Evaluation of effective material properties of randomly distributed short cylindrical fiber composites

12 Pages·2008·0.99 MB·English

Journal of
Mechanics of
Materials and Structures
EVALUATION OF EFFECTIVE MATERIAL PROPERTIES OF
RANDOMLY DISTRIBUTED SHORT CYLINDRICAL FIBER
COMPOSITES USING A NUMERICAL HOMOGENIZATION
TECHNIQUE
Harald Berger, Sreedhar Kari, Ulrich Gabbert,
Reinaldo Rodríguez Ramos, Julian Bravo Castillero
and Raúl Guinovart Díaz
Volume 2, Nº 8 October 2007
mathematical sciences publishers

JOURNALOFMECHANICSOFMATERIALSANDSTRUCTURES
Vol.2,No.8,2007
EVALUATION OF EFFECTIVE MATERIAL PROPERTIES OF RANDOMLY
DISTRIBUTED SHORT CYLINDRICAL FIBER COMPOSITES USING A
NUMERICAL HOMOGENIZATION TECHNIQUE
HARALD BERGER, SREEDHAR KARI, ULRICH GABBERT, REINALDO RODRÍGUEZ RAMOS,
JULIAN BRAVO CASTILLERO AND RAÚL GUINOVART DÍAZ
Inthispapereffectivematerialpropertiesofrandomlydistributedshortfibercompositesarecalculated
with a developed comprehensive tool for numerical homogenization. We focus on the influence of
changeinvolumefractionandlength/diameteraspectratiooffibers. Twotypesoffiberalignmentsare
considered: fiberorientationswitharbitraryanglesandparallelorientedfibers. Thealgorithmisbased
onanumericalhomogenizationtechniqueusingaunitcellmodelinconnectionwiththefiniteelement
method. Togeneratethethree-dimensionalunitcellmodelswithrandomlydistributedshortcylindrical
fibers, a modified random sequential adsorption algorithm is used, which we describe in detail. For
verificationofthealgorithmandcheckingtheinfluenceofdifferentparameters,unitcellswithvarious
fiberembeddingsarecreated. Numericalresultsarealsocomparedwiththosefromanalyticalmethods.
1. Introduction
Shortfibercompositescanbeeasilyproducedandhavegoodmechanicalproperties. Sincethemixture
ofshortfibersandliquidresincan bemanufacturedby injectionorcompressionmolding, theproduction
ofpartswithnearlyarbitraryandverycomplicatedshapesispossible. Compositesconsistingofspatially
distributedshortfibershavebecomepopularinawidevarietyofapplications. Moreover,usingspatial
short fibers as reinforcing elements in a controlled manner can provide more balanced properties, which
leadtoanimprovedthrough-the-thicknessstiffness/strength.
A classical problem in solid mechanics is the determination of effective elastic properties of a com-
posite material made up of a statistically isotropic random distribution of isotropic and elastic short
cylindrical fibers embedded in a continuous, isotropic and elastic matrix. Even though analytical and
semianalyticalmodelshavebeendevelopedtohomogenizefibercomposites,theyareoftenapplicable
onlytospecificcases. Numericalmodelsseemtobeawell-suitedapproachtodescribethebehaviorof
these materials, because there are no restrictions on the geometry, on material properties, on the number
of phases (constituents) and on size. In order to obtain realistic predictions of a new material, micro-
macroconsiderationsaretheappropriateapproach. Inthiscontextthefiniteelementmethodhasbeen
usedtodetermineeffectivepropertiesoftheshortfibercompositesbasedonunitcellmodels.
Keywords: finiteelementmethod,unitcell,representativevolumeelement,homogenization,shortfibers,randomsequential
adsorptionalgorithm,effectivematerialproperties.
ThisworkwassupportedbyDFGGermany,Graduiertenkolleg828Micro-MacroInteractionsinStructuredMediaandParticle
Systems.Thissupportisgratefullyacknowledged.
1561
1562 BERGER,KARI,GABBERT,RODRÍGUEZ,BRAVOANDGUINOVART
A number of classical micromechanics theories have been developed. Using variational principles,
Hashin[1962]andHashinandShtrikman[1963]establishedboundsonmaterialsthatcouldbeconsid-
ered as mechanicalmixturesofanumberofdifferentisotropicandhomogeneouselasticphases which
are then treated as statistically isotropic and homogeneous. These two-point bounds were improved by
three-pointbounds[Milton1982;MiltonandPhan-Thien1982],whichincorporateinformationaboutthe
phasearrangementthroughthestatisticalcorrelationparameters. Thediluteapproximationcanbeused
tomodeladilutesuspensionofsphericalelasticparticlesincontinuouselasticphases. Theinteraction
betweenparticlesisneglected. Sothealgorithmreducestothatofsolvingtheproblemofasphericalin-
clusioninaninfinitematrixsubjectedtohydrostaticloadingatinfinity. Eshelby[1957;1959]considered
the problem of an ellipsoidal inclusion in an infinite isotropic matrix, assuming a well-defined matrix.
That, however,isnotalwaystrue inpolycrystalline materials. Avarietyof propertiescanbe exhibited,
butthereisnoclearlydefinedmatrixphase. Inthesecasestheinteractionsbetweenparticlesaremore
significant. The Mori–Tanaka method [Mori and Tanaka 1973] was designed to calculate the average
internal stress in the matrix containing precipitates with eigenstrains. Benveniste [1987] reformulated
it so that it could be applied to composite materials. He considered isotropic phases and ellipsoidal
phases. Recently, Segurado and Llorca [2002] and Bo¨hm et al. [2002] have assessed the effective co-
efficients of randomly distributed spherical particles using random sequential adsorption method and
comparedthemwithHashin–Shtrikmanboundsandotherresultsfromliterature. Gusevetal.[2000]and
Lustietal.[2002] performed experiments of randomly distributed short cylindrical fiber composites and
foundgoodagreementwithnumericalresults. However,duetothelackofliteraturewhichdealswith
randomlydistributedshortcylindricalfibersandtherestrictiontolowvolumefractionsoffibers,wehave
beenmotivatedtodevelopanumericalhomogenizationtoolwhichextendsthelimitsandprovidesthe
basisforinvestigationofcompositeswitharbitraryinclusions. Inouropinionmicro-macromechanical
approachesoffernewinsightsinthematerial behaviorofsuch fibercomposites, and may result innew
procedurestodeveloprealisticmaterialmodelsfordesignandoptimizationpurposes.
2. Numericalhomogenization
2.1. Basicprocedure. Themechanicalandphysicalpropertiesoftheconstituentmaterialsarealways
regarded as a small-scale/micro structure. To predict the overall behavior of the structure on a macro
level, the knowledge of effective material properties is necessary. One of the most powerful tools to
estimate such effective properties is the homogenization method. The main idea is to find a globally
homogeneous medium equivalent to the original composite, such that the strain energy stored in both
systemsisapproximatelythesame. Thecommonapproachtomodelthemacroscopicpropertiesoffiber
composites is to create a unit cell or a representative volume element (RVE) that captures the major
featuresoftheunderlyingmicrostructure.
The RVE can generally be considered as a periodic part of the heterogeneous structure that is suffi-
cientlylargetobeastatisticallyrepresentativeofthecomposite,thatis,toeffectivelyincludeasampling
of all microstructural heterogeneities that occur in the composite [Kanit et al. 2003]. To obtain the
homogenizedeffectivematerialproperties,periodicitymustbeensuredforthemechanicalbehaviorof
theRVEbyintroducingperiodicboundaryconditionsbetweenoppositesurfaces. Byconstructingseveral
loadcaseswithselectedtractionloadsandselectedshearloadsinonedirectionandpreventingstrainsin
MATERIALPROPERTIESOFRANDOMLYDISTRIBUTEDSHORTCYLINDRICALFIBERCOMPOSITES 1563
theotherdirections,alleffectiveelasticitycoefficientscanbecalculatedfromtheconstitutiverelations
byanaveragingtechnique. Thisprocedureis describedindetail in[Bergeretal.2005a;2005b] andshall
notbeexplainedhere.
2.2. Fibergenerationbyrandomsequentialadsorptionalgorithm. CreationofanRVEwithrandomly
distributed cylindrical short fibers which fulfill certain restrictions, such as nonoverlapping, ensuring
periodicity and the like, is a difficult task. Due to the statistical distribution of inclusions, the RVE
can be modeled as a cube with unit size. For automatic generation of such RVEs, a modified random
sequential adsorption algorithm [Hinrichsen et al. 1986] is used. Several input parameters can be given,
includingthesizeofRVE,diameterandlengthrangeoffibers,minimumdistancebetweenneighboring
fibers, and desiredvolumefraction. Thealgorithmstartsby creating the cylinder axisof the firstfiber
atarandomposition,withrandomlengthandwithrandomangle. Subsequentlynewfibersarecreated
with random distribution values. If the new fiber matches the restriction of nonoverlapping and sufficient
distancetotheearlierone,itisaccepted;otherwiseitisdeleted. Furthermore,toensureperiodicity,if
anysurfaceofthecylindercutsanyofthecubicRVEsurfacesitiscopiedtotheoppositesurfacewith
theRVEsizelength. Inthiscaseonealsochecksalltherestrictions;ifitfails,theoriginalandcopied
fibersaredeleted. Concerningthelaterfiniteelementgeneration,wewouldalsoliketoensurethatsome
(cid:3)
practical limitations are fulfilled. For instance, the cylinder surfaces should not be very close to the
RVEsurfaceaswellastocornersoftheRVEinordertoavoidhighlydistortedfiniteelementsduring
(cid:3) meshing. The generation of new fibers is repeated until the desired volume fraction is reached or no
morefiberscanbeplacedduetotheaforementionedrestrictions. Figure1showsasampleofgenerated
fibers before cuttin(cid:3)g on the RVE surface, after cutting, and an ensemble of four RVEs which demonstrate
thatperiodicityismaintained.
By modifying the input parameters it is possible to create RVEs with different fiber arrangements,
suchas,forexample,fibersofsamediameter,fibersofsamelength,oronlyparallelalignmentoffibers.
CombiningthesearrangementsopensthepossibilityofgeneratingRVEswhichrepresentdifferenttypes
offiberreinforcedcompositessuchasthosepresentedinthispaper. Thepossiblemaximum fibervolume
fractionplaysanimportantrole. Ingeneral,forfibersofidenticalsizethealgorithmcangenerateupto
(a) (b) (c)
Figure1. Generationofrandomlydistributedshortfibers: (a)uncutfibers,(b)cutfibers,
(c)periodicitydemonstratedwithfourRVEs.
(cid:3)
(cid:3)
1564 BERGER,KARI,GABBERT,RODRÍGUEZ,BRAVOANDGUINOVART
X2(cid:3) X2(cid:3)
X3(cid:3) X1(cid:3) X3(cid:3) X1(cid:3)
(a) (b)
Figure2. Typesofcomposites: (a)ROFand(b)POF.
(cid:3) 25% fiber volume fraction(cid:3). For higher volume fractions, one must use fibers of different sizes, which
canbegeneratedbycreatingfiberswithsubsequentlydescendingdiameters. Usingthisapproach,fiber
volumefractionupto40%canbeachievedwithminimumdistortionofthefiniteelements.
Forcalculatingeffectivematerialpropertiesofrandomlydistributedshortfibercomposites,weinvesti-
gatetwotypes offiber arrangements: randomly oriented fibers(ROF) and parallel-oriented fibers(POF);
see Figure 2. For POF composites, the fibers in the models are aligned along the x -axis. This is denoted
3
asthelongitudinaldirection,whiletheperpendicular x x planearethetransversedirections.
1 2
2.3. Finiteelementmodeling. AllfiniteelementcalculationswereperformedwiththecommercialFE
packageANSYS.Thematrixandthefibersweremeshedwith10nodetetrahedronelementswithfull
integration. For the calculation of geometry of the fibers by random sequential adsorption algorithm,
aspecialpreprocessorwasdevelopedinFORTRANprogramminglanguage,whichproducesapartial
input file for ANSYS. Cutting of the fibers on the RVE surfaces is carried out by geometrical modeling
featuresofANSYS.Figures3and4showsamplesofmeshedRVEsforROFandPOFmodels.
Toapply the periodic boundary conditionson the RVE, identicalmeshesarenecessaryonopposite
surfaces. For this purpose a surface of the RVE is first meshed with blind plane elements; then this
(a) (b)
Figure3. SampleformeshedRVEforROF:(a)onlyfibers,(b)fibersandmatrix.
(cid:3) (cid:3)
MATERIALPROPERTIESOFRANDOMLYDISTRIBUTEDSHORTCYLINDRICALFIBERCOMPOSITES 1565
(a) (b)
Figure4. SampleformeshedRVEforPOF:(a)onlyfibers,(b)fibersandmatrix.
element configuration is copied to the opposite surface. Based on all meshed surfaces, three-dimensional
meshing is carried out. In order to apply periodic boundary conditions, we must generate constraint
equations between opposite nodal pairs. Here the ANSYS Parametric Design Language (APDL) is
used to automate this process. This script language allows the common nodal pairs to be identified
automaticallybytheircoordinates. Furthermore,APDLisusedtocollectaveragedstressesandstrains
fromelementsolutionaswellastocalculatetheeffectiveelasticconstants.
ThecombinationoftheFORTRANpreprocessor,APDLandANSYSbatchprocessingletsusauto-
matethewholeprocess. Italsoprovidesapowerfultoolforthefastcalculationofhomogenizedmaterial
propertiesforcompositeswithagreatvarietyofinclusiongeometries.
3. Testmodels
Wehaveinvestigatedtwotypesofshortfibercomposites: ROFandPOF.Inordertotesttheinfluence
ofvariousparameters,differentRVEsweregenerated. Furthermoresothatwecanobtainstatistically-
averagedresultsforeveryconfiguration,fiveRVEswithdifferentstartingvaluesfortherandomalgorithm
were generated. The material properties of the constituents used for the analysis to evaluate the effective
materialpropertiesweretakenfromliterature[Bo¨hmetal.2002]toverifythedevelopedmethodwith
othersolutions. Table1containsYoung’smoduliandPoisson’sratiosformatrixandfibers.
The calculated results were compared with different analytical methods such as Hashin–Shtrikman
two-pointbounds(HS)[HashinandShtrikman1963],Mori–Tanakaestimates(MTM)[MoriandTanaka
1973],theself-consistentmethod(SCM)[LiandWang2005],andthegeneralizedself-consistentmethod
(GSCM)[ChristensenandLo1979]. Wealsoperformedstudiestodeterminetheinfluenceofaspectratio
length/diameteroffibersontheeffectivematerialpropertiesofthesecomposites.
Constituent Young’smodulus Poisson’sratio
MatrixAl2618-T4 70GPa 0.3
FiberSiC 450GPa 0.17
Table1. Materialconstantsforconstituentsofthecomposite.
1566 BERGER,KARI,GABBERT,RODRÍGUEZ,BRAVOANDGUINOVART
4. Resultsanddiscussion
4.1. Variationofvolumefraction. Effective values of Young’s modulus E, shear modulus G and Pois-
son’sratioν wereevaluatedfordifferentvolumefractionsfrom10%to40%instepsof10%;seeFigure5.
FivesamplesofRVEmodelswithrandomlydistributedshortfibers(randomangleoforientation,random
diameterandlengthinacertainrange)weregeneratedforeachvolumefraction. Insixparticularload
casestheRVEsweresubjectedtouniaxialtensionaswellassheardeformationalongthethreecoordinate
axes[Bergeretal.2005a; 2005b]. Fromtheseloadcasesnine materialconstantswerecalculated: E ,
11
E , E , G , G , G ,ν ,ν ,ν . Becauseofthestatisticallyisotropymeanvalues E, G andν from
22 33 12 13 31 12 23 31
alldirectionswereusedforcomparisonwithothermethods. Furthermore,duetotherandomdistribution
180
ROF ROF
70
MTM MTM
160 GSCM GSCM
SCM SCM
HS−L 60 HS−L
a)140 HS−U a) HS−U
P P
G G
E (120 G (50
100 40
80 30
10 20 30 40 10 20 30 40
Volume fraction Volume fraction
(a) (b)
0.3
0.28
n0.26
ROF
MTM
0.24 GSCM
SCM
HS−L
HS−U
0.22
10 20 30 40
Volume fraction
(c)
Figure5. VariationofeffectivematerialpropertiesforROFcompositeswithchangein
volumefractionandcomparisonwithdifferentanalyticalresults: (a)Young’smodulus,
(b)shearmodulus,(c)Poisson’sratio.
MATERIALPROPERTIESOFRANDOMLYDISTRIBUTEDSHORTCYLINDRICALFIBERCOMPOSITES 1567
offibersineachfive samples,acertainvariancecanbe observedfortheeffectivevalues. Theboundsof
thisvariancearemarkedinthefigureswithverticalbarsinthesenseofastandarddeviation.
Theeffectivematerialproperties,whichwereobtainedforROFcompositesusingthepresentednu-
merical homogenization technique, lie within the lower (HS-L) and upper (HS-U) Hashin–Shtrikman
bounds. The results of the analytical methods MTM and GSCM are always the same and are nearly
identicalwithHS-L.Theresultsofoursolution(ROF)arenearertotheself-consistentmethod(SCM)
forallvolumefractions. ThemaximumdifferencebetweenROFandSCMisabout3%.
ToshowthenearlyisotropicbehavioroftheROFcomposite,inFigure6weploteffectiveYoung’s
moduliin allcoordinatedirectionsasmeanvalues fromthe fiverandomsamplesfordifferent volume
fractions. EffectiveYoung’smoduli,whichwereobtainedforthethreecoordinatedirections,arenearly
thesameoverthefullinvestigatedrangeofvolumefraction;themaximumdifferenceislessthan1.5%.
Thisindicatesanearlyisotropicmacrobehavioroftheshortfibercompositewithrandomlydistributed
fiberorientation.
EffectivematerialpropertiesobtainedforPOFcompositeswerecomparedwithROFcomposites. Fig-
ure7showsthevariationofeffectiveYoung’smoduliforPOFcompositeswithchangeinvolumefraction
in three coordinate directions, and compares it with the results for ROF composites. The transverse
Young’s moduli of POF composites have slightly lower values compared to ROF composites. Never-
theless,alongthelongitudinaldirectionthePOFeffectivematerialpropertieshavehighervalueswhen
compared with ROF composites. Thisis obvious because in case of POF composites, fibers are aligned
along the longitudinal direction, which results in higher stiffness relative to the transverse directions.
From Figure 7 it can also be seen that the effective Young’s moduli E and E are nearly the same;
11 22
thisfactexpressestransverseisotropy.
4.2. Variationoffiberaspectratio. Wehaveinvestigatedtheinfluenceofaspectratiolength/diameter
L/D of fibers on their effective material properties for ROF and POF composites. As L/D increases,
thecompositetendstoalongfibercomposite. Theeffectivematerialpropertieswerecalculatedat10%
volumefractionoffibers. Table2representsthevariationofeffectivematerialconstants E , E and
11 22
160
E11
E22
E33
140
a)
P
G120
E (
100
80
10 20 30 40
Volume fraction
Figure6. IsotropyofeffectivematerialpropertiesexpressedbyYoung’smoduliinthree
directionsforROFcomposites.
1568 BERGER,KARI,GABBERT,RODRÍGUEZ,BRAVOANDGUINOVART
160 160
ROF ROF
POF POF
140 140
a) a)
P P
G 120 G 120
(1 (2
1 2
E E
100 100
80 80
10 20 30 40 10 20 30 40
Volume fraction Volume fraction
(a) (b)
180
ROF
POF
160
a) 140
P
G
(3
E3120
100
80
10 20 30 40
Volume fraction
(c)
Figure7. VariationofeffectiveYoung’smoduliforallthreedirectionswithchangein
volumefractionforROFandPOFcomposites.
E with change in aspect ratio L/D of the fibers for ROF and POF composites. From Table 2 it can
33
be observed that with respect to change in aspect ratio of fibers, there are no significant variations in
effectiveYoung’smodulialongthethreecoordinatedirectionsforROFcomposites. Thisisnottruefor
POF composites, which show a significant variation in E with the increase in aspect ratio of fibers.
33
Alongthetransversedirection, E and E ofPOFcompositesareslightlylessthantheseparameters
11 22
for ROF composites, but variations in the transverse Young’s moduli with respect to the aspect ratio of
fibersarenotsignificant.
5. Conclusions
Numericalhomogenizationtoolshavebeendevelopedandpresentedfortheevaluationoftheeffective
materialpropertiesofshortfiberreinforcedcomposites. Theeffectivematerialpropertiesofrandomly
orientedfiber(ROF)andparallel-orientedfiber(POF)compositeswereobtainedusingthesetoolsand