesgiverisetodifferentstatisticsofinitialmisalignment.Oncethe
imperfectionsignature isknown, thenthevaluesof thatareassociatedwithaspecificmanufacturingprocess canbe extracted during the inspection (using techniques such as ultrasonic methods, xray
techniquesetc.)phaseofhardware.Forprepregbasedlaminates nominalvaluesintherangeof=0.5020isreasonable.
Throughouttheanalyses,thefiber(IM7)isassumedtobelinearlyelastic(seeTable1),andthematrix
(9773 toughened epoxy) property within the laminate (insitu) is evaluated from a +45/45 laminated
coupontest.Thistestisdescribedinthenextsection.Thus,theinsitunonlinearstressstrainresponseof
the matrixbehavior is incorporated in the present analysis. The matrix ismodeled as aJ2 incremental
flowtheorysolidwithisotropichardening,andthisisalsoverifiedthroughmeasurement.
III.InSituMatrixCharacterization
To evaluate the material properties of the matrix within the laminate (insitu matrix properties),
a ( coupon test (Figure5)atroom temperaturewasperformed following theprocedureofASTMD351876[ASTMStandard,(1982)]
)45 ns8. Theelasticpropertiesofthelaminaandthecompleteshearstress
shearstrainbehaviorofanIM7/9773intheprincipalmaterialcoordinatesystemwereobtainedfromthis
test.Theprocedureusedtogeneratethedataconsistsofsubjectinga ( )45ns
angleplylaminatetouniaxial
compression and measuring the laminate strains and yyxx and the applied remote stress on the
laminate.Notethat,forthistest, xy xx yy = .ThedatashowninFigure6fromthistestcanbeusedto
extract the complete nonlinear shear stress shear strain response of the insitu matrix (9773) as
discussedbelow.Beginbyassuming thatthe9773matrixmaterialcanbemodeledasanelasticplastic
solidobeyingthesmallstrainJ2flowtheoryofplasticity[Lubliner,(1998)]9.Then,fromtheelastic(linear)
portionof thecurve inFigure6, the inplane lamina shear modulus is firstobtained. In the lamina
principalcoordinates,onehas,
12G
11 22
12 12
2
2
xx yy
xx yy
+= =
= = (1)
and
11
22
12
2
2
2
xx
xx
xx
=
=
=
(2)
Thus,using(1)and(2)aboveandthedefinitionsofequivalentstress, andequivalentplasticstrain
increment, [Lubliner,(1998)]pd 9,thedatainFigure7canbeusedtoconstructaplotof against .p
According to theJ2 flow theory of plasticity with a MisesHenky yield condition, the ratio of the
increment of each plastic strain component to its corresponding deviatoric stress component remains
constant[Lubliner,(1998)]9,
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p
ij
ij
dd
= (3)
( )12Using (3),and therelationbetween and , thenonlinearportionof theshearstressp
d shear
strain ( )12 responsecurveforasinglelaminacanbeextracted.Thecurve thusobtained forIM7/9773
system
is
shown
in
Figure
8.
The
instantaneous
slope
of
the
curve
in
Figure
8
is
the
tangent
shear
modulusofthelamina, ( )12 12T
G ,whiletheratioofshearstresstoshearstrainisthesecantshearmodulus.
TheapproximateHalpinTsairelations[DanielandIshai,(1994)]10areusedtoextractthevariationofthe
insitumatrixshearmodulus ( )12S
mG ,
1
2 2
12
2
12
2
12 2
1
1
fS S
m
f
f m
f m
vG G
v
G G
G G
+
=
=+
(4)
with for random packing of fibers. With2
1 = ( )12SmG so obtained, the insitu matrix shear stress
shearstraincurveisasshowninFigure9.Inasimilarmanner,theinsituuniaxialstressstraincurvein
compressionforthematrixisalsoobtained(Figure10).Thisuniaxialresponsecurvewasdiscretizedand
used in the input data file to represent the matrix material in the microregion for the micromechanics
basedfiniteelementanalysiscarriedoutusingABAQUS.
IV.FEAResultsandDiscussion
A typical load responsebehavior of a microregion within a cross ply laminate model is shown in
Figure11andaseriesofdeformed plotsof the microregionshowing the initiationandpropagationof
damageintheformofkinkbandingisasshowninFigure12,foraspecimenwithaholesize0.25inch
radius,withafibermisalignmentof=0.87.
The
microregion
response
follows
a
linear
path
up
to
pointc
.
Although
local
matrix
yielding
(in
areasofthemicroregionnearthecutout)isindicatedpriortotheattainmentofpointd,thetotalintegrity
of this region is not affected much from the matrix yieldingbecause, (1) the area of yielding is small
compared to theoverallsizeof themicroregionand (2) the fiberrotationsaresmallup to thepointof
maximum load (pointe). The sudden load drop is defined as the failure initiation point (the
correspondingload,therefore,isthefailureinitiationload). Asloadingincreases,thefibersintheareas
wherethematrixhasbecomesofterstarttorotate,resultinginadropoftheresultantforce.Oncethe
loaddropinitiates(characterizedbyhighlocalizeddeformationattheholecenter),thebandoflocalized
deformation continues to propagate into the unkinked material until it reaches the outer micromodel
boundary.
Simulations corresponding to several different fiber misalignment angles were carried out and the
resultsfor
maximum
load
so
obtained
were
plotted
against
imperfection
amplitude
in
order
to
determine
theupperboundstrengthcorrespondingtoperfectlystraightfibers(Figure13).
Anotherfailuremechanismobservedwithlaminatesthatincludeadominantnumberofoffaxisplies
was matrix shearing eventually leading to fiber/matrix interface fracture. An analysis that includes
incorporationofcohesivezonemodelsmuchinthespiritof[SongandWaas(1995)]11,isrelegatedtothe
future. For now, we observe the good agreement between the farfield load corresponding to the
maximumloadpredictionandtheexperimentallymeasuredfailureloadsforkinkbandingasshownin
Figure13,especiallyforthecaseofsmallimperfection(=0.45o)andaholesizeof0.25inchradius.The
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9. LublinerJacob,PlasticityTheory,Text,PrenticeHall,1998.
10. Daniel, Isaac M. and Ori Ishai, EngineeringMechanics of CompositeMaterials, Oxford University
Press,NewYork,1994.
11. Song,S.J.andWaas,A.M.,AnEnergyMethodbasedModelforMixedModelFailureofLaminated
Composites,AIAAJournal,vol.33,No.4,pg.739745,1995.
12. Soutis,C., Fleck, N. and Smith, F.,Failure Prediction Technique for CompressionLoadedCarbon
FiberEpoxyLaminatewithanOpenHole,J.CompositeMaterials,25,pg.1476 1498,1991
13. Starnes,J.andWilliams,J.G.(1982),FailurecharacteristicsofGraphite/Epoxystructuralcomponents
loadedincompression,NASATM84552.
14. Khamseh A. and Waas A., Failure Mechanisms of Composite Plates with a Circular Hole under
remoteBiaxialPlanarCompressiveLoads,ASMEJ.MaterialsandTechnology,vol.119,pg.5664,1997
15. Ahn,Junghyun and Waas, A.M., Stress Gradient Effects on Notched Composite Failure using a
LocalGlobalApproach,45thAIAAASMESDMConferenceAIAA20041844
VII.TablesandFigures
MaterialE11
(Msi)
E22
(Msi)
G12
(Msi)
Thicknes
s(in)12
IM7/9773
Epoxy23.5 1.21 0.72 0.30 0.0052
2.756E
04IM7Fiber 42 0.25
9773
Epoxy
2.444E
040.7 0.34
Table1Zeroplymaterialpropertiesofthe48plygraphite/977 3epoxycomposites
Hole Radius (in) 0.0625 0.125 0.25
Max. Load (noimperfection) 163 123 75
(ksi)
Max. Load (1.5 degimperfection)
Table2AnalysesResultsusingFullModel(fromAhnandWaas,200415)
(ksi)78 63
44 (matchingtest result)
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x
yz Multi-Ply
Uni-Ply
x
y
z
x
yz Multi-Ply
Uni-Ply
x
y
z
x
y
z
y
z
Micro-Region
Figure 1. Problem configuration for Crossp ly Laminates
B
CD
J
Figure 2. Full Model
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E1 F1E
H G
F
CD
A BJ
Figure 3. Extended Model
Eigenvalue Analysis(Model Perturbation)
Static Analysis(Force Field)
A
D C
B
X
Y
Nonlinear Response Analysis
Response of Microregion
0.125R
0
5
10
15
20
25
30
35
40
0 .0 E +0 0 1 .0 E+ 04 2 .0 E+ 04 3 .0 E +0 4 4 .0 E+ 04 5 .0 E+ 04 6 .0 E +0 4 7 .0 E+ 04 8 .0 E+ 04 9 .0 E +0 4
Far Field Load (LB)
LocalRF1(LB)
5E-5
7.5E-5
10E-4
Extrapolation to Perfect Case
Extrapolation (0.25R)
y = -241.19x3 + 4509.7x2 - 26153x + 74550
R2 = 1
0.0E+00
1.0E+04
2.0E+04
3.0E+04
4.0E+04
5.0E+04
6.0E+04
7.0E+04
8.0E+04
9.0E+04
0.0E+00 5.0E-01 1.0E+00 1.5E+00 2.0E+00 2.5E+00 3.0E+00 3.5E+00
Fiber Angle (deg)
P(LBF)
Figure 4(a)Imperfection Sensitivit y Analysis Procedure
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D C
A B
EigenvalueAnalysis
CL
( )1tan =
EigenvalueAnalysis
CL
( )1tan =
Figure 4(b)Details of initial imperfection
Figure 5. +45/-45 coupon test of a specimentaken from the cross p ly laminates
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Figure6.+45/45compressionexperimentresult.
Equivalentstressvs.equivalentplasticstrain.
Figure 7. Equivalent stress vs. equivalentplastic st rain (room temperature).
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Figure 8 Shear Stress,12
vs. Shear Strain,12
curve for IM7 / 977-3 based on J2 Flow Theory ofPlasticity and the data in Figure 7.
In-Situ Matrix Shear Response
0
2500
5000
7500
10000
12500
15000
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
Shear Strain
ShearStress
(psi)
Figure 9. The in-situ Shear Stress vs.Shear Strain response of the matrix.
In-Situ Matrix Axial Response
0
5000
10000
15000
20000
25000
0 0.01 0.02 0.03 0.04 0.05
Strain
Stress
(psi)
Figure 10. The in-situ uniaxial stress vs.uniaxial strain response of the matrix.
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Load vs. Deflection
0
5000
10000
15000
20000
25000
30000
35000
0.0E+00 5.0E-05 1.0E-04 1.5E-04 2.0E-04 2.5E-04 3.0E-04 3.5E-04 4.0E-04
Disp (in)
FarFieldL
oad(lbf)
e
cd
Figure 11. Typical Response Curve of the microregion (0.25R)
c
Limit Load
Max. Load ~ 39 ksi
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