ti

Kuamp

C. Computational modeling

LFT)at abeusptionydr

predicted by the model are easily incorporated into micromechanics models to predict the stressstrain

ites (oer thees thation med in t

LFT pellets are usually prepared by a pultrusion process, inwhich continuous bers with a polymer matrix surrounding themare pulled through a circular die. As the resulting ber-lled strandsolidies, it may be sliced into individual pellets. These pellets aretypical 23 mm in diameter and 1013 mm long, with 1013 mmlong bers aligned along their length. This is a dramatic contrast to

ion occurs beforeerimentalarametersnd nozzle

Bailey and Kraft [6] and Vu-Khanh et al. [12] found thatlong glass bers in a polyamide matrix were reduced to lenapproximately 1 mm by the end of the molding process, with meanber lengths no greater than 1.6 mm at the screw exit (i.e., beforeentering the mold). However, moldings made with a polypropylenematrix had mean ber lengths two to three times greater than withpolyamide matrices. In an edge-gated disk molding of unspeciedthickness, Bailey and Kraft [6] found mean ber lengths in the in-ner core to be three times those of the skin layer, a phenomenonattributed to the high shear rates in the skin during mold lling.

Corresponding author. Tel.: +1 217 244 3822.

Composites: Part A 51 (2013) 1121

Contents lists available at

te

evE-mail address: ctucker@illinois.edu (C.L. Tucker III).carbonate or polyamide 6,6 with polytetrauoroethylene [14].Mechanical property improvements over short-ber plastics havebeen signicant, and the ability to injection mold these materialshas been preserved.

Bailey and Kraft [6] found that signicant attritthe polymer melt enters the mold. Thus, most expexamine ber breakage as a function of process pscrew back pressure, injection speed, gate size, a1359-835X/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compositesa.2013.04.002studiessuch asdesign.10 mmgths ofrial Chemical Industries of the United Kingdom, and initially con-sisted of a polyamide 6,6 matrix reinforced with glass bers[1,2]. To date, glass remains the most common reinforcing mate-rial, but carbon is seen as well [35]. Other matrix materials haveincluded polypropylene [68,4,911], polyethylene terephthalate[7,12], polybutylene terephthalate [12], polyphthalamide [13,14],polycarbonate [14], polyphenylene sulde [14], and blends of poly-

developed to better capture some of the quantitative details of LFTber orientation [18].

Regarding ber length, a number of experimental studies haveexamined changes in ber length in LFT materials. The bers inan LFT initially have the length of the pellet, 1013 mm, but few -bers with this length survive in an injection-molded part. While -bers may break during any stage of the injection molding process,E. Injection moulding

1. Introduction

Long-ber thermoplastic composbetween short- and continuous-boffering better mechanical propertibut retaining the ability to be injectrade name of Verton, were developbehavior of molded LFT materials. 2013 Elsevier Ltd. All rights reserved.

r LFTs) bridge the gaprmoplastic composites,n short-ber materialsolded. LFTs, under thehe mid-1980s by Impe-

the 0.20.4 mm long, randomly oriented bers in short-ber pel-lets [2].

Processing affects the microstructure of LFTs, just as it doeswith other composite materials. For LFTs the main microstructuralvariables are ber orientation and ber length. Fiber orientation forLFTs can be modeled using conventional moment tensor equations[1517], and at least one special ber orientation model had beenKeywords:A. Discontinuous reinforcementB. Microstructures

to predict spatial and temporal variations in ber length distribution in a mold cavity during lling.The predictions compare well to experiments on a glassber/PP LFT molding. Fiber length distributionsA model for ber length attrition in injeclong-ber composites

Jay H. Phelps a, Ahmed I. Abd El-Rahman a, VlastimilaDepartment of Mechanical Science and Engineering, University of Illinois at Urbana-ChbOak Ridge National Laboratory, Oak Ridge, TN, USA

a r t i c l e i n f o

Article history:Received 20 January 2013Received in revised form 1 April 2013Accepted 2 April 2013Available online 12 April 2013

a b s t r a c t

Long-ber thermoplastic (carbon reinforcing bers thmold lling degrades theattrition in a owing ber snode. A conservation equabuckling of bers due to h

Composi

journal homepage: www.elson-molded

nc b, Charles L. Tucker III a,aign, Urbana, IL 61801, USA

composites consist of an engineering thermoplastic matrix with glass orre initially 1013 mm long. When an LFT is injection molded, ow duringr length. Here we present a detailed quantitative model for ber lengthension. The model tracks a discrete ber length distribution at each spatialfor total ber length is combined with a breakage rate that is based onodynamic forces. The model is combined with a mold lling simulation

SciVerse ScienceDirect

s: Part A

ier .com/locate /composi tesa

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ites:Mean ber lengths on the order of 1 mmwere also observed by Vu-Khanh et al. [12] for 6 mm thick rectangular plate moldings(124 mm 67 mm) with polybutylene terephthalate and polyeth-ylene terephthalate matrices. ORegan and Akay [19] and LaFran-che et al. [20] also give typical ber lengths for both platemoldings and more complex production parts. ORegan and Akayalso found skin/core differences in ber length, and the ber lengthdecreased along the ow path, in some cases by as much as 30%.

Signicant ber length degradation also occurs in compoundingequipment, as shown by a number of experimental studies. Thisliterature is nicely reviewed by Inceoglu et al. [21]. One key obser-vation is that changes in ber length are primarily dependent ontotal strain, at least within a modest range of stress levels[22,21]. Also, if the stress level is substantially reduced, then theaverage ber length will degrade more slowly [22].

In contrast to the large number of experiments, very little hasbeen done to model changes in ber length during processing.Shon et al. [23] proposed a model for compounding devices inwhich an average ber length L decreased toward an equilibriumvalue L1 according to

dLdt

kf L L1 1

Shon and co-workers did not specify how the average length L is de-ned, nor how one might predict the model parameters L1 and kf.They used the kinetic constant kf as a tting parameter, and re-ported kf values for different compounding devices as a way ofdescribing the performance of those devices.

Inceoglu et al. [21] provided experimental data showing howthe weight-average ber length Lw progresses with time or strain,verifying the exponential decrease in ber length predicted byShon et al.s model. They also suggested that specic mechanicalenergy (SME) is a useful parameter for correlating the rates of berlength degradation in different devices. They proposed an exten-sion of Shon et al.s model,

dLwdt

K 00SMELw Lw1 2

Using SME allowed Inceoglu et al. to collapse data from both lab-scale and production-scale twin-screw extruders onto a singlecurve, but a ten-fold change in the rate constant K00 was requiredto represent the data for a batch mixer, which operates at muchlower stress levels.

A much more sophisticated model for ber length attrition wasrecently reported by Durin et al. [24], while this article was in ini-tial review. That model and the one we present here are closely re-lated, and should produce similar results. We will discuss themodel of Durin et al. in more detail in Section 2.6, after presentingour model.

In this study, we develop a quantitative, mechanistic model topredict changes in ber length distribution during the molding ofLFT composites, and compare the predictions of that model tomolding experiments. Having a model that accounts for the rolesof stress and total strain is particularly important for injectionmolding, where the stress and shear rate levels vary dramaticallyacross the cavity thickness. Also, having a model that predictsthe entire ber length distribution allows more accurate predictionof mechanical properties, especially strength and fracturetoughness.

We apply our model to study changes in ber length caused byow in the runner system and mold cavity, and defer the modelingof the injection unit. In most manufacturing situations the screwselection and plasticizing conditions are relatively stable across a

12 J.H. Phelps et al. / Composvariety of parts, while the mold cavity geometry will vary greatly.Thus, being able to model the effect of part geometry on berlength is a good rst goal in developing predictive models for berlength. Focusing on the cavity ow also allows us to take a contin-uum approach, which is appropriate for a rst model. We recog-nize that some special, non-local treatment of the melting zoneof the screw, and of geometric features like gates and sharp bendsin runners, may have to be added to create a complete model forber length.

2. Theory

The main steps in developing the theory are to select appropri-ate variables to describe the microstructure; write the balanceequation for these variables; and develop constitutive equationsas needed to describe the process dynamics. The resulting theorycan then be implemented in a mold lling simulation.

2.1. Microstructural variables

The microstructure at any position within the ow is describedby a discrete approximation to the ber length distribution. Choosea nite increment of ber length D, and let i = iD be a set of dis-crete length values. i ranges from 1 to n, where n is large enoughthat n is greater than or equal to the initial ber length in the pel-lets. One should also choose D = 1 to be smaller than the shortestber that will break during the ow. This is not a strict require-ment, but without this the model may not conserve total berlength.

Now let Ni represent the expected value of the number of bersof length i in a sample taken from a small region. Alternately, theNi values can be regarded as proportional to the expected numberof bers of length i per unit volume. The set of Ni values, i = 1 to n,represents the local ber length distribution. While it is commonto normalize the ber length distribution function Ni, we do not as-sume any particular normalization of this function in what follows.

An alternate way to describe ber length distributions is to usea weight-based distribution function, wi. This is related to thenumber-based distribution function by

wi jNii 3where j is any convenient normalization constant. Often j is cho-sen to make

Pwi 1.

Length distribution data is often summarized by giving an aver-age length value. Here one must be careful to distinguish how theaverage is computed. The number-average ber length Ln is

Ln P

NiiPNi

; 4

while the weight-average (or length-average) ber length Lw is

Lw P

Ni2iP

Nii: 5

It is always true that LwP Ln, with the equality applying when allbers have the same length. In analogy with the polydispersity in-dex for polymer molecular weights, one could characterize thebreadth of the ber length distribution by the ratio Lw/Ln, thoughthis idea has not been widely used.

In studying models for the ber length distribution it is alsohelpful to dene a total ber length:

Lt X

Nii 6The value of Lt is arbitrary, depending on the normalization of Ni,but any model for the ber length distribution should maintain aconstant value of Lt.

Part A 51 (2013) 1121Fig. 1 shows an experimental ber length distribution for one ofthe samples in our study. The distribution of lengths is quite broad,with a few bers of 12 mm or longer remaining in the sample, even

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ites:though the majority of the bers are less than 2 mm long. Verysimilar experimental distributions have been shown in the litera-ture [22,21]. The number-average and weight-average berlengths are also shown in this gure.

Micromechanics theories can predict the properties of LFTs andother composites as a function of the ber length, volume fraction,and orientation, as well as the ber and matrix properties. Suchtheories can either use a single average value of length, or theycan use the complete length distribution Ni [25]. Since the averagelength values can easily be calculated from the Ni values, eithertype of micromechanics calculation can use the results from our -ber length model.

2.2. Conservation of total length

The basic equation that governs the dynamics of the ber lengthdistribution expresses how Ni changes as bers break. To write this

0 2 4 6 8 10 12 140

50

100

150

200

250

Fiber Length, mm

Fibe

r Len

gth

Dis

tribu

tion

Ni

Fig. 1. Experimental ber length distribution at position A for sample AF3D.Vertical lines indicate the number-average ber length Ln = 1.58 mm and theweight-average ber length Lw = 3.05 mm. (For interpretation of the references tocolor in this gure legend, the reader is referred to the web version of this article.)

J.H. Phelps et al. / Composequation we must dene the rate at which parent bers break, andthe corresponding rate at which new, shorter children bers areformed. Consider a time interval Dt, and dene the breakage ratePi such that PiDt is the probability that a ber of length i will breakover the time interval Dt. Similarly, dene the child generation rateRji such that the probability of generating a child ber of length jover the interval Dt by breaking a parent ber of length i is RjiDt.

We assume that all breakage events involve a single ber break-ing into two smaller bers. Consider a ber of length i that breaksinto two children, with lengths j and k. Since total ber length isconserved, we must have j + k = i, or j + k = i. The probability ofthese two child bers being created must be the same, so the childgeneration rate must satisfy

Rki Riki 7

For example, R6,10 = R4,10. Also, the sum of the rates of creating allpossible child bers from parents of a given length, say i, is deter-mined by the probability of breaking that parent ber. Accountingfor the fact that each break of a parent creates two children, this re-quires thatXj

Rji 2Pi 8

Because shorter bers never re-combine to produce longer bers,we also know that Rji = 0 for all jP i.Now consider the bers of length i in a small volume of mate-rial. Within some normalization factor there are Ni such bers.Over a time Dt we would expect PiNiDt of these bers to break,thus decreasing the value of Ni. Ni could also increase, when longerbers break to produce children of length i. The contribution ofparent bers with length k to children of length i would be RikNkDt. The total rate of child generation is formed by summing this overall possible parent lengths k.

Combining the loss due to breakage with the growth due togeneration of children gives a basic conservation equation for Ni:

dNidt

PiNi Xk

RikNk 9

This is the basic conservation equation for ber length.Using Eq. (9) one can write an equation for the rate of change of

total length, Lt (see Eq. (6)). One can then show that if Eqs. (7) and(8) hold, dLt/dt = 0, and total ber length is conserved [26].

2.3. Fiber breakage rate

The next step is to develop a constitutive equation that givesthe breakage rate Pi as a function of ow conditions and ber prop-erties. Our initial model is based on hydrodynamic loading of a sin-gle ber. In an LFT, as in most practical ber composites, each berwill also have contact or near-contact with a number of neighbor-ing bers, and this may affect the breakage rate. Some models existfor the frequency and number of short-range contacts [27], and forsliding forces at berber contacts [28]. However, at presentthere is not a good model for normal forces due to berber con-tacts. Also, the effect of the contact forces on total ber loading isunclear. While hydrodynamic loading can place a ber in compres-sion (causing buckling, the presumed mode of failure) or tension, itis not known if the individual contact forces increase or decreasethe effects of the hydrodynamic loading. For instance, normalforces from contacts may promote bending in bers, but theseforces also drive the formation of ber networks, which may sup-port individual bers and prevent breakage. What can be safely as-sumed, however, is that the presence of even slight contact forcesmay trigger buckling from the compressive forces of a hydrody-namically loaded ber. With this assumption, we choose as a rststep to focus on the hydrodynamic forces acting on a ber. This ap-proach is consistent with the ndings of von Turkovich and Erwin[29], who suggested that dilute-suspension models of ber buck-ling can account for observed data on ber breakage.

Prior research into the effect of hydrodynamic loadings on -bers in suspension has largely been focused on calculating bulkstress and suspension viscosity. In this context, Dinh and Arm-strong [30] used continuum mechanics arguments to develop amodel for the effects of hydrodynamic loading of bers on bulk vis-cosity. Shaqfeh and Fredrickson [31] also developed a stress modelusing particle scattering theory, and this model has been preferredin more recent studies [27]. However, we are most concerned witha model for intra-ber forces, and the classical work of Dinh andArmstrong is convenient for our needs.

Consider a ber of length i whose long axis is oriented parallelto the unit vector p. Following Dinh and Armstrong [30] and otherber suspension models, the ber will be loaded in tension or com-pression, in proportion to the stretching or contraction rate parallelto the ber axis. This rate is given by D: pp, where D is the rate-of-deformation tensor, Dmn 12 @vm=@xn @vn=@xm.

In regions of simple shear ow, like the shell region of injection-molded composites, individual bers in suspension reach a near

Part A 51 (2013) 1121 13steady-state orientation where the axis of the ber is cantedslightly in relation to the ow direction, and where D:pp > 0. In thisorientation, the ber is in tension. Then, through a process known

Bi i=Lub4 18To translate these ideas about ber buckling into a breakage prob-ability, we rst assume that Pi is proportional to the shear rate _c,and to fi, which is the fraction of bers of length i that have an ori-entation p such that Fi(p)/FcritP 1.

Pi const: _cfi 19We adopt the proportionality to shear rate based on the following:Consider two experiments at different shear rates, with the viscos-ities adjusted so that the shear stress is the same in both experi-ments. This gives the both experiments the same value of Bi, andwe would expect to see the same amount of breakage in the twoexperiments if they have the same value of total shear strain. Theproportionality of Pi to _c ensures that the model will behave in thisway.

The function fi depends on the probability distribution functionfor ber orientation, w(p). To guide model development, we calcu-lated ber orientation distributions using a discrete version of theFolgarTucker orientation model [36]. Details of the orientationcalculation are give in Appendix B. Several different values of therotary diffusion parameter were used, and the results are shownhere in terms of the steady-state alignment in the ow direction,

ites:as a Jeffery orbit, the individual bers periodically ip, and may ro-tate through a region in orientation space where D:pp < 0, placingthe ber in compression. Thus, collections of suspended bersundergoing simple shear deformation are primarily loaded in ten-sion, but individual bers will periodically be placed in compres-sion. This is the mechanism by which bers buckle in a simpleshear ow [29,3235]. For a brittle ber, buckling will cause the -ber to break.

Using the slender-body analysis of Dinh and Armstrong [30],one can show that the magnitude of the compressive force at thecenter of a ber oriented in direction p is

Fip fgm2i

8D : pp 10

Details are given in Appendix A. Here f is a dimensionless drag coef-cient and gm is the viscosity of the polymer matrix. Note that Fi(p)is positive when D: pp < 0 and the ber is in compression.

From classical Euler buckling theory, the critical force to causean end-loaded ber to buckle is

Fcrit p3Ef d

4f

642i11

Here Ef is the elastic modulus of the ber and df is the berdiameter.

We expect that a ber will buckle if Fi > Fcrit. Combining Eqs.(10) and (11), we expect buckling if

FipFcrit

Bi2bD : ppP 1 12Here Bi is a dimensionless variable for ber buckling, dependent onthe ber length i and dened as

Bi 4fgm_c4i

p3Ef d4f

13

Also, _c is the scalar magnitude of D, and bD is a dimensionless rate-of-deformation tensor,bD D= _c 14We can see in Eq. (12) that Bi combines many of the physical param-eters of the problem into a single dimensionless variable. Note thatthe shear stress gm _c appears in the numerator, and that the beraspect ratio i/df is raised to the fourth power. The elastic modulus Efis the only ber property that appears, because buckling is an elasticinstability governed by the stiffness of the ber, rather than by itsstrength.

The 2bD : pp term in Eq. (12) depends on of the type ofdeformation (shear ow, elongational ow, etc.) and the orienta-tion of the ber relative to that deformation. The orientationdependence of this term for simple shear ow is shown in Fig. 2.Fibers experience maximum compression when p lies in theow/gradient plane and is oriented at 45 or 135 to the owdirection (the red zones in the gure). The value of 2bD : ppat this orientation is unity. Thus, for Bi < 1, there is no direction pin which the ber satises the buckling criterion, and the ber can-not be broken. For any given set of parameters, the ber length be-low which Bi < 1 is an unbreakable length.

The more interesting case occurs when Bi > 1. Now there is a setof orientations at which a ber can be broken. The size of this re-gion in Fig. 2 will increase as Bi increases, though the region neveroccupies more than half the area of the sphere.

With this in mind, there are several ways to interpret Bi physi-

14 J.H. Phelps et al. / Composcally. If we imagine varying the shear rate while holding all otherparameters constant, then we can dene the critical shear rate thatis just able to break a ber of length i as_ccrit;i p3Ef d

4f

4fgm4i

15

and we see that Bi is the ratio between the actual shear rate and thiscritical shear rate,

Bi _c= _ccrit;i 16Alternately, if we think of the ber length as being variable while allother parameters are held constant, then the longest ber lengththat cannot be broken in the ow (the unbreakable length) is

Lub p3Ef d

4f

4fgm _c

" #1=417

and Bi is determined by the ratio of the actual ber length to thisunbreakable length,

Fig. 2. Sphere of all possible ber directions p, colored by the value of (D: pp) forthe simple shear ow vx _cz. Negative values (red to yellow colors) indicateorientations where the ber is in compression. Points on the sphere are a sample ofber orientations at steady state for this ow, calculated using the FolgarTuckermodel.

Part A 51 (2013) 1121A11. We focus the analysis on steady-state distributions in simpleshear ow. This is reasonable, since most mold-lling ows arevery nearly simple shear ow, and the orientation near the mold

walls, where stresses are highest and the most ber breakage willoccur, is typically close to the steady-state orientation for simpleshear.

For each steady-state orientation distribution, and for each gi-ven value of Bi, we examine each ber in our discrete orientationdistribution function. Using Eq. (12), we evaluate the ratio of thecompressive force on the ber to the critical force for bucklingforce. The fraction of bers for which this ratio is greater than orequal to unity is the value of fi. This calculation is repeated overa range of Bi values.

The results, shown in Fig. 3, indicate that fi equals zero for Biequal to unity, and that fi increases as Bi increases. fi saturates atlarge values of Bi, with saturation values in the range of 0.100.05, depending on the degree of ber alignment at steady state.This is consistent with our qualitative understanding from Fig. 2:even for very large values of Bi, only a fraction of the bers willhave an orientation in which the ber is in compression. This frac-tion will decrease as the degree of ber alignment increases, andfewer bers are found in the regions of orientation space where

R a N ; k ; S

22

iable i with mean k/2 and standard deviation Sk, and ak is a scalefactor. The scale factor ak is chosen to normalize Rik in order to sat-isfy Eq. (8). This scales the child generation rates to match thebreakage rates, and ensures conservation of total ber length.

The variable S is a dimensionless tting parameter that controlsthe shape of the Gaussian breakage prole. A small value of S cor-responds to a high probability of breakage occurring near the bercenter, while a large S value distributes the probable breakagepoints more evenly along the ber length. This completes thedevelopment of our model for ow-induced changes in the berlength distribution.

2.5. Inuence of model parameters

Our model has three parameters: CB, f, and S. The breakage coef-cient CB controls the overall rate of change of ber length, how-ever the shapes that the ber length distribution takes over timeare independent of CB, at least at constant shear rate. f is a hydro-dynamic drag coefcient. It inuences the force on the berthrough the expression for Bi, and thus it affects the length belowwhich bers do not break. S affects the shape of the ber lengthdistribution by affecting the distribution of child bers produced

J.H. Phelps et al. / Composites:they are loaded in compression.Fig. 4 replots this same data, scaling each curve by its saturation

value fmax and altering the horizontal axis to scaled ber lengthusing Eq. (18). The data falls more closely together now, suggestingthat it would be reasonable to choose a single function fi/fmax ver-sus Bi for different ber volume fractions and aspect ratios (whichgenerate different steady-state values of A11). Note that fmax de-pends on the ber volume fraction and aspect ratio. In what fol-lows we use a simple exponential function,

fi fmax1 exp1 Bi 20This function is shown in Fig. 4 as the solid line. Clearly this functionreproduces only the general trend of the calculated fi values. Whilea better t of the calculated fi values could certainly be found, wehave used Eq. (20) for two reasons. First, at this early stage of modeldevelopment there is great uncertainty regarding many aspects ofber breakage phenomena, so it makes sense not to over-renethe details of the model. And second, numerical experimentation[26] shows that the overall model results are not sensitive to the de-tails of this function.

Substituting Eq. (20) into Eq. (19), we combine fmax with theremaining constant of proportionality, calling the product of thesetwo factors CB, the breakage coefcient. Our nal model for thebreakage probability Pi for bers of length i is then

100 101 102 1030

0.02

0.04

0.06

0.08

0.1

0.12

Bi

f i

A11 = 0.605

A11 = 0.711

A11 = 0.796

A11 = 0.850

A11 = 0.894Fig. 3. Fraction fi of bers that have an orientation in which they can buckle, as afunction of the buckling parameter Bi, for various steady-state orientations insimple shear ow.ik k PDF i 2 k

where NPDF() is the normal probability density function for the var-Pi 0 for Bi < 1CB _c1 exp1 Bi for Bi P 1

21

2.4. Child generation rates

Some assumptions must be made in order to develop a func-tional form for the child generation rates Rik. First, the center ofthe ber is the most likely location for breakage. Thus, the functionRik should be maximum at i = k/2. Second, due to the effects of -berber interactions and material inconsistencies within the -ber, other locations along the ber should have non-zeroprobabilities for breakage. From these requirements, it is reason-able to assume a Gaussian breakage prole, or

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

li /Lub

f i/f m

ax

A11 = 0.605

A11 = 0.711]

A11 = 0.796

A11 = 0.850

A11 = 0.894

Model

Fig. 4. Fraction fi of bers that have an orientation in which they can buckle, scaledby the maximum fraction at that orientation state fmax, as a function of scaled berlength. The line labeled Model is Eq. (20).

Part A 51 (2013) 1121 15when bers break.The inuence of these parameters is illustrated in Figs. 5 and 6.

Four different cases are shown here, and we use normalized

weight-based length distributions to facilitate comparison. All berlengths are normalized by Lmax, the maximum initial length. The CBparameter is accounted for by reporting results in terms of adimensionless time, t CB _ct. S values of 0.2 and 1.0 bound themost useful values for this parameter. The value of f is combinedwith the ow stress and ber properties to create a value for theunbreakable length (see Eq. (17)). Lub/Lmax = 0.1 represents a severedegradation of the initial ber length, while Lub/Lmax = 0.5 is a verymodest degradation of ber length. The initial condition for allcases has a uniform weight distribution in the range 0.9Lmax toLmax.

Fig. 5 shows data at t = 2, an intermediate time at which the -ber length distributions are changing rapidly. A portion of the ini-tial distribution is apparent on the right side of the gure for allfour cases. This demonstrates that our model can readily capturebimodal ber length distributions. The inuence of Lub is clear,with peaks in the distributions developing around Lub for all cases.The larger value of S tends to produce sharper peaks and a linearrise in the early part of the distribution, while the smaller value

fect of S is mainly visible for the case of Lub/Lmax = 0.5. For this casemost of the bers have experienced only one breakage event. For

Both the Weibull and Gaussian distributions are somewhatarbitrary choices; their use is justied by the fact that they pro-duce ber length distributions very similar to the ones observedexperimentally.

The same analysis of hydrodynamic forces on the bers andber buckling is used in both papers. Durin et al. use a dragcoefcient f based the Burgers [38] slender-body theory for adilute suspension, while we maintain f as a model parameter.Our choice seems more appropriate for a highly concentratedsuspension, but it does introduce an additional tting parame-ter to the model. In both models, the elastic modulus Ef is theonly mechanical property of the ber that appears, and onemight ask why the ber strength is not present. Durin et al.[24] perform a detailed analysis of the stresses on a ber withan initial curvature, which could potentially break before reach-ing the buckling load. They show that for properties typical ofglass bers this does not happen. Thus, bers fracture by buck-ling, and the tensile strength of the ber does not affect berbreakage.

The overall rate of ber breakage in the Durin model is gov-erned by the Jeffery orbit period of a ber in a dilute suspension.

16 J.H. Phelps et al. / Composites:the other cases the bers tend to have experienced a series ofbreakage events, which leads to the smoother-shaped lengthdistributions.

As these examples show, f controls the location of the peak inthe near-steady-state ber length distribution, while CB controlsthe time required to reach that state. S inuences the exact shapeof the ber length distribution.

2.6. Relationship to model of Durin et al.

A ber-length model very similar to the one presented here wasrecently reported by Durin et al. [24]. The twomodels are very sim-ilar, and indeed they share a common framework [26,37]. Bothmodels calculate the time evolution of the full ber length distri-bution, and both models assume that the mechanism of ber

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Scaled Fiber Length, l i / Lmax

wi

Lub/Lmax = 0.1, S = 0.2

Lub/Lmax = 0.1, S = 1.0

Lub/Lmax = 0.5, S = 0.2

Lub/Lmax = 0.5, S = 1.0produces more rounded peaks. In our model bers cannot breakif i < Lub, so the bers in that range are all child bers accumulatedfrom the breakage of bers with i > Lub.

In Fig. 6 we see the ber length distributions nearly at steadystate, t = 10. The initial condition has disappeared, and most ofthe bers now have lengths below Lub. The length distributionslook much more like experimental length distributions, and the ef-Fig. 5. Weight-based ber length distributions at intermediate time, t = 2, fordifferent values of the model parameters S and Lub in steady simple shear ow.t CB _ct.breakage is buckling under hydrodynamic forces that load the berin compression. A quantitative comparison of the two models isbeyond the scope of this paper, but we can outline the similaritiesand differences between the models here.

The model of Durin et al. starts with a conservation equationthat is equivalent to Eq. (9). Their main equation is posed interms of the weight-based ber length distribution (wi in ournotation) rather than the number-based distribution Ni, butconversion between the two forms is straightforward usingEq. (3). Thus, any differences between the two models mustreside in the expressions for the breakage rate Pi and the childgeneration rate Rij.

Durin et al. build Rij using a Weibull distribution of breakageprobability along the ber, while here we use a Gaussian distri-bution. They recommend using rather large values of the Wei-bull parameter, and in that range the Weibull distribution andthe Gaussian distribution have very similar shapes. Their rec-ommended range of the Weibull parameter is mP 3, whichcorresponds closely to S 6 0.8 for our Gaussian distributions.Thus, the two models are actually quite similar in this aspect.

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Scaled Fiber Length, l i / Lmax

wi

Lub/Lmax = 0.1, S = 0.2

Lub/Lmax = 0.1, S = 1.0

Lub/Lmax = 0.5, S = 0.2

Lub/Lmax = 0.5, S = 1.0

Fig. 6. Weight-based ber length distributions at long time, t = 10, for differentvalues of the model parameters S and Lub in steady simple shear ow. t CB _ct.

Part A 51 (2013) 1121This provides a physical motivation for making the breakagerate Pi proportional to _c, just as it is in our model. The Jefferyperiod for an average ber also provides the overall breakage

rate (roughly equivalent to CB in our model), though theirmethod of choosing the average ber aspect ratio is notexplained in detail.

In our model a ber whose breakage parameter is less thanunity (Bi < 1, or i < Lub) cannot break. In contrast, Durin et al.assign a non-zero, probability of breakage to bers with Bi < 1.No explicit expression for Pi is given in their paper, but they cer-tainly have Pi > 0 for some values of Bi < 1. This has little effecton the short-time behavior, and both models should producerather similar results in the initial period of rapid degradationof ber length. After this rapid initial decrease, ber length willdecrease more slowly in both models, but more rapidly in themodel of Durin et al. than in our model.

moved. This isolates a region that samples uniformly the entire

J.H. Phelps et al. / Composites:Overall the two models are very similar, and they provide asound framework upon which future efforts can build.

2.7. Implementation in a mold lling simulation

To implement our model in a mold lling simulation, one mustcalculate the ber length distribution (a full set of Ni values) ateach node in the mesh. We expect that ber length will degrademore rapidly near the cavity walls, where the stresses are high,than near the midplane of the cavity, so ber length data mustbe calculated at each node or layer across the thickness of themold, as well as at each in-plane location.

A typical mold-lling simulation solves governing equations ona xed mesh or grid. Thus, in Eq. (9) the time derivative d/dt mustbe replaced by the material derivative, and on a xed grid we solve

@Ni@t

v rNi PiNi Xk

RikNk 23

At each time step the results of the ow simulation provide the lo-cal shear rate, and viscosity. These are used to calculate the break-age rates and child generation rates. The ber length distribution istypically updated using explicit time integration. This may be doneon a smaller time step than the lling step in the moldingsimulation.

We have implemented our ber length model in conjunctionwith the ORIENT mold lling code [16]. This code uses the Hele-Shaw formulation [39] and solves for lling, heat transfer, and berorientation in two simple mold geometries: an end-gated strip anda center-gated disk (Fig. 7). We use the suspension viscosity, deter-mined from experimental measurements, in place of the matrixviscosity gm when calculating breakage rates. At this stage ofdevelopment we ignore any changes in suspension viscosity dueto changes in ber length. Thus, the ber length calculations donot affect the lling or ber orientation calculations. Also, the berFig. 7. Geometry and dimensions of center-gated disk moldings prepared by PacicNorthwest National Laboratory.thickness of the molding. The epoxy column is then burned awayso that the bers can be separated, and the sample bers are ana-lyzed as in the standard technique. Even this method can preferen-tially select long bers, since the diameter of the epoxy column issmaller than the length of the longest bers. To account for this, acorrection function is used in order to provide an unbiased berlength distribution [41].

For each sample, ber length measurements are made at loca-tions A, B, and C, as shown in Fig. 7. These positions are centered3.2. Measurement of ber length

The usual approach for measuring the ber length distributionis to burn off the polymer matrix, separate the bers, and measurethe bers under a microscope, usually with the aid of image anal-ysis software. While this approach is standard, the means of select-ing the sample of bers from the molding has not beenstandardized. It is likely that common methods, such as usingtweezers to extract a group of bers from the ber mat that re-mains after burn-off, may preferentially sample the longer bersor bers from some portion of the thickness of the part, and thusproduce a biased measurement.

To obtain an unbiased sample of bers, we used a two-stepsampling technique [41]. In the rst step, the polymer matrix isburned away while the sample is constrained to its original samplesize is a perforated metal container. This constraint prevents the -ber network from expanding, which it would otherwise do becauseof stored elastic energy in bent bers. A hypodermic needle is theninserted into the constrained network, and a thin column of epoxyresin is injected. The epoxy is allowed to solidify, after which thesurrounding bers not encased in the epoxy column are gently re-orientation results do not affect the calculation of ber lengths.This allows the ber length calculation to be implemented as apost-processor to ORIENT. In this study we use a crude fountain-ow model, in which ber-length data from the midplane nodeat the ow front is copied into nodal locations near the wall when-ever a new column of nodes is lled by the advancing front. Thisalgorithm probably overestimates the effect of the fountain ow.Additional details of the mold lling and ber length calculationsare given by Phelps [26].

3. Experimental

3.1. Molding geometries, materials, and conditions

LFT samples were injection molded by Dr. James Holbery of thePacic Northwest National Laboratory. The moldings featurematrices of polypropylene with reinforcement of 40% by weightglass ber (MTI PP40G). These materials were compounded specif-ically for this study by Montsinger Technologies. The geometrieswere a 90 mm long 80 mm wide end-gated ISO plaque [40]and a 180 mm diameter center-gated disk.

We focus here on sample AF3D, which is a center-gated diskmolding with a fast lling speed. The geometry is shown inFig. 7, the lling time was 0.65 s, the melt temperature was510 K, and the mold wall temperature was 344 K. Material proper-ties necessary for mold-lling simulations were either measuredby Moldow Plastics Labs, Kilsyth, Australia, or taken from theMoldow Plastics Insight material library. Details are reported byPhelps [26]. Glass ber properties for the breakage model are bermodulus Ef = 73 GPa and ber diameter df = 17 lm.

Part A 51 (2013) 1121 17approximately 15 mm, 45 mm, and 75 mm from the gate, respec-tively. Each region is three millimeters thick (the full thickness asthe sample) and approximately 10 mmwide. At least 2300 individ-

ual bers were measured at each location. The resulting data areaverages of the ber length distribution across the thickness ofthe molding. Our model predicts the ber length distribution atmultiple nodes across the cavity thickness, but in the comparisonsthat follow we average the predictions across the cavity thickness.

4. Results and discussion

The calculations use D = 0.1 mm and n = 130 length values torepresent the ber length distribution. This is a very detailed rep-

to 3.0 mm. This indicates that the ber length is highly degraded inthis particular molding. In other injection-molding experiments, tobe reported at a later date, we have seen signicant numbers oflonger bers. The ber length distributions in these samples are bi-modal, a feature that is readily reproduced in our model, as dem-onstrated in Fig. 5.

A quantitative comparison between the experiment and ourpredictions appears in Fig. 11, which shows the average lengths

100

150

200

250

r Len

gth

Dis

tribu

tion

Ni

gure legend, the reader is referred to the web version of this article.)

0 2 4 6 8 10 12 140

50

100

150

200

250

Fiber Length, mm

Fibe

r Len

gth

Dis

tribu

tion

Ni

Fig. 10. Experimental (bars) and predicted (line with dots) ber length distributionat position C for sample AF3D. (For interpretation of the references to color in thisgure legend, the reader is referred to the web version of this article.)

18 J.H. Phelps et al. / Composites: Part A 51 (2013) 1121resentation, which come at the cost of longer computing times. Weuse it here to ensure that the results are not distorted by a coarsediscretization of the length distribution.

The length distribution Eq. (23) needs an inlet boundary condi-tion for Ni. Here we use the experimental length data at location A(x = 1.5 cm from the gate) as the inlet condition at all points acrossthe cavity thickness, and predict the results downstream of that.This is done because the ORIENT code cannot model ow in thegate region. A more sophisticated lling code could start the berlength calculation further upstream, perhaps at the beginning ofthe runner system, using some information about the length distri-bution exiting the screw and nozzle.

All of the following results use f = 0.55, CB = 0.025, and S = 1.0.These values were determined empirically by adjusting them toproduce a reasonable t to the experimental data. The valuesthemselves are physically reasonable in that we expect f 1 andCB 1. Also, S = 1.0 distributes the breakage location along each -ber, while retaining a preference for breaking bers near theircenters.

Fig. 8 shows a portion of the ber length data calculated by ourmodel for this molding. In this gure, the length distribution ateach node has been used to calculate nodal weight-average lengthsLw according to Eq. (5). The resulting values are used to draw thecontours in the gure. We see that the bers remain long (red col-ors in the gure) near the midplane (z = 0), due to the low shearstress and low accumulation of shear strain there. Closer to themold wall we see shorter ber lengths (blue colors), owing to thehigher shear stresses and greater accumulated shear strain. The re-gion with shorter bers becomes thicker with increasing distancefrom the gate. Very close to the cavity wall the bers remain fairlylong; this is material that traveled along the midplane, caught upto the ow front, and was then pushed to the mold walls by thefountain ow. It froze there quickly, so that it experienced very lit-tle shear strain at the high stress levels near the wall. The thicknessof this region is also greater the farther away one is from the gate.

Figs. 9 and 10 compare the measured ber length distributionsat positions B and C with the predictions of the model. The pre-dicted length distributions are very similar in character to the mea-sured distributions. Note that both the experimental and predicteddistributions are averaged across the cavity thickness.

As Figs. 1, 9 and 10 show, our samples have very few bers thatare longer than 6 mm, and the average ber lengths range from 1.5Fig. 8. Contours of weight-average ber length Lw versus position in the mold cavity, predpath, starting just downstream of the gate. z measures distance from the cavity midplan0 2 4 6 8 10 12 140

50

Fiber Length, mm

Fibe

Fig. 9. Experimental (bars) and predicted (line with dots) ber length distributionat position B for sample AF3D. (For interpretation of the references to color in thisicted for sample AF3D. Colorbar gives Lw in mm. xmeasures distance along the owe. The arrows indicate the velocity prole at x = 4 cm.

as a function of distance along the ow path. The predicted curvesare at for x < 1.5 cm, because the ber length calculations actuallystart at this location, which is point A. Average ber length (again,averaged across the part thickness) initially decreases with owlength. This is because bers farther down the ow path haveexperienced greater shear strains, and thus have had more oppor-tunity to break. The weight-average length decreases more rapidlythan the number-average length, because it is more sensitive to thenumber of very long bers, and these bers are the most likely tobreak. Closer to the end of the ow path both of the averagelengths increase. This is a result of the fountain effect, as discussedin connection with Fig. 8. The model is quite a good t for thisexperimental data, though the changes in average ber lengthare fairly modest in this particular example.

5. Conclusions

In long-ber thermoplastic composites, changes in the berlength distribution due to processing can be signicant, and can af-fect the mechanical properties of the nal material. We have devel-oped a quantitative model than can predict these changes. Local

2.5

h (J.H. Phelps et al. / Composites:0 2 4 6 80

0.5

1

1.5

2

Axial Distance (cm)

Aver

age

Fibe

r Len

gt

Lw predictedLw experimentalLn predictedLn experimental

Fig. 11. Experimental and predicted average ber lengths as a function of owdistance for sample AF3D. Predictions begin at point A (x = 1.5 cm), using themicrostructure is represented by a discrete version of the berlength distribution. A balance equation for the length distributionis written. A model for the breakage rate is then developed, usingthe concept that bers break by buckling under hydrodynami-cally-induced compressive forces in certain unfavorable orienta-tions relative to the ow. The resulting model can beimplemented in a conventional injection mold lling simulation.Preliminary results show very good agreement with experimentson a glass ber/polypropylene LFT, molded in a center-gated diskgeometry.

In the experiments presented here, the effect of changes in berlength on the stiffness of the molded part is quite modest. Usingthe micromechanics calculations described in Nguyen et al. [25],the difference in elastic modulus between the maximum berlength of 13 mm and the observed average ber length of approx-imately 1.5 mm is only about 5%. This difference is small becauseeven at 1.5 mm average ber length the average ber aspect ratiois approximately 90, and elastic modulus is not sensitive to beraspect ratio in this range. However, tensile strength does continueto improve with ber length in this range, and toughness improves

3

3.5

mm

)measured length distribution there as the initial condition. (For interpretation ofthe references to color in this gure legend, the reader is referred to the web versionof this article.)empirical knowledge could be incorporated into the theoreticalframework developed here.

The model we have developed also provides some insights intothe way that simpler models like those of Shon et al. [23], Inceogluet al. [21] might be improved. In our model, the overall rate ofreduction of ber length scales with CB _c. Replacing the terms kfor K00SME in Eqs. (1) and (2) with an expression like CB _cwould pro-vide the desired scaling with shear rate. The effect of ow stresscould then be introduced by making the equilibrium lengths L1or Lw1 depend on stress in a manner similar to Lub in Eq. (17). Ofcourse these models only predict an average ber length, whereasour model and the model of Durin et al. [24] predict the entire berlength distribution.

Our model as it stands can be implemented with almost anyow solver, whether for injection molding (Hele-Shaw or true 3-D), or in software for other processes such as extrusion compound-ing. The large number of variables at each node used to representthe length distribution does make the computations expensive,particularly for large parts. One approach to resolving this issueis to use a coarser discretization of the length distribution, leadingto fewer Ni values at each nodel. Another possibility is to developan approach similar to that used for ber orientation, where afew parameters are used to characterize the distribution function,and equations are derived for those parameters starting from thegoverning equations of the distribution function [15]. This remainsas a potential task for future research.

Acknowledgements

This research was sponsored by the US Department of Energy,Ofce of Energy Efciency and Renewable Energy, Vehicle Technol-ogies Program, as part of the Lightweight Materials Program undercontract DE-AC05-00OR22725 with UT-Battelle, LLC. Commentsand suggestions from Drs. Mark T. Smith, Ba Nghiep Nguyen, andJames D. Holbery of Pacic Northwest National Laboratory, andfrom the industrial advisory board of the project, are gratefullyacknowledged. We also thank Drs. Xiaoshi Jin and Jin Wang ofAutodesk Moldow for their helpful insights and suggestions.

Appendix A

To derive Eq. (10) for the hydrodynamic force acting to com-press a ber, consider a ber with a unit orientation vector p anda local coordinate s along the ber length. Let s = 0 at the centroidof the ber, so s ranges from i/2 to i/2. Following Dinh and Arm-strong [30], the increment of force df on an increment of berlength ds due to hydrodynamic loading is

df feff v _rds 24where feff is a tensorial anisotropic drag coefcient, v is the unper-even more [11]. This highlights the value of a model that calculatesthe entire ber length distribution, and can capture any very longbers that survive the processing.

Our present model is very promising. Many renements of themodel are possible within the framework that we have developedhere. The basic conservation equation will not change, but onecould introduce more detailed modeling and perhaps additionalphysics into the breakage rate model for Pi. Breakage rate could in-clude a dependence on the ber orientation state relative to theow, or it could incorporate the loading from berber contacts.Some careful, basic experiments to measure directly the breakagerates in a well-dened ow would be highly desirable, and such

Part A 51 (2013) 1121 19turbed uid velocity at the given point along the ber axis, and _r isthe rate of change of the position vector r, which extends from theorigin to the given material point on the ber.

Fi fgm2i

8D : pp 36

which is Eq. (10).

Appendix B

To evaluate fi in Eq. (19) we use a discrete approximation to the

0.03 0.0387 0.7958 0.00776

ites: Part A 51 (2013) 1121Dening rc to be the position vector of the centroid of the ber,we can write r as:

r rc sp 25Allowing vo to represent the velocity of the (arbitrarily selected) ori-gin, the unperturbed uid velocity v at any point along the ber isthen

v vo L r 26where L is the velocity gradient tensor (Lij = @ vi/@xj), and we haveassumed that v is approximately linear in r over the ber length.

If only hydrodynamic forces act then the centroid of the bermoves with the unperturbed uid velocity of rc [30], so the rateof change of rc is

_rc vo L rc 27Combining Eqs. (25)(27) and simplifying gives

v _r sL p _p 28Note that _p describes the rate of rotation of the ber axis. For thiswe use Jefferys equation,

_p W p nD p D : ppp 29where W 12 L LT is the vorticity tensor and D 12 L LT is therate-of-deformation tensor. The scalar n is the shape parameter,which approaches unity for slender bers. In the limit of n? 1,Eq. (29) may be recast in terms of L, to give

_p L p L : ppp 30Note that W: pp = 0, owing to the anti-symmetry of W.

Combining (30) with Eqs. (24) and (28) gives the increment offorce on a slice ds of the ber as

df feff pD : pp s ds 31Dinh and Armstrong [30] note that the drag tensor feff may bedecomposed into a component transverse to the ber axis and acomponent parallel to the ber axis, or:

feff ftI pp fppp 32where ft and fp are scalar drag coefcients. Substituting this into Eq.(31) shows that only the parallel drag coefcient fp matters for forceon the ber. Again following Dinh and Armstrong, both drag coef-cients scale with the matrix viscosity gm, so we write fp = fgm,where f (no subscript) is a dimensionless drag coefcient. Using thisin (31) gives an explicit expression for the increment of force on asegment of a ber,

df fgmpD : pp s ds 33The force df acts parallel to the ber axis; it is proportional to theuid stretch rate parallel to that axis, (D: pp), and to the distances from the centroid of the ber.

Since the ber center is the most likely location for buckling tooccur, we integrate from s = 0 to s = i/2 to nd the force vector fc atthe ber centroid.

fc Z si=2s0

df 34

Substituting Eq. (33) and evaluating the integral, we nd

fc fgm2i

8p D : pp 35

For the analysis of ber buckling we need a scalar value of force at

20 J.H. Phelps et al. / Composthe centroid Fi, dened such that Fi is positive when the ber is incompression. Since fc is directed along the p axis, this means thatFi fc p, and we haveber orientation distribution, wi(p). This is formed using a largenumber of unit orientation vectors pk, with k = 1 to np. Typically,np = 10,000. These vectors are initialized in some convenient way,in this case random in 3-D space, and are then subject to a timehistory that corresponds to their motion in simple shear ow. Aftera sufcient amount of time the family of vectors reach a statisticalsteady state, and they form a representative sample of the orienta-tion distribution wi(p).

Here we use the original model of Folgar and Tucker [36], whichhas an isotropic rotary diffusivity, and use the limit of high particleaspect ratio, n? 1. In this limit the deterministic part of the bermotion gives a new orientation for any nite time step Dt as

pk F pkjF pkj

37

where F is the deformation gradient tensor over the time step. For13 simple shear ow, this is

F 1 0 _cDt0 1 00 0 1

264375 38

Our calculation uses a small time step _cDt 0:05, and introducesthe rotary diffusion term stochastically by adding a small randomperturbationDpk to each orientation vector. Thus, in our calculationthe new value of orientation at each time step is calculated from theprevious value pk by rst calculating pk using Eq. (37), and thenadding a random component Dpk according to

pnewk pk Dpkjpk Dpkj

39

The random perturbation Dpk is chosen to be uniformly distributedon a small circular disk normal to the original vector. For each pk wedetermine mutually orthogonal unit vectors sk and tk, and then ndthe perturbation as

Dpk rkcos/ksk sin/ktk 40Here the angles /k are uniformly distributed on [0,2p) and the sca-lar magnitudes rk are given by

rk Qqk

p 41Q is a scale factor that is constant for the entire calculation, and eachqk is uniformly distributed on [0,1). With this calculation we expectthat the rotary diffusivity CI _c will scale approximately with Q2/Dt.

Our calculations use 10,000 p vectors and _cDt 0:05. The initialorientation state is random in 3-D, and the calculation is run for1500 steps (75 strain units) to reach a steady state orientation. Anadditional 500 time steps are run, and the results of all 500 steps

Table 1Correspondence between orientation perturbation parameter Q, steady-state orien-tation in the ow direction, and corresponding interaction coefcient CI using the OREclosure, for the ber orientation distribution calculations in Appendix B.

Q2= _c Dt Q A11,steady CI,ORE

0.30 0.1225 0.6051 0.04430.10 0.0707 0.7111 0.01840.01 0.0224 0.8504 0.003770.003 0.0122 0.8938 0.0176

are combined to get the steady-state orientation tensor. A series of Bivalues are then considered, and the set of orientation vectors fromthe nal time step are used to compute fi, the fraction of bers thatcould buckle and break, for each Bi according to Eq. (12).

The results are shown in Fig. 3, labeled according to the steady-state orientation in the ow direction, A11. Table 1 shows the cor-respondence between the Q values used in each calculation and thesteady-state values of A11. Also shown here are the CI values thatgive the same steady-state value of A11 when used in the mo-ment-tensor equations for ber orientation [15], with the ORE ver-sion of the orthotropic closure approximation [42,43]. Note thatother closure approximations will require different values of CI toget the same A11 value.

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[20] LaFranche E, Krawczak P, Ciolczyk JP, Maugey J. Injection moulding of longglass ber reinforced polyamide 66: processing conditions/microstructure/exural properties relationship. Adv Polym Technol 2005;24:11431.

[21] Inceoglu F, Ville J, Ghamri N, Pradel JL, Durin A, Valette R, et al. Correlationsbetween processing conditions and ber breakage during compounding ofglass ber-reinforced polyamide. Polymer Compos 2011;32(11):184250.

[22] Shimizu Y, Arai S, Itoyama T, Kawamoto H. Experimental analysis of thekneading disk region in a co-rotating twin screw extruder: Part 2. Glassberdegradation during compounding. Adv Polym Technol 1997;16(1):2532.

[23] Shon K, Liu D, White JL. Experimental studies and modeling of development ofdispersion and ber damage in continuous compounding. Int Polym Proc2005;20(3):32231.

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A model for fiber length attrition in injection-molded long-fiber composites1 Introduction2 Theory2.1 Microstructural variables2.2 Conservation of total length2.3 Fiber breakage rate2.4 Child generation rates2.5 Influence of model parameters2.6 Relationship to model of Durin et al.2.7 Implementation in a mold filling simulation

3 Experimental3.1 Molding geometries, materials, and conditions3.2 Measurement of fiber length

4 Results and discussion5 ConclusionsAcknowledgementsAppendix A Appendix B References