MethodologyforCharacterisationofGlassFibreCompositeArchitecture
J.Zangenberg1,2,J.B.Larsen2,R.C.stergaard2,andP.Brndsted11MaterialsResearchDivision,RisDTU,Frederiksborgvej399,4000Roskilde,Denmark2LMWindPower,Jupitervej6,6000Kolding,DenmarkSeptember4,2012AbstractThe
present study outlines a methodology for micro-structural
char-acterisationof brereinforcedcomposites containingcircular
bres.Digital micrographsof
polishedcross-sectionsareusedasinputtoanumerical image processing
tool that determines a spatial mapping andradii detection of the
bres. The information is used for dierent
anal-ysestoinvestigateandcharacterisethebrearchitecture.
Asanex-ample,themethodology isapplied toglass bre reinforced
compositeswithvarying brecontent.
Thedierentbrevolumefractionsaectsthenumberof contactpointsperbre,
thecommunal bredistance,andthelocal brevolumefraction.
Thebrediameterdistributionandpackingpatternremainsomewhatsimilarfortheconsideredma-terials.The
methodology is a step towards a better understanding of the
com-posite micro-structure, and can be used to evaluate the
interconnectionbetweenthebrearchitectureandcompositeproperties.Keywords:
Glassbres;Scanningelectronmicroscopy(SEM);Compositemicro-structure;Microstructuralcharacterisation.1
IntroductionThe bre volume fraction (FVF) is considered being the
most crucial micro-structural
parameterfordescribingabrereinforcedcomposites. Thepa-This paperis
part of aspecial issueonDeformationandfractureof polymers
andtheircompositeCorresponding author. Tel.: +45 5138 8407. Fax:
+45 4677 5758.
Email:[email protected]@lmwindpower.com1rameterforms the
basis forthe well-knownrule of mixtures fordeterminingthe in-plane
stiness of a composite material,see e.g. Jones [1].
Commonly,theFVFisdeterminedonbasisoftheweightfractionsandthedensitiesoftheindividual
constituents, andyieldsanaveragelaminateparameter
fortheentirecomposite.
However,onalocalscaletheFVFattainslargerval-uese.g.
insideindividual bundles wherethebre
concentrationisenlarged.Itisevidentthatthereexistsothermicro-structuralparametersthatchar-acterise
a bre reinforcedcomposite e.g. packing pattern,
neighbouringdistance, clustering, etc. Informationabout
theseparameters canbeex-tracted from measurements on cross-sections
of the given composite.
Digitalimage-analysisand-processingaretoolstobeusedtoidentifythematerialmicro-structure.
Attempts have been made in order to characterise the full
3dimensionalstructureof(glass)brecomposites(FVF,bremisalignment,waviness,bre
curvature, etc.) using a sectioning approachbased on micro-scope
images. This analysis requires great accuracy in the sectioning
processandinindetingtheindividualbres.
Themethodistimeconsuming,butserveswellinordertodescribethelocalappearanceofthemicro-structure.This
3D characterisationhas been carried out by severalresearchers, see
forinstancePaluch[2]orClarkeetal. [3].
Thesestudiesalsoincludedetermi-nationofthebre
misalignmentangleinspiredbytheideasofYurgartis[4]wherethemisalignmentangleisdeterminedbymeasurementsoftheellip-tical
bre-shapethatappearsonaninclinedcrosssectioncomparedtotheprincipalbredirection.
Usingnumericalroutines,Creightonetal. [5]andKratmann et al. [6]
determined the misalignment angle for much lower
reso-lutionimages,andthussavingtheprocessingtime.
Thedierencebetween2Dand3Dcharacterisationisoutlinedinthe,tosomeextend,personalac-countofExner[7].Basedondigital
micrographs of aplanar andpolishedcross-sectionof auni-directional
brecomposite, this studypresents
amethodologywhichdetectsthespatialdistributionandradiiofthebres.
Thismappingofthebres isusedtodene
anumberofparametersthatareconsideredrelevantforamicro-structural
characterisationof brereinforcedcomposites. Themethodis capableof
analysinglocal regions as well as entirebundles orcross-sections.
Forillustratingtheapplicability,themethodologyisappliedtodierentglassbrereinforcedcompositeswithavaryingbrecontent.Theideaspresentedareinspiredbythereviewworkof
Guild&Summer-scales[8] whodiscussanumberof
dierentapproachesforimageanalysisofbrecomposites,
Pyrz[9]whoquantitativelyinvestigatedthecompositemicro-structurebystatistical
tools,
andPaluch[2]whomadea3Dcharac-terisationbasedon2Dimagetechniques.
Thisstereological
approachforinvestigatingthe3Dmicro-structurebasedon2Dimageshasbeenusedex-tensively,but
it seems that the methods and ideas have not been brought toany
practicalexperiences. Forinstance when the composite
micro-structureisinvestigatednumericallyusingrepresentativevolumeelements
(RVEs),2general assumptions are made on the bre distribution and
packing pattern,communal bredistance, boundaryconditions, etc.
Wongsto&Li [10] in-vestigatedthe eect of boundary condition on
the RVE and made numericalsimulation of a random distributed bre
arrangement. The bre randomnesswasfurtherinvestigatedbyMelroet. al
[11]
whodevelopedastatisticallycharacterisedalgorithmtogeneraterandomcompositecross-sections.
Thepredictions fromthealgorithminterms of eective
compositepropertiesagreedwell withexperimental data.
TheideaofrandombredistributionwasalsousedbyMishnaevsky&Brndsted[12]who,inanumericalstudy,considered
fatigue damage of unidirectional bre reinforced composites.
On-going work will extend the knowledge of the interplay
betweenfatiguedam-ageandbrearchitecturesincedetailsinthebrearchitecturehasproventobedetrimentalforthefatigueperformance.2
MethodShapedetectionalgorithmsarewell-knownwithintheeldofdigitalimageanalysis,
andthereexistsnumerousalgorithmsdependingonthepurpose.AcommonprocedureistheHoughtransformation,
Hough[13], whocameup with the idea of shape detection of lines
based on a parameter space. Thework was further developedby Duda
& Hart [14] where the shape detectionwas extended to include
circular objects. This method is commonly referredto as the
Circular Hough Transformation, and a more profound
presentationcanbefoundinShapiroetal. [15].
Themagnitudeoftheparameterspacefor the shape detectiondepends on
the shapes considered. For instance, linedetectionuses a2parameter
space(slopeandintersection),
circledetec-tion3(centre(x,y)andradius),
andellipses5(minor/majoraxis, centre(x,y), andorientation).
Increasingtheparameterspaceheavilyincreasesthecomputational
requirements.
ThecurrentmethodisbasedontheCir-cularHoughTransformationfordetectionofcircularshapes.The
analysis is split up into three dierent steps: pre-processing,
processing,andpost-processingaspresentedinFig.
1andthefollowingsub-sections.31Specimenpreparation2.1Image
preparation andacquisition2.2Shapedetectionandbremapping3Geometry
analysesPre-processingProcessingPost-processingFigure1:
Flowchartfordetectionofcompositemicro-structure.2.1
Pre-processingThepreparationof thetestsamplesiscrucial
inordertoobtainthebestpossibleimagequality.
Standardmicroscopespecimensareusedwherearepresentativematerialsampleofthecompositelaminateiscutandcastinanepoxyresintoformacylinderwithadiameterof
30mm.
Thereupon,thespecimensarepolishedinagrinderwithabrasivepapervaryingfrom#250to#4000ingrainsize.
Thepolishingtimeisadjustedaccordingtothegrainsize.
Itisevidentthatthesamplesurfaceappearsasplaneandsmoothaspossible.2.2
ProcessingTo obtain the best image quality,it is recommended to use
a Scanning Elec-tronMicroscope(SEM), anEnvironmental
ScanningElectronMicroscope(ESEM), or a Low Vacuum Scanning Electron
Microscope (LVSEM) for im-ageacquisition.
IfaSEMisused,thespecimensneedtobecoatedduetotheir
non-conductingsurface. Itispossibletouseanoptical microscope,but
this requires evenmoresensitivityinthepre-processing stepdue
tothelowerresolution, lowerdepth-of-eld,
andpoorerlightingconditionsinanoptical microscope compared to an
electron microscope. In the present
workaLVSEMisused.Themagnicationof
themicroscopeshouldbeadjustedsothemisinter-pretationintheradiidetectionisreduced,seefurtherdiscussioninSection2.4.
Tominimisecomputationalrequirements,theshapedetectionisbasedon8-bitgreyscaleimages.
Uponacquisitionofthemicrographs,theshapedetectionisdividedintotwoindependentsteps:
aCannyedgedetectionofthe image,Canny [16], and a circle
detectionbased on these edges using theCircularHoughTransformation.
TheCannyedgedetectionlocates edgesby searching for a local maxima
of the colour gradient within the consideredimage.
AnexampleisillustratedinFig. 2wherethegreylevel
intensityisillustratedalongagivenpathintheimage.
Itisevidentthattheedgeof4thebresaredetectableduetotheobviouscolourgradient.50m(a)
SEMmicrograph of glass bres (lightpart) andmatrix(darker part)
inaUDre-inforcedcomposite. Thearrowindicatesthedirectionfor the
greylevel intensityinFig.2(b).0 10 20 30 40 50050100150200Distance
[m]Grey value [](b)Greylevelintensityalongthe pathshowninFig. 2(a).
Whitelevel is 255, blackis
0.Notethattheinterfacialregiongivesrisetoalargergreyvaluethanthebre.Figure2:
Identicationof breedgesinauni-directional
(UD)brerein-forcedcompositebasedonaSEMmicrograph.Oncetheedges of
thebres aredetected, theCircular HoughTrans-formationis
appliedbyavotingproceduretoidentifythebest matchingcircles.
Theinputparametersforthetransformationcanbeadjustedde-pending onthe
imagequality(thresholdlevel,searchrange,andmagnitudeofsearchlter).
Theresultsofthedetectionarethein-planebrelocation(x,y-coordinates)andthebreradius,r.Duetothenumerical
processingpowerthereisalimitfortheimagesizethatcanbe analysedby
theCircular HoughTransformation.
Therefore,forlargerimagesizes(approximatelylargerthan1000
1000pixels)theentireimageissplitupintoanumberofuserdenedsegmentsthatareanalysedseparately,
andtheresultsarestoredforeachsegment. Anaveragevaluefor the
entireimage is evaluatedonce allsegments havebeen analysed.
Thesegmentation procedure can also be used to characterise the
micro-structureacrossabrebundle,eectsofclusteringduetostitchingtension,etc.2.3
Post-processingCirclesetsonaplanearedenedif thepositionsandradii of
thecirclesareknown.
Characterisingthesecirclesetsrequiresknowledgeaboutthenumberofcircles,circlediameterdistribution,communaldistancebetweenthecircles,
clustering, packingpattern, andcirclesincontact.
Whentheseparametersareknownitispossibletomakeafull
characterisationof thecircleset.
ThefollowinganalysesareconsideredbasedonthebrecentreandradiiobtainedfromtheCircularHoughTransformation:5
globalbrevolumefraction voidcontent brediameterdistribution
numberofcontactpointsperbre nearestneighbourdistance
breclusteringparameter
numberofneighboursandlocalbrevolumefraction.Eachoftheaboveanalysesareoutlinedinthefollowing.FibrevolumefractionThe
FVF is considered a fundamental
parameter,whichisusedincharacterisingandcalculatingmechanical
propertiesofbrereinforcedcomposites.
Inthepresentwork,theFVFisdeter-minedusingthreedistinctprocedures:1.
Knowingtheindividualbreradiiandthetotalimagesize,itispossibletodeterminetheFVFof
thedetectedbres. Itisas-sumed that the position and diameter of the
bres do not vary inthe normal directionof the bres. Hence,the FVF
is determinedas:FVF =N
i=1Vi,fibreVtotal=N
i=1Ai,fibreAimage=N
i=1r2iAimage, (1)whereV is thevolume, Ais thearea, Nis thenumber
of -bres, andriis theindividual breradius. This
procedurefordeterminingtheFVFisreferredtoasFVF1.2. Asimple
procedure toevaluatethe FVFisby athresholdanaly-sis.
Theimageistransformedintoablack/whiteimagewhereastheresinappearsasblackpixelsandthebresaswhitepixels.TheFVFisfoundastheratiobetweenthewhitepixelsandthetotalnumberofpixels.
ThisprocedureisreferredtoasFVF2.3.
ItislikelythattheCircularHoughTransformationdoesnotde-tect every
bre inthe cross-section especially near the imageboundary
wherethebres arecuto. This proceduresimply
re-movesthedetectedbres(usedinprocedureone)andperformsathresholdanalysis(asinproceduretwo)ontheremainingun-detectedbres.
ThisprocedureisreferredtoasFVF3.ItisgiventhatthesumoftheFVFsfromproceduresoneandthreeshouldmatchproceduretwo.6Voidcontent
Voidsmaybepresentinthematerials,
andthesevoidsareknowntoinuencethemechanical properties.
Sincevoidsappearasblackregionsinthemicrographs,
theseareidentiedifapixel valueis lowerthan a user dened threshold.
The totalvoidcontentis
foundasthesumofthesepixelsinrelationtothetotalnumberofpixels.FibrediameterdistributionItis
oftenassumedinnumerical analysesof
brereinforcedcompositesthatthebrediameterdistributionisuniform,
whichisoftennotthecaseinpractice. The bre diameterismeasured by the
Circular Hough Transformation and assumed normaldistributed meaning
that it is characterised by the mean value and
thestandarddeviation.NumberofcontactpointsperbreContactbetweensurfaces
leads tostressconcentrations; therefore, contact
betweenbresisconsideredasapotentialzoneforcrackinitiation,seee.g.
Wongsto&Li[10].
Acontactconditionbetweentwocircles(bres)isdenedifthecentre-to-centre
distance between two adjacent circles i and j, dij, is less
thanthesumof theirindividual radii, ri + rj. However,
duetothebresurface roughness, interface, and inaccuracy in the
circle detection, thecontact condition is given as dij (ri+rj)
where the factor = 1.01follows from the work of Mishnaevsky &
Brndsted [12]. Denoting thetotal number of circle contact
conditions as c, the total number
ofcontactpoints,CPtotal,isdeterminedas:CPtotal=c N2,
(2)wherethefactoroftwoisincludedtoavoidrepeatedcontactpoints.Since
the total number of contact points in Eq. (2) is dependent on
thenumber of inclusions, the totalnumber of contactpoints is
normalisedwiththenumberofdetectedbres,N,inordertogetthenumberofcontactpointsperbre,CP.Fortwocommonbrepackingpatterns,thenumberofcontactpointsperbreandFVFisillustratedinFig.3foraninnitebrearray.7(b)Hexagonal
array.2rRFVF=23
rR
2(a)Squarearray.FVF=
rR
2R2rCP=2(R = 2r)CP=3(R = 2r)Figure 3: Typical bre packing
patterns in a uni-directional composite
alongwithbrevolumefraction(FVF) andnumberof contact
pointsperbre(CP).NearestneighbourdistanceSeparation distance
between the bres mayminimise the stress concentration, and this
analysis determines the dis-tancetothenearestneighbour.
TheNeighbouringDistancebetweenbres i andj is denedas NDij=dij ri
rj, andtheshortestdistanceisfoundbythesolutiontotheclassical
TravellingSalesmanProblem(TSP)usingaTSP-algorithm.
FurtherinformationontheTSPcanbefounde.g. inLawleretal. [17].
Theresultoftheanaly-sisistheshortestroutethroughallbres,andtherebytheNearestNeighbouringDistances,
NND. Forquantication,
thenearestneigh-bourdistanceisassumedtofollowalog-normalbehaviourwithmeanandstandarddeviationmeaningthatthelimitsareexpressedasexp
(log() log()).FibreclusteringClusteringof bres leads toregions that
inuencethemechanical behaviour. Inordertoestimatethebreclustering,
thesimple relationdue toClark & Evans [18] is used asa measure
for thebreclustering. Originally, thetheorywas usedintermsof
spatialrelationsin populations, but it is directly appliedin the
current studyeventhoughtherearesomelimitations.
Theclusteringparameter,R,isexpressedas:R = rA rE, rA=
NNDNand rE=12(3)with rAbeing the mean of the series of distances
to the nearest neigh-bour,and
rEisthemeandistancetothenearestneighbourexpectedinaninnitelylargerandomdistributionwithdensity.
Inthecur-rentcase, thesumof thenearestneighbouringdistances,
NND,
isfoundastheshortestroutebetweenbrecentrementionedabove.Thebredensityisfoundas
=N/Aimage.8The convenient thing about the clustering parameter is
its easily acces-sible interpretationsince it is bound by the
limits 0 R 2.15wherethe lower limit follows fromanintermediate
neighbouringdistanceequal
tozero(fullyclustered)andtheupperlimitisforahexagonalarraywithequidistantdistancetootherneighbours.
ForR=1thepackingpatternisrandom.Numberofneighboursandlocal
brevolumefractionOftenwhenmodelling the composite micro-structure
an idealised bre packing
pat-ternisassumedtobeinthearraysshowninFig. 3. Thesepackingpatterns
are seldomlyfoundinpractice, andthe
followinganalysisestimatesthepackingpatternintermsof thenumberof
neighboursand a local bre volume fraction. More sophisticated
methods may beusedtocharacterisethepackingpatterne.g.
thesecondorderinten-sityfunctionasproposedbyPyrz[9]andGhostetal.
[19].ThenumberofneighboursisfoundfromaDelaunaytriangulationofthebrecentrepoints,
andthelocal FVFisdeterminedasthebreareainrelationtotheareaof
Voronoi cell
associatedtoeachbre.Inbrief,aVoronoitessellationisadecompositionofascatter(inthiscasepointsontheplane)intoacellstructureeachcontainingexactlyonepoint.
Thepropertyofthecellisthatanypointinsidethecellisclosertothatpointthantoanyothersite.
ADelaunaytriangulation,ontheotherhand, is sortofthedual problem
totheVoronoitessella-tion such that no point sets are inside the
circumcircle of any triangle.The two principles are illustrated for
a generatedbre arrangement inFig. 4.
ForamoreprofoundpresentationseeOkabe[20].Fibre
centreVoronoiDelaunayFibreFigure4:
SketchoftheVoronoitessellationandDelaunaytriangulationforarandompointsetintheplane.9Inthepresentanalysis,
theVoronoi cellsareusedtodeterminealo-cal
FVF(referredtoFVF4)foreachbre, sincetheareaof thecell,Avoronoi, can
be considered as an area parameter associated to each -bre with
area, Afibre. Thereby, the local FVF is determined
asAfibreAvoronoi.Theareaof theVoronoi cells
alongtheimageboundarycannot beevaluated explicitly,which is why the
boundary bres are disregarded.Thelocal FVFisassumedtofollowanormal
distributiondescribedbythemeanvalueandthestandarddeviation.
Thecentrepointsofthe detectedbres areusedas input for the
triangulationinordertondthenumberof neighbours,
herepresentedasthemeanvalueandstandarddeviation,respectively.
Similaranalyseshavebeenusedpreviouslye.g. intheworkofPaluch[2].2.4
ImplementationThe programming language MATLAB is used for the
implementationof themethodology.
TheCircularHoughTransformationandtheTSP-algorithmcanbefoundattheMathWorksFileExchange[21].The
algorithm and implementation have been tested on two selected bres
onseveralimageswithdierentmicroscopemagnicationinordertoestimatethe
accuracy of the circle detection. The bres are shown in Fig. 5(a)
whereFibre A is completelyisolated,and Fibre B is surrounded by
severalothers.TheresultfromtheanalysisisshowninFig.
5(b)wherethedetectedbrediameterisplottedasfunctionofmicroscopemagnication.10100mFibreAFibreB(a)Fibresconsideredintheaccuracyanalysis.0
5 10 15141516171819Pixels per micronFiber diameter [m] Fibre AFibre
B(b) Detected bre diameter as afunction of microscope
magnica-tion.Figure5:
Accuracyanalysisfordetectionofthebreradius.For low magnication
images below 5 pixels per micron there is a scatterin the measured
bre diameters. However, in order to get sucient accuracyandamountof
bresperimage, avalueof
1.64pixelspermicronisusedinthefollowingwellawarethatthisgivesrisetoanon-negligibledeviationindeterminationof
the bre volumefractioninprocedure FVF1mentionedabove.Depending on
the image shape, image size, and number of bres, the
numberofcontactpointsisaected. Therefore,
atestiscarriedouttoinvestigatetheimageshape/sizesensitivity.
Thetestiscarriedoutforasquarebrepackingarrangementwithnointermediatedistancebetweenthebres.
Foravaryingimageshape/size, itturnsoutthatforalow numberof
bres(say, lessthan50bresinanarrowshapedimage)thenumberof
contactpointsisaected.113 ApplicationTwo-layered glass/polyester
composites with varying bre content were
man-ufacturedusingtheVacuumAssistedResinTransferMoulding(VARTM)process,
andthespecimens areanalysedbythemethodologyoutlinedinSection2. All
imagesareacquiredwithinabundlewithoutanyresinrichzonesneartheedges.
AtypicalresultofthecircledetectionispresentedinFig. 6.(a)Basis for
image analysis: SEM micrographofapolishedcross-section.(b) Spatial
mapping of the bres with de-tectedradii
usingtheCircularHoughTrans-formation.Figure 6: Typical images for
analysing the micro-structure of a uni-directional
glassbrereinforcedcomposite.
Thepresentedimagesareseg-mentsoftheanalysedimages.Close examination
of the detected circles shown in Fig. 6(b) reveals
thatthealgorithmdetects anumber of non-existingbres; nonetheless,
thesebres are removedinthe post-processingstepbyevaluationof the
greylevelintensityatthecentreofthe(mis)detectedbre.
Thepost-processinganalyses are carried out as mentioned in the
previous section and the resultsarepresentedinTable1andFigs. 7and8.
anddenotethemeanvalueandstandarddeviation,andalldataareassumedtobenormal/log-normaldistributed
and independent. Table 1 presents the number of detected
bresandthebrediameterdistribution.12Table1: Resultof imageanalysis:
numberof detectedbresandbredi-ameterdistribution.
Meanandstandarddeviation.No. NoFaFD/[-] [m/m]1 4745 17.19/1.462
4605 17.17/1.453 4797 17.21/1.574 3335 16.96/1.445 7679
17.32/1.50aNoF NumberofFibresFD FibreDiameterFig.
7presentstheobtainedFVFsforthedierentanalyses, andtheresults are
normalisedwiththevalue fromFVF3(thresholdanalysis)
insamplenumberone.1 2 3 4 50.9511.051.11.15Sample no.Norm. FVF []
FVF1FVF2FVF3FVF4Figure7: Resultsofimageanalyses:
brevolumefractionbasedondier-entprocedures. FVF1: radii of bres.
FVF2: boundaryanalysis. FVF3:thresholdanalysis. FVF4:
local(Voronoi).Fig. 8showsthenumberof contact pointsperbre, CP,
thenearestneighbour distance, NN, the clusteringparameter, ClP,
andthe numberofneighbours,NoNasafunctionoftheobtainedFVF. Again,
theFVFisnormalised with the value from FVF3 (threshold analysis) in
sample numberone.130.95 1.00 1.05 1.10 1.15
1.200.50.550.60.650.70.750.8Norm. FVF []CP [](a)0.95 1.00 1.05 1.10
1.15 1.20147101316Norm. FVF []NN [m](b)0.95 1.00 1.05 1.10 1.15
1.201.961.971.981.9922.012.02Norm. FVF []ClP [](c)0.95 1.00 1.05
1.10 1.15 1.2055.566.57Norm. FVF []NoN [](d)Figure8:
ResultsofimageanalysesfordierentFVFs. (a)Contactpointsper bre, CP.
(b) Nearest neighbour distance, NN. (c) Clustering
parameter,ClP.(d)Numberofneighbours,NoN.Thevariationinthebrediametersissmall,andthemeasurementsarealmost
constant in the range around df 17m, which is also the
prescribeddiameterbythemanufacturer.As mentioned in the previous
section, the sum of the FVFs determined fromprocedures
FVF1andFVF2shouldbeequal totheFVFdeterminedbyprocedureFVF3.
ThisisillustratedinFig. 7wherethedeviationbetweenthemethods is
within 1.2%, whichis regardedas asucient accuracyinthe shape
detection. It is worthnoticingthat the local bre volumefraction,
FVF4,
isconsistentlylargerthantheFVFsdeterminedfromtheotheranalyses.
Novoidsarefoundinthesamplesinvestigated.For a heavier bre
compaction, it is found that the number of Contact Pointsper
bre(CP) increases as well as theNearest Neighbour
distance(NN)decreases, seeFig. 8.
ThepackingpatternremainsthesameindependentoftheFVF,whichisreectedintheNumberofNeighbours(NoN)andtheClustering
Parameter (ClP) approaching the upper limit of 2.15. This
meansthatthepackingpatternconvergesagainstapseudo-hexagonalarray.144
DiscussionFromthenumberofbresdetected(seee.g.
Table1)andtheimagesizesused,theimagesizesensitivityinrelationtothenumberofcontactpointsper
bre is limited for the samples considered. The number of contact
pointsperbreincreasesforincreasingbrevolumefraction,whichinaccordancetowhat
isreportedby Mishnaevsky &
Brndsted[12]inanumericalstudy.Eventhoughtheactual brepackinginFig.
6(a) doesnotappeartobesystematic,
thebrestendtoarrangeinwhatisreferredtoasapseudo-hexagonal
packingpattern. UsingtheDelaunaytriangulation,
Paluch[2]madeasimilarconclusioninrelationtothepackingpattern.Pyrz[9]carriedoutaquantitativestudy
onthemicro-structureofcompos-ites based on statisticalanalyses. In
specic, a probability
investigationwasmadebetweenthenearestneighbouringdistancefordierentFVFs.
ThetrendisobviouslythatthelargerFVF,
thesmallerameannearestneigh-bouring distance and standard
deviationis observed. The same ndings
areconcludedinthepresentstudy.The clustering parameterdescribed in
the work of Clark & Evans [18] is
de-terminedbasedonthebrecentrewithoutinformationregardingthebreradius.
Therefore, thebasisof
usingthisanalysisismisleadingsincetheunderlyingstatistical
theoryisbasedonpointsetsratherthancircles. Ithas not been possible
at this stage of the study to nd a suitable
descriptionforclusteringofcircleswheretheindividualradiiareincluded.It
is apparent that it wouldbemore accurate todeterminethe
Voronoidiagramsintermsof
circlesetsonaplaneratherthanpointsduetothecircularcrosssectionofthebres,seee.g.
presentationbyKimetal. [22].However,
basedontheworkofPaluch[2]andtoeasetheimplementation,thepresentedmethodisconsideredtobesucient.The
local FVF (FVF4)predicts a largervalue comparedto the
globalFVF,whichisincontradictiontowhatcouldbe expectedfrom the
idealisedbrearrangementsinFig. 3thatproducessimilarresultsforalocal
andglobalFVF. Still, themagnitudesof
thedeviationisinsimilarrangetowhatisfound from Ghosh etal. [19]
based onsimulationsof uniform bre
distribu-tionswithFVFintherangeupto32.4%.Theoptimumsolutiontothetravellingsalesmanproblemhasbeeninves-tigated
by several researchers, and there exist numerous of dierent
solutiontechniquesfortheproblem. However,
theproblemofndingtheoptimumsolutionisfarfromtrivial.
InthecaseofNbres,thenumberofdierentroutes is given as(N1)!2, which
indicates the increasing problem complexityfor largeN. Therefore,
approximatedmethodsareoftenusedwheretheroutemightnotbetheoptimumonebutsomewhatcloseto.
Suchanalgo-rithm is used in the present,and as a result the
shortest route might not betheoptimum.155
ConclusionAnewmethodologyispresenteddealingwithshapedetectionandmicro-structuralanalysisofcompositematerialscontainingcircularbres.
Basedon digital micrographs and numerical image processing, the bre
architecture/micro-structure isevaluatedfora number of dierent
parameters. Fordemonstra-tion, themethodologyis shownuseful
toidentifyandtocomparemicro-structural parameters for dierent glass
bre reinforcedcomposites withvaryingbre content. The
followingmicro-structural parameters are af-fected by changes in
the bre volume fraction: the number of contactpointsper bre,
nearest neighbour distance, andthelocal
brevolumefraction(thebreareainrelationtotheareaof
theassociatedVoronoi cell). Forincreasingbrevolumefraction,
thenumberof
contactpointperbrein-creaseswhereasthenearestneighbourdistancedecreases.
Thelocal brevolumefractionissomewhatlarger thantheglobal
brevolumefraction.Voidsarenotfoundinthesamples.
Independentofthebrevolumefrac-tion, the bres are arranged in a
pseudo-hexagonal array with
approximatelysixneighboursperbre.References[1] R. M.
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