non uniform spatial arrangement of fibres

11
A c t a mater. Vol.45,No.7,pp. 3059-3069, 1997 G 1997ActaMetallurEica Inc. Pergamon PublishedbyElsevier Sci&ceLtd PII: S1359-6454(96)00394-1 PrintedinGreatBritain.Allrightsreserved 1359-6454/97 $17.00+0,00 MODELING OF NON-UNIFORM SPATIAL ARRANGEMENT OF FIBERS IN A CERAMIC MATRIX COMPOSITE S. YANG, ASIM TEWARI and ARUN M. GOKHALE Department of Materials Scienceand Engineering,Georgia Institute of Technology,Atlanta, GA 30332-0245, U.S.A. (Received19 July 1996) Abstract-In the unidirectional fiberreinforcedcomposites, the spatialarrangement of fibersis often non-uniform. Thesenon-uniformitiesare linkedto the processingconditions,and they affectthe properties of the composite. In this contribution, a recently developeddigital image analysis technique is used to quantify the non-uniform spatial arrangement of Nicalon fibers in a ceramic matrix composite (CMC). These quantitative data are utilized to developa six parameter computer simulatedmicrostructure model that is statistically equivalent to the non-uniform microstructure of the CMC. The simulated microstructure can be utilized as a RVE for the micro-mechanicalmodeling studies. 0 1997 Ac?a A4etallurgicaInc. INTRODUCTION The properties and performance of unidirectional aligned fiber reinforced composites depend on the metric geometric attributes of the fibers such as volume fraction, number density, size distribution, etc., and spatial arrangement of fibers in the matrix. Usually, the spatial arrangement of fibers is not uniform. Figure 1 shows the microstructure of a ceramic matrix composite (CMC) containing uni- directional aligned SiC fibers (Nicalon) in a glass ceramic (MAS) matrix. The spatial distribution of fibers appears to be non-uniform, and there is a distribution of fiber sizes.The non-uniform nature of the spatial distribution of fibers is usually linked to the processing techniques used for making the composites. Recently, Brockenbroughet al. [1]showedthat, in the plastic regime, the transverse deformation of metal matrix composites is significantlyaffected by non-periodic spatial distribution of fibers. These authors concludedthat the theories and modelsbased on periodic spatial distribution of brittle fibers in a ductile matrix cannot correctly predict the transverse deformation behavior of the composites having non-uniform spatial distribution of fibers. Sorensen and Talreja [2] studied various non-uniformities of fiber distributions systematicallyand found that the local residual stresses due to cool-down during processing depend significantly on deviations from uniformity of spatial distributions of fibers.They also showed that in assessing the effects of coatings and interphases such non-uniformitiesmust be taken into account [3]. Pyrz [4,5] observed that in a polymer matrix composite, the transverse constitutive behav- ior and fracture path tortuosity are sensitive to the spatial arrangement of fibers,which in turn depends on the material’sprocessingparameters. Whited et al. [6] observed that micro-crack damage evolution during thermal cyclingof MMC is quite sensitiveto the inter-fiber distances. The above discussion shows that: (1) the micro- structure of compositesoften consistsof non-uniform spatial distributions of fibers;(2)the non-uniformities in the spatial arrangement of fibers are governed by the techniques and parameters used for material processing; and (3) plastic deformation, damage evolution and fracture of composites depend on the spatial distribution of fibers. Therefore, it is of interest to develop quantitative models for the microstructure of fiber composites that account for the non-uniform spatial distribution of fibers. An appropriate microstructural model can provide an important input for the virtual integrated prototyping of the composite manufacturing process. In this context, it is important to point out that the spatial distribution of fibers may vary from batch to batch, or from one plate to another plate of the same batch, and some times, from one region to another region in the same plate. The current approaches to the micro-mechanical modeling of the behavior of composites do not consider the non-randomnessand non-uniformity of the spatial distribution of fibers. Therefore, there is a need to develop a micro-mechanical modeling approach, where the representative volume element (RVE) represents the real composite microstructure. Such a modeled microstructural volume element should have the same volume fraction, number density and size distribution of fibers as in the real composite, and the spatial distribution of fibers should be statistically equivalent to that in the 3059

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Page 1: Non uniform spatial arrangement of fibres

A c t amater.Vol.45,No.7,pp.3059-3069,1997G 1997ActaMetallurEicaInc.Pergamon

PublishedbyElsevierSci&ceLtdPII: S1359-6454(96)00394-1 Printedin GreatBritain.Allrightsreserved

1359-6454/97$17.00+ 0,00

MODELING OF NON-UNIFORM SPATIAL ARRANGEMENTOF FIBERS IN A CERAMIC MATRIX COMPOSITE

S. YANG, ASIM TEWARI and ARUN M. GOKHALEDepartment of Materials Scienceand Engineering,Georgia Institute of Technology,Atlanta,

GA 30332-0245,U.S.A.

(Received19July 1996)

Abstract-In the unidirectionalfiberreinforcedcomposites,the spatialarrangementof fibersis oftennon-uniform.Thesenon-uniformitiesare linkedto the processingconditions,and theyaffectthe propertiesof the composite. In this contribution, a recently developeddigital image analysis technique is used toquantify the non-uniform spatial arrangement of Nicalon fibers in a ceramic matrix composite (CMC).These quantitative data are utilizedto developa six parameter computer simulatedmicrostructuremodelthat is statistically equivalent to the non-uniform microstructure of the CMC. The simulatedmicrostructure can be utilized as a RVE for the micro-mechanicalmodeling studies. 0 1997Ac?aA4etallurgicaInc.

INTRODUCTION

The properties and performance of unidirectionalaligned fiber reinforced composites depend on themetric geometric attributes of the fibers such asvolume fraction, number density, size distribution,etc., and spatial arrangement of fibers in the matrix.Usually, the spatial arrangement of fibers is notuniform. Figure 1 shows the microstructure of aceramic matrix composite (CMC) containing uni-directional aligned SiC fibers (Nicalon) in a glassceramic (MAS) matrix. The spatial distribution offibers appears to be non-uniform, and there is adistribution of fiber sizes.The non-uniformnature ofthe spatial distribution of fibers is usually linkedto the processing techniques used for making thecomposites.

Recently, Brockenbroughet al. [1]showedthat, inthe plastic regime, the transverse deformation ofmetal matrix composites is significantlyaffected bynon-periodic spatial distribution of fibers. Theseauthors concludedthat the theoriesand modelsbasedon periodic spatial distribution of brittle fibers in aductile matrix cannot correctlypredict the transversedeformation behavior of the composites havingnon-uniform spatial distribution of fibers. Sorensenand Talreja [2] studied various non-uniformities offiber distributions systematicallyand found that thelocal residual stresses due to cool-down duringprocessing depend significantlyon deviations fromuniformityof spatial distributions of fibers.They alsoshowed that in assessing the effectsof coatings andinterphases such non-uniformitiesmust be taken intoaccount [3]. Pyrz [4,5] observed that in a polymermatrix composite, the transverse constitutive behav-ior and fracture path tortuosity are sensitive to the

spatial arrangement of fibers,which in turn dependson the material’sprocessingparameters. Whitedet al.[6] observed that micro-crack damage evolutionduring thermal cyclingof MMC is quite sensitivetothe inter-fiber distances.

The above discussion shows that: (1) the micro-structure of compositesoften consistsof non-uniformspatial distributions of fibers;(2)the non-uniformitiesin the spatial arrangement of fibers are governed bythe techniques and parameters used for materialprocessing; and (3) plastic deformation, damageevolution and fracture of composites depend onthe spatial distribution of fibers. Therefore, it is ofinterest to develop quantitative models for themicrostructure of fiber composites that account forthe non-uniform spatial distribution of fibers. Anappropriate microstructural model can provide animportant input for the virtual integratedprototypingof the composite manufacturing process. In thiscontext, it is important to point out that the spatialdistribution of fibers may vary from batch to batch,or from one plate to another plate of the same batch,and some times, from one region to another regionin the same plate.

The current approaches to the micro-mechanicalmodeling of the behavior of composites do notconsider the non-randomnessand non-uniformityofthe spatial distribution of fibers. Therefore, there isa need to develop a micro-mechanical modelingapproach, where the representative volume element(RVE) represents the real composite microstructure.Such a modeled microstructural volume elementshould have the same volume fraction, numberdensity and size distribution of fibers as in the realcomposite, and the spatial distribution of fibersshould be statistically equivalent to that in the

3059

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YANG et al.: NON-UNIFORM ARRANGEMENT OF FIBERS 3061

then be investigated through the micro-mechanicalmodeling approach.

It is the purpose of this contribution to present amethodology that enables the development of aquantitative microstructure model for unidirectionalaligned fiber reinforced CMC, where the fibers arenon-uniformly distributed. The next section givesa brief background to two important statisticaldistributions that quantify the spatial arrangement offibers. The subsequent section describes the exper-imental technique for an unbiased estimation of thesedistribution functions, and that is followed by theinterpretation of the measured spatial descriptors.Finally, a geometric model is developed for thenon-uniform spatial arrangement of fibers observedin a CMC, and it is shown that the spatialarrangement of fibers in the modeled microstructureis statistically equivalent to that in the CMC.

BACKGROUND

The theoretical statistics literature contains a largenumber of contributions that deal with the statisticsof spatial point patterns (not finite size particles orfibers), and the quantitative descriptors that reflectvarious attributes of these patterns [7,8].There are alarge number of distance distribution functions thatcan be used to quantify differentaspectsof the spatialarrangement of fibercenters. In the present work, thenearest neighbor distribution function and radialdistribution function are utilized to quantify thespatial arrangement of fiber centers.

The nearest neighbor distribution function pro-videsquantitative information on the spatial arrange-ment of fiber centers in the immediate vicinity of atypical fiber [9–11]. The radial distribution function[7,8, 12-14] reflects quantitative information con-cerning the long range, the intermediate range andthe short range spatial arrangement as a function ofdistance from a typical fiber. These descriptors areuseful to characterize overall spatial clustering,repulsion or randomness of the population of fibers.

The nearest neighbordistribution function [9–11]isgiven by the probability density function i(r) suchthat ~(r)dr is equal to the probability that there is nofiber center in a circle of radius r around a typicalfiber, and there is at least one fiber center in thecircular shell of radii r and (r + dr). The nearestneighbor distribution function can be experimentallymeasured by measuring the nearest neighbordistances of a large number of fibers sampled in anunbiased manner.

Let N be the number of fiber centers in themetallographic plane of area A. The auerage numberof fiber centers per unit area NAis equal to [N/A].Draw a circular shell of radii r and (r+ dr) aroundthe ith fiberwhosecentroid is at coordinates (X,, Y,),and count the number of fiber centers in this circularshell.Repeat the measurementby placing the circularshell around a large number of fibers (say, i = 1,

2,. . ., N). Calculate the average number of fibercenters in a circular shell of radii r and (r + dr)around a typicalfiber,and divideit by ~2nrNAdr]: theratio is equal to the value of the radial distributionfunction at a distance of r. The radial distributionfunction at difference values of distances r can beestimatedby repeatingthe measurementsfor the shellof various sizes. Formally, the radial distributionfunction, G(r) is defined as follows [7,8, 12-14]

1 dK(r)G(r) = -“T. (1)

dK(r) is the average number of fiber centers in acircular shell of radii r and (r + dr) around a typicalfiber. Note that for randomly distributed points (notfibers of finite size) having number density N., theaveragenumber of points in the circular shellaroundany point is equal to [27rrNAdr]. Therefore, for a setof randomly distributed points, G(r) is equal to 1. Itis important to point out that for a population offinite size fibers, a value of G(r) higher than 1 doesnot necessarily imply that the fibers are clustered.This is because, as demonstrated in a subsequentsection, the radial distribution function of the finitesizefibers depends on the spatial arrangement of thefiber centers, and the fiber volume fraction, averagesize, etc. It is also important to point out that theradial distribution function for any arbitrary spatialarrangement approaches 1, as r+ m. This is becauseat sufficientlylarge distances, the shell area [2m dr]becomesso large that it automatically averages overall possiblevariations, consequently,in equation (l),dK(r)+NA[2mdr], as r+cn, and therefore, G(r)+ 1,as r + m.

EXPERIMENTALMEASUREMENTOF SPATIALDISTRIBUTIONS

The experimentalmeasurementof nearest neighbordistribution function or radial distribution functioninvolves measurements of distances between fibercentersin a metallographicplane perpendicularto thefibers.For this purpose, it is necessaryto observethemicrostructure at a reasonably high magnification,where the individual fibers can be clearly resolved,and the fiber sizesand the inter-fiberdistancescan beaccurately measured. At such magnifications (typi-cally from 100 x to 500 x depending on averagefiber size)the microstructure is observed one field ofview at a time; each field of view or microstructuralframe represents a small area of the metallographicplane. Unfortunately, there are serious problemsassociated with reliable and unbiased estimation ofthe spatial distribution functions when the micro-structure is observedfield by field, i.e. one field at atime. Some difficultiesare listed below.

(1) The nearest neighbor of a given fiber may notbe in the same field of view [see Fig. 2(a)].Therefore, measuring the nearest neighbordistancesfrom individualmicrostructural fields

Page 4: Non uniform spatial arrangement of fibres

3062 YANG et al.: NON-UNIFORM ARRANGEMENT OF FIBERS

(a)

Fig. 2. (a) Exampleof a fiber that has a nearest neighborinthe adjoining field. (b) Due to the presence of edges, thefibers on the frame boundaries are partly-in–partly-out;

their centers may be in the adjoining field. -

(2)

(3)

or micrographs cannot provide the data forreliable and unbiased estimation of the nearestneighbor distribution function. The error andbias are even larger for the higher orderneighbors.Some fibers are alwayspresent on the “edges”of the microstructural field or micrograph.Such fibersare “partly in and partly out”, andtheir centroids may not be in the field of viewunder observation[seeFig. 2(b)].Ignoringsuchfibers can create a serious systematic error(bias), in both the radial distribution andnearest distribution function, because largerfibers are more likely to intersect the frameboundaries.Measurement on individual fields restricts theestimation of radial distribution function to thedistances typically less than one fourth ofthe size of the microstructural frame. There-fore, reliable information about intermediateand long range spatial patterns cannot beobtained from such data.

The above difficultiesare due to the “edge effect”,which can be elimiriated by elim~natingthe edges!Digital image analysis offers a practical solution tothis problem. It is possible to create a “montage” of

IThe process is equivalent to (but a lot more efficientandaccurate than) taking (say) 25–50 micrographs ofcontiguousmicrostructural frames,matchingthe borders,and pasting them together on a large board.

perfectlymatching (withinone pixel)large number ofadjoining contiguous microstructural fields (say, 100or more microstructural frames) in the memory ofthe digital image analyzert. The software basedprocedure for creating such an image montage isdescribedin detail in another contribution [12].Oncesuch a montage is created, the “edges” are eliminatedfor all practical purposes, and it is possible to obtainthe (A’,Y)centroid coordinates [referredto the same(O,O)origin] and size of the fibers in the montage,through digital imageanalysisprocedures.The imagesegmentation procedure to separate the touchingfibers and to correctly identify each individual fiberis describedelsewhere[15].From the data on (X, 1’)centroid coordinates of the fibers in the montagereferred to the same (O, O) origin, one can thencalculate the distance between any two fibers in themontage, that may or may not be in the samemicrostructural field (see Fig. 3).

Calculation of spatial distribution functions fromimage analysis data

The raw data from image analysis consist of astring of numbers for each fiber representing (X, Y)centroid coordinates, size, shape, perimeter, etc.Typically,each set will consist of these data on about500-1000fibers.The radial distribution function andnearest neighbor distribution function can becalculated from such a data set through appropriatecomputer codes. These calculations involve oper-ations on reasonably large data sets, and requireextensive number crunching. For example, tocalculate the radial distribution function, a circularshell of radii r and (r + Ar) is placed at the center ofthe ith fiber and the number of other fiber centroidsinsidethis shellare counted. This essentiallyamountsto identifying all the fiber centers that are in thedistance range r to (r + Ar) from the given fiberlocated at (xi, YJ. The test shell is then placed at thecenter of the next fiberand the count is repeated. Theprocedure is then continued for all the fibers (say,i = 1–1000);the average value of this quantity yieldsthe radial distribution function for that value of r.The procedure is then repeated for different r valuesto generate the completeradial distribution function.

[ 1

. . . . . . -’.

. . . . . . . .

------ ,- - - - - - ,

,.

. .

. .

(f>Yj)

Fig. 3. Creation of montage eliminates edge effects for allpractical purposes,and distancesbetweenfibersthat may be

in different microstructural fields can be measured.

Page 5: Non uniform spatial arrangement of fibres

YANG et al.: NON-UNIFORM ARRANGEMENT OF FIBERS 3063

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0I3t

5 ‘7 9’11’13” 15’17’19’ 21’23’25”27Fiberdiameter@m)

Fig. 4. Size distribution of SiC fibers (Nicalon) in ahistogram form.

Spatial distribution functions of SiCjbers in a ceramicmatrix composite

The above experimental procedure was appliedto estimate the radial distribution function and thenearest neighbor distribution function of uni-directional aligned SiC fibers in a CMC, whosemicrostructure is shownin Fig. 1. Figure 4 showsthesizedistribution of the fibers.The volume fraction ofthe fibers is 0.35, the averagefiber diameter is 14pm,and the size distribution can be represented by anormal distribution. The spatial distributions wereestimated from the “montage” of 60 fields of viewobserved at 500 x . The “montage” covered an areaof 1.25mmz on the metallographic specimen in theform of a strip. The montage contained about 2500fibers.Each data point on the radial distribution is anaverage obtained by placing the circular shell aroundat least 1000fibers. It is important to emphasizethatextensive sampling is necessary to obtain a statisti-cally representative and reliable radial distributionfunction. Figure 5 shows the nearest neighbordistribution function of the SiC fibers, and Fig. 6presents the radial distribution function of the SiCfibers. Note that the first peak of the radialdistribution functionis quite high, and there is a smallbut consistent dip below 1 at large distances. It willbe shown in the next section that these observationsimply short range clustering and long range scarcityof fibers.

COMPUTERSIMULATIONOF HARD-CORERANDOMFIBERARRANGEMENTSAND

INTERPRETATIONOF SPATIALDISTRIBUTIONS

To quantify the deviations of spatial arrangementof fiber centers from uniformity and randomness, itis necessary to compare the experimental spatialarrangement with that for a “uniform” distributionof fibers.Strictly speaking, the spatial distribution of

0.!o

n

,

5 10 15 20 25 30 35 40 45Distance@)

Fig. 5. Experimentallymeasured nearest neighbor distri-bution function of the SiC fiber centers in the CMC.

fiber centers cannot be random (i.e. Poisson) if thefibers are of non-zero (finite) size. This is because,for a completely random spatial distribution, theprobability of findinga fibercenter should be exactlythe same in any infinitesimal region of the micro-structure. This condition cannot be satisfiedin a realmicrostructurehavingfibersof non-zerosize,becausethe probability of findinga fibercenter insideanotherfiber is obviouslyzero, and it is not zero outside thefiber. Therefore, there can be a random distributionof zero dimensional points only. For a randomdistribution of points in .a plane, the radialdistribution function is equal to one, and the nearestneighbor distribution can be calculated analytically[11].For the fibers of finite size, the “base line” ormost uniform spatial distribution is what is called the“hard-core random” distribution, which can becomputer simulated, but there are no analytical

2

1.8- 1

1.6- 1

1.4-

1.2- 1

1- v

0.8-

0.6-

0.4-

0.2-

00 20 40 60 80 100 120 140 160

Fig. 6. Experimentallymeasuredradial distributionfunctionof SiC fiber centers in the CMC.

Page 6: Non uniform spatial arrangement of fibres

3064 YANG et al.: NON-UNIFORM ARRANGEMENT OF FIBERS

equations for the radial distribution function or thenearest neighbor distribution function for such aspatial arrangement.

The hard-core random distribution of finite sizeunidirectional aligned fibers can be simulated in astraight-forward manner. A random number gener-ator is used to select a random point in a planeperpendicular to fibers, and a fiber of given size is“placed” at that location. Another random point isselected,if this point does not lie inside the first fiber,then a second fiber is placed at this location. If thesecond random point lies inside the first fiber then itis discarded, and another random point is generated.The process is repeated until the required numberdensity, volume fraction and size distribution offibers are generated. The nearest neighbor distri-bution and radial distribution functions of suchsimulated hard-core random microstructure canbe calculated by averaging over a large number ofrealizations.

The spatial distribution functions of the simulatedhard-core random population of fibers dependsignificantly on the fiber volume fraction, numberdensity, average size and standard deviation of thesize distribution of fibers. Figure 7(a) shows thecalculated ra-dialdistribution of a computer simu-lated hard-core random structure for different fibervolume fractions. For this simulation, all the fibersare of 14#m diameter (which is the average size ofSiC fibers in the CMC). Note that the radialdistribution function varies significantly with thefiber volume fraction. The height of the first peakincreases with the increase in the fiber volumefraction. Figure 7(b) depicts the nearest neighbordistribution function for the same hard-core randomsimulations. Note that the nearest neighbor distri-bution depends significantly on the fiber volumefraction. Figure 8(a)illustrates the effectof the spreadin the fiber sizes(standard deviation a) on the radialdistribution function for hard-core random arrange-ment of fibers of normal size distribution, samevolume fraction (0.35) and the same average size(14pm). The height of the first peak and the widthare particularly sensitiveto the standard deviation(o)of the normal fiber size distribution. The nearestneighbor distribution of fibers [Fig. 8(b)] is alsosensitive to the standard deviation of the fiber sizedistribution. Figure 9 shows three distinctly differentfiber size distributions that have the same averagefiber size and standard deviation: these size distri-butions have differentskewness.Figure 10showsthespatial distributions of the simulated hard-corerandom arrangements corresponding to the sizedistributions in Fig. 9. The radial distribution and thenearest neighbor distribution do not vary with theskewnessof the fiber size distribution!

The simulation studies clearly demonstrate that toquantify and interpret the deviations from uniform-ity, it is imperative that the experimental spatialdistributions be compared with the simulated

1.4

1.2

1~QQ”0.81L Fiberdianwei=14ym

II ● VV=O.450.6 QVV=O.40

0.4 ■ VV=O.35❑ VV=O.15

0.2H

-0 10 20 20 40 50 60 70 60 90 100110120

Distances(jun)

/

(b)0.3 fl

FiberdiamcteEl@m

.20.25 --● VW13,45

Jo VV+.40

a o.2--H

■ VV=O.35

jg o.15- -

❑ VV=O.15

o.1-

omo5--

0+12 14 16 16 20 22 24 26 26 : )

Fig. 7. Spatial distribution functions of randomly dis-tributed monosize fibers in simulated microstructure withhard-core model. (a) Radial distribution functions ofrandomly distributed monosize fibers at different fibervolume fractions. (b) Nearest neighbor distribution func-tions of randomly distributed monosize fiber at different

volume fractions.

hard-core random spatial distributions that have thesamevolumefraction, averagefibersizeand standarddeviation of the fiber size distribution. Any con-clusions regarding the deviations from randomnessthat are not based on such a comparison need to bere-examined. Figure 1l(a) compares the experimen-tally radial distribution function of the SiC fibers inthe CMC with the correspondingradial distributionfunction for simulated hard-core random fiberpopulation having the same volume fraction and sizedistribution of fibers.Note that the first peak in theexperimentallymeasured distribution is significantlyhigher than that in the hard-core random structure,although the peak positions are approximately thesame. Let P. be the height of the first peak of theexperimentallymeasured radial distribution functionof the SiC fibers, and let P, be the height of the firstpeak for the corresponding simulated hard-corerandom distribution. To quantify the extent of

Page 7: Non uniform spatial arrangement of fibres

YANG et al,: NON-UNIFORM ARRANGEMENT OF FIBERS 3065

short-range clustering, a clusteringparameter C, canbe defined as follows.

c, = Pe/Pr. (2)

For the present composite, C, is equal to 1.33.Therefore, in the CMC, at short distances, there areabout 33°/0 more fiber centers in a givenarea arounda typical fiber, as compared to that in the hard-corerandom structure.

For hard-core random structure, the radial distri-bution is equal to 1.0, soon after the first peak,irrespective of the fiber volume fraction or sizedistribution of fibers [see Figs 7(a), 8(a) and IO(a)].The experimentallymeasured distribution is consist-ently below 1.0 for distances ranging from 50 to140pm, and then it increasesand settlesat 1.0at verylarge distances. The long distance behavior of theradial distribution function is governedby long rangespatial arrangements only; it does not depend on thevolume fraction of size distribution of fibers. Itfollowsthat in the experimentaldistribution, the dip

1.67 I

● 0=0.00.6 o CJ=l.o

■ 0=1.50.4

❑ 0=3.0

o.2- A CT=30

o0 5 10 15 20 25 30 35 40 45 50

Distances@n)

0.2gg

EO.15.3

#

A 0.1

0.05

06 8 10 12 14 16 18 20 22 24 26 28 30

Distances(p)

Fig. 8. Effect of the variation in standard deviation (o) ofthe fiber size distribution on the spatial arrangement of thesimulatedhard-core fiberpopulation. (a) Radial distributionfunctions of randomly distributed fiber centers with normalsizedistribution. (b)Nearest neighbordistributionfunctions

of randomly distributed fiber centers.

0446 8 10 12 14 16 16 20 22 :

sizeoffeature(yrn)

Fig. 9. Three distinctly different fiber size distributionshaving the same average size and same standard deviation,

but different skewness.

below 1.0 in the distance range of 50–140pmindicates that there are fiber poor regions at largedistances from a typical fiber.

Figure 1l(b) compares the experimentallymeasured nearest neighbor distribution with thecorresponding distribution for the simulated hard-core random spatial arrangement having the samevolume fraction and sizedistribution of fibers.Thesedistributions also differ significantly. The peakheights are different,although the peak positions areapproximately the same. The average nearestneighbor distance is 17pm both in the CMC and thesimulated hard-core random structure. Figure 12depictsa typicalmicrostructural fieldof the simulatedhard-core random microstructure having the samevolumefraction and sizedistribution as the SiCfibersin the CMC. Comparison of Figs 12and 1illustratesthe differencesbetween the spatial arrangements inthe two microstructure.

From the comparisons of the radial distributionfunction and nearest neighbor distribution functionof the SiCfibersin CMC and the simulatedhard-corerandom structure, it can be concluded that: (1) thespatial distribution of SiC fibers is not uniform;(2) the spatial arrangement of the SiC fibers exhibitsclustering at small distance ranges; and (3) the SiCfiber population exhibit scarcity of fibers (comparedto hard-core random) at large distances. This isconsistentwith the qualitative microstructural obser-vations (see Fig. 1).

MODELINGOF NON-UNIFORMSPATIALARRANGEMENTOF FIBERSIN CMC

The present CMC was manufactured by a slurryinfiltration type process [16]. In this process, resinbinder and glass powder coated Nicalon (SiC) fibersare wound on a mandrel to form a tape. The tapesare stacked together, burned out to remove the resin

Page 8: Non uniform spatial arrangement of fibres

3066 YANG et al.: NON-UNIFORM ARRANGEMENT OF FIBERS

binder, and then hot-pressedto densifythe laminate.In this manner, the CMC is produced in the form ofstrips. Such a processmay lead to alternate fiber-richand fiber-poor regions along the thickness of thestrip. A very low magnification micrograph of theCMC, shown in Fig. 13 illustrates these micro-structural variations along the thickness of the strip.There are no sharp boundaries betweenthe fiber-richand fiber-poor regions, but qualitatively the numberdensity of fibers does appear to vary. Theseobservations are consistent with the conclusionsdrawn from the analysis of the radial distributionfunction of the fibers in the last section.

The above discussionshowsthat it is reasonable tomodel the spatial arrangement of SiC fibers asalternate strips of fiber-rich and fiber-poor regionsstacked together. It is assumedthat the sizefrequencyfunction of the fibersis the same in the fiber-richandfiber-poorregions.The number densitiesof the fibers(and therefore, the fibervolumefraction) are differentin the fiber-richand fiber-poorregions.Let (~A)f and(~& be the number densities of the fibers in thefiber-rich and fiber-poor regions, respectively.Let Aand LPbe the widths of the fiber-richand fiber-poorstrips. In each strip, the spatial distribution of fibersis

i & )

(a)1.2

1

0.8-~

0.6-

0.4-

0.2--

00 20 40 60 80 100

0.15-

0.12..EJ

3 Q.Qg.$

%-0.08-E

o.03-

0-

distances(~)

(b)

distamxs(pm)

Fig. 10. Spatial distribution functions of simulatedhard-core fiber populations having the three different sizedistributions shown in Fig. 9. Note that the spatialdistribution functions are not sensitive to the skewnessofthe fibersizedistributions. (a) Radial distributionfunctions.

(b) Nearest neighbor distribution function.

~ (a)

J● CM(:

1.8-0 Raniom cistrib Jtion

1.6-

1.4- {’1

~ 1.2-%1

0.8 -

0.6 -

0.4

0.2 -

00 20 40 60 80 100 120 140 160

(b)

0’24~● CMC

o.2- f o Randomdistribution.--$~ 0.16-- 48

g~0,12- -

tA 0.08--

0.04- -

o-i0 5 10 15 20 25 30 35 40 45 50

Distance@m)

Fig. 11.Comparisonof the experimentallymeasuredspatialdistribution functionsof the SiCfibers in the CMC and thespatial distribution functions for the corresponding simu-lated hard-core random fiber population having the samevolume fraction, number density and size distribution offibers.(a) Radial distributionfunction.(b) Nearest neighbor

distribution function.

assumedto be hard-core random; the non-uniformityis represented by different number densities of thefibers in the fiber-rich and fiber-poor strips. Themodel and parameters are schematicallyillustrated inFig. 14.Let NAbe the overallaveragenumber densityof the fibers,whichis equal to 2.3 x 103per (mm)zforthe present CMC. One can write,

A<NA)~ + l,p(N.)p = [If+ IP]NA. (3)

Therefore, out of the four model parameters, (NA)f,(NA)P, 1, and &, only three can be variedindependently once the overall average numberdensity of fibers is specified. The four modelparameters are essentiallygovernedby the processingtechniques and parameters. These model parametersgovern the extent of non-uniformities in the spatial

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YANG et al.: NON-UNIFC)RMARRANGEMENT OF FIBERS 3067

Fig. 12. A typical microstructural field of the computersimulated hard-core random microstructure having thesame volume fraction, same number density and same size

distribution of fibers as in the CMC.

distribution, and thereby affect the mechanicalproperties of the composite. .4s there are nowell-defined boundaries between the fiber-rich andfiber-poorregionsin the compositemicrostructure, itis not possible to directly measure these modelparameters. Therefore, the followingtrial and errorschemeis used to estimate the model parameters. Thevalues of the parameters are assumed and themicrostructure is simulated. The radial distributionfunction and the nearest neighbor distribution of thefibers is calculated for the simulated structure byaveraging over a large number of statisticalrealizations of the model. These spatial distributionsare then compared with the corresponding exper-

Fig. 14.Schematicillustration of non-uniformdistributionof fibers in strip model.

imentallymeasuredspatial distributions for the CMC(Figs 5 and 6), and the model parameters are varieduntil the spatial distributions of the simulatedstructure agree well with the corresponding exper-imentally measured distributions.

Figure 15 compares the experimentallymeasuredradial distribution function of the SiC fibers in theCMC with that for the computer simulated stripmodel.Observethat the firstpeak position and heightin the two distributions match very well, and the tworadial distributions match at the larger distances aswell. Both the distributions show a dip below 1 inapproximately the same distance range. The nearestneighbor distribution functions are compared inFig. 16.The nearest neighbordistribution of the stripmodel matches perfectly with the experimentallymeasured distribution for the strip model. Figure 17

Fig. 13. Low magnificationmicrograph of CMC shows the alternate fiber-rich and fiber-poor regions.

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3068 YANG et al.: NON-UNIFORM ARRANGEMENT OF FIBERS

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0,4

0.2

00 20 40 60 80 100 120 140 160

Fig. 15. Comparison of the radial distribution of the SiCfibers in the CMC compositeand the simulatedmicrostruc-

ture with strip model.

shows two typical microstructural fields of thesimulated strip model microstructure at low and highmagnification. Note that microstructure in Fig. 17are not schematic; they are computer generatedmicrostructure from the simulated strip model.Compare these computer simulated microstructurewith Fig. 1, which depicts the actual microstructureof the CMC at the same magnifications. It isinteresting to observe that the computer simulatedand real microstructure look very similar.

The best agreement between the experimentaland strip model spatial distributions is obtainedfor the following values of the model parameters:(~~)~= 3500 per (mm)’, (~.)~ = Ilt)o per (mm)’,If= 80~m and 1P= 75pm. The overall volumefraction, number density and size distribution offibers in the strip model are exactlyequal to those ofthe SiC fibers in the CMC. It may be concludedthatthe simulated microstructure of the strip model withthese parameters is statistically equivalent to themicrostructure of the CMC under investigation.

u

● CMCo Simulated

O 5 10 15 20 25 30 35 40 45 50

Distance@m)

Fig. 16. Comparison of the nearest neighbor distributionfunctionof the SiCfibersin the compositeand the simulated

microstructure with strip model.

Fig. 17. Computer simulated microstructure with the stripmodel. Compare with Fig. 1 under the same magnification.(a) Low magnification (100 x ). (b) High magnification

(500x).

Therefore, the model represents all the statisticalattributes of the CMC microstructure: it hasstatistically the same spatial distribution of fibers,and the same average volume fraction, numberdensity and size distribution of the fibers. Thenon-uniformmicrostructure of the CMC is thereforecompletely described in terms of a few numbers.

DISCUSSION

A quantitative model is developed for thenon-uniform microstructure of a CMC containingunidirectional aligned Nicalon fibers in a glassceramic matrix. The development of the model isbased on detailed quantitative characterization of thenon-uniform spatial arrangement of the Nicalonfibers and computer simulation of microstructure.

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YANG ef al.: NON-UNIFORM ARRANGEMENT OF FIBERS 3069

The methodology is quite general, and it should beapplicable to any non-uniform microstructure of theunidirectional aligned fiber composite.

The “strip” model of three non-uniform CMCmicrostructureconsistsof alternate strips of fiber-richand fiber-poor regions. Within each region, the fibercenters have a hard-core random spatial arrange-ment. It is assumed that the frequency sizedistribution of fibers(not number distribution) is thesame in the fiber-richand fiber-poorregions,and thesize distribution of the fibers is normal (see Fig. 14).There are eight parameters in the model: fibervolumefraction, overall number density of fibers, averagefiber size, standard deviation of the fiber sizedistribution, number densities of the fibers in thefiber-rich and fiber-poor regions [(~A)f and (NA)P],

and the widths of fiber-rich and fiber-poor strips (Jrand IP). However, the last four parameters, and theoverall average fiber number density are relatedthrough equation (3). Further, the volumefraction ofthe fibers is equal to their area fraction in themetallographic plane, and therefore, the volumefraction (v,), overallaveragenumber densityof fibers(N,), average fiber size (R.) and the standarddeviation of the fiber sizedistribution (o) are relatedas follows:

Vv= n~A{[Ra]2– ~’} (4)

Therefore, the model has six independent micro-structural parameters. The parameters, (~,4)f, (~.4)p

and 2P govern the non-uniformity of the spatialarrangement of the fibers, and V,, NA and R, (howmuch, how many and how big) govern the metricattributes of the microstructure. The computersimulated microstructure with these six inputparameters is statistically equivalent to the micro-structure of the CMC. The microstructural modelparameters are governed by the geometry and sizedistribution of the Nicalon fibers and the compositeprocessing parameters, for a given manufacturingtechnique. It is important to emphasize that thespatial distribution of the fibers may vary from onebatch to another, or from one plate to another platein the same batch, or even from one region to anotherregion in a given plate (these details will be presentedin another contribution). These variations reflect thefluctuations in the process parameters. Therefore, themicrostructural model parameters can be correlatedto the process parameters, to develop very detailedprocessing–microstructure correlations. The resultscan be utilized to examine the changes in themicrostructure when the processing parameters arechanged.

The simulated microstructure can be used togenerate the RVE for the micro-mechanical modelingof the mechanical response of the material. Thesimulated microstructure can also be utilized todetermine the smallest RVE that is statistically

representative of the material microstructure. Forthis purpose, the radial distribution function can becomputed as a function of the volume of thesimulated microstructure. The smallest RVE is thesmallest simulated microstructural volume, whoseradial distribution, fiber size distribution and fibervolume fraction are comparable to the correspondingexperimentally measured attributes.

SUMMARYANDCONCLUSIONS

A digital image analysis technique is utilized forquantitative characterization of the spatial arrange-ment of Nicalon fibers in a glass ceramic matrixcomposite. These quantitative data and computersimulations are utilized to develop a quantitativemodel for the non-uniform microstructure of theCMC. The model requires six independent inputmicrostructural parameters, and it generates amicrostructure that is statistically equivalent to themicrostructure of the CMC. The simulated micro-structure can be utilized to generate the RVE formicro-mechanicalmodelingof the mechanicalbehav-ior of the composite.

Acknowledgements—Thisresearchwork is supportedby theU.S. National Science Foundation through the researchgrant DMR-9301986and its extension DMR-9642073,DrB. MacDonald is the project monitor. The financialsupportis gratefully acknowledged.

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