Rheol Acta (2009) 48:59–72DOI 10.1007/s00397-008-0309-9

ORIGINAL CONTRIBUTION

Investigation of the rheological properties of short glassfiber-filled polypropylene in extensional flow

Julien Férec · Marie-Claude Heuzey ·José Pérez-González · Lourdes de Vargas ·Gilles Ausias · Pierre J. Carreau

Received: 7 January 2008 / Accepted: 15 September 2008 / Published online: 10 October 2008© Springer-Verlag 2008

Abstract The behavior of short glass fiber–polypro-pylene suspensions in extensional flow was investigat-ed using three different commercial instruments: theSER wind-up drums geometry (Extensional RheologySystem) with a strain-controlled rotational rheometer,a Meissner-type rheometer (RME), and the Rheotens.Results from uniaxial tensile testing have been com-pared with data previously obtained using a planar slitdie with a hyperbolic entrance. The effect of three ini-tial fiber orientations was investigated: planar random,fully aligned in the stretching flow direction and per-pendicular to it. The elongational viscosity increasedwith fiber content and was larger for fibers initiallyoriented in the stretching direction. The behavior at lowelongational rates showed differences among the vari-ous experimental setups, which are partly explained bypreshearing history and nonhomogenous strain rates.However, at moderate and high rates, the results arecomparable, and the behavior is strain thinning. Finally,

J. Férec · M.-C. Heuzey (B) · P. J. CarreauCenter for Applied Research on Polymers and Composites(CREPEC), Chemical Engineering Department,École Polytechnique de Montréal, PO Box 6079,Stn Centre-Ville, Montreal, QC H3C3A7, Canadae-mail: marie-claude.heuzey@polymtl.ca

J. Pérez-González · L. de VargasLaboratorio de Reología,Escuela Superior de Física y Matemáticas,Instituto Politécnico Nacional,México D. F., México

J. Férec · G. AusiasLaboratoire d’Ingénierie des MATériaux de Bretagne(LIMATB), Université Européenne de Bretagne,rue de St Maudé, 56325 Lorient, France

a new constitutive equation for fibers suspended into afluid obeying the Carreau model is used to predict theelongational viscosity, and the predictions are in goodagreement with the experimental data.

Keywords Fiber-filled thermoplastics · Elongationalviscosity · Fiber suspensions · Constitutive equation

Introduction

Fiber reinforced polymer materials have received at-tention from many researchers in the last decades.These composites improve considerably stiffness,strength, and heat distortion resistance in comparisonwith unfilled polymers. Given that the filler particleshave geometrically anisotropic shapes, the reinforce-ment of the solidified materials may depend also onfiber orientation. Therefore, it is of interest to developrelationships between flow properties and fiber orien-tation during polymer processing.

In common processing techniques such as extrusion,injection molding, and blow molding, the material issubjected to a combination of shear and elongationalflows. While many papers discuss the rheological prop-erties of fiber suspensions in shear flow (Laun 1984;Kamal and Mutel 1989; Ausias et al. 1992; Ramazaniet al. 2001; Sepehr et al. 2004a, b), a few reports dealwith the studies of fiber-filled polymers in extensionalflow, and there are many contradictions in the literatureabout these properties. Metzner (1985) published aninteresting review on polymer suspensions, mentioningthat fiber suspensions show large resistance to strainin extensional flow. One of the pioneering work wasdone by Chan et al. (1978), who investigated the elon-

60 Rheol Acta (2009) 48:59–72

gational viscosity of high-density polyethylene (HDPE)and polystyrene (PS) melts filled, respectively, with 20and 40-wt.% of glass fibers. They observed that theelongational viscosity, ηE, of the composites was alwayslarger than that of the matrices. This was confirmed byseveral authors (Laun 1984; Kobayashi et al. 1995; Ooiand Sridhar 2004; Thomasset et al. 2005) and especiallyby Takahashi et al. (1999) who used a low-density poly-ethylene (LDPE) filled with glass beads and fibers andstudied the composite behavior in uniaxial elongationwith a Meissner-type elongational rheometer. How-ever, using a Gottfert tensiometer (Rheotens), Lin andHu (1997) surprisingly observed that above a criticalfiber content, the elongation viscosity of a glass fiberreinforced polypropylene (PP) dropped. Also at lowstrain rate, Chan et al. (1978) showed that the elon-gational viscosity of composites showed large valuesbut decreased with increasing deformation rates. Chanet al. (1978) noted that the Trouton ratio ηE/η0 wasclose to 3 for both unfilled polymers at low strain rate(PS and HDPE), whereas it increased at high Henckyrates for a PS matrix. Moreover, the ratio ηE

/ηcomposite

was larger for a 20-wt.% fiber-filled system than for a40-wt.% content. They explained this particularity bythe fact that higher fiber concentrations could inducelarger shear rates for the fluid trapped between fibersand consequently result in lower shear viscosities dueto the shear-thinning character of the matrix. Similarresults have been published (Creasy et al. 1996a, b;Creasy and Advani 1997) for long discontinuous carbonfiber in poly-ether-ketone-ketone (PEKK). For longglass fibers (L = 3.1 mm) suspended in a polybutene(Newtonian fluid), Mewis and Metzner (1974) showedthat the elongational viscosity was independent of thestrain rate and ηE

/ηmatrix

0 increased by a factor of 156.Weinberger and Goddard (1974) found a constant elon-gational viscosity that was 26 and 23 times larger thanthe shear viscosity for glass fibers suspended in a highlyviscous silicone oil and in a polybutene, respectively.In their paper on the resistance to uniaxial exten-sional flow of fibers suspended in a Newtonian epoxy(Epikote) and Boger fluid (polyisobutylene + poly-butene + kerosene) matrices, Ooi and Sridhar (2004)found the ratio ηE

/ηmatrix

0 to be larger than 3 for bothcomposites at low Hencky strain. These measurementswere performed using a filament stretching technique,ensuring that no necking occurred. They found thatfiber addition had a marginal effect after the onset ofstrain hardening for the Boger fluid.

Using a Rheometrics Elongational Rheometer(RER), Kamal et al. (1984) observed that the elonga-tional viscosity of glass fiber-filled PP increased withfiber content and decreased with strain rate. From mea-

surements in an online slit rheometer mounted on aninjection molding press, Mobuchon et al. (2005) evalu-ated the apparent planar extensional viscosity of fiber-filled PP from the pressure drop in a hyperbolic die. Forstrain rates varying between 0.1 and 200 s−1, they deter-mined that the Trouton ratio, ηE

/ηmatrix, dropped from

40 to 4 with shear rate for the unfilled PP, and from42 to 4.2 and 54 to 5.4 for the 10- and 30-wt.% filled PP,respectively. Takahashi et al. (1999) found that a LDPEfilled with glass beads exhibited a weak strain hardeningand showed a strain-softening behavior when filled withfibers. Kobayashi et al. (1995) investigated the influenceof 10 vol.% of potassium titanate whiskers suspendedin a PS matrix and of their orientation during uniaxialelongational tests (using a Meissner-type elongationalrheometer). They concluded that the whiskers pre-vented the strain-hardening behavior. Kobayashi et al.(1995) concluded that the elongational viscosity wasalmost independent of the filler orientation, which isin contradiction with the Batchelor theory for fibersin Newtonian fluids (Batchelor 1971) and the Goddardanalysis in non-Newtonian fluids (Goddard 1976). Also,Kobayashi et al. (1995) used X-ray diffraction patternsto show that the initially random oriented whiskersbecame remarkably aligned in the stretching direction.Zhang et al. (2001) analyzed the changes in fiber orien-tation using scanning electron microscopy of quenchedsamples of glass fiber-filled PP melts subjected to biax-ial extensional flows. They found an appropriate rangeof strain rate, for which the degree of fiber alignment inthe flow direction increased.

The behavior in elongational flow is in contrastwith that in shear, where significant stress overshootshave been observed due to fiber orientation (Sepehret al. 2004a). That is why Kobayashi et al. (1995) sug-gested increasing the filler content, filler length, andaspect ratio in order to highlight the influence of theorientation changes of nonspherical particles on theelongational viscosity. Model predictions (made fromLipscomb et al. 1988 constitutive equation, where theNewtonian viscosity was replaced by a Carreau expres-sion) for isotropic and aligned fiber orientations withrespect to flow have been presented by Thomasset et al.(2005). They found that their measured apparent elon-gational viscosity data lied between the two limitingcases (3D random and fully aligned fiber orientations).On the other hand, Wagner et al. (2003) did not ob-serve any significant strain hardening for 12-wt.% fibersin a polyamide-6 although the fibers oriented froma random state to a completely aligned orientation,as observed from selected micrographs taken duringthe tests. Ooi and Sridhar (2004) assumed that fiberorientation occurred under elongational flow but no

Rheol Acta (2009) 48:59–72 61

strain-hardening was observable for the suspensions ina Newtonian matrix, whereas the tensile stress for thesuspensions in a Boger fluid (PIB) grew with strainfrom a stress-free condition to a plateau, and thenthe behavior became strain-hardening before reachinga second plateau. Takahashi et al. (1999) concludedthat when particles were small and their aspect ratiolarge (length >> diameter), strain hardening becameinsignificant. For glass fiber-filled HDPE composites,Laun (1984) found that the highest tensile stress wasobtained when the fibers were aligned along the elon-gational flow. The same value of the stress was reachedwhen the fibers were initially oriented perpendicu-larly to the principal axis of the strain, suggesting thatthe fiber alignment most probably increased duringelongation. However, no strain hardening phenomenawere noticeable in the tensile stress versus strain plot.Finally, Kobayashi et al. (1995), as Wagner et al. (2003),noted that the orientation of nonspherical fillers wasstrain dependent.

A few theoretical expressions have been derived topredict the stress generated by nondilute fiber suspen-sions in elongational flow. Batchelor (1971) derived anexpression to relate the force imposed by a particle ona Newtonian surrounding medium to the particle char-acteristics. From first-order singularities in the forcedistributions, the specific elongational viscosity ηSP

E forperfectly aligned fibers can be obtained:

ηSPE = ηE − 3ηmatrix

0

3ηmatrix0

= 4φr2

9 ln(π

/φ) , (1)

where φ is the fiber volume fraction and r is the aspectratio, L/D (length over diameter). Batchelor (1971)extrapolated the formula to reach the dilute domain.Petrie (Petrie 1999) modified this extrapolation to ob-tain a better correlation expressed as:

ηSPE = 2φr2

9[ln 2r − ln

(1 + 2r

√φ/(

πe3)) − 1.5

] , (2)

where e is an adjusting parameter. Later, Lubanskyet al. (2005) extended the original Batchelor theoryto take into account the cylindrical shape of fibers ashigher-order singularities along the principal axis. Forthe semidilute regime of constant circular cross-sectionparticles, they obtained:

ηSPE = 2φr2

[ln

(π

/4φ

) + 8/

3]

9 ln(π

/φ) [

ln(π

/φ) − 3

] . (3)

In summary, there are a lot of controversies and con-tradictions in the literature on the rheological behavior

of fiber suspensions in polymers under elongationalflow. Open questions are: do fibers cause strain harden-ing in elongational flow, and is fiber orientation respon-sible for the strain hardening? In most of the previousstudies, the fibers were initially aligned in the stretchingdirection due to sample preparation. What would bethe effect of other fiber orientations? Therefore, themain objective of this work is to determine the influ-ence of the initial fiber orientation on the elongationalbehavior of short glass fiber-filled polypropylene. Inaddition, the Dinh and Armstrong governing equationis used to analyze the trends in the transient regime.We also report elongational data for short glass fiber-filled PP obtained using three different instruments.The results for uniaxial elongation are compared withdata previously obtained in planar elongation using aslit die and new data obtained using a Rheotens. Weunderstand that the flow in a hyperbolic convergingdie is not purely elongational as corrections have tobe made for shear components due to the presence ofthe walls. Also, the elongational rate is not controlledas in a Meissner-type rheometer or with the SER unitused in this work. The same is true for the Rheotens.Nevertheless, both instruments simulate well industrialprocessing conditions and the data are believed to berelevant. Finally, an extension of the Carreau modelfor suspended fibers in a polymeric fluid is proposed todescribe the steady uniaxial elongational data.

Modeling

In this section, the equations that describe the evolutionof a fiber population with respect to the matrix flow arepresented. We also propose a new constitutive equationfor fibers suspended into a fluid obeying the Carreaumodel.

Orientation tensors

A unit vector p, directed along the principal axis ofthe particle, is generally used to describe the fiberorientation. A probability distribution function ψ canrepresent a fiber population in an elementary cell fluid,and second- and a fourth-order orientation tensors canbe defined as (Advani and Tucker 1987):

a2 =∫

pppψd p, (4)

a4 =∫

pppppψd p, (5)

62 Rheol Acta (2009) 48:59–72

where a2 is a positive and symmetric tensor with aconstant trace equals to one, which states that the fiberlength does not change under flow.

Fiber motion

At low Reynolds numbers, Jeffery (1922) derived theellipsoid motion p in a Newtonian medium for dilutesuspensions. Later, Folgar and Tucker (1984) havetaken into account a randomizing effect of fiber inter-actions by adding a diffusion term (−CI |γ | ∇ ln ψ) tothe Jeffery equation. CI is an interaction coefficient, |γ |is the effective deformation rate, and ∇ represents thenabla (or del) operator. Moreover, ψ may be regardedas a convected quantity in the sense that when a fiberleaves one orientation, it must appear in another one(Bird et al. 1987). The combination of the continuityrelation with the Jeffery equation yields the equationof change for ψ :

Dψ

Dt= −∇ · (

pJefferyψ) + CI |γ | ∇2ψ, (6)

where D/Dt represents the material derivative andreduces to partial derivative in the case of homoge-nous flows. Defined by λ = (

r2 − 1)/(

r2 + 1), the form

factor λ appears implicitly in Eq. 6. In the scope ofthis work, Eq. 6 was discretized using a finite volumemethod. Hence, no closure approximation for thefourth-order orientation tensor was needed. Detailsabout this numerical procedure are presented in Férecet al. (2008).

Constitutive equation

A constitutive equation for rigid cylindrical particlessuspended in a Newtonian fluid has been proposed byDinh and Armstrong (1984). Their expression may bewritten as:

σ = −Pδ + η0γ + η0φr2

3 ln (2h/D)

∫

p

γ : ppppψdp, (7)

where P is the hydrostatic pressure, δ is the identitytensor, η0 the Newtonian fluid viscosity, and γ is therate of strain tensor. The average spacing h between afiber and its neighbors is given by (Doi and Edwards1978a, b):

h = D2

√π

/φ, for aligned fibers, (8)

h = Dπ

4φr, for 3D random fibers. (9)

Souloumiac and Vincent (1998) have derived a con-stitutive equation for rigid cylindrical particles sus-pended in a shear-thinning matrix with a viscosityrepresented by a power-law. Using a cell model ap-proach, the total stress is the sum of the contributionof the matrix and the fibers:

σ = −Pδ + K |γ |m−1 γ + Kφrm+1

2m−1 (m + 2)

×⎡

⎢⎣

1 − m

m(

1 − (D

/2h

) 1−mm

)

⎤

⎥⎦

m

×∫

p

γ : pppp∣∣γ : pp

∣∣m−1ψdp, (10)

where K is the consistency index, and m is the power-law index.

By considering Eqs. 7 and 10, it seems possible toexpress the total stress of fiber suspensions in a gener-alized Newtonian fluid (GNF) described by the Carreaumodel (1997). After some straightforward calculations,we obtain:

σ = −Pδ + η0[1 + (λm |γ |)2

] m−12 γ

+ η0φμ2

∫

p

γ : pppp[1 + (

λ f γ : pp)2

] m−12

ψdp,

(11)

with the coupling coefficient μ2 expressed as:

μ2 = rm+1

2m−1 (m + 2)f(D

/2h

)

= rm+1

2m−1 (m + 2)

⎡

⎣ 1 − m

m[1 − (

D/

2h)(1−m)/m

]

⎤

⎦

m

. (12)

We note that when m tends to be 1, μ2 =r2

/3 ln

(2h

/D

), which is the Dinh and Armstrong

expression. In their paper, Souloumiac and Vincent(1998) assumed a similar power-law index for the sus-pension and for the unfilled matrix. Also, as done byYasuda with the Carreau model, an additional expo-nent parameter can be added to improve the fit withthe experimental data (Carreau et al. 1997).

Extensional viscosity

To determine the transient elongational viscosity η+E of

a Newtonian fluid, the equation of change for ψ (Eq. 6)

Rheol Acta (2009) 48:59–72 63

is coupled with the expression for the bulk stress tensor(Eq. 7) developed by Dinh and Armstrong (1984).It is convenient to define η+

E for fiber suspensions as(Batchelor 1971; Mewis and Metzner 1974; Mongrueland Cloitre 2003):

η+E = σ11 − 1

2 (σ22 + σ33)

ε, (13)

where index “1” denotes the principal stretching direc-tion. Appendix 1 gives the transient elongational stresscomponents for the Dinh and Armstong expressionaccording to the fourth-order orientation tensor com-ponents. Appendix 2 presents the elongational stresscomponents for the new Carreau-based model (Eq. 11)for fibers perfectly aligned in the stretching direction.

Experiments

Materials

A linear unfilled polypropylene (PP0, Hostacom G3N01L) and two filled polypropylenes, containing, re-spectively, 10- and 30-wt.% glass fibers (PP10 andPP30), especially prepared for this work by Basell,have been studied. An intermediate fiber content of20 wt.% (PP20) was obtained by blending equal quan-tities of the two last products. The filler consists oftreated glass fibers with a density of ρ f = 2500 kg/m3,nominal diameter D = 16 μm and average length L =250–320 μm (fiber breakage during sample preparationis discussed below). In order to investigate the effectof particle shape, a model suspension of microbeads(PPB11) was prepared using the same base PP with avolume fraction corresponding to the glass fiber contentof sample PP30 (i.e., φ = 11.5 vol.%). The microbeadswere untreated hollow glass beads (D = 10 μm) witha density of ρb = 1,100 kg/m3. To reduce the ther-mal degradation, 1-wt.% of stabilizer (Irganox B225,Ciba Specialty Chemical) was added to the materi-als during sample preparation (Sepehr et al. 2004a).Density measurements of PP0, PP10, and PP30 wereperformed on a Thermo Haake PVT100 equipment.The sample nomenclature, fiber volume concentrationand density of the composites are presented in Table 1.Three regimes of fiber concentrations are proposed inthe literature according to fiber dimensions (Doi andEdwards 1986): dilute, in which φ < D2/L2; semidi-lute D2/L2 < φ < D/L and concentrated φ > D/L. Thesuspensions’ respective regimes are also presented inTable 1. PP10 is in the semidilute regime, whereas PP20and 30 are in the concentrated one.

Table 1 Sample nomenclature, fiber content, concentrationregime, density, and zero-shear viscosity

Sample φ(%) Regime ρ (kg/m3 ) η0 (kPa.s)

PP0 0 – 760 27.0PP10 3.3 Semidilute 817 34.0b

PP20 7.1 Concentrated 883a 41.0b

PP30 11.5 Concentrated 961 69.0b

aCalculatedbPresheared samples

Elongational devices

The elongational experiments were carried out usinga SER Universal Testing Platform (Xpansion Instru-ments LLC, model SER-HV-A01) specifically designedfor a controlled rate rheometer (ARES, TA Instru-ments; Sentmanat 2004; Sentmanat et al. 2005). It iscomposed of two wind-up drums, which ensure a uni-form extensional deformation during experiments. Thedevice is incorporated into the oven of the ARES andallows performing the tests at high temperature undera nitrogen atmosphere. The experiments were carriedout at 200◦C. All results obtained with the SER geom-etry were averages of three tests.

A Göttfert Rheotens device was also used to testthe composites in uniaxial extension at 200◦C for theunfilled, 10- and 30-wt.% filled polypropylenes. In theRheotens experiment, a filament extruded at constantshear rate is stretched by two counter-rotating wheelswhose velocity is increased with time, starting from thatat which the tensile force is zero, and the resultingtensile force is recorded as a function of time. Byknowing the average extrusion velocity (v0) and that ofthe wheels at a given time (v), the draw ratio (DR) iscalculated as DR = v/v0. The spinline length and theacceleration factor were fixed at 0.09 m and 0.005 m/s2,respectively. A capillary with a length to diameter(L/Dc) of 20 with Dc = 0.002 m was used to extrude thecomposites using a Göttfert Rheotester 1000 capillaryrheometer. The extrusion rates in this work were in anappropriate range to keep a quasi-isothermal flow ofthe extrudate along the spinline, as well as to avoid meltfracture, namely, between 200 and 300 s−1.

Finally, a Meissner-type uniaxial extensional rheo-meter (RME, Rheometric Scientific), equipped withrotary clamping devices, was used to verify the defor-mation uniformity of the filled PP. Using a video acqui-sition system, the analysis of the sample width versustime data gives the true Hencky strain rate applied onthe sample. The PP specimens were supported withan air bed, which is fed through a fritted table. Thetemperature chamber was set at 200◦C. As for the SER

64 Rheol Acta (2009) 48:59–72

unit, the RME results were averages of three tests. Alot of care has been devoted when using the device asprescribed by Schweizer (2000).

The planar elongational results reported for the slitdie with a hyperbolic entrance (Mobuchon et al. 2005)were obtained using the Binding analysis (Binding1988).

Simple shear measurements

Measurements of the shear properties were carriedout using a stress controlled rheometer (CSM, BohlinInstruments) under nitrogen atmosphere at 200◦C. Theparallel-plate geometry was used instead of the cone-and-plate geometry to avoid wall effects, and the latterare not significant if the parallel-plate gap is greaterthan three times the fiber length (Attanasio et al. 1972).For all experiments, the gap was around 1.5 mm with aplate diameter of 25 mm. Frequency sweep tests wereperformed from the highest to the lowest frequencyin the linear domain (γ = 0.01). Before the dy-namic measurements, the samples were presheared at1,000 Pa during 1,000 s to pre-align the fibers in theshear direction.

Samples

For the SER unit, rectangular samples (1.8 cm×1.3 cm×0.08 cm) were used. Given that the sample length andwidth are larger compared to the thickness, the fillerwas assumed to take a 2D planar orientation. Samplesfor three different initial fiber orientations were pre-pared as shown in Fig. 1:

– The first case consisted in obtaining a planar ran-dom orientation (PRO) as illustrated in Fig. 1a.Firstly, the materials were mixed into a Brabenderinternal mixer at 40 RPM at 200◦C under a nitrogenatmosphere for 5 min. Then, the blends were com-pressed at 200◦C into a rectangular mold of a thick-ness close to that of the final samples to minimize

fiber orientation during compression. For samplePP30, the average fiber length was evaluated to beL = 320 μm after blending and calcination of thematrix at 500◦C for 40 min. Consequently, the fiberaspect ratio was r ≈ 20, as opposed to r ≈ 23 forPP10.

– In the second case, samples with nearly fully alignedfibers were prepared. Strands were produced usinga Leistritz twin screw extruder (type ZSE18HP-40D) and a converging slit die. Afterwards, twolayers of the extruded tape were compressionmolded at 200◦C. The final samples were cut par-allel (PARA, Fig. 1b) and perpendicular (PERP,Fig. 1c) to the extrusion flow direction. For theseprocessing conditions, the average fiber length forPP30 was reduced to L ≈ 250 μm: therefore, theaspect ratio became r ≈ 16. It was verified that thesamples had a uniform cross-section and were freeof voids.

For the RME experiments, samples of 75 mm ×7 mm × 2.1 mm were obtained by compression mold-ing of the strands produced in twin screw extrusion.Samples with parallel (PARA, Fig. 1b) and with per-pendicular (PERP, Fig. 1c) fiber orientations wereprepared.

In the case of the Rheotens experiments, pellets wereused as received. The initial fiber orientation couldnot be controlled; however, it is expected that largeextrusion shear rates in the capillary rheometer willinduce a high alignment of the fibers before the sampleis stretched by the Rheotens device.

Elongational rheometry

During elongational deformation, a tensile force F(t)appears in the sample. The RME measures directlyF(t), whereas for the SER unit mounted on the ARESrheometer, F(t) is determined according to the mea-sured torque and the drum radius. The tensile stress

(a) (b) (c)

Fig. 1 Schematic representation of the three different initialfiber orientations used with the SER unit: a PRO planar randomorientation, b PARA fibers aligned parallel to the extensional

flow direction, and c PERP fibers oriented perpendicularly to theelongational flow direction. The arrows indicate the stretchingflow direction

Rheol Acta (2009) 48:59–72 65

growth function η+E(t) at a constant Hencky rate ε is

expressed as:

η+E (t) = σ (t)

ε= F (t)

εA (t), (14)

where A(t) is the instantaneous cross-sectional area,which evolves exponentially with time as:

A (t) =(

ρ0

ρ

)2/3A0 exp (−εt) . (15)

In Eq. 15, ρ and ρ0 are the densities at the testand room temperature, respectively, and A0 is the ini-tial cross-sectional surface of the unstretched specimenbefore sample loading. Assuming a similar uniformdeformation in the thickness and the width (w), Eq. 15yields the width of the specimen being stretched as afunction of time:

w = w0 exp(−εt

/2), (16)

where w0 is the initial width at the test temperature.Figure 2 reports the true Hencky strain rate applied

in a typical RME test by analysing the width evolutionversus time obtained by video camera. The results arepresented for the unfilled matrix (PP0) and the30-wt.% filled polypropylene (PP30) in the initialPARA and PERP configurations. The nominal strainrate applied is 0.1 s−1, and for the unfilled matrix (PP0),

t [s]0 5 10 15 20 25

ln(w

idth

) [

a.u.

]

3.6

3.8

4.0

4.2

4.4

4.6

A-BC-DE-F

t [s]

0 10 20 30 40

ln(w

) [

a.u.

]

2.0

2.5

3.0

3.5

4.0

4.5PP0PP30 - PARAPP30 - PERP

ε = 0.103 s-1

ε = 0.088 s-1

ε = 2*slope = 0.068 s-1.

.

.

Fig. 2 Hencky strain rate as measured by a video camera forthe unfilled matrix (PP0), for 30-wt.% fiber-filled polypropylenewith fibers aligned parallel to the flow (PP30-PARA) and per-pendicular (PP30-PERP) to the stretching direction. The nominalHencky strain rate was 0.1 s−1. Also shown in the inset are thewidth changes for PP30-PERP in the middle of the sample (A-B)and close to the left (C-D) and right (E-F) of the rotary clamps

it is well reproduced as two times the slope gives thetrue Hencky rate (cf. Eq. 16). In the case of the filledPP large differences are observed, probably due toslip between the rotary clamps and the material as thetensile force increases considerably for filled systems.Surprisingly, the lowest true Hencky rate was observedfor the PP30-PERP sample (i.e., more slip), whereas thelargest tensile stress was obtained for aligned fibers inthe stretching direction (PP30-PARA sample). It seemsthat for sample PP30-PERP, the width deformation wasnot completely uniform. The inset in Fig. 2 depicts thewidth decrease for sample PP30-PERP as measured inthe middle (A–B), on the left (C–D), and on the right(E-F) side of the sample, close to the rotary clamps,respectively. It indicates that the width deformationalong the specimen length was uniform. However, astraight mark initially placed along the width of thePP30-PERP sample deformed nonhomogeneously withtime, showing more pronounced strains at the widthedges than in the center of the sample. Hence, ourRME data for the PARA samples may have beenobtained under nonuniform deformation conditions.However, it is nearly impossible to assess the effectof the nonhomogeneity on the measured elongationalviscosity. Finally, for the SER data, we were unable toverify if the deformation was uniform for containingfibers samples.

Results and discussion

Transient elongational viscosity

The transient elongational viscosities, η+E, of the unfilled

matrix PP0 as well as that of the model suspensionof beads in PP (PPB11), determined with the SERunit, are shown in Fig. 3. Also shown is the predictedlinear viscoelastic (LVE) behavior determined fromthe shear relaxation modulus after a step strain forPP0 and PPB11. For PP0, the curves are superposedfor all applied strain rates at short times and coincidewith the predictions of the linear viscoelastic regime;no strain hardening can be observed at any strain rate.For PPB11, again a good superposition in the linear vis-coelastic regime at different Hencky rates is observed.As expected, the presence of glass beads induces nostrain hardening, but increases slightly the elongationalviscosity compared to the neat polypropylene.

Figure 4 presents the transient elongational viscos-ity for samples PP30-PARA and PP30-PERP obtainedwith the SER unit at Hencky strain rates of 0.05 and0.1 s−1. Figure 4 also reports the transient elonga-tional viscosity obtained with the RME at true Hencky

66 Rheol Acta (2009) 48:59–72

t [s]

10-2 10-1 100 101 102 103

η E+

[Pa

.s]

103

104

105

106

107

0.05 s-1

0.1 s-1

0.3 s-1

1 s-1

5 s-1

10 s-1

20 s-1

LVE PP0LVE PPB11

x 10

εH.

Fig. 3 Transient elongational viscosity of the unfilled matrix PP0and the glass bead-filled polypropylene PPB11 (values multipliedby 10) at 200◦C for Hencky strain rates ranging from 0.05 to20 s−1 obtained with the SER unit. Also shown is the predictedLVE behavior as determined from the shear relaxation modulusafter a step strain

rates of 0.088 s−1 for PP30PARA and 0.068 s−1 forPP30PERP, respectively. For both samples, the datafrom the RME are located between the boundariesdelimited by the SER data, as expected considering theapplied strain rates. For PP30-PERP, both the RMEand SER data show a slight strain-hardening (this be-havior is explained below). Except for the initial dataat 0.1 s−1 for PP30-PARA, these results are in goodagreement and confirm that no slip occurred in theSER unit.

t [s]10-1 100 101 102

η E+

[P

a.s]

104

105

106

107

SER εH = 0.05 s-1

SER εH = 0.1 s-1

RME εH = 0.088 s-1

SER εH = 0.05 s-1

SER εH = 0.1 s-1

RME εH = 0.068 s-1

x 10

PP30 - PERP

PP30 - PARA

.

.

.

.

.

.

t [s]10-1 100 101 102

η E+

[P

a.s]

104

105

106

107

SER εH = 0.05 s-1

SER εH = 0.1 s-1

RME εH = 0.088 s-1

SER εH = 0.05 s-1

SER εH = 0.1 s-1

RME εH = 0.068 s-1

x 10

PP30 - PERP

PP30 - PARA

.

.

.

.

.

.

Fig. 4 Comparison of the transient elongational viscosity ofthe 30-wt.% fiber-filled polypropylene with fiber parallel (PP30-PARA) and perpendicular (PP30-PER) to the stretching direc-tion obtained with the SER unit (at Hencky rates of 0.05 and0.1 s−1) and with the RME (at the corrected Hencky rates)

Figure 5 reports the transient elongational viscosityof fiber-filled sample PP30 for three different initialfiber orientations (shown in Fig. 1). These samples havebeen stretched at a constant Hencky strain rate of0.05 s−1. Independently of the initial fiber orientation,the transient viscosity of the melt is strongly increasedby the presence of the fibers. As expected, the largestvalues are observed when the fibers are oriented par-allel to the principal axis of the strain (fibers resist tothe stretching). A slight strain-hardening behavior isobserved for the PRO and PERP initial orientations,which could be explained by a dynamic fiber alignmentduring the elongation test. The PRO sample showsa transient behavior between that of the PARA andPERP samples, as initially some fibers were alreadyaligned and contributed to the largest stress (Batchelor1971; Dinh and Armstrong 1984). Also, the value of thetransient elongational viscosity for PRO sample tendsto reach that of the preoriented sample, i.e., PP30-PARA, while the PERP sample needs a much longertime to reach a similar orientation. If the deformationis assumed to be uniform during the tests, this meansthat the planar random orientation is transformed intoa fully aligned orientation as the elongational test isperformed. However, as mentioned previously, we can-not rule out a possible effect of the nonhomogeneity ofthe flow.

To describe the observed trend of the elongationalviscosity, the equation of change for the probability

t [s]10-2 10-1 100 101 102

η E+

[Pa

.s]

103

104

105

106

PARAPROPERPPPB11LVE PP0

εH = 0.05 s-1.

Fig. 5 Transient elongational viscosity at a Hencky strain rateof 0.05 s−1 for PP30 with different initial fiber orientations:parallel to the stretching direction, PARA, planar random ori-entation, PRO, and perpendicular to the stretching, PERP. Alsoshown is the transient elongational viscosity for PPB11 and theLVE regime of PP0 for the purpose of comparison. The dashedlines represent the model predictions (Eqs. 5, 6, and 20) for thedifferent initial fiber orientations

Rheol Acta (2009) 48:59–72 67

εH [-]

10-3 10-2 10-1 100 101

η E+

[P

a.s]

103

104

105

106

0.01 s-1

0.05 s-1

0.1 s-1

0.3 s-1

1 s-1

εH

.

Fig. 6 Transient elongational viscosity as a function of Henckydeformation at different strain rates for PP30 with the initial fiberorientation perpendicular to the stretching direction

distribution function (Eq. 6) was combined with theDinh and Armstrong governing equation (Eq. 7) for aNewtonian matrix in the transient case (the transientcase has not been developed for the Carreau-basedmodel). Appendix 1 presents the transient elongationalviscosity obtained with the Dinh and Armstrong formu-lation, for which no closure approximation is needed.Details for setting the three initial probability functions(i.e., PARA, PRO, and PERP) are given in Appendix 3.The matrix viscosity was taken as the zero-shear viscos-ity of the neat polypropylene (i.e., η0 = 27 kPa.s) andthe fitting parameters racine, μ2 and CI , were chosen tobe equal to 85 and 0.001/

√3 ≈ 0.00058, respectively, for

these calculations. The results are represented in Fig. 5by the short dashed lines. At short times, the effectof the matrix elasticity is not described by the modelformulated for a Newtonian matrix, but the purpose ofthese simulations is to highlight the effect of fiber ori-entation at large deformations. As expected, a plateauis predicted for fully aligned fibers in the stretchingdirection, which is in good agreement with the PP30-PARA data. In the case of the initial planar random(PRO) fiber orientation, a slight strain hardening ispredicted, followed by a plateau when the fibers aretotally aligned at long times or large Hencky strains.This is qualitatively in agreement with the data. Forthe initial perpendicular orientation (PERP) case, aslight decrease in the elongational viscosity is predicted,followed by a strong strain-hardening and then thesteady plateau. Except for the initial decrease, the pre-dictions are also in qualitative agreement with the data.Therefore, the model predictions confirm the ob-served transient behavior of the filled polypropylenes.

Interestingly, the strain hardening in the PRO andPERP samples appears at the same Hencky strain, i.e.,εH = 0.6. Strain hardening has been observed for theinitial PERP orientation at different elongational ratesand always appear at the same Hencky deformation(εH = 0.6), as shown in Fig. 6. As the applied elonga-tional rate is increased, the strain hardening becomesbarely visible. This is a similar strain behavior as ob-served for the overshoot in shear flow of fiber filledpolypropylenes (Ausias et al. 1992) and for reversibleshear flow of model fiber suspensions (Sepehr et al.2004b).

“Steady” elongational viscosity

In this section, we report “steady” elongational viscos-ity data as final values (plateau region) of the transientelongational viscosity curves, which may not be the realsteady-state regime in elongation. Figure 7 comparesthe shear and elongational viscosities of various sam-ples evaluated at the effective deformation rate γ =√

3ε where γ and ε are, respectively, the shear andHencky rates. The “steady” values of the fiber filled PPpresented in Fig. 7 were obtained with PARA samplesfor which closest values to steady state were obtained.For comparison, the complex viscosity and steady-sheardata for the neat PP (PP0) are also reported. It canbe observed that for PP0, the Cox–Merz rule appliesquite well. The zero-shear viscosity of PP0 was deter-mined using the Carreau-Yasuda model, for which thebest fitting parameters were η0 = 27 kPa.s, λm = 0.99 s,m = 0.28, and a = 0.51. At low Hencky rates, the

γ, ω, εH [s

-1]

10-2 10-1 100 101 102

η , η

∗ , η

E

[Pa.

s]

102

103

104

105

106

PP0 - ηE

PP10 - ηE

PP20 - ηE

PP30 - ηE

PP0 - ηPP0 - η∗

. .3

Fig. 7 Shear and elongational viscosities of PP0. Also shownare the elongational viscosity data of fiber-filled PP when thefibers are aligned (obtained from PARA data in elongation). Thedashed lines represent the model predictions

68 Rheol Acta (2009) 48:59–72

Table 2 Values of modelparameters for Eq. 11 andrelated quantities

ηmatrix0 η0 λm m λ f μ2 ηE0 Tr

(kPa.s) (kPa.s) (s) (–) (s) (–) (kPa.s) (–)

PP0 27.0 27.0 3.64 0.71 – – 81.0 3.0PP10 27.0 34.0 3.06 0.59 23.96 116.9 289 8.5PP20 27.0 41.0 2.19 0.51 38.94 185.9 793 19.3PP30 27.0 69.4 1.30 0.46 30.95 194.3 1290 18.6

Trouton ratio (Tr = ηE/η0) for PP0 reaches the theoret-ical value of 3. With increasing strain rate, the elonga-tional viscosity of PP0 shows a strain-thinning behavior,observed as well for the fiber-filled systems. The strain-thinning appears at lower strain rate as the fiber contentincreases. This phenomenon has also been observed forglass beads (Poslinski et al. 1988) and short glass fibers(Crowson and Folkes 1980) suspensions in shear flow.Also shown in Fig. 7 are the new Carreau-based modelpredictions (Eq. 11), where the fiber orientation waschosen as perfectly aligned in the stretching direction.Details of the calculations for the elongational floware reported in Appendix 2. The fits are in very goodagreement with the data and allow us to extrapolatethe elongational viscosity at low strain rate. Table 2presents the values of the parameters used, along withthe evaluated Trouton ratios for the fiber suspen-sions. The presence of fibers increases considerably the

Trouton ratio defined by ηE0

/η

composite0 where ηE0 is the

elongational viscosity at low Hencky rates. The zero-shear viscosity of the suspensions η

composite0 was esti-

mated by fitting frequency sweep data for pre-shearedsamples with the Carreau-Yasuda model. Table 2 showsthat the Trouton ratio increases from three for unfilledPP to 19.3 and 18.6 for PP20 and PP30, respectively.The slightly smaller Trouton ratio for PP30 comparedto PP20 is possibly explained by small differences infiber alignment in simple shear for PP30 compared toPP20. For a similar 30 wt.% filled PP in planar exten-sion, Mobuchon et al. (2005) found a Trouton ratio ofabout 54 at a deformation rate of 0.2 s−1 that decreasesto 5.4 at 200 s−1.

From the use of Eq. 12 for the coupling coefficientμ2, it is possible to determine the average distance hbetween fibers for the different fiber volume fractionsand with the model parameters used to fit the data

Table 3 Calculated average distance between fibers

PP10 PP20 PP30

h (μm) From Eq. 8 (aligned) 78.1 53.2 41.8From Eq. 12 9.67 8.35 8.18

of Fig. 7 (μ2 and m). Table 2 lists the determinedparameters and Table 3 compares the average distancesbetween fibers calculated using Eq. 12 with the the-oretical values for perfectly aligned fibers from Eq. 8(Doi and Edwards 1978a, b). The results show thatthe fitted distances h are much smaller than the theo-retical values for perfectly aligned fibers. A plausibleexplanation comes from Laun (1984), who suggeststhat macroscopic uniaxial elongation induces a com-plex flow between the rigid fibers on a microscopicscale (shear and elongation), as illustrated in Fig. 8.Equation 8 was developed for static conditions and, inelongation, the average distance h is strongly influencedby hydrodynamic interactions and the matrix elasticity(a strong adhesion between the matrix and the fillersis expected as glass fibers are treated with silane).Therefore, developing an appropriate and meaningfultheoretical expression for the coupling coefficient μ2

remains a challenging task.The specific elongational viscosity, ηSP

E , defined as(ηE0 − 3ηmatrix

0

)/3ηmatrix

0 , is presented in Fig. 9. ηSPE

shows a high dependence on fiber volume fraction. TheBatchelor expression (Eq. 1) underestimates the data inthe concentrated regime. A good agreement is obtainedfor PP10 and PP20 when the Petrie expression (Eq. 2)is used with e = 1.24. The formulation of Lubansky et al.(Eq. 3) deviates considerably as the fiber concentrationincreases. The cited theories have been developed forsemidilute systems, and additional work is required toextend them to cover the concentrated regime.

(a) (b)

Fig. 8 Schematic illustration of microscopic distortions pro-duced by aligned fibers in uniaxial elongation: a initial state withlength L0 and b elongated state with L = 2L0 (taken from Laun1984)

Rheol Acta (2009) 48:59–72 69

φ0.00 0.02 0.04 0.06 0.08 0.10 0.12

η ESP

[-]

-2

0

2

4

6

8

10

12

14

16Uniaxial elongationBatchelor [Eq. (1)]Petrie (e=1.24) [Eq. (2)]Lubansky et al. [Eq. (3)]

Fig. 9 Specific elongational viscosity of polypropylene systems.Also shown are the predictions of various models for the uniaxialelongational viscosity

Figure 10 presents the new Carreau-based model pa-rameters, i.e., characteristic time related to the matrixλm and power-law index m, as functions of the fibervolume fraction. A decreasing linear dependence ofλm with φ is observed, which indicates that the strain-thinning behavior appears more rapidly with increasingfiber content. A quadratic correlation gives a good ap-proximation of the variation of m with the fiber volumefraction.

The Rheotens experiments were performed for dif-ferent shear rates under stable extrusion conditions.The elongational viscosity was calculated from the ten-sile force versus draw ratio using the Wagner et al.analysis (1998). For this purpose, the tensile force ver-

φ0.00 0.02 0.04 0.06 0.08 0.10 0.12

λ m [

s]

1.0

1.5

2.0

2.5

3.0

3.5

4.0

m [-]

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

Fig. 10 Matrix characteristic time and power-law index as func-tions of fiber volume fraction. The lines represent the best fits(linear for λm and quadratic for m)

εH, 2εH [s-1]

10-3 10-2 10-1 100 101 102 103

η E/η

0Tr

[-]

0

1

2

3

4

5

6 Rheotens (200 s-1)Rheotens (300 s-1)Planar hyberbolic dieSER unitTrouton ratio (=3)

..3

(a)

εH, 2εH [s-1]

10-3 10-2 10-1 100 101 102 103

η E/η

0Tr

[-]

0

1

2

3

4

5

6 Rheotens (200 s-1)Rheotens (300 s-1)Planar hyberbolic dieSER unit

..3

(b)

εH, 2εH [s-1]

10-3 10-2 10-1 100 101 102 103

ηη E/η

0Tr

[-]

0

1

2

3

4

5

6 Rheotens (200 s-1)Rheotens (300 s-1)Planar hyperbolic dieSER unit

. .3

(c)

Fig. 11 Comparison of the reduced “steady” elongational vis-cosity of PP and its composites from measurements using threedifferent instruments: a PP0, b PP10, and c PP30

70 Rheol Acta (2009) 48:59–72

sus draw ratio curves were divided into two regions,namely, a linear one characteristic of the viscoelasticbehavior at low draw ratios, and a non-linear onerelated to a purely viscous behavior of the unfilled PPand the suspensions at high draw ratios. Results for twodifferent extrusion rates, 200 and 300 s−1, are shownin Fig. 11 and compared with data obtained with theSER (from PARA samples) and the planar hyperbolicdie (Mobuchon et al. 2005) for samples PP0, PP10,and PP30. The effective deformation rate based on thesecond invariant of the rate-of-deformation tensor isused to report the data for different flows. Also, inorder to compare planar and uniaxial elongational re-sults, the elongational viscosity is normalized accordingto the Trouton ratio (three for uniaxial and four forplanar elongation) using the zero-shear viscosity valuesof the materials (listed in Table 2). Firstly, we notefrom Fig. 11 that the elongational viscosity increaseswith increasing fiber content, as expected. Secondly, thedata from the three different instruments are in verygood agreement from moderate to high strain rates,while discrepancies are observed at low rates for allsamples. The Rheotens data cover lower elongationalrates than those from the other two rheometers, andthe low rate data for the neat PP (Fig. 11a) is somewhathigher than the theoretical value expected from theTrouton relation. The range for the extrusion rate whenoperating the Rheotens is quite limited since extrusionat shear rates lower than 200 s−1 would induce non-isothermal conditions and premature filament crystal-lization, while for rates larger than 300 s−1, extrusioninstabilities such as melt fracture would be observed.The Rheotens experiments have been repeated manytimes for the neat PP, confirming the trend reported inFig. 11a.

Again, it is interesting to note from Fig. 11 thatthe qualitative trend of the data obtained with theRheotens, the hyperbolic die, and the SER are verysimilar, except for the much larger elongational viscos-ity values for the unfilled PP obtained at lower defor-mation rates with the planar hyperbolic die (Fig. 11a)and the very large values for the PP30 data obtainedat low rates with the SER unit (Fig. 11c). The trendsdepicted by both the Rheotens and the hyperbolicdie data are coherent, suggesting that the elongationalbehavior is first strain-hardening at low strain rates,and then becomes strain-thinning at larger strain rates.However, the SER data rule out the possible strainhardening at moderate rates (>0.1 s−1) with a smoothdecrease of the elongational viscosity with strain rate.The observed discrepancies between the SER resultsand those of the other techniques may be explained bythe different flow kinematics prevailing in the converg-

ing die and in the Rheotens experiments. Finally, wenote that the trend for the elongational behavior of thesuspensions is similar to that of the unfilled matrix, withan almost constant viscosity value at low extensionalrates and a strain-thinning behavior at high elongationrates.

Concluding remarks

In this work, the elongational flow behavior of shortglass fiber-filled polypropylenes was investigated fordifferent fiber contents and initial fiber orientations.The presence of the fibers increased considerably theelongational viscosity, and the dynamic fiber orienta-tion during tensile tests induced a slight strain harden-ing. The transient elongational viscosity was observedto be strongly dependent on the initial fiber orientationand considerably larger transient values were obtainedfor initially aligned fibers in the flow direction. TheDinh and Armstrong governing equations coupled witha finite volume method was used to solve the prob-ability distribution function and predict the transientelongational viscosity of the materials. The predictionswere qualitatively in good agreement with the experi-mental data.

We have compared the “steady” elongational vis-cosity data as a function of the effective deformationrate obtained from three different rheometers: a SERunit mounted on an ARES, a slit die with a hyper-bolic entrance and a Rheotens. At low strain rates,discrepancies were observed in the data from the threerheometers; however, the superposition of the data wasastonishingly good at moderate to high strain rates forall materials investigated. The elongational behaviorof the composites was found to be similar to thatof the polymer matrix over the range of strain ratesinvestigated, with a plateau at low elongational ratesand strain-thinning at large rates. A novel constitutiveequation was proposed to calculate the steady-stateelongational viscosity of fiber suspensions in a general-ized Newtonian fluid (GNF) described by the Carreaumodel. This model predicts fairly well the fiber-filledpolypropylene behavior over the range of Hencky rateinvestigated in this study.

Acknowledgements This work was supported by the France-Québec collaboration program and funded by NSERC (Nat-ural Science and Engineering Research Council of Canada—CIAM program), and by CONACYT (CIAM 51837) and SIP-IPN (20060296-20070642) in México. The authors wish to thankDr. G. Krotkine from Basell who kindly provided the materialsused in this study. We are also thankful to Ms. M.J. Ramirez-Moreno, from the Instituto Politécnico Nacional in Mexico City

Rheol Acta (2009) 48:59–72 71

(Mexico) for the Rheotens measurements, to Mr. J.R. R.-Siffert,from Pontificia Universidade Católica-RJ in Rio de Janeiro(Brazil) for helpful sample preparation and preliminary tests, andto Christophe Mobuchon for the Binding analysis of the slit diedata. Finally, the authors wish to acknowledge Dr. S. Dagreouand Dr. F. Leonardi at the University de Pau for the use oftheir RME.

Appendix 1

From the Dinh and Armstrong expression (Dinh andArmstrong 1984), the normal stress components aredefined as:

σ11 = 2η0ε + η0φμ2(2a1111 − a1122 − a1133

)ε, (17)

σ22 = −η0ε − η0φμ2(−2a1122 + a2222 + a2233

)ε, (18)

σ33 = −η0ε − η0φμ2(−2a1133 + a2233 + a3333

)ε. (19)

According to the definition of Eq. 13, the transientelongational viscosity is written as:

η+E = 3η0 + η0φμ2

(2a1111 − 2a1122 − 2a1133 + a2233

+ 1

2a2222 + 1

2a3333

). (20)

Thanks to the equation of change for ψ , the fourth-order orientation tensor components are determinedeasily and the transient extensional viscosity is ex-pressed without the need of a closure approximation forthe fourth-order tensor.

Appendix 2

Equation 11 is used to express the steady extensionalflow in the case where fibers are perfectly aligned in theprincipal stretching direction (x1). Therefore, the orien-tation distribution function is simply a Dirac function:

ψ = δ(p − x1

). (21)

Then the normal stress components are given by:

σ11 = 2εη0

[1 +

(λm

√3ε

)2] m−1

2

+ 2εη0φμ2

[1 + (

λ f 2ε)2

] m−12

, (22)

σ22 = σ33 = −εη0

[1 +

(λm

√3ε

)2] m−1

2

. (23)

Consequently, the steady elongational viscosity iswritten as:

ηE = 3η0

[1 +

(λm

√3ε

)2] m−1

2

+ 2η0φμ2

[1 + (

λ f 2ε)2

] m−12

, (24)

where λm, m, μ2, and λ f are used as fitting parameters.

Appendix 3

In order to achieve model predictions for the tran-sient elongational viscosity, initial orientation functionshave to be defined to take into account the three ini-tial fiber orientations of Fig. 1. Following Advani andTucker (1987), highly oriented states (i.e., parallel andperpendicular to the stretching direction) are obtainedby a nonconventional distribution function having theform:

ψ (ϕ, θ) = K⊥ sinp θ sinq ϕ, for PERP, (25)

ψ (ϕ, θ) = K= sinp θ cosq ϕ, for PARA, (26)

where K⊥ and K= are constants that satisfy the nor-malization condition. Both p and q are taken equal to60 to give aligned fiber orientation. The planar randomorientation state (not to be confused with a biplanarorientation obtained by adding Eqs. 25 and 26 withrespect to the normalization condition) is reached withthe following distribution function:

ψ (ϕ, θ) = K×δ(θ − π

2

), for PRO, (27)

where δ is the Dirac function and K× is introduced torespect the normalization condition.

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