MMF Laminate Combine

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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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